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Aspects of Gauge/Gravity Dualities
Aspects of Gauge/Gravity Dualities
Marcus K. Benna
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of Physics
Adviser: Igor R. Klebanov
September 2009
c Copyright by Marcus K. Benna, 2009.
All Rights Reserved
Abstract
This dissertation is composed of several studies of gauge/gravity dualities aimed at
providing evidence for the dualities we investigate, and exploring possible applications.
First we discuss the universal scaling function f (g), which appears in the dimensions
of high-spin operators of the N = 4 Super Yang-Mills theory. We study numerically an
integral equation that implements a resummation of the complete planar perturbative
expansion, and find a smooth function which for large coupling constant g matches with
high accuracy the asymptotic form predicted by string theory. Furthermore, we give an
exact analytic solution of the strong coupling limit of the integral equation.
Next, we study the N = 1 supersymmetric warped deformed conifold, which has
two bosonic massless modes, a scalar and a pseudoscalar, that are dual to the modulus
and phase of the baryonic condensates in the cascading SU(k(M + 1)) × SU(kM ) gauge
theory. We generalize both perturbations to include non-zero 4-dimensional momentum,
and find the mass spectra of S P C = 0+− and S P C = 0−− glueballs. We argue that these
massive modes belong to 4-dimensional axial vector or vector supermultiplets.
Extending our discussion to the complete baryonic branch of the gauge theory, we
show that the action of a Euclidean D5-brane wrapping all six deformed conifold directions measures the baryon expectation values. We demonstrate that this is consistent
with its coupling to the scalar and pseudoscalar massless modes, and reproduces the
scaling dimension of baryon operators. We also derive an expression for the variation of
the baryon expectation values along the supergravity dual of the baryonic branch.
Finally we turn to 3-dimensional gauge theories with N = 3 supersymmetry, and
calculate the non-abelian R-charges of BPS monopole operators. In the UV limit they
are described by classical backgrounds, and this allows us to find their exact SU(2)R
charges in a one-loop computation, by quantizing an SU(2)/U(1) collective coordinate.
We show that monopole operators with vanishing scaling dimensions exist in the ABJM
theory, which is essential for matching its spectrum with supergravity on AdS4 × S7 /Zk .
iii
Acknowledgements
I am very grateful to my principal adviser, Igor R. Klebanov, whose patient explanations as well as sharp reasoning have guided my thinking and countless times have
helped my research back onto the right track. His intuition and talent for generating
innovative ideas have been an inspiration to me. Igor has always been generous with his
time and knowledge, and encouraged me to freely explore my research interests.
I would like to thank my collaborators Luis Fernando Alday, Gleb Arutyunov, Sergio
Benvenuti, Anatoly Dymarsky, Burkhard Eden, Thomas Klose, Antonello Scardicchio,
Mikael Smedbäck and Alexander Solovyov. I have learned much from them, and without
their invaluable contributions this dissertation would not have been possible.
Moreover, I am indebted to the faculty of the High Energy Theory groups at the
Department of Physics, in particular Chris Herzog and Herman Verlinde, and at the
Institute for Advanced Study, especially Juan Maldacena, for generously sharing their
insights as well as many fruitful discussions.
Finally I’d like to express my deep appreciation of my fellow students, some of whom
I have the honor of calling my friends. Without them life in Princeton would have
been unimaginably poorer. Especially Andrea, Filip, Natalia, Juan and Sabrina have
given me much in the way of joyful memories as well as profound conversations. Most
importantly, Catherine has always been there for me, and shared her life and dreams
with me throughout these years.
iv
To my parents,
who put up with my stubbornness,
and never tried to dissuade me from the path I chose,
even when there was no telling where it would lead.
v
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
1 A Brief Review of Select Topics in Gauge/Gravity Dualities
1
1.1
A Sketch of the AdS/CFT Correspondence . . . . . . . . . . . . . . . . .
2
1.2
Generalizations to Lower Supersymmetry and the Non-Conformal Case .
10
1.2.1
Reducing Supersymmetry and the Conifold . . . . . . . . . . . . .
10
1.2.2
Deformation of the Conifold . . . . . . . . . . . . . . . . . . . . .
13
1.2.3
Massless Modes of the Warped Throat . . . . . . . . . . . . . . .
18
1.2.4
The Baryonic Branch . . . . . . . . . . . . . . . . . . . . . . . . .
23
Superconformal Chern-Simons Theories and an AdS4 /CFT3 Duality . . .
27
1.3.1
BLG Theory in Component and N = 2 Superspace Formulation .
29
1.3.2
ABJM U(N ) × U(N ) Gauge Theory in Superspace . . . . . . . .
36
1.3
2 On the Strong Coupling Scaling Dimension of High Spin Operators
42
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.2
Numerical Study of the Integral Equation
. . . . . . . . . . . . . . . . .
45
2.3
Discussion of Results and Further Investigations . . . . . . . . . . . . . .
49
2.4
Fluctuation Density at Strong Coupling . . . . . . . . . . . . . . . . . . .
53
2.4.1
Analytic Solution at Strong Coupling . . . . . . . . . . . . . . . .
55
2.4.2
An Alternative Derivation . . . . . . . . . . . . . . . . . . . . . .
58
vi
2.5
2.4.3
Fluctuation Density in the Rapidity Plane . . . . . . . . . . . . .
60
2.4.4
Subleading Corrections . . . . . . . . . . . . . . . . . . . . . . . .
64
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3 On Normal Modes of a Warped Throat
68
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.2
Radial Excitations of the GHK scalar . . . . . . . . . . . . . . . . . . . .
70
3.2.1
Equations of Motion for NSNS- and RR-Forms . . . . . . . . . . .
72
3.2.2
Einstein Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.2.3
Two Coupled Scalars . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.2.4
Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.3
Pseudoscalar Modes from the RR Sector . . . . . . . . . . . . . . . . . .
84
3.4
Organizing the Modes into Supermultiplets . . . . . . . . . . . . . . . . .
87
3.5
Effects of Compactification . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4 Baryonic Condensates on the Conifold
4.1
4.2
4.3
4.4
97
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.1.1
D-Branes, κ-Symmetry and Killing Spinors of the Conifold
. . . 101
4.1.2
Branes Wrapping the Angular Directions . . . . . . . . . . . . . . 104
Derivation of the First-Order Equation for the Worldvolume Gauge Bundle106
4.2.1
κ-Symmetry of the Lorentzian D7-Brane . . . . . . . . . . . . . . 106
4.2.2
An Equivalent Derivation Starting from the Equation of Motion . 109
4.2.3
κ-Symmetry of the Euclidean D5-Brane . . . . . . . . . . . . . . . 113
Euclidean D5-Brane on the KS Background . . . . . . . . . . . . . . . . 115
4.3.1
The Gauge Field and the Integrated Form of the Action . . . . . 115
4.3.2
Scaling Dimension of Baryon Operator . . . . . . . . . . . . . . . 117
4.3.3
The Pseudoscalar Mode and the Phase of the Baryonic Condensate 119
Euclidean D5-Brane on the Baryonic Branch . . . . . . . . . . . . . . . . 122
vii
4.5
4.4.1
Solving for the Gauge Field and Integrating the Action . . . . . . 122
4.4.2
Baryonic Condensates . . . . . . . . . . . . . . . . . . . . . . . . 126
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5 Charges of Monopole Operators in Chern-Simons Yang-Mills Theory 131
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3
N = 3 Chern-Simons Yang-Mills Theory . . . . . . . . . . . . . . . . . . 138
5.4
5.5
5.6
5.3.1
Action and Supersymmetry Transformations . . . . . . . . . . . . 139
5.3.2
Classical Monopole Solution . . . . . . . . . . . . . . . . . . . . . 146
U(1)R Charges from Normal Ordering . . . . . . . . . . . . . . . . . . . . 150
5.4.1
Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.4.2
Application to N = 3 Gauge Theory . . . . . . . . . . . . . . . . 155
SU(2)R Charges and Collective Coordinate Quantization . . . . . . . . . 157
5.5.1
Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.5.2
Application to N = 3 Gauge Theory . . . . . . . . . . . . . . . . 164
Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Appendix: Useful Formulae, Notations and Conventions
171
A.1 The Type IIB Supergravity Equations . . . . . . . . . . . . . . . . . . . . 171
A.2 Chern-Simons Field Theory Notations and Conventions . . . . . . . . . . 172
A.3 N = 3 Chern-Simons Yang-Mills on R1,2 . . . . . . . . . . . . . . . . . . 175
A.4 Monopole Spinor Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 177
Bibliography
179
viii
Chapter 1
A Brief Review of Select Topics in
Gauge/Gravity Dualities
While string theory is often thought of primarily as a candidate theory of everything,
aiming to unify all fundamental interactions including gravity in a single quantum theory,
it has transpired during the past twelve years that it is also a framework in which
quantum field theory and general relativity appear as complementary, dual descriptions
of the same physics. This allows us to use gravity to study strongly coupled field theories,
or conversely, employ field theory ideas to define what we mean by quantum gravity.
Let us begin by reviewing some well-known facts about gauge/gravity dualities. We
shall first discuss the prototype of such dualities, the celebrated N = 4 AdS5 /CFT4
correspondence [1, 2, 3], before we move on to examples with less supersymmetry and
without conformal symmetry. Finally, we shall turn to some more recent developments
concerning AdS4 /CFT3 dualities and the worldvolume theory of M2-branes.
This introductory chapter is based on material from work in collaboration with
I. R. Klebanov, T. Klose and M. Smedbäck [4, 5]. For further background on the
topics touched upon in Section 1.1 we refer the reader to the reviews [6, 7, 8, 9], while
summaries of much of the material discussed in Section 1.2 can be found in [10, 11, 12].
1
1.1
A Sketch of the AdS/CFT Correspondence
Even though the elementary excitations of perturbative string theory are (as the name
suggests) 1+1 dimensional objects, whose world-sheet dynamics is governed by the
Nambu-Goto area action
SNG
1
=−
2πα0
Z
d2 σ
p
−det ∂a X µ ∂b Xµ ,
(1.1)
it was realized a long time ago that supersymmetric theories of closed strings also contain higher dimensional objects [13]. While closed superstrings move freely in the 9+1
dimensional spacetime, open strings are bound to end on p+1 dimensional hyperplanes
known as Dirichlet p-branes, usually denoted as Dp-branes. This terminology refers to
the Dirichlet boundary conditions which open string have to satisfy at their endpoints
in the 9−p directions orthogonal to the D-brane, while Neumann boundary conditions
apply for the p+1 coordinate directions parallel to it.
This seemingly innocent observation has profound consequences, because the worldvolumes of D-branes support gauge theories. If we consider a single brane, open strings
beginning and ending on it have massless excitations identical to those of a maximally
supersymmetric U(1) gauge theory in p+1 dimensions. There are 9−p scalar Goldstone
bosons, since the brane breaks translation invariance in the transverse directions, and
a massless vector which arises from the string excitations in the p+1 directions parallel
to the brane. Together with the appropriate fermions they combine to form a vector
supermultiplet.
Extending these considerations to multiple parallel branes, one finds that their pairwise separations are determined by the expectation values of the transverse scalar fields
associated with strings stretching between two different branes. In particular, if all
scalar expectation values vanish, the branes are stacked on top of each other. If this
is the case, then N parallel branes support N 2 massless modes (the number of distinct
2
choices of beginning and endpoint for an open string), which transform in the adjoint
representation of U(N ). The upshot is that there is a p+1 dimensional U(N ) gauge
theory with 16 supercharges living on such a collection of branes.
The gravitational background of a stack of D3-branes.
For large N a stack
of parallel Dp-branes is a heavy object and will backreact onto the spacetime around
it. The so-called black p-branes, gravitational backgrounds corresponding to such situations, were originally discovered as supersymmetric soliton solutions of the low energy
supergravity theory, which carry electric charge with respect to the p+1 form RamondRamond (RR) potential Cp+1 [14]. In type IIA string theory (and the associated supergravity theory) p is even, while for type IIB p is odd.
Since we will mostly consider 3+1 dimensional U(N ) gauge theory we will be particularly interested in stacks of D3-branes. This case is special insofar as the dilaton
field of the type IIB theory is constant in such a background, and the RR field strength
F̃5 the brane couples to is self-dual (the supergravity equations relevant to this case are
reviewed in Appendix A.1). The metric of this classical solution is given by [14]
"
ds2 = h−1/2 (r) −f (r)(dx0 )2 +
3
X
#
(dxi )2 + h1/2 (r) f −1 (r)dr2 + r2 dΩ25 ,
(1.2)
i=1
where dΩ25 is the metric of a unit 5-dimensional sphere S5 and
h(r) = 1 +
L4
,
r4
f (r) = 1 −
r04
.
r4
(1.3)
We would like to consider two successive limits of this black 3-brane background.
First we take the extremal limit r0 → 0 of the solution (1.2), in which the area of the
event horizon at constant radial coordinate r = r0 vanishes. We have f (r) → 1 and thus
ds2 = h−1/2 (r)ηµν dxµ dxν + h1/2 (r) dr2 + r2 dΩ25 .
3
(1.4)
In this limit the mass (per unit volume) of the 3-brane saturates a BPS-type bound and
it can be shown that the extremal solution preserves 16 of the 32 supersymmetries of
the type IIB theory, as it must if it is to describe a stack of parallel D3-branes.
When r is large compared to L it is easy to see that (1.4) simply approaches 9+1
dimensional flat space. In the opposite, near horizon limit r → 0 we find that, defining
a new variable z = L2 /r, the metric (1.4) simplifies to
L2
ds = 2 dz 2 + ηµν dxµ dxν + L2 dΩ25 .
z
2
(1.5)
This metric describes the direct product of a five-sphere S5 , and the Poincaré wedge of
the 5-dimensional Anti-de Sitter space, AdS5 , both with radius of curvature L.
By comparing the tension of the extremal 3-brane solution, parameterized by the
length scale L, to that of a stack of N D3-branes, characterized by the string length
√
scale α0 , one finds that
2
L4 = gYM
N α02 ,
(1.6)
where gYM is the coupling constant of the supersymmetric Yang-Mills theory that the
brane supports. The five-form flux piercing the S5 that is required to satisfy the supergravity equations of motions is then simply
Z
∗F˜5 = N ,
(1.7)
S5
i.e. the number of flux units equals the number of colors in the gauge theory picture.
2
The combination λ ≡ gYM
N that appears
Large N and the ‘t Hooft coupling.
on the right hand side of (1.6) is known as the ‘t Hooft coupling, and is familiar to field
theorists from ‘t Hooft’s generalization of SU(3) gauge theory (i.e. QCD) to SU(N ) gauge
group [15]. Increasing the number of colors, while keeping λ constant, implies that each
Feynman diagram acquires a topological factor of N raised to the power of the Euler
4
characteristic of the graph. Hence in the large N limit Feynman graphs that can be
drawn on a sphere (called planar) dominate, since they carry a factor of N 2 , while those
that can be drawn on a torus are subleading (of order N 0 ) and those requiring higher
genus surfaces even more suppressed. Thus making N large significantly simplifies field
theory; if in addition λ is small one can perform calculations perturbatively.
It has been argued that the sum over Feynman graphs with a given Euler characteristic can be thought of as a sum over string world-sheets. In such an interpretation the
string coupling constant would be proportional to N −1 , such that large N would lead to
a weakly coupled string theory. While this argument had never been made precise before
the arrival of the AdS/CFT correspondence, one can indeed check that in the full string
theory the string loop corrections to the classical solution considered above proceed in
powers of N −2 . Thus taking N to infinity allows us to use a classical description of
string theory on AdS5 × S5 , and for the remainder of this subsection we will assume the
large N limit with λ held fixed.
Now from (1.6) we see that the curvature radius L of the near-horizon geometry is
equal to λ1/4 in units of the string scale. Since the supergravity approximation to type
√
IIB string theory is only applicable for weakly curved spaces we require L α0 , which
implies λ 1. Hence the gravitational calculation is reliable precisely in the limit in
which field theory computations are not perturbative and therefore difficult.
The AdS5 /CFT4 correspondence with N = 4 supersymmetry.
At this point
we have arrived at two superficially completely different descriptions of 3-branes. On
the one hand, we have a field theory picture of a stack of N D3-branes in terms of
N = 4 supersymmetric SU(N ) gauge theory that is most useful for small λ. In the
low-energy limit the field theory decouples from the bulk closed string theory, and the
massless modes of the open strings living on the branes adequately describe the situation.
On the other hand, the supergravity picture of a black 3-brane is applicable for large
5
λ, and here the AdS5 × S5 region (at r L) decouples from the asymptotically flat
large r spacetime in the low-energy limit, as confirmed by investigations of so-called
greybody factors describing the cross-section for the absorption of radiation by the brane
[16, 17, 18]. This second description is purely gravitational, i.e. in terms of the massless
excitations of closed strings.
These two complementary points of view are at the core of the AdS/CFT correspondence. Maldacena [1] made the seminal conjecture that type IIB string theory on
AdS5 × S5 , of radius L as in (1.6), is dual to the N = 4 SU(N ) SYM theory.1 This
can be understood as a duality between open and closed string descriptions of the same
phenomena. Crucially, it is also a weak-strong duality, since the two pictures apply for
different regimes of λ. This renders the AdS/CFT correspondence very hard to prove
rigorously, but at the same time exceedingly useful as a computational tool.
The original conjecture of the AdS/CFT correspondence was partially motivated by
symmetry considerations [1]. The isometry supergroup of the AdS5 × S5 background is
PSU(2, 2|4), which perfectly matches the N = 4 superconformal symmetry. The maximal bosonic subgroup of this supergroup is SO(2, 4) × SO(6). The first factor is nothing
but the conformal group in 3+1 dimensions, which coincides with the isometry group
of AdS5 , since Anti-de Sitter space is the maximally symmetric space with (constant)
negative curvature. The second factor of SO(6) ∼ SU(4) appears as the R-symmetry
of the N = 4 SYM theory, while in the geometry it is realized simply as the isometry
group of the S5 .
The development of a detailed dictionary that allows one to translate from conformal
field theory to string theory language was initiated in [2, 3]. Each gauge invariant
operator in the conformal field theory corresponds to a particular field in the AdS5 × S5
background (or in some cases, as e.g. for the baryonic operators we will discuss below,
to an extended object such a D-brane), in such a way that its scaling dimension ∆ is
1
In its strongest form the conjecture states that this holds even for finite N .
6
related to the mass of the dual field. E.g. scalar operators satisfy ∆(∆ − 4) = m2 L2 .
For the fields in AdS5 that come from the type IIB supergravity modes, including
the Kaluza-Klein excitations on the 5-sphere, the masses are of order 1/L. Hence, it is
consistent to assume that their operator dimensions are independent of L, and therefore
independent of λ. This is due to the fact that such operators commute with some of
the supercharges and are thus protected by supersymmetry. Perhaps the simplest such
operators are the chiral primaries which are traceless symmetric polynomials in the six
real scalar fields. These operators are dual to spherical harmonics on S5 which mix
the graviton and RR 4-form fluctuations. Their masses are m2k = k(k − 4)/L2 , where
k = 2, 3, . . .. These masses reproduce the operator dimensions ∆ = k which are the
same as in the free theory. The situation is completely different for operators dual to
the massive string modes: m2n = 4n/α0 . In this case the AdS/CFT correspondence
predicts that the operator dimension grows at strong coupling as 2n1/2 λ1/4 .
In order to compute correlation functions of operators in the conformal field theory
using its dual fields ϕ one identifies a certain gauge theory quantity W with a string
theory quantity Zstring [2, 3]
W [ϕ0 (~x)] = Zstring [ϕ0 (~x)] .
(1.8)
Here W is the generator of the connected Euclidean Green’s function of a gauge theory
operator O,
Z
W [ϕ0 (~x)] = hexp
d4 x ϕ0 Oi .
(1.9)
On the other hand, Zstring is the string theory path integral calculated as a functional
of ϕ0 , the boundary condition on the field ϕ related to O by the AdS/CFT duality. In
the large N limit the string theory becomes classical, which implies
Zstring ∼ e−S[ϕ0 (~x)] ,
7
(1.10)
where S[ϕ0 (~x)] is the extremum of the classical string action which is again a functional
of ϕ0 . If we are further interested in correlation functions at very large ‘t Hooft coupling,
the problem of extremizing the classical string action reduces to solving the equations
of motion in type IIB supergravity.
Long operators and the cusp anomalous dimension.
Operators with large
quantum numbers provide a simple setting in which the AdS/CFT correspondence can
be tested very efficiently. E.g. the single-trace operators examined in [19] carry a large
R-charge (dual to a string angular momentum J on the compact space S5 ) and the dual
states are almost point-like closed strings moving rapidly on the five-sphere [20]. If we
consider such a long operator with a second large quantum number the dual object may
be a macroscopic semi-classical string [21], which allows for quantitative checks of the
correspondence (and in addition strongly suggests that string theory on AdS5 × S5 and
N = 4 Super Yang-Mills theory are integrable [22, 23]).
We can also consider operators with large spin S, such as the twist-2 operator [24, 25]
S−2
Tr F+µ D+
F+µ .
(1.11)
The scaling dimension ∆ of such twist-2 operators at large values of the spin S is
characterized by the universal scaling function (or cusp anomalous dimension) f (g):
∆ − S = f (g) ln S + O(S 0 ) ,
with g =
(1.12)
p
gY2 M N /4π. The logarithmic dependence of the dimension on large Lorentz
spin is a generic feature that has been independently observed for both N = 4 SYM and
its string theory dual, see e.g. [26, 27, 28]. Importantly, such a scaling behavior stems
[29, 30] from the large spin limit of the Bethe equations [31, 32, 33, 34] which underlie
the integrable structures of gauge and string theories.
8
Due to the universal role played by the function f (g), one would like to compute it
in the N = 4 SYM theory. In this case we can consider operators in the SL(2) sector,
of the form
Tr DS Z J + . . . ,
(1.13)
where Z is one of the complex scalar fields, the R-charge J is the twist, and the dots
serve as a reminder that the operator is a linear combination of such terms with the
covariant derivatives acting on the scalars in all possible ways. The object dual to such
a high-spin twist-2 operator is a folded string [20] spinning around the center of AdS5 ;
its generalization to large J was found in [28]. The result (1.12) is generally applicable
when J is held fixed while S is sent to infinity [29].
Though the function f (g) for N = 4 SYM is not the same as for QCD, its perturbative
expansion is in fact related by the conjectured transcendentality principle [35], which
states that each expansion coefficient has terms of definite degree of transcendentality
(namely the exponent of g minus two), and that the QCD result contains the same terms
(in addition to others which have lower degree of transcendentality).
An interesting problem is to smoothly match the explicit predictions of string theory
for large g to those of gauge theory at small g. During the past few years methods
of integrability in AdS/CFT [36, 37, 38] have led to major progress in addressing this
question. In an impressive series of papers [30, 39, 40] a linear integral equation has
been derived, which allows one to compute the universal scaling function f (g) to any
desired order in perturbation theory. We will refer to it as the BES equation.
It was obtained from the asymptotic Bethe ansatz for the SL(2) sector by considering
the limit S → ∞ with J finite, and extracting the piece proportional to ln S, which is
manifestly independent of J. Taking the spin to infinity, the discrete Bethe equations
can be rewritten as an integral equation for the density of Bethe roots in rapidity space.
In Chapter 2 we shall discuss how to extract the strong coupling behavior of f (g)
from the BES equation, and show that it agrees with the predictions from string theory.
9
1.2
Generalizations to Lower Supersymmetry and
the Non-Conformal Case
1.2.1
Reducing Supersymmetry and the Conifold
To obtain an AdS/CFT duality with less than the maximal N = 4 supersymmetry, we
consider a stack of D3-branes located at the singularity of a 6-dimensional Ricci-flat
cone [41, 42, 43, 44]. The metric is then given by
ds2 = h−1/2 (r)ηµν dxµ dxν + h1/2 (r) dr2 + r2 ds2Y
.
(1.14)
where ds2Y is the metric of the 5-dimensional compact space Y5 , which is an Einstein
manifold (i.e. Rab = 4gab ) and forms the base of the cone. The geometry dual to the
conformal field theory supported by the D3-branes at the tip of the cone emerges in
the near horizon limit. It is simply AdS5 × Y5 , which still has the SO(2, 4) conformal
symmetry manifest in the geometry, even though the R-symmetry group which must be
contained in the isometries of Y5 will in general be smaller than SU(4).
In order to find gauge theories with N = 1 superconformal symmetry the Ricci-flat
cone must be a Calabi-Yau 3-fold [44, 45] whose base Y5 is called a Sasaki-Einstein space.
Among the simplest examples of these is Y5 = T 1,1 . The corresponding Calabi-Yau cone
is called the conifold.
The conifold is a singular non-compact Calabi-Yau three-fold [46]. Its importance
arises from the fact that the generic singularity in a Calabi-Yau three-fold locally looks
like the conifold, described by the quadratic equation in C4 :
z12 + z22 + z32 + z42 = 0 .
(1.15)
This homogeneous equation defines a real cone over the 5-dimensional manifold T 1,1 .
10
The topology of T 1,1 can be shown to be S2 × S3 and its metric [47] is
ds2T 1,1 =
1
(dψ + cos θ1 dφ1 + cos θ2 dφ2 )2
9
1
1
+ (dθ12 + sin2 θ1 dφ21 ) + (dθ22 + sin2 θ2 dφ22 ) .
6
6
(1.16)
T 1,1 is a homogeneous space, and can be described as the coset SU(2) × SU(2)/U(1).
The metric on the cone is then ds26 = dr2 + r2 ds2T 1,1 .
Symmetries and field theory interpretation.
We can introduce different complex
coordinates on the conifold, ai and bj , as follows:




 z3 + iz4 z1 − iz2   a1 b1 a1 b2 
=

Z =
 


a2 b 1 a2 b2
z1 + iz2 −z3 + iz4


i
i
 −c1 s2 e 2 (ψ+φ1 −φ2 ) c1 c2 e 2 (ψ+φ1 +φ2 ) 
 ,
=r 


i
i
−s1 s2 e 2 (ψ−φ1 −φ2 ) s1 c2 e 2 (ψ−φ1 +φ2 )
3
2
(1.17)
where ci = cos θ2i , si = sin θ2i (see [46] for more details on the z and angular coordinates).
The equation defining the conifold is now det Z = 0.
The ai , bj coordinates above will be of particular interest to us because the symmetries
of the conifold are most apparent in this basis. The conifold equation has SU(2)×SU(2)×
U(1) symmetry since under these symmetry transformations,
det LZR† = det eiα Z = 0.
(1.18)
This is also a symmetry of the metric presented above where each SU(2) acts on θi , φi , ψ
(thought of as Euler angles on S3 ) while the U(1) acts by shifting ψ. This symmetry can
be identified with U(1)R , the R-symmetry of the dual gauge theory, in the conformal
case. The action of the SU(2) × SU(2) × U(1)R symmetry on ai , bj defined in (1.17) is
11
given by
SU(2) × SU(2) symmetry :
R-symmetry :
a1
a1
→L
,
a2
a2
b1
b1
→R
,
b2
b2
α
(ai , bj ) → ei 2 (ai , bj ) ,
(1.19)
(1.20)
(1.21)
i.e. a and b transform as (1/2, 0) and (0, 1/2) under SU(2) × SU(2) and with R-charge
1/2 each. We can thus describe the singular conifold as the manifold parametrized by
a and b, but from (1.17), we see that there is some redundancy in the a, b coordinates.
Namely, the transformation
ai → λ ai ,
bj →
1
bj ,
λ
(λ ∈ C) ,
(1.22)
results in the same z coordinates in (1.17). Thus we impose the additional constraint
|a1 |2 + |a2 |2 − |b1 |2 − |b2 |2 = 0 to fix the magnitude in the above transformation. To
account for the remaining phase, we describe the singular conifold as the quotient of the
a, b space with the above constraint by the relation a ∼ eiα a, b ∼ e−iα b.
The importance of the coordinates ai , bj is that in the gauge theory on D3-branes at
the tip of the conifold they are promoted to chiral superfields. The low-energy gauge
theory on N D3-branes is an N = 1 supersymmetric SU(N ) × SU(N ) gauge theory
with bifundamental chiral superfields Ai , Bj (i, j = 1, 2) in the (N, N) and (N, N)
representations of the gauge groups, respectively [44, 45]. The superpotential for this
gauge theory is
W ∼ Tr det Ai Bj = Tr (A1 B1 A2 B2 − A1 B2 A2 B1 ) .
(1.23)
The continuous global symmetries of this theory are SU(2) × SU(2) × U(1)B × U(1)R ,
where the SU(2) factors act on Ai and Bj respectively, U(1)B is a baryonic symmetry
12
under which the Ai and Bj have opposite charges, and U(1)R is the R-symmetry with
charges of the same sign RA = RB = 21 . This assignment ensures that W is marginal,
and one can also show that the gauge couplings do not run. Hence this theory is
superconformal for all values of gauge couplings and superpotential coupling [44, 45].
Resolution of the conifold.
A simple way to understand the resolution of the
conifold is to deform the modulus constraint above into
|b1 |2 + |b2 |2 − |a1 |2 − |a2 |2 = u2 ,
(1.24)
where u is a real parameter which controls the resolution. The resolution corresponds to
a blow up of the S2 at the bottom of the conifold. In the dual gauge theory turning on u
corresponds to a particular choice of vacuum [48]. After promoting the a, b fields to the
bifundamental chiral superfields of the dual gauge theory, we can define the operator U
as
U=
1
Tr(B1† B1 + B2† B2 − A†1 A1 − A†2 A2 ) .
N
(1.25)
Thus, the singular conifolds correspond to gauge theory vacua where hUi = 0, while the
warped resolved conifolds correspond to vacua where hUi =
6 0. In the latter case, some
VEVs for the bifundamental fields Ai , Bj must be present. Since these fields are charged
under the U (1)B symmetry, the warped resolved conifolds correspond to vacua where this
symmetry is broken [48]. In this thesis we shall not discuss the resolved conifold further
(except in the context of the baryonic branch), but we shall be interested instead in the
deformation of the conifold singularity, which is the subject of the following subsection.
1.2.2
Deformation of the Conifold
We have seen above that the singularity of the cone over T 1,1 can be replaced by an
S2 through resolving the conifold (1.15) as in (1.24). An alternative supersymmetric
13
blow-up, which replaces the singularity by an S3 , is the deformed conifold [46]
z12 + z22 + z32 + z42 = ε2 .
(1.26)
To achieve the deformation, one needs to turn on M units of RR 3-form flux. This
modifies the dual gauge theory to N = 1 supersymmetric SU(N ) × SU(N + M ) theory with chiral superfields A1 , A2 in the (N, N + M) representation, and B1 , B2 in the
(N, N + M) representation.
The necessary 3-form field strength linking the three-cycle of the conifold can be
turned on by wrapping M D5-branes on the S2 at the tip of the cone. They are often
referred to as fractional branes. Since their number is dual to the difference in the ranks
of the two gauge groups, the deformation, unlike the resolution, cannot be achieved in
the context of the SU(N ) × SU(N ) gauge theory.
The KS solution.
The 10-dimensional metric takes the following form [49]:
ds210 = h−1/2 (τ ) ηµν dxµ dxν + h1/2 (τ ) ds26 ,
(1.27)
where h(τ ) is a warp factor to be discussed below, and ds26 is the Calabi-Yau metric of
the deformed conifold:
ds26 =
h
τ ε4/3
K(τ ) sinh2
(g 1 )2 + (g 2 )2
2
2
τ 2
i
1
5 2
+ cosh2
(g 3 )2 + (g 4 )2 +
dτ
+
(g
)
,
2
3K 3 (τ )
(1.28)
with K(τ ) ≡ (sinh τ cosh τ − τ )1/3 / sinh τ .
For τ 1 we may introduce another radial coordinate r defined by
r2 =
3
25/3
ε4/3 e2τ /3 ,
14
(1.29)
and in terms of this coordinate we find ds26 → dr2 + r2 ds2T 1,1 .
The basis one-forms g i in terms of which this metric is diagonal are defined by
e2 − 2
e1 − 1
√
, g2 ≡ √
,
2
2
e2 + 2
e1 + 1
g3 ≡ √
, g4 ≡ √
,
2
2
g 5 ≡ 3 + cos θ1 dφ1 ,
g1 ≡
(1.30)
(1.31)
(1.32)
where the ei are one-forms on S2
e1 ≡ dθ1 ,
e2 ≡ − sin θ1 dφ1 ,
(1.33)
1 ≡ sin ψ sin θ2 dφ2 + cos ψdθ2 ,
(1.34)
2 ≡ cos ψ sin θ2 dφ2 − sin ψdθ2 ,
(1.35)
3 ≡ dψ + cos θ2 dφ2 .
(1.36)
and the i a set of one-forms on S3
The NSNS two-form is given by
B2 =
τ τ i
gs M α0 τ coth τ − 1 h
sinh2
g 1 ∧ g 2 + cosh2
g3 ∧ g4 ,
2
sinh τ
2
2
(1.37)
and the RR fluxes are most compactly written as
M α0
F3 =
2
sinh τ − τ 1
3
4
5
3
2
4
g ∧g ∧g +d
(g ∧ g + g ∧ g ) ,
2 sinh τ
F̃5 = dC4 + B2 ∧ F3 = (1 + ∗) (B2 ∧ F3 ) .
(1.38)
(1.39)
Note that the complex three-form field of this BPS supergravity solution is imaginary
15
self-dual:
∗6 G3 = iG3 ,
G3 = F3 −
i
H3 ,
gs
(1.40)
where ∗6 denotes the Hodge dual with respect to the unwarped metric ds26 . This guarantees that the dilaton is constant, and we set φ = 0.
The above expressions for the NSNS- and RR-forms follow by making a simple ansatz
consistent with the symmetries of the problem, and solving a system of differential
equations, which owing to the supersymmetry of the problem are only first order [49].
The warp factor is then found to be completely determined up to an additive constant,
which is fixed by demanding that it go to zero at large τ :
h(τ ) = (gs M α0 )2 22/3 ε−8/3 I(τ ) ,
Z ∞
x coth x − 1
1/3
I(τ ) ≡ 2
dx
(sinh x cosh x − x)1/3 .
2
sinh
x
τ
(1.41)
(1.42)
For small τ the warp factor approaches a finite constant since I(0) ≈ 0.71805. This
implies confinement because the chromo-electric flux tube, described by a fundamental
string at τ = 0, has tension
Ts =
2πα0
1
p
.
h(0)
(1.43)
The KS solution [49] is SU(2) × SU(2) symmetric and the expressions above can be
written in an explicitly SO(4) invariant way. It also possesses a Z2 symmetry I, which
exchanges (θ1 , φ1 ) with (θ2 , φ2 ) accompanied by the action of −I of SL(2, Z), changing
the signs of the three-form fields.
Examining the metric ds26 for τ = 0 we see that it degenerates into
dΩ23
1 4/3
1 5 2
1/3
3 2
4 2
= ε (2/3)
(g ) + (g ) + (g ) ,
2
2
(1.44)
which is the metric of a round S3 , while the S2 spanned by the other two angular
coordinates, and fibered over the S3 , shrinks to zero size. In the ten-dimensional metric
16
(1.27) this appears multiplied by a factor of h1/2 (τ ), and thus the radius squared of
the three-sphere at the tip of the conifold is of order gs M α0 . Hence for gs M large, the
curvature of the S3 , and in fact everywhere in this manifold, is small and the supergravity
approximation reliable.
The field theory interpretation of the KS solution exhibits some unusual features.
The deformation breaks the conformal symmetry of the singular conifold and thus the
gauge couplings now run. At a certain point along the RG flow, the coupling of the
higher rank SU(N + M ) group diverges and one is forced to perform a Seiberg duality
transformation [50].
Interchanging the two gauge groups, the theory becomes SU(Ñ ) × SU(Ñ + M ) with
the same bifundamental field content and superpotential, but with a reduced number of
colors Ñ = N − M . Since this otherwise looks exactly like field theory we started with,
the procedure can be iterated many times, gradually reducing the rank of the gauge
groups along the RG flow. For a careful field theoretic discussion of this quasi-periodic
RG flow, see [12]. In particular, if N = (k +1)M for some integer k, the so-called duality
cascade stops after k steps, resulting in an SU(M ) × SU(2M ) gauge theory. This IR
field theory exhibits a number of interesting effects reflected in the dual supergravity
background, which include confinement and chiral symmetry breaking.
The KT solution.
Let us briefly note here some formulas describing the KT solu-
tion [51], which corresponds to the large τ limit of the more general KS solution. For
simplicity we take gs = α0 = 1, M = 2 and N = 0. In terms of the radial coordinate
r ∼ ε2/3 eτ /3 the KT background is given by
1
ds2 = p
h(r)
ηµν dxµ dxν +
p
h(r)(dr2 + r2 ds2T 11 ) ,
3
dr ∧ ω2 ,
r
r
B2 = 3 log ω2 ,
F3 = ω 3 ,
r∗
i
r h
54
F̃5 = (1 + ∗) B2 ∧ F3 = 3 log
ω2 ∧ ω3 − 2 5 d4 x ∧ dr .
r∗
hr
H3 =
17
(1.45)
(1.46)
(1.47)
The warp factor differs from the AdS case by an additional logarithmic term
81
h(r) = 4
8r
r
1 + 4 log
,
r∗
(1.48)
and the metric of the base of the conifold can be expressed concisely as
4
1
1X i 2
ds2T 11 = (g 5 )2 +
(g ) .
9
6 i=1
(1.49)
√ 5
hr 4
d x ∧ ω2 ∧ ω3 ∧ dr .
vol =
54
(1.50)
The volume form is given by
In the expressions above we have introduced two harmonic forms,
1
1
ω2 = (g 1 ∧ g 2 + g 3 ∧ g 4 ) = (sin θ1 dθ1 ∧ dφ1 − sin θ2 dθ2 ∧ dφ2 ) ,
2
2
(1.51)
and ω3 = ω2 ∧ g 5 .
1.2.3
Massless Modes of the Warped Throat
As we shall see below, the U(1) baryonic symmetry of the warped deformed conifold is in
fact spontaneously broken, since baryonic operators acquire expectation values. The corresponding Goldstone boson is a massless pseudoscalar supergravity fluctuation which
has non-trivial monodromy around D-strings at the bottom of the warped deformed
conifold [52, 53]. Like fundamental strings they fall to the bottom of the conifold (corresponding to the IR of the field theory), where they have non-vanishing tension. But
while F-strings are dual to confining strings, D-strings are interpreted as global solitonic
strings in the dual cascading SU(M (k + 1)) × SU(M k) gauge theory.
Thus the warped deformed conifold naturally incorporates a supergravity description
18
of the supersymmetric Goldstone mechanism. Below we review the supergravity dual of
a pseudoscalar Goldstone boson, as well as its superpartner, a massless scalar glueball
[52, 53]. In the gauge theory they correspond to fluctuations in the phase and magnitude
of the baryonic condensates, respectively.
The Goldstone mode.
A D1-brane couples to the three-form field strength F3 , and
therefore we expect a four-dimensional pseudoscalar p(x), defined so that ∗4 dp = δF3 ,
to experience monodromy around the D-string.
The following ansatz for a linear perturbation of the KS solution
δF3 = ∗4 dp + f2 (τ ) dp ∧ dg 5 + f20 (τ ) dp ∧ dτ ∧ g 5 ,
(1.52)
4/3
δ F̃5 = (1 + ∗)δF3 ∧ B2 = (∗4 dp −
ε
h(τ ) dp ∧ dτ ∧ g 5 ) ∧ B2 ,
6K 2 (τ )
where f20 = df2 /dτ , falls within the general class of supergravity backgrounds discussed
by Papadopoulos and Tseytlin [54]. The metric, dilaton and B2 field remain unchanged.
This can be shown to satisfy the linearized supergravity equations [52, 53], provided
that d ∗4 dp = 0, i.e. p(x) is massless, and f2 (τ ) satisfies
d
8
(gs M α0 )2
τ
2
4
0
− [K sinh τ f2 ] +
f2 =
(τ coth τ − 1) coth τ −
.
dτ
9K 2
3ε4/3
sinh2 τ
(1.53)
The normalizable solution of this equation is given by [52, 53]
2c
f2 (τ ) = − 2
K sinh2 τ
Z
τ
dx h(x) sinh2 x ,
(1.54)
0
where c ∼ ε4/3 . We find that f2 ∼ τ for small τ , and f2 ∼ τ e−2τ /3 for large τ .
As we have remarked above, the U(1) baryon number symmetry acts as Ak → eiα Ak ,
Bj → e−iα Bj . The massless gauge field in AdS5 dual to the baryon number current
originates from the RR 4-form potential δC4 ∼ ω3 ∧ Ã [44, 55].
19
The zero-mass pseudoscalar glueball arises from the spontaneous breaking of the
global U(1)B symmetry [56], as seen from the form of δ F̃5 in (1.52), which contains a
term ∼ ω3 ∧ dp ∧ dτ that leads us to identify à ∼ dp.
If N is an integer multiple of M , the last step of the cascade leads to a SU(2M ) ×
SU(M ) gauge theory coupled to bifundamental fields Ai , Bj (with i, j = 1, 2). If the
SU(M ) gauge coupling were turned off, we would find an SU(2M ) gauge theory coupled
to 2M flavors. In this Nf = Nc case, in addition to the usual mesonic branch, there exists
a baryonic branch of the quantum moduli space [57]. This is important for the gauge
theory interpretation of the KS background [49, 56]. Indeed, in addition to mesonic
operators (Nij )αβ ∼ (Ai Bj )αβ , the IR gauge theory has baryonic operators invariant under
the SU(2M ) × SU(M ) gauge symmetry, as well as the SU(2) × SU(2) global symmetry
rotating Ai , Bj :
α
α
(1.55)
α
α
(1.56)
A ∼ α1 α2 ...α2M (A1 )α1 1 (A1 )α2 2 . . . (A1 )αMM (A2 )1 M +1 (A2 )2 M +2 . . . (A1 )αM2M ,
B ∼ α1 α2 ...α2M (B1 )α1 1 (B1 )α2 2 . . . (B1 )αMM (B2 )1 M +1 (B2 )2 M +2 . . . (B1 )αM2M .
These operators contribute an additional term to the usual mesonic superpotential:
W = λ(Nij )αβ (Nk` )βα ik j` + X det[(Nij )αβ ] − AB − Λ4M
2M
,
(1.57)
where X can be understood as a Lagrange multiplier. The supersymmetry-preserving
vacua include the baryonic branch:
X=0;
Nij = 0 ;
AB = −Λ4M
2M ,
(1.58)
where the SO(4) global symmetry rotating Ai , Bj is unbroken. In contrast, this global
symmetry is broken along the mesonic branch Nij 6= 0. Since the supergravity background of [49] is SO(4) symmetric, it is natural to assume that the dual of this back-
20
ground lies on the baryonic branch of the cascading theory. The expectation values of the
baryonic operators spontaneously break the U(1) baryon number symmetry Ak → eiα Ak ,
Bj → e−iα Bj . The KS background corresponds to a vacuum where |A| = |B| = Λ2M
2M ,
which is invariant under the exchange of the A’s with the B’s accompanied by charge
conjugation in both gauge groups. This gives a field theory interpretation to the I
symmetry of the warped deformed conifold background. As noted in [56], the baryonic
branch has complex dimension one, and it can be parametrized by ζ as follows
i
B = Λ2M
.
ζ 2M
A = iζΛ2M
2M ,
(1.59)
The pseudoscalar Goldstone mode must correspond to changing ζ by a phase, since this
is precisely what a U(1)B symmetry transformation does.
Thus the non-compact warped deformed conifold exhibits a supergravity dual of the
Goldstone mechanism due to breaking of the global U(1)B symmetry [52, 53, 56]. On
the other hand, if one considered a warped deformed conifold throat embedded in a
flux compactification, U(1)B would be gauged, the Goldstone boson p(x) would combine
with the U(1) gauge field to form a massive vector, and therefore in this situation we
would find a manifestation of the supersymmetric Higgs mechanism.
The scalar zero-mode.
By supersymmetry the massless pseudoscalar is part of
a massless N = 1 chiral multiplet. and therefore there must also be a massless scalar
mode and corresponding Weyl fermion, with the scalar corresponding to changing ζ by a
positive real factor. This scalar zero-mode comes from a metric perturbation that mixes
with the NSNS 2-form potential.
The warped deformed conifold preserves the Z2 interchange symmetry I. However,
the pseudoscalar mode we found breaks this symmetry: from the form of the perturbations (1.52) we see that δF3 is even under the interchange of (θ1 , φ1 ) with (θ2 , φ2 ),
while F3 is odd; similarly δ F̃5 is odd while F̃5 is even. Therefore, the scalar mode must
21
also break the I symmetry because in the field theory it breaks the symmetry between
the expectation values of |A| and of |B|. The necessary translationally invariant perturbation that preserves the SO(4) but breaks the I symmetry is given by the following
variation of the NSNS 2-form and the metric:
δB2 = χ(τ ) dg 5 ,
δG13 = δG24 = λ(τ ) ,
(1.60)
where, for example δG13 = λ(τ ) means adding 2λ(τ ) g (1 g 3) to ds210 . To see that these
components of the metric break the I symmetry, we note that
(e1 )2 + (e2 )2 − (1 )2 − (2 )2 = g 1 g 3 + g 3 g 1 + g 2 g 4 + g 4 g 2 .
(1.61)
Defining λ(τ ) = h1/2 K sinh τ z(τ ) one finds [52, 53] that all the linearized supergravity equations are satisfied provided that
(K sinh τ )2 z 0
(K sinh τ )2
0
=
8 1
4 cosh τ
2+
−
6
9K
3 K3
z
,
sinh2 τ
(1.62)
and
sinh 2τ − 2τ
1
.
χ0 = gs M z(τ )
2
sinh2 τ
(1.63)
The solution of (1.62) for the zero-mode profile is remarkably simple:
z(τ ) = s
(τ coth τ − 1)
,
(sinh 2τ − 2τ )1/3
(1.64)
where s is a constant. Like the pseudoscalar perturbation, the large τ asymptotic is
again z ∼ τ e−2τ /3 . We note that the metric perturbation has the very simple form
δG13 ∼ h1/2 (τ coth τ − 1). The perturbed metric ds̃26 differs from the metric of the
deformed conifold (1.28) by terms ∼ (τ coth τ − 1)(g 1 g 3 + g 3 g 1 + g 2 g 4 + g 4 g 2 ), which
grow as ln r in the asymptotic radial variable r.
22
The scalar zero-mode is actually an exact modulus; there is a one-parameter family
of supersymmetric solutions which break the I symmetry but preserve the SO(4) (an
ansatz with these properties was found in [54], and its linearization agrees with (1.60)).
These backgrounds, the resolved warped deformed conifolds, will be reviewed below.
We add the word resolved because both the resolution of the conifold, which is a Kähler
deformation, and these resolved warped deformed conifolds break the I symmetry. In the
dual gauge theory turning on this mode corresponds to the transformation A → (1+s)A,
B → (1+s)−1 B on the baryonic branch. Therefore, s is dual to the I-breaking parameter
of the resolved warped deformed conifold.
The presence of these massless modes is a further indication that the infrared dynamics of the cascading SU(M (k + 1)) × SU(M k) gauge theory, whose supergravity dual is
the warped deformed conifold, is richer than that of the pure glue N = 1 supersymmetric SU(M ) theory. The former incorporates a Goldstone supermultiplet, which appears
due to the U(1)B symmetry breaking, as well as solitonic strings dual to the D-strings
placed at τ = 0 in the supergravity background.
In Chapter 3 we will discuss massive modes of the warped throat, which arise from
generalizing the massless scalar and pseudoscalar modes by adding 4-dimensional momentum, as well as some massive vector excitations related to them by supersymmetry.
1.2.4
The Baryonic Branch
Since the global baryon number symmetry U(1)B is broken by expectation values of
baryonic operators, the spectrum contains the Goldstone boson found above. The
zero-momentum mode of the scalar superpartner of the Goldstone mode leads to a
Lorentz-invariant deformation of the background which describes a small motion along
the baryonic branch. In this subsection we shall extend the discussion from linearized
perturbations around the warped deformed conifold solution to finite deformations, and
describe the supergravity backgrounds dual to the complete baryonic branch. These are
23
the resolved warped deformed conifolds, which preserve the SO(4) global symmetry but
break the discrete I symmetry of the warped deformed conifold.
The full set of first-order equations necessary to describe the entire moduli space of
supergravity backgrounds dual to the baryonic branch was derived and solved numerically in [58] (for further discussion, see [59]). This continuous family of supergravity
solutions is parameterized by the modulus of ζ (the phase of ζ is not manifest in these
backgrounds). The corresponding metric can be written in the form of the PapadopoulosTseytlin ansatz [54] in the string frame:
2
−1/2
ds = h
µ
ν
ηµν dx dx + e
x
ds2M
−1/2
=h
dx21,3
+
6
X
G2i ,
(1.65)
i=1
where
eg
cosh τ + a (x+g)/2
e
e2 +
e(x−g)/2 (2 − ae2 ) , (1.66)
sinh τ
sinh τ
g
e
cosh
τ + a (x−g)/2
G3 ≡ e(x−g)/2 (1 − ae1 ) , G4 ≡
e(x+g)/2 e2 −
e
(2 − ae2 ) , (1.67)
sinh τ
sinh τ
G1 ≡ e(x+g)/2 e1 ,
G2 ≡
G5 ≡ ex/2 v −1/2 dτ ,
G6 ≡ ex/2 v −1/2 g 5 .
(1.68)
These one-forms describe a basis that rotates as we move along the radial direction,
and are particularly convenient since they allow us to write down very simple expressions
for the holomorphic (3, 0) form
Ω = (G1 + iG2 ) ∧ (G3 + iG4 ) ∧ (G5 + iG6 ) ,
(1.69)
and the fundamental (1, 1) form
J=
i
ih
(G1 +iG2 )∧(G1 −iG2 )+(G3 +iG4 )∧(G3 −iG4 )+(G5 +iG6 )∧(G5 −iG6 ) . (1.70)
2
While in the warped deformed conifold case there was a single warp factor h(τ ),
24
now we find several additional functions x(τ ), g(τ ), a(τ ), v(τ ). The warp factor h(τ ) is
deformed away from (1.41) when |ζ| 6= 1.
The background also contains the fluxes
B2 = h1 (1 ∧ 2 + e1 ∧ e2 ) + χ (e1 ∧ e2 − 1 ∧ 2 ) + h2 (1 ∧ e2 − 2 ∧ e1 ) , (1.71)
1
F3 = − g5 ∧ 1 ∧ 2 + e1 ∧ e2 − b (1 ∧ e2 − 2 ∧ e1 )
2
1
(1.72)
− dτ ∧ b0 (1 ∧ e1 + 2 ∧ e2 ) ,
2
F̃5 = F5 + ∗10 F5 ,
with F5 = −(h1 + b h2 ) e1 ∧ e2 ∧ 1 ∧ 2 ∧ 3 ,
(1.73)
parameterized by functions h1 (τ ), h2 (τ ), b(τ ) and χ(τ ). In addition, since the 3-form flux
is not imaginary self-dual for |ζ| 6= 1 (i.e. ∗6 G3 6= iG3 ), the dilaton φ now also depends
on the radial coordinate τ .
The functions a and v satisfy a system of coupled first order differential equations
[58] whose solutions are known in closed form only in the warped deformed conifold
and the Chamseddine-Volkov-Maldacena-Nunez (CVMN) [60, 61, 62] limits. All other
functions h, x, g, h1 , h2 , b, χ, φ are unambiguously determined by a(τ ) and v(τ ) through
the relations
h = γ U −2 e−2φ − 1 ,
γ = 210/3 (gs M α0 )2 ε−8/3 ,
(1.74)
2
e2x =
(b C − 1)
e2g+2φ (1 − e2φ ) ,
4(a C − 1)2
τ
,
S
(1.76)
h1 = −h2 C ,
(1.77)
e2g = −1 − a2 + 2a C ,
h2 =
b=
e2φ (b C − 1)
,
2S
(1.75)
χ0 = a(b − C)(a C − 1) e2(φ−g) ,
(1.78)
2
φ0 =
(C − b) (a C − 1) −2 g
e
,
(b C − 1) S
(1.79)
where C ≡ − cosh τ, S ≡ − sinh τ , and we require φ(∞) = 0. In writing these equations
we have specialized to the baryonic branch of the cascading gauge theory by imposing
25
appropriate boundary conditions at infinity [59], which guarantee that the background
asymptotes to the warped deformed conifold solution [51]. The full two-parameter family of SU(3) structure backgrounds discussed in [58] also includes the CVMN solution
[60, 61, 62], which however is characterized by linear dilaton asymptotics that are qualitatively different from the backgrounds discussed here. The baryonic branch family of
supergravity solutions is labelled by one real resolution parameter U [59]. While the
leading asymptotics of all supergravity backgrounds dual to the baryonic branch are
identical to those of the warped deformed conifold, terms subleading at large τ depend
on U . As required, this family of supergravity solutions preserves the SU(2) × SU(2)
symmetry, but for U 6= 0 breaks the Z2 symmetry I.
On the baryonic branch we can consider a transformation that takes ζ into ζ −1 , or
equivalently U into −U . This transformation leaves v(τ ) invariant and changes a(τ ) as
follows
a→−
a
.
1 + 2a cosh τ
(1.80)
It is straightforward to check that a e−g is invariant while (1 + a cosh τ ) e−g changes sign.
This transformation also exchanges eg + a2 e−g with e−g and therefore it is equivalent to
the exchange of (θ1 , φ1 ) and (θ2 , φ2 ) involved in the I symmetry.
The baryonic condensates can be calculated on the string theory side of the duality
by identifying the Euclidean D5-branes wrapped over the resolved warped deformed
conifold, with appropriate gauge fields turned on, as the objects dual to the baryonic
operators in the sense of gauge/string duality. The details of this identification will be
the subject of Chapter 4.
26
1.3
Superconformal Chern-Simons Theories and an
AdS4/CFT3 Duality
Let us now turn to an example of the AdS/CFT correspondence involving 2+1 dimensional field theory. A long-standing open problem in this area had been to find the
gauge theory dual to the near-horizon geometry of a stack of M2-branes, the AdS4 × S7
background of M-theory.
Significant progress towards this goal was made in the work of Bagger and Lambert
[63, 64, 65], and the closely related work of Gustavsson [66], who succeeded in finding a
2+1 dimensional superconformal Chern-Simons theory with the maximal N = 8 supersymmetry and manifest SO(8) R-symmetry. These papers were inspired in part by the
ideas of [67, 68], whose original motivation was the search for a theory describing coincident M2-branes. An interesting clue emerged in [69, 70] where it was shown that, for a
specially chosen level of the Chern-Simons gauge theory, its moduli space coincides with
that of a pair of M2-branes at the R8 /Z2 singularity. The Z2 acts by reflection of all 8
coordinates and therefore does not spoil the SO(8) symmetry. However, initial attempts
to match the moduli space of the Chern-Simons gauge theory for arbitrary quantized
level k with that of M2-branes led to a number of puzzles [69, 70, 71]. These puzzles were
resolved by a very interesting modification of the Bagger-Lambert-Gustavsson (BLG)
theory proposed by Aharony, Bergman, Jafferis and Maldacena (ABJM) [72] which, in
particular, allows for a generalization to an arbitrary number of M2-branes.
The original BLG theory is a particular example of a Chern-Simons gauge theory
with gauge group SO(4), but the Chern-Simons term has a somewhat unconventional
form. However, van Raamsdonk [71] rewrote the BLG theory as an SU(2) × SU(2) gauge
theory coupled to bifundamental matter. He found conventional Chern-Simons terms
for each of the SU(2) gauge fields although with opposite signs, as noted already in [73].
A more general class of gauge theories of this type was introduced by Gaiotto and Wit-
27
ten (GW) [74] following [75]. In this formulation the opposite signs for the two SU(N )
Chern-Simons terms are related to the SU(N |N ) supergroup structure. Although the
GW formulation generally has only N = 4 supersymmetry, it was recently shown how
to enlarge the supersymmetry by adding another hypermultiplet [72, 76]. In particular,
the maximally supersymmetric BLG theory emerges in the SU(2) × SU(2) case when
the matter consists of two bifundamental hypermultiplets. Furthermore, the brane constructions presented in [72, 74] indicate that the relevant gauge theories are actually
U(N ) × U(N ). The presence of the extra interacting U(1) compared to the original BLG
formulation is crucial for the complete M-theory interpretation [72].
Below we will present the BLG theory using N = 2 superspace formulation in 2+1
dimensions, which is quite similar to the familiar N = 1 superspace in 3+1 dimensions.
In such a formulation only the U(1)R symmetry is manifest, while the quartic superpotential has an additional SU(4) global symmetry. For a specially chosen normalization
of the superpotential, the full scalar potential is manifestly SO(8) invariant. In Section
1.3.1, where we establish the superspace formulation of the BLG theory after briefly reviewing its components formulation, we demonstrate how this happens through a special
cancellation involving the F- and D-terms.2
In Section 1.3.2 we study its generalizations to U(N ) × U(N ) gauge theory found by
ABJM [72]. The quartic superpotential of this 2+1 dimensional theory has exactly the
same form as in the 3+1 dimensional theory on N D3-branes at the conifold singularity
[44]. For general N , its global symmetry is SU(2) × SU(2) but for N = 2 it becomes
enhanced to SU(4) [77] (in this case the theory becomes equivalent to the BLG theory
with an extra gauged U(1) [72]). For N > 2 ABJM showed that this theory possesses
N = 6 supersymmetry [72]. In the N = 2 superspace formulation, this means that, for
a specially chosen normalization of the superpotential, the global symmetry is enhanced
2
This phenomenon is analogous to what happens when the N = 4 SYM theory in 3+1 dimensions
is written in terms of an N = 1 gauge theory coupled to three chiral superfields. While only the
U(1)R × SU(3) symmetry is manifest in such a formulation, the full SU(4) ∼ SO(6) symmetry is found
in the potential as a result of a specific cancellation between the F- and D-terms.
28
to SU(4)R . We demonstrate explicitly how this symmetry enhancement happens in
terms of the component fields, once again due to a special cancellation involving F- and
D-terms.
1.3.1
BLG Theory in Component and N = 2 Superspace Formulation
Here we review the BLG theory in van Raamsdonk’s product gauge group formulation
[71], which rewrites it as a superconformal Chern-Simons theory with SU(2) × SU(2)
gauge group and bifundamental matter. It has a manifest global SO(8) R-symmetry
which shows that it is N = 8 supersymmetric.
We use the following notation. Indices transforming under the first SU(2) factor of
the gauge group are a, b, . . ., and for the second factor we use â, b̂, . . .. Fundamental
indices are written as superscript and anti-fundamental indices as subscript. Thus, the
gauge and matter fields are Aa b , Ââ b̂ , X a b̂ , and Ψa b̂ . The conjugate fields have indices
(X † )â b and (Ψ† )â b . Most of the time, however, we will use matrix notation and suppress
gauge indices. Lorentz indices are µ = 0, 1, 2 and the metric on the world volume is
gµν = diag(−1, +1, +1). SO(8) vector indices are I, J, . . .. The fermions a represented
by 32-component Majorana spinors of SO(1, 10) subject to a chirality condition on the
world-volume which leaves 16 real degrees of freedom. The SO(1, 10) spinor indices are
generally omitted.
The action is then given by [71]
Z
S=
h
d3 x tr −(Dµ X I )† Dµ X I + iΨ̄† Γµ Dµ Ψ
(1.81)
8f 2
2if † IJ I J†
Ψ̄ Γ X X Ψ + X J Ψ† X I + ΨX I† X J −
tr X [I X †J X K] X †[K X J X †I]
3
3
i
1
2i
1 µνλ
2i
+ µνλ (Aµ ∂ν Aλ + Aµ Aν Aλ ) −
(µ ∂ν Âλ + µ Âν Âλ ) ,
2f
3
2f
3
−
29
where the covariant derivative is
Dµ X = ∂µ X + iAµ X − iX µ .
(1.82)
The Chern-Simons level k is contained in
f=
2π
.
k
(1.83)
The bifundamental scalars X I are related to the original BLG variables xIa with
SO(4) index a through
1
X I = (xI4 1 + ixIi σ i ) ,
2
(1.84)
where σ i are the Pauli matrices. It is important to note that the scalars satisfy the
reality condition
X ∗ = −εXε ,
(1.85)
where ε = iσ2 . This condition can only be imposed for the gauge group SU(2) × SU(2),
which might seem to present an obstacle for generalizing the theory to rank N > 2. This
obstacle was overcome by using complex bifundamental superfields [72], and this will be
reviewed in Section 1.3.2.
Finally we note the form of the SU(2) × SU(2) gauge transformations
Aµ → U Aµ U † − iU ∂µ U † ,
X → U X Û † ,
µ → Û Âµ Û † − iÛ ∂µ Û † ,
X † → Û X † U † ,
where U, Û ∈ SU(2).
30
(1.86)
Superfield formulation. Let us now write the BLG theory (1.81) in N = 2 superspace. Of the SO(8)R symmetry this formalism leaves only the subgroup U(1)R × SU(4)
manifest. However, we will demonstrate how the SO(8) R-symmetry is recovered when
the action is expressed in terms of component fields. Our notations and many useful
superspace identities are summarized in Appendix A.2.
The gauge fields A and  become components of two gauge vector superfields V and
V̂. Their component expansions in Wess-Zumino gauge are
V = 2i θθ̄ σ(x) − 2 θγ m θ̄ Am (x) +
√ 2 †
√
2i θ θ̄χσ (x) − 2i θ̄2 θχσ (x) + θ2 θ̄2 D(x) , (1.87)
and correspondingly for V̂. Here σ and D are auxiliary scalars, and χσ and χ†σ are
auxiliary fermions. The matter fields X and Ψ are accommodated in chiral superfields Z
and anti-chiral superfields Z̄ which transform in the fundamental and anti-fundamental
representation of SU(4), respectively. Their SU(4) indices Z A and Z̄A will often be
suppressed. The component expansions are
√
2θζ(xL ) + θ2 F (xL ) ,
√
Z̄ = Z † (xR ) − 2θ̄ζ † (xR ) − θ̄2 F † (xR ) .
Z = Z(xL ) +
(1.88)
(1.89)
The scalars Z are complex combinations of the BLG scalars
Z A = X A + iX A+4
for A = 1, . . . , 4 .
(1.90)
We define two operations which conjugate the SU(2) representations and the SU(4)
31
representation, respectively, as3
Z ‡A := −ε(Z A )T ε = X †A + iX †A+4 ,
(1.91)
Z̄A := −ε(Z A )∗ ε = X A − iX A+4 .
(1.92)
Separating these two operations in possible only for gauge group SU(2) × SU(2), since
for gauge groups of higher rank there is no reality condition analogous to (1.85). In
these cases only the combined action, which is the hermitian conjugate Z † ≡ Z̄ ‡ , makes
sense. The possibility to conjugate the SU(4) representation independently from the
SU(2) × SU(2) representation allows us to invert (1.90):
XA =
1
2
Z A + Z̄A
X A+4 =
,
1
2i
Z A − Z̄A .
(1.93)
The superspace action S = SCS + Smat + Spot consists of a Chern-Simons part, a
matter part and a superpotential given by
h
i
1
dt tr V D̄α etV Dα e−tV − V̂ D̄α etV̂ Dα e−tV̂ , (1.94)
SCS = −iK d x d θ
0
Z
3
4
Smat = − d x d θ tr Z̄A e−V Z A eV̂ ,
(1.95)
Z
Z
Spot = L d3 x d2 θ W(Z) + L d3 x d2 θ̄ W̄(Z̄) ,
(1.96)
Z
3
4
Z
with
W=
1
ABCD tr Z A Z ‡B Z C Z ‡D
4!
,
W̄ =
1 ABCD
‡
.
tr Z̄A Z̄B‡ Z̄C Z̄D
4!
(1.97)
In terms of SO(4) variables Za , which are related to the SU(2) × SU(2) fields according
3
We should caution that the bar denoting the anti-chiral superfield Z̄ is just a label and does not
mean that the component fields are conjugated by (1.92). In fact, the components of Z̄ are the hermitian
conjugates, see (1.89).
32
to (1.84), it assumes the form
W=−
1
ABCD abcd ZaA ZbB ZcC ZdD .
8 · 4!
(1.98)
This superpotential possesses only a U(1)R × SU(4) global symmetry as opposed to the
SO(8)R symmetry of the BLG theory. We will show in the following that when the
normalization constants K and L are related as K =
1
,
L
then the R-symmetry of the
model is enhanced to SO(8). If we furthermore set L = 4f , we recover precisely the
action (1.81).
The gauge transformations are given by [78]
ˆ
ˆ
etV → eiΛ etV e−iΛ̄ , etV̂ → eiΛ̂ etV̂ e−iΛ̄ , Z → eiΛ Ze−iΛ̂ , Z̄ → eiΛ̄ Z̄e−iΛ̄ , (1.99)
ˆ are chiral and anti-chiral superfields, respectively.
where the parameters Λ, Λ̂ and Λ̄, Λ̄
Their t dependence is determined by consistency of the transformation law for V and V̂.
In order to preserve the WZ gauge, these fields have to be simply
Λ = λ(xL ) ,
Λ̄ = λ(xR ) ,
Λ̂ = λ̂(xL ) ,
ˆ = λ̂(x ) .
Λ̄
R
(1.100)
with λ and λ̂ real. These transformations reduce to the ones given in (1.86) when we
set U (x) ≡ eiλ(x) and Û (x) ≡ eiλ̂(x) .
Expressions in components. We will now show that the above superspace action
describes the BLG theory by expanding it into component fields. The Chern-Simons
action then reads
Z
SCS = K
h
d3 x tr
2i
2i
Aµ Aν Aλ ) − 2µνλ (µ ∂ν Âλ + µ Âν Âλ )
3
3
i
+ 2iχ†σ χσ − 2iχ̂†σ χ̂σ − 4Dσ + 4D̂σ̂ ,
(1.101)
2µνλ (Aµ ∂ν Aλ +
33
and the matter action becomes4
Z
Smat =
h
/ + F † F + Z † DZ − Z † Z D̂
d x tr − (Dµ Z)† Dµ Z + iζ † Dζ
3
+ iZ † χσ ζ + iζ † χ†σ Z − iZ † ζ χ̂σ − iζ † Z χ̂†σ
i
− Z † σ 2 Z − Z † Z σ̂ 2 + 2Z † σZ σ̂ − iζ † σζ + iζ † ζ σ̂ .
(1.102)
The gauge covariant derivative is defined in (1.82). The superpotential contains the
following interactions of the component fields
Spot
L
=−
12
Z
d3 x tr
h
ABCD ζ A ζ ‡B Z C Z ‡D − ζ ‡A ζ B Z ‡C Z D + ζ A Z ‡B ζ C Z ‡D
‡
+ ζ̄A‡ Z̄B ζ̄C‡ Z̄D
+ ABCD ζ̄A‡ ζ̄B Z̄C‡ Z̄D − ζ̄A ζ̄B‡ Z̄C Z̄D
i
+ 2ABCD F A Z ‡B Z C Z ‡D − 2ABCD F̄A‡ Z̄B Z̄C‡ Z̄D .
(1.103)
Integrating out auxiliary fields. The fields D and D̂ are Lagrange multipliers for
the constraints
σn =
1
tr tn ZZ †
4K
,
σ̂ n =
1
tr tn Z † Z ,
4K
(1.104)
where tn are the generators of SU(2). The equations of motion for the χσ ’s are
1
tr tn Zζ † ,
2K
1
χ̂nσ = −
tr tn ζ † Z ,
2K
1
tr tn ζZ † ,
2K
1
(χ̂†σ )n = −
tr tn Z † ζ ,
2K
(χ†σ )n = −
χnσ = −
4
(1.105)
(1.106)
Let us remind the reader that our notation suppresses indices in standard positions, e.g.
† α A
† â
b
A c
tr Z † χσ ζ ≡ tr ZA
χσ ζα ≡ (ZA
) b (χα
σ ) c (ζα ) â .
The standard position of an index is defined when the field is introduced, and those for spinor indices
are explained in Appendix A.2.
34
and the ones for F are
L
F A = − ABCD Z̄B Z̄C‡ Z̄D
6
L
FA† = + ABCD Z ‡B Z C Z ‡D .
6
,
(1.107)
Using these relations one finds the following action
Z
S=
h
d3 x 2K µνλ tr Aµ ∂ν Aλ +
2i
A A A
3 µ ν λ
− µ ∂ν Âλ − 2i3 µ Âν Âλ
i
/ − Vferm − Vbos .
− tr(Dµ Z)† Dµ Z + i tr ζ † Dζ
(1.108)
The quartic terms Vferm are interactions between fermions and bosons, and the sextic
terms Vbos are interactions between bosons only. Separated according to their origin we
have
h
i
i
tr ζ A ζA† Z B ZB† − ζA† ζ A ZB† Z B + 2ζ A ZA† Z B ζB† − 2ZA† ζ A ζB† Z B , (1.109)
4K
h
i
L
VFferm = ABCD tr ζ A ζ ‡B Z C Z ‡D − ζ ‡A ζ B Z ‡C Z D + ζ A Z ‡B ζ C Z ‡D
12
i
L ABCD h ‡
‡
‡
‡
‡
‡
(1.110)
+ tr ζ̄A ζ̄B Z̄C Z̄D − ζ̄A ζ̄B Z̄C Z̄D + ζ̄A Z̄B ζ̄C Z̄D ,
12
VDferm =
and
h
i
1
† A † B † C
† B † A † C
A † B † C †
tr
Z
Z
Z
Z
Z
Z
+
Z
Z
Z
Z
Z
Z
−
2Z
Z
Z
Z
Z
Z
, (1.111)
A
B
C
A
B
C
A
B
C
16K 2
L2
VFbos = − ABCG DEF G tr Z ‡A Z B Z ‡C Z̄D Z̄E‡ Z̄F .
(1.112)
36
VDbos =
When substituting in (1.90) we find that for K =
1
L
all sextic interactions can be joined
together to
V bos =
L2
tr X [I X †J X K] X †[K X J X †I] .
6
(1.113)
Furthermore, setting L = 4f , this is precisely the scalar potential of the BLG theory
(1.81). With this choice also the other coefficients match exactly.
35
1.3.2
ABJM U(N ) × U(N ) Gauge Theory in Superspace
As remarked above, it is not immediately obvious how to generalize van Raamsdonk’s
formulation of the BLG theory to higher rank gauge groups. This difficulty is also evident
in our superspace formulation, since the manifestly SU(4) invariant superpotential is
gauge invariant only for SU(2) × SU(2) gauge theory.
The way to circumvent this difficulty is the generalization proposed by ABJM [72].
Their key idea is to give up the manifest global SU(4) invariance by forming the following
complex combinations of the bifundamental fields:
Z 1 = X 1 + iX 5 ,
W1 = X 3† + iX 7† ,
(1.114)
Z 2 = X 2 + iX 6 ,
W2 = X 4† + iX 8† .
(1.115)
Promoting these fields to chiral superfields, the superpotential of the BLG theory (1.97)
may be written as [72]
Z
Spot = L
3
Z
2
d3 x d2 θ̄ W̄(Z̄, W̄) ,
(1.116)
1
W̄ = AC BD tr Z̄A W̄ B Z̄C W̄ D .
4
(1.117)
d x d θ W(Z, W) + L
with
1
W = AC BD tr Z A WB Z C WD
4
,
This form of the superpotential is exactly the same as for the theory of D3-branes on
the conifold [44] and it generalizes readily to SU(N ) × SU(N ) gauge group. This superpotential has a global symmetry SU(2) × SU(2) and also a “baryonic” U(1) symmetry
Z A → eiα Z A
,
WB → e−iα WB .
(1.118)
In the 3+1 dimensional case this symmetry is originally gauged, but far in the IR
36
it becomes global [44]. However, in the present 2+1 dimensional example this does not
happen, so it is natural to add it to the gauge symmetry [72]. Including also the trivial
neutral U(1), we thus find the U(N ) × U(N ) Chern-Simons gauge theory at level k. The
gauging of the symmetry (1.118) seems important for obtaining the correct M-theory
interpretation for arbitrary k and N . Since this symmetry corresponds to simultaneous
rotation of the 4 complex coordinates of C4 transverse to the M2-branes, this space
actually turns into an orbifold C4 /Zk [72]. Because of this gauging, even for N = 2 the
ABJM theory is slightly different from the BLG theory.
Let us summarize the properties of the ABJM theory [72] and explicitly prove that
its U(1)R × SU(2) × SU(2) global symmetry becomes enhanced to SU(4)R . The fields
Z and W transform in the (2, 1) and the (1, 2̄) of the global SU(2) × SU(2) and in
the (N, N̄) and the (N̄, N) of the gauge group U(N ) × U(N ), respectively. We use the
following conventions for SU(2)×SU(2) indices: Z A , Z̄A , WA , W̄ A and for U(N )×U(N )
indices: Z a â , Z̄ â a , W â a , W̄ a â . The gauge superfields have indices V a b and V̂ â b̂ . The
component fields for Z, Z̄ and V are as previously in (1.88), (1.89) and (1.87). The
components of W and W̄ will be denoted by
√
2θω(xL ) + θ2 G(xL ) ,
√
W̄ = W † (xR ) − 2θ̄ω † (xR ) − θ̄2 G† (xR ) .
W = W (xL ) +
(1.119)
(1.120)
The Chern-Simons action is formally unaltered (1.94), the matter part (1.95) splits into
Z
Smat =
h
i
d3 x d4 θ tr −Z̄A e−V Z A eV̂ − W̄ A e−V̂ WA eV ,
(1.121)
and the superpotential is given by (1.116). The symmetry enhancement to SU(4)R
requires the normalization constants in (1.94) and (1.116) to be related as K = L1 .
37
Expressions in components. The component form of the Chern-Simons action has
been computed in (1.101) and the matter action involving Z looks identical to (1.102)
where now Z, ζ, F have only two components. The matter action for W is analogously
given by
W
Smat
Z
=
h
/ + G† G + W † D̂W − W † W D
d x tr − (Dµ W )† Dµ W + iω † Dω
3
+ iW † χ̂σ ω + iω † χ̂†σ W − iW † ωχσ − iω † W χ†σ
i
− W † σ̂ 2 W − W † W σ 2 + 2W † σ̂W σ − iω † σ̂ω + iω † ωσ , (1.122)
where Dµ W = ∂µ W + iµ W − iW Aµ . The superpotential expands to
Spot
L
=
4
Z
h
d3 x tr AC BD 2F A WB Z C WD + 2Z A WB Z C GD − 2ζ A WB Z C ωD
−2ζ A ωB Z C WD − Z A ωB Z C ωD − ζ A WB ζ C WD
AC
− BD 2FA† W †B ZC† W †D + 2ZA† W †B ZC† G†D + 2ζA† W †B ZC† ω †D
i
+2ζA† ω †B ZC† W †D + ZA† ω †B ZC† ω †D + ζA† W †B ζC† W †D .(1.123)
Integrating out auxiliary fields. The auxiliary fields can be replaced by means of
the following equations:
1
tr T n ZZ † − W † W ,
4K
1
=−
tr T n Zζ † − ω † W ,
2K
1
=−
tr T n ζ † Z − W ω † ,
2K
L AC
= + BD W †B ZC† W †D ,
2
L
= − AC BD WB Z C WD ,
2
σn =
χnσ
χ̂nσ
FA
FA†
1
tr T n Z † Z − W W † ,
4K
1
(χ†σ )n = −
tr T n ζZ † − W † ω ,
2K
1
(χ̂†σ )n = −
tr T n Z † ζ − ωW † ,
2K
L
†
GA = − AC BD ZB† W †C ZD
,
2
L
G†A = + AC BD Z B WC Z D .
2
σ̂ n =
38
(1.124)
(1.125)
(1.126)
(1.127)
(1.128)
Then the complete action reads
Z
S=
3
h
d x 2K µνλ tr Aµ ∂ν Aλ +
2i
A A A
3 µ ν λ
− µ ∂ν Âλ −
2i
  Â
3 µ ν λ
/ + i tr ω † Dω
/
− tr(Dµ Z)† Dµ Z − tr(Dµ W )† Dµ W + i tr ζ † Dζ
i
− Vferm − Vbos ,
(1.129)
with the potentials
VDferm =
(1.130)
h
B †
† B
i
i
† A
A †
†A
†B
†A
†B
tr ζ ζA − ω ωA Z ZB − W WB − ζA ζ − ωA ω
ZB Z − WB W
+
4K
h
i
i
tr ζ A ZA† − W †A ωA Z B ζB† − ω †B WB − ZA† ζ A − ωA W †A ζB† Z B − WB ω †B ,
2K
VFferm =
(1.131)
h
i
L
AC BD tr 2ζ A WB Z C ωD + 2ζ A ωB Z C WD + Z A ωB Z C ωD + ζ A WB ζ C WD +
4
h
i
L AC
BD tr 2ζA† W †B ZC† ω †D + 2ζA† ω †B ZC† W †D + ZA† ω †B ZC† ω †D + ζA† W †B ζC† W †D ,
4
and
h
1
tr
Z A ZA† + W †A WA Z B ZB† − W †B WB Z C ZC† − W †C WC
(1.132)
2
16K
+ ZA† Z A + WA W †A ZB† Z B − WB W †B ZC† Z C − WC W †C
−2ZA† Z B ZB† − W †B WB Z A ZC† Z C − WC W †C
i
−2W †A ZB† Z B − WB W †B WA Z C ZC† − W †C WC ,
L2 h †A † †C
bos
(1.133)
VF = − tr W ZB W WA Z B WC − W †A ZB† W †C WC Z B WA
4
i
+ZA† W †B ZC† Z A WB Z C − ZA† W †B ZC† Z C WB Z A .
VDbos =
Let us note that VFbos and VDbos are separately non-negative. Indeed, the F-term
39
contribution is related to the superpotential W through
VFbos
∂W 2 ∂W 2
= tr F † F A + G†A GA ,
= A + A
∂Z
∂WA (1.134)
with F A and GA from (1.127) and (1.128). The D-term contribution may be written as
VDbos = tr NA† N A + M †A MA ,
(1.135)
where N A = σZ A − Z A σ̂ and MA = σ̂WA − WA σ. Thus, the total bosonic potential
vanishes if and only if
F A = GA = N A = MA = 0 .
(1.136)
SU(4) invariance. If the coefficients of the Chern-Simons action and the superpotential are related by K =
1
,
L
then the R-symmetry of the theory is enhanced to SU(4).5
In order to make this symmetry manifest we combine the SU(2) fields Z and W into
fundamental and anti-fundamental representations of SU(4) as
Y A = {Z A , W †A } ,
YA† = {ZA† , WA } ,
(1.137)
where the index A on the left hand side now runs from 1 to 4. Then the potential can
be written as [72]
V bos = −
L2 h A † B † C †
tr Y YA Y YB Y YC + YA† Y A YB† Y B YC† Y C
48
+ 4Y
A
YB† Y C YA† Y B YC†
− 6Y
A
YB† Y B YA† Y C YC†
i
.
(1.138)
The fermions have to be combined as follows
ψA = {AB ζ B e−iπ/4 , −AB ω †B eiπ/4 } ,
ψ A† = {−AB ζB† eiπ/4 , AB ωB e−iπ/4 } , (1.139)
5
This SU(4)R symmetry should not be confused with the global SU(4) of the BLG theory. The latter
is not manifest in the ABJM theory, but should nevertheless be present for k = 1 and k = 2 [72].
40
and we can write fermionic interactions in the manifestly SU(4) invariant way
V
ferm
iL h † A B†
=
tr YA Y ψ ψB − Y A YA† ψB ψ B† + 2Y A YB† ψA ψ B† − 2YA† Y B ψ A† ψB
4
i
− ABCD YA† ψB YC† ψD + ABCD Y A ψ B† Y C ψ D† .
(1.140)
Thus, the U(1)R × SU(2) × SU(2) global symmetry is enhanced to SU(4)R symmetry,
with the U(1)R corresponding to the generator
1
2
diag(1, 1, −1, −1). This shows that the
theory in general possesses N = 6 supersymmetry.
In [72] it was proposed that this U(N ) × U(N ) Chern-Simons theory at level k
describes the world volume of N coincident M2-branes placed at the Zk orbifold of C4
where the action on the 4 complex coordinates6 is y A → e2πi/k y A . This action preserves
the SU(4) symmetry that rotates them, which in the gauge theory is realized as the
R-symmetry. The N = 6 supersymmetry of this orbifold can be checked as follows. The
generator of Zk acts on the spinors of SO(8) as
Ψ → e2πi(s1 +s2 +s3 +s4 )/k Ψ ,
(1.141)
where si = ±1/2 are the spinor weights. The chirality projection implies that the sum
of all si must be even, producing an 8-dimensional representation. The spinors that
P
are left invariant by the orbifold have 4i=1 si = 0 (mod k). This selects 6 out of the 8
spinors; therefore, the theory on M2-branes has 12 supercharges in perfect agreement
with the Chern-Simons gauge theory with general level k. This constitutes just one of
many pieces of evidence that strongly suggest that the theory reviewed in this section is
indeed dual to M-theory on AdS4 × S 7 /Zk with N units of flux. For k = 1 or 2 there is
further enhancement to N = 8 supersymmetry, which is subtle in the gauge theory [72]
and requires the introduction of monopole operators, as we will discuss in Chapter 5.
6
Let us note that these coordinates are not the same as the complex coordinates z A natural for the
superspace formulation of BLG theory in Section 1.3.1. They are related through y 1 = z 1 , y 2 = z 2 , y 3 =
z̄ 3 , y 4 = z̄ 4 .
41
Chapter 2
On the Strong Coupling Scaling
Dimension of High Spin Operators
2.1
Introduction
The dimensions of high-spin operators are important observables in gauge theories. It
is well-known that the anomalous dimension of a twist-2 operator grows logarithmically
for large spin S,
∆ − S = f (g) ln S + O(S 0 ) ,
p
gY2 M N
.
with g =
4π
(2.1)
This was demonstrated early on at one-loop order [24, 25] and at two loops [79] where
a cancellation of ln3 S terms occurs. There are solid arguments that (2.1) holds to all
orders in perturbation theory [80, 81, 82], and that it also applies to high-spin operators
of twist greater than two [29]. The function of coupling f (g) also measures the anomalous
dimension of a cusp in a light-like Wilson loop, and is of definite physical interest in
QCD.
There has been significant interest in determining f (g) in the N = 4 SYM theory.
This is partly due to the fact that the AdS/CFT correspondence [1, 2, 3] relates the
42
large g behavior of f to the energy of a folded string spinning around the center of a
weakly curved AdS5 space [20]. This gives the prediction that f (g) → 4g at strong
coupling. The same result was obtained from studying the cusp anomaly using string
theory methods [83]. Furthermore, the semi-classical expansion for the spinning string
energy predicts the following correction [28]:
f (g) = 4g −
3 ln 2
+ O(1/g) .
π
(2.2)
It is of obvious interest to confirm these explicit predictions of string theory using extrapolation of the perturbative expansion for f (g) provided by the gauge theory.
Explicit perturbative calculations are quite formidable, and until a few years ago
were available only up to three-loop order [35, 84]:
88
8
f (g) = 8g 2 − π 2 g 4 + π 4 g 6 + O(g 8 ) .
3
45
(2.3)
Kotikov, Lipatov, Onishchenko and Velizhanin [35] extracted the N = 4 answer from
the QCD calculation of [85] using their proposed transcendentality principle stating that
each expansion coefficient has terms of the same degree of transcendentality.
Since then, the methods of integrability in AdS/CFT1 [36, 37, 38], prompted in part
by [19, 20], have led to dramatic progress in studying the weak coupling expansion.
In the beautiful paper by Beisert, Eden and Staudacher [40], which followed closely
the important earlier work in [30, 39], an integral equation that determines f (g) was
proposed, yielding an expansion of f (g) to an arbitrary desired order. The expansion
coefficients obey the KLOV transcendentality principle. In an independent remarkable
paper by Bern, Czakon, Dixon, Kosower and Smirnov [89], an explicit calculation led to
1
For earlier work on integrability in gauge theories, see [86, 87, 88], for reviews see [22, 23].
43
a value of the four-loop term,
−16
73 6
2
π + 4ζ(3) g 8 ,
630
(2.4)
which agrees with the idea advanced in [40, 89] that the exact expansion of f (g) is
related to that found in [30] simply by multiplying each ζ-function of an odd argument
by an i, ζ(2n + 1) → iζ(2n + 1). The integral equation of [40] generates precisely this
perturbative expansion for f (g).
A crucial property of this integral equation is that it is related through integrability
to the “dressing phase” in the magnon S-matrix, whose general form was deduced in
[32, 90]. In [40] a perturbative expansion of the phase was given, which starts at fourloop order, and at strong coupling coincides with the earlier results from string theory
[39, 32, 91, 92, 93]. An important requirement of crossing symmetry [94] is satisfied by
this phase, and it also satisfies the KLOV transcendentality priciple. Therefore, this
phase is very likely to describe the exact magnon S-matrix at any coupling [40], which
constitutes remarkable progress in the understanding of the N = 4 SYM theory, and of
the AdS/CFT correspondence.
The papers [40, 89] thoroughly studied the perturbative expansion of f (g) which
follows from the integral equation. Although the expansion has a finite radius of convergence, as is customary in certain planar theories (see, for example, [95]), it is expected
to determine the function completely. Solving the integral equation of [40] is an efficient
tool for attacking this problem. In this chapter we solve the integral equation numerically at intermediate and strong coupling, and show that f (g) is a smooth function that
approaches the asymptotic form (2.2) predicted by string theory for g > 1. The two
leading strong coupling terms match those in (2.2) with high accuracy. This constitutes
a remarkable confirmation of the AdS/CFT correspondence for this non-supersymmetric
observable.
44
This chapter is based on the papers [96, 97] coauthored with L. F. Alday, G. Arutyunov, S. Benvenuti, B. Eden, I. R. Klebanov and A. Scardicchio, and its structure is as
follows. The BES integral equation is reviewed and solved numerically in Section 2.2. An
interpretation of these results and their implications for the AdS/CFT correspondence
are given in Section 2.3, where we also summarize further investigations which are the
subject of the remainder of this chapter. In Section 2.4 we derive the exact fluctuation
density that solves the integral equation in the strong coupling limit, before we conclude
in Section 2.5.
2.2
Numerical Study of the Integral Equation
The cusp anomalous dimension f (g) can be written as [30, 40, 98]
f (g) = 16g 2 σ̂(0) ,
(2.5)
where σ̂(t) obeys the integral equation
t
σ̂(t) = t
e −1
Z
2
K(2gt, 0) − 4g
∞
0
0
0
dt K(2gt, 2gt )σ̂(t )
,
(2.6)
0
with the kernel given by [40]
K(t, t0 ) = K (m) (t, t0 ) + 2K (c) (t, t0 ) .
(2.7)
The main scattering kernel K (m) of [30] is
K
(m)
J1 (t)J0 (t0 ) − J0 (t)J1 (t0 )
(t, t ) =
,
t − t0
0
45
(2.8)
and the dressing kernel K (c) is defined as the convolution
0
(c)
K (t, t ) = 4g
2
∞
Z
dt00 K1 (t, 2gt00 )
0
t00
K0 (2gt00 , t0 ) ,
e −1
t00
(2.9)
where K0 and K1 denote the parts of the kernel that are even and odd, respectively,
under change of sign of t and t0 :
K0 (t, t0 ) =
∞
tJ1 (t)J0 (t0 ) − t0 J0 (t)J1 (t0 )
2 X
(2n − 1)J2n−1 (t)J2n−1 (t0 ) , (2.10)
=
2
02
0
t −t
t t n=1
∞
t0 J1 (t)J0 (t0 ) − tJ0 (t)J1 (t0 )
2 X
K1 (t, t ) =
= 0
(2n)J2n (t)J2n (t0 ) .
t2 − t02
t t n=1
0
(2.11)
We find it useful to introduce the function
et − 1
σ̂(t) ,
t
s(t) =
(2.12)
in terms of which the integral equation becomes
s(t) = K(2gt, 0) − 4g
2
Z
∞
dt0 K(2gt, 2gt0 )
0
t0
s(t0 ) .
e −1
t0
(2.13)
Again, f (g) = 16g 2 s(0).
Both K (m) and K (c) can conveniently be expanded as sums of products of functions
of t and functions of t0 :
K
(m)
∞
2 X
(t, t ) = K0 (t, t ) + K1 (t, t ) = 0
nJn (t)Jn (t0 ) ,
t t n=1
0
0
0
(2.14)
and
K (c) (t, t0 ) =
∞ X
∞
X
8n(2m − 1)
Z2n,2m−1 J2n (t)J2m−1 (t0 ) .
0
tt
n=1 m=1
46
(2.15)
This suggests writing the solution in terms of linearly independent functions as
s(t) =
X
sn
n≥1
Jn (2gt)
,
2gt
(2.16)
so that the integral equation becomes a matrix equation for the coefficients sn . The
desired function f (g) is now
f (g) = 8g 2 s1 .
(2.17)
It is convenient to define the matrix Zmn as
∞
Z
Zmn ≡
dt
0
Jm (2gt)Jn (2gt)
.
t(et − 1)
(2.18)
Using the representations (2.14) and (2.15) of the kernels and (2.16) for s(t), the integral
equation above is now of the schematic form
sn = hn −
X
(m)
(c)
Knm
+ 2Knm
sm ,
(2.19)
m≥1
whose solution formally is
s=
1
1+
K (m)
+ 2K (c)
h.
(2.20)
The matrices appearing in (2.19) are
(m)
Knm
= 2(N Z)nm ,
(2.21)
(c)
Knm
= 2(CZ)nm ,
(2.22)
Cnm = 2(P N ZQN )nm ,
(2.23)
where Q = diag(1, 0, 1, 0, ...), P = diag(0, 1, 0, 1, ...), N = diag(1, 2, 3, ...) and the vector
h can be written as h = (1 + 2C)e, where e = (1, 0, 0, ...)T . The crucial point for the
numerics to work is that the matrix elements of Z decay sufficiently fast with increasing
47
m, n (they decay like e−max(m,n)/g ). For intermediate g (say g < 20) we can work with
moderate size d by d matrices, where d does not have to be much larger than g. The
integrals in Znm can be obtained numerically without much effort and so we can solve
for the sn . We find that the results are stable with respect to increasing d.
Even though at strong coupling all elements of Znm are of the same order in 1/g,
those far from the upper left corner are numerically small (the leading terms in 1/g are
suppressed by a factor [(m − n − 1)(m − n + 1)(m + n − 1)(m + n + 1)]−1 for m 6= n ± 1).
This last fact makes the numerics surprisingly convergent even at large g and moreover
gives some hope that the analytic form of the strong coupling expansion of f (g) could
be obtained from a perturbation theory for the matrix equation.
Therefore, when formulated in terms of the Zmn , the problem becomes amenable to
numerical study at all values of the coupling. We find that the numerical procedure
converges rather rapidly, and truncate the series expansions of s(t) and of the kernel
after the first 30 orders of Bessel functions.
The function f (g) is the lowest curve plotted in Figure 1. For comparison, we also
plot fm (g) which solves the integral equation with kernel K (m) [30], and f0 (g) which
solves the integral equation with kernel K (m) + K (c) . Clearly, these functions differ at
strong coupling. The function f (g) is monotonic and reaches the asymptotic, linear
form quite early, for g ' 1. We can then study the asymptotic, large g form easily and
compare it with the prediction from string theory. The best fit result (using the range
2 < g < 20) is
f (g) = (4.000000±0.000001)g−(0.661907±0.000002)−
0.0232 ± 0.0001
+. . . .
g
(2.24)
The first two terms are in remarkable agreement with the string theory result (2.2),
while the third term is a numerical prediction for the 1/g term in the strong coupling
expansion.
48
Figure 2.1: Plot of the solutions of the integral equations: fm (g) for the ES kernel K (m)
(upper curve, red), f0 (g) for the kernel K (m) + K (c) (middle curve, green), and f (g)
for the BES kernel K (m) + 2K (c) (lower curve, blue). Notice the different asymptotic
behaviors. The inset shows the three functions in the crossover region 0 < g < 1.
2.3
Discussion of Results and Further Investigations
A very satisfying result of this investigation is that the BES integral equation yields a
smooth universal function f (g) whose strong coupling expansion is in excellent numerical
agreement with the spinning string predictions of [20, 28]. This provides a highly nontrivial confirmation of the AdS/CFT correspondence.
The agreement of this strong coupling expansion was anticipated in [40] based on a
similar agreement of the dressing phase. However, some concerns about this argument
were raised in [89] based on the slow convergence of the numerical extrapolations. Luckily, our numerical methods employed in solving the integral equation converge rapidly
and produce a smooth function that approaches the asymptotics (2.2). The cross-over
49
region of f (g) where it changes from the perturbative to the linear behavior lies right
around the radius of convergence, gc = 1/4, corresponding to gY2 M N = π 2 .
Analytic Structure. The qualitative structure of the interpolating function f (g) is
quite similar to that involved in the circular Wilson loop, where the conjectured exact
result [99, 100] is
ln
I1 (4πg)
2πg
.
(2.25)
The function (2.25) is analytic on the complex plane, with a series of branch cuts along
the imaginary axis, and an essential singularity at infinity. The function f (g) is also
expected to have an infinite number of branch cuts along the imaginary axis, and an
essential singularity at infinity [40].
Let us compare this with the exact anomalous dimension of a single giant magnon
of momentum p [31, 93, 101, 102, 103]:
r
−1 +
1 + 16g 2 sin2
p
2
.
This function has a single branch cut along the imaginary axis, going from g =
(2.26)
i
4 sin( p2 )
to infinity, and no essential singularity at infinity. Observables of the gauge theory are
composites of giant magnons with various momenta p ∈ (0, 2π). We thus expect the
anomalous dimension of a generic unprotected operator to have a superposition of many
cuts along the imaginary axis. This should endow a generic multi-magnon state with an
analytic structure similar to that of f (g).
We found numerically the presence of the first two branch cuts of f (g) on the imaginary axis, starting at g = ± ni
, n = 1, 2. The first of them, which also occurs for the
4
giant magnon with maximal momentum p = π, agrees with the summation of the perturbative series [40]. The full structure of f (g) is expected to contain an infinite number
of branch cuts accumulating at infinity, where an essential singularity is present.
50
Fluctuation Density. It is remarkable that the integral equation of [40] allows the
universal scaling function f (g), which is not a BPS quantity, to be solved for. This
amounts to evaluating the fluctuation density σ̂(t) at the origin, t = 0, but of course
the full function σ̂(t) is of interest in its own right, and in the following section we shall
derive an exact analytic expression for it in the strong coupling limit.
If one insists on truncating the BES equation at leading order for large g the kernel
becomes degenerate and therefore a unique solution for σ̂(t) cannot be found without
additional assumptions. However, for finite values of g the solution is unique and can
be found numerically with high precision. These two statements are consistent because
expanding the BES equation in a power series in 1/g we find a second equation at
subleading order that removes any ambiguity in the strong coupling solution σ̂(t).
To gain some intuition our approach will be to first investigate the solution numerically for finite values of the coupling, which will suggest a simple additional requirement
that should be imposed on the underdetermined, truncated BES equation to single out a
unique solution in the strong coupling limit which is the convergence point of our numerics. After this additional requirement is identified, we find an analytic solution for σ̂(t)
in the strong coupling limit. We then show that this solution is in fact completely determined by taking into account the subleading contributions to the BES equation, thus
confirming analytically the validity of the auxiliary condition suggested by the numerics
and of our strong coupling solution σ̂(t).
Upon performing the Fourier transform to the rapidity u-plane the corresponding
density σ̂(u) reads


v
θ(|u| − 1) u
1 
u1 + q 1
√
σ̂(u) =
1 −
t
2
4πg
2
1−
1
u2

,
(2.27)
where θ(u) is the step function. Thus, on the rapidity plane the leading density is an
algebraic function which is constant in the interval |u| < 1. We see that, in contrast to
51
Figure 2.2: Plot of the analytic solution for the leading density 4πg 2 σ̂(u).
the weak coupling solution of the BES equation [40], the strong coupling density exhibits
a gap between [−∞, −1] and [1, ∞], see Figure 2.2. Remarkably, this is reminiscent of
the behavior of the corresponding solutions describing classical spinning strings.
Below we will analyze the matrix form of the BES equation at strong coupling, first
numerically and then analytically. We compute the coefficients in the expansion of σ̂(t)
in terms of Bessel functions and based on this numerical analysis we make a guess of
how these coefficients could be (partially) related to each other at strong coupling. In
the strong coupling limit we obtain an analytic equation for the coefficients which turns
out to have a degenerate kernel. Supplementing this equation with the proposed relation
among the coefficients allows us to find an exact analytical solution for σ̂(t). We then
show that in fact no guess is necessary if one expands the BES equation to higher order
in 1/g, which leads to a second equation that singles out our solution as the correct one.
We proceed to find an analogous pair of equations for the subleading coefficients at
strong coupling. Again they appear at different orders in the inverse coupling constant,
each of them individually being a degenerate, half-rank equation. We identify some
constraints on the subleading value of σ̂(t), but as we shall see this case is more subtle
than the leading solution.
52
2.4
Fluctuation Density at Strong Coupling
In this section we study the density of fluctuations σ̂(t) for large values of the coupling
constant. First we argue, by performing a numerical analysis, that σ̂(t) obeys a certain
additional requirement. Then we consider the matrix BES equation at leading order
in the large g expansion, use the previously found requirement to solve for the density
and confirm by expanding the BES equation further that this solution is in fact complete determined analytically. Finally, we derive and briefly discuss constraints on the
subleading value of σ̂(t).
As discussed in Section 2.2, the key observation is that for intermediate values of
g one can approximate the infinite-dimensional matrices entering the BES equation by
matrices of finite rank d, with d not much larger than g. With matrices of finite rank it
is possible to solve numerically for the coefficients sk for different values of the coupling
constant and to find the best fit result for an expansion of the type
1
1
sk = s`k + 2 ss`
+ ... .
g
g k
(2.28)
As the numerical analysis indicates, the finite rank approximation is valid for computing
the coefficients sk with k d. In the table below the values for a few leading coefficients
s`k are exhibited.
k
s`2k−1
s`2k
100|(s`2k−1 − s`2k )/s`2k |
1
0.500006
0.499993
0.003
2
-0.75005
-0.74977
0.038
3
0.93727
0.93676
0.055
4
-1.09281
-1.09415
0.12
5
1.2239
1.2333
0.77
53
Here we have solved numerically equation (2.19) for numerous points in the range
2 < g < 20, using d = 50. Some comments are in order. First, we stress again that the
value for s`1 is in perfect agreement with the value predicted from string theory, s`1 = 1/2,
and confirmed numerically above (with a precision higher than the one presented here).
Second, notice that the difference between s`2k−1 and s`2k is in all the cases smaller than
1%, and gets bigger as k increases; this is related to the fact that the rank of the matrices
used is finite.
Thus, the numerical analysis suggests that in the limit of infinite rank matrices the
following relation holds for the leading coefficients in the strong coupling expansion
s`2k−1 = s`2k .
(2.29)
As we will see later on, this condition will allow one to solve analytically for the coefficients s`k and the values obtained will be in perfect agreement with the ones computed
numerically.
Further evidence comes from the fact that one can approximate the matrix elements
Zmn by their analytic values at strong coupling (see next subsection). Therefore, one
can consider matrices of much higher rank, fix a sufficiently large value of g and compute
(numerically) the coefficients s`k . Below we present the results for g = 10000 and d = 250.
k
s`2k−1
s`2k
100|(s`2k−1 − s`2k )/s`2k |
1
0.49993
0.49991
0.0049
2
-0.74943
-0.74938
0.0073
3
0.93585
0.93577
0.0085
4
-1.09033
-1.09023
0.0089
5
1.22455
1.22444
0.0087
54
As d increases, we see that the difference between s`2k−1 and s`2k (for k = 2, 3, . . . )
decreases considerably. Also, the difference is approximately constant for small values
of k. We should stress, however, that a priory there is no reason to expect the results
obtained by keeping in the BES equation only the leading term for Z to be valid, since as
we will see the subleading terms in the matrix elements Zmn are necessary to fix uniquely
the leading order solution for sk . Surprisingly, one still obtains a good approximation
to the large g solution in this fashion. Nevertheless, we regard the present computation
as less robust.
To conclude, requiring continuity in g, the leading coefficients s`k in the strong coupling expansion of the function σ̂(t) exhibit the relation (2.29), which constraints the
form of σ̂(t).
2.4.1
Analytic Solution at Strong Coupling
For finite real values of g the matrix element Zmn is given by a convergent integral.
However, it is not obvious if it is possible to express the result of integration as a power
series in 1/g. Indeed, expanding the integrand in (2.18) as
Z
Zmn =
0
∞
Jm (t)Jn (t)
dt
t
t
2g 1
− +
+ ...
t
2 24g
,
(2.30)
leads to a power series (with coefficients given essentially by the Bernoulli numbers)
which converges only for |t/(2g)| < 2π. We see that only the first two leading terms
in this expansion can be integrated, while already in the third term divergent integrals
appear. It is therefore natural to assume that the expansion of Z develops as
Z = gZ ` + Z s` + . . . ,
55
(2.31)
where the first two terms, Z ` and Z s` , are given by the convergent integrals mentioned
above. Explicitly, they are
8
cos((m − n)π/2)
,
π (m + n + 1)(m + n − 1)(m − n + 1)(m − n − 1)
1 sin((m − n)π/2)
s`
Zmn
=−
.
π
m2 − n2
`
Zmn
=−
(2.32)
(2.33)
Assumption (2.31) is well supported by the numerics. We have checked that numerical
values for Zmn for large g are in a very good agreement with the analytic expressions
(2.32) and (2.33). On the other hand, the status of the higher order terms in (2.31) is
not obvious. In what follows, we largely restrict our investigation of the BES equation
to the first two leading terms.
With this word of caution we proceed to investigate (2.19) in the strong coupling
limit. It is not hard to see that it implies the following equation for the leading vector s`
K (c) ` s` = C ` e ,
(2.34)
where the leading matrices K (c) ` and C ` are obtained by keeping the leading contribution
Z ` only. It turns out that for even values of d the kernel K (c) ` has rank d/2, hence from
the equation above it is possible to solve only for half of the components of s` . However,
imposing the condition (2.29) together with (2.34) allows one to uniquely determine the
vector s` .
In order to find the solution in the limit of infinite d it is convenient to express the
leading equation in the following way
1
K o so + K e se = e ,
2
(2.35)
where so , se are vectors of length d/2 comprising the odd and even components of s` respectively: so = (s`1 , s`3 , s`5 , ...)T , se = (s`2 , s`4 , s`6 , ...)T and we recall that e = (1, 0, 0, ...)T .
56
It is easy to check that equations (2.34) and (2.35) are equivalent provided that
`
`
(K o )mn = Z2m−1,2n−1
+ Z2m+1,2n−1
,
`
`
(K e )mn = Z2m−1,2n
+ Z2m+1,2n
.
(2.36)
Then the unique solution of (2.35) satisfying the relation so = se turns out to be
s`2k−1 = s`2k = (−1)k+1
Γ(k + 12 )
.
Γ(k)Γ( 12 )
(2.37)
This remarkably simple expression for the coefficients s`k is the main result of this section.
By using the following identities (true in the limit of infinite rank)
(K e
−1
)mn = −4(−1)m+n+1 mn2 ,
for n ≤ m ,
(2.38)
(K e
−1
)mn = −4(−1)m+n+1 m3 ,
for n > m ,
(2.39)
(K e
−1
K o )mn = (−1)m−n
m3
32
,
π (4m2 − (1 − 2n)2 )(1 − 2n)2
(2.40)
one can check explicitly that the coefficients (2.37) indeed solve (2.35). Thus, we found
that restricting the coefficients sk to the leading order expressions s`k the function s(t) is
∞
1
1X
k+1 Γ(k + 2 ) J2k (2gt) + J2k−1 (2gt)
s(t) =
(−1)
+ ... ,
g k=1
2gt
Γ(k)Γ( 21 )
(2.41)
where we have omitted the subleading contributions. This expression can be considered
the leading term in the large g expansion of the density s(t, g) with g t kept finite.
As is clear from (2.6), we are only interested in values of s(t) for t ≥ 0. The series
can be summed and for this range of t the result expressed in terms of the confluent
hypergeometric function of the second kind U (a, b, x):
i
2igt
3
1
5
1
e
Γ(
)
U
(−
,
0,
−4igt)
+
Γ(
)
U
(
,
0,
−4igt)
4
4
4
4
8πg 2 t
i
−2igt
3
1
5
1
+
e
Γ(
)
U
(−
,
0,
4igt)
+
Γ(
)
U
(
,
0,
4igt)
.
4
4
4
4
8πg 2 t
s(t) = −
57
(2.42)
The leading density (2.42) has a rather complicated profile. For g t → 0 it perfectly
reproduces the desired result s(t) → 1/(4g). On the other hand, the asymptotics of s(t)
for g t → ∞ exhibit a highly oscillatory behavior.
2.4.2
An Alternative Derivation
Let us now show that the result (2.37) can in fact be derived without resorting to
any auxiliary conditions obtained from numerical arguments. The degenerate equation
appearing at leading order, which can be used to express one half of the coefficients sn in
terms of the other, can be supplemented by another half-rank equation from subleading
terms in the BES equation. Together they determine a unique solution.
Writing out the integral equation (2.6) in the basis (2.16) with explicit matrix indices
we find
sn
Jn (2gt) J1 (2gt)
J2n (2gt)
Jn (2gt)
=
+ 8nZ2n,1
− 2nZnm sm
2gt
2gt
2gt
2gt
J2n (2gt)
−16n(2m − 1)Z2n,2m−1 Z2m−1,r sr
,
2gt
(2.43)
where all indices are summed over from 1 to ∞. Now we split up the integral equation
according to powers of g and into odd and even rows (indices of Bessel functions).
At O(g) the odd equation is trivial and the even one reads
1
`
`
`
2(2m − 1)Z2n,2m−1
Z2m−1,r
s`r = Z2n,1
= δn,1 .
4
(2.44)
This is precisely equation (2.34) employed in the previous subsection. At O(1) the odd
rows lead to the condition
1
`
Z2m−1,r
s`r = δm,1 .
2
(2.45)
Actually this equation implies the previous one. It determines one half of the coefficients
58
s`n in terms of the other. Expanding further, at O(1) the even equation is given by
s`
`
`
`
8nZ2n,1
− 4nZ2n,m
s`m − 16n(2m − 1)Z2n,2m−1
Z2m−1,r
ss`
r
`
s`
s`
`
−16n(2m − 1)Z2n,2m−1
Z2m−1,r
s`r − 16n(2m − 1)Z2n,2m−1
Z2m−1,r
s`r = 0 .
(2.46)
To determine the other half of the coefficients s`n we need to eliminate ss`
n from this
equation. To do this we examine the odd equation at O(1/g)
`
s`
`
`
−2(2m − 1)Z2m−1,r
ss`
r − 2(2m − 1)Z2m−1,r sr = s2m−1 .
(2.47)
Now we use (2.45) and (2.47) to simplify (2.46), which gives
`
`
`
Z2n,m
s`m − 2Z2n,2m−1
s`2m−1 = Z2n,m
(−1)m s`m = 0 .
(2.48)
This is the second equation we were looking for, which together with (2.45) completely
determines s`n . If we define a matrix Z̃nm which is identical to Znm except for a sign flip
when both n is even and m is odd, we can combine the two conditions into the full-rank
equation for the strong coupling solution
1
`
Z̃nm
s`m = δn,1 .
2
(2.49)
Note that the additional minus signs in the definition of Z̃nm arise precisely because
of the introduction of the dressing kernel, and would be absent if the kernel consisted
solely of the main scattering part. Writing out (2.49) explicitly shows that the s`n have
to satisfy
∞
X
k=1
4(−1)n+k+1 s`2k−1
(2n − 2k − 1)(2n − 2k + 1)(2n + 2k − 3)(2n + 2k − 1)π
=−
s`2n−2
s`2n
1
−
+ δn,1 ,
8n(2n − 1) 8(n − 1)(2n − 1) 4
59
(2.50)
∞
X
k=1
4(−1)n+k+1 s`2k
(2n − 2k − 1)(2n − 2k + 1)(2n + 2k − 1)(2n + 2k + 1)π
=
s`2n−1
s`2n+1
+
,
8n(2n + 1) 8n(2n − 1)
(2.51)
for all n ≥ 1 (where it is understood that the second term on the right hand side of
(2.50) is absent for n = 1). Indeed these equations are obeyed by
s`2n−1 = s`2n =
(−1)n−1 (2n − 1)!!
,
2n (n − 1)!
(2.52)
which is precisely the solution (2.37) found in the previous subsection. To show this,
note that the coefficients s`2n−1 are generated by the Taylor expansion of (1 + x)−3/2 ,
which makes it easy to perform the above sums as integrals over functions of the form
xm (1 − x2 )−3/2 for some appropriate power m chosen to generate the necessary terms in
the denominators of the left hand sides of (2.50) and (2.51).
2.4.3
Fluctuation Density in the Rapidity Plane
To get more insight into the structure of the leading solution, we find it convenient to
perform the (inverse) Fourier transform of the density σ̂(t) → σ̂(u):
1
σ̂(u) =
2π
Z
∞
dt ei 2gtu e|t|/2 σ̂(|t|) .
(2.53)
−∞
We recall that u is a rapidity variable originally used to parameterize the Bethe root
distributions of gauge and string theory Bethe ansätze [31, 32], which gives another
reason for studying the density of fluctuations on the u-plane. Thus, substituting into
(2.53) the power series expansion for σ̂(t) we get
Z ∞
∞
1 X
|t| Jn (2g|t|)
σ̂(u) =
sn
dt ei 2gtu e−|t|/2
+ ... .
2π n=1
1 − e−|t| 2g|t|
−∞
60
(2.54)
For large values of g this expression can be well approximated as
Z ∞
∞
1 X
Jn (2g|t|)
σ̂(u) =
sn
dt ei 2gtu e−|t|/2
+ ... .
2π n=1
2g|t|
−∞
(2.55)
The last integral is computed by using the following formula [30]
Z
∞
dt e
±2giut −t/2 Jn (2gt)
e
0
2gt
−n
p
(2g)n−1 ± 2
±
2
=
u 1 + 1 + 4g /(u )
,
n
(2.56)
with u± = 1/2 ∓ 2igu.
In this way we obtain the following series representation for the density σ̂(u)
∞
1 X
sn f n + . . . ,
σ̂(u) =
2πg n=1
(2.57)
where
fn =
p
p
−n
−n i
(2g)n h +
u (1 + 1 + 4g 2 /(u+ )2 )
+ u− (1 + 1 + 4g 2 /(u− )2 )
. (2.58)
2n
In what follows it is convenient to introduce the expansion parameter = 1/(2g).
Our considerations above suggest that the density σ̂(u) expands starting from the second
order in :
σ̂(u) = 2 σ̂ ` (u) + 3 σ̂ s` (u) + . . . .
(2.59)
To find the leading contribution σ̂ ` (u) we have to develop the large g expansion of the
functions fn . The result is not uniform, it depends on whether n is even or odd and also
on the value of u . For n even we find
f2k


k

 (−1) T2k (u) + O()
2k
=
q

k

 (−1) u 1 + 1 −
2k
for |u| < 1 ,
1
u2
61
−2k
(2.60)
+ O()
for |u| > 1 .
Here T2k (u) are the Chebyshev polynomials of the first kind. For n odd we obtain
f2k−1 =


k√

 − (−1) 1 − u2 U2k−2 (u) + O()
2k−1
for |u| < 1 ,



for |u| > 1 ,
(2.61)
0 + O()
where U2k−2 (u) are the Chebyshev polynomials of the second kind. We recall that the
Chebyshev polynomials of the first and the second kind form a sequence of orthogonal polynomials on the interval [−1, 1] with the weights (1 − u2 )−1/2 and (1 − u2 )1/2 ,
respectively.
To find the leading density σ̂ ` (u) inside the interval |u| < 1 we can use the trigonometric definition of the Chebyshev polynomials which corresponds to choosing parametrization u = cos θ with 0 ≤ θ ≤ π. Thus, taking the limit g → ∞ we obtain for the leading
density the following expression
∞
2 X `
s2k−1 f2k−1 (θ) + s`2k f2k (θ) ,
σ̂ (u) =
π k=1
`
(2.62)
where
f2k (θ) = (−1)k
cos(2kθ)
,
2k
f2k−1 (θ) = −(−1)k
sin((2k − 1)θ)
.
2k − 1
(2.63)
Given the result (2.37) for the coefficients s`k , we can now sum the series for σ̂ ` (u) and
obtain
σ̂ ` (u) =
1
.
π
(2.64)
Thus, inside the interval [−1, 1] the density σ̂ ` (u) is constant.
Further, it is easy to sum up the series defining the leading density for |u| > 1. The
62
Figure 2.3: The left figure is a plot of a numerical solution for the leading density πσ̂ ` (u).
The right figure shows the exact analytic solution for the same quantity.
result is
√
2

σ̂ ` (u) =
1 
2− q
2π
√
1 − u(u ∓ u2 − 1)

.
(2.65)
Here the minus and plus signs in the denominator corresponds to the regions u > 1 and
u < −1, respectively. The plot of the complete analytic solution for σ̂ ` (u) is presented
in Figure 2.3. It should be compared to the plot of the numerical solution obtained by
using in the series representation for the density with the coefficients s`k obtained from
our numerical analysis. The numerical plot corresponds to taking g = 40 and truncating
the series at k = 20.
Finally, we mention the expression for the scaling function f (g) in terms of the
density σ̂(u)
f (g) = 32g
3
Z
∞
Z
∞
du σ̂(u) = 8g
−∞
du σ̂ ` (u) + . . . = 4g + . . . ,
(2.66)
−∞
which confirms analytically the leading term of the numerical results of Section 2.2. This
completes our discussion of the leading order analytic solution of the BES equation in
the strong coupling limit.
63
2.4.4
Subleading Corrections
Here we will investigate the first subleading correction to the leading coefficients s`k . By
expanding the BES equation we have already obtained equation (2.47) which expresses
the subleading density ss`
n in terms of the leading one:
`
s`
`
Z2m−1,r
ss`
r = −Z2m−1,r sr −
1
s`2m−1 .
2(2m − 1)
(2.67)
Again this equation is degenerate and needs to be supplemented by a second one that
appears at higher order in 1/g in the BES equation. Substituting the explicit form of
the coefficients s`n obtained above, the right hand side of (2.67) is seen to vanish (i.e.
K (c) ` ss` = 0), which already implies tight restrictions on the subleading corrections ss` .
As discussed above, this equation allows us to solve for half of the components. For
instance, using (2.35) and (2.40) we can solve for the even components in terms of the
odd ones
ss`
2m = −
∞
X
(−1)m−n
n=1
32
m3
ss` .
π (4m2 − (1 − 2n)2 )(1 − 2n)2 2n−1
(2.68)
To obtain another equation constraining the subleading solution, we examine corrections to the BES equation as follows. From the O(1/g) contribution to the even rows of
the BES equation (2.43) we find
s`
`
ss`
`
− 4nZ2n,m
ss`
(2.69)
s`2n = 8nZ2n,1
m − 4nZ2n,m sm
h
ss`
`
`
s`
s`
`
`
`
− 16n(2m − 1) Z2n,2m−1
Z2m−1,r
sss`
r + Z2n,2m−1 Z2m−1,r sr + Z2n,2m−1 Z2m−1,r sr
i
s`
`
s`
s`
`
ss`
`
`
+Z2n,2m−1
Z2m−1,r
ss`
+
Z
Z
s
+
Z
Z
s
r
2n,2m−1 2m−1,r r
2n,2m−1 2m−1,r r .
To eliminate the term in sss` we turn to the odd rows of the O(1/g 2 ) equation
i
h
`
s`
`
s`
s`
ss`
+
Z
s
+
Z
s
−2(2m − 1) Z2m−1,r
sss`
r
2m−1,r r
2m−1,r r = s2m−1 .
64
(2.70)
Using this together with (2.45) and (2.47) to simplify (2.69) we obtain
`
s`
m `
Z2n,m
(−1)m ss`
m = −Z2n,m (−1) sm −
1 `
s .
4n 2n
(2.71)
This can be combined with (2.67) into the full rank equation
s` `
`
ss`
Z̃nm
m = −Z̃nm sm −
1 `
s ,
2n n
(2.72)
where again the right hand side vanishes when evaluated on the leading solution s` found
above, i.e. the subleading corrections appear to satisfy a homogeneous equation.
Thus complications arise at higher order in g which suggest that the equations for
the subleading terms in sn have to be supplemented by additional constraints. Luckily,
the equations for the leading solution derived in Subsection 2.4.2 are not affected by this
complication. This clearly cannot be the whole story, which is not surprising in view
of the divergence of the O(1/g) correction to Znm that we have ignored here by naively
expanding to third order.
Subsequently this problem was solved, and the complete asymptotic expansion of
f (g) determined in an impressive paper [104] (for further work, see [105]). This expansion
obeys its own transcendentality principle. In particular, the coefficient of the 1/g term
in (2.2) was shown to be given by
−
K
≈ −0.0232 ,
4π 2
(2.73)
where K is the Catalan constant, in agreement with the numerical result (2.24). As a
further check, this coefficient was reproduced analytically from a two-loop calculation
in string sigma-model perturbation theory [106, 107].
65
2.5
Conclusions
In this chapter we have addressed the problem of extrapolating the perturbative expansion controlled by the BES equation, which describes the universal scaling function of
high spin operators in N = 4 gauge theory, to large values of the coupling constant and
shown that the resulting expression is consistent with string theory predictions.
First, we devised a numerical method to solve the BES equation around the strong
coupling point and demonstrated that the numerical solution is in perfect agreement
with equation (2.2), providing a non-trivial test of the AdS/CFT correspondence. Thus,
the cusp anomaly f (g) is an example of an interpolation function for an observable not
protected by supersymmetry that smoothly connects weak and strong coupling regimes.
The final form of f (g) was arrived at using inputs from string theory, perturbative gauge
theory, and the conjectured exact integrability of planar N = 4 SYM.
Secondly, we have analytically studied the strong coupling limit of the BES equation
and demonstrated that expanding it in inverse powers of the coupling constant leads to
two equations for the large g solution, one appearing at leading, the other at subleading
order in 1/g. Together they determine a unique solution in the g → ∞ limit whose exact
analytic form we present.2
As was shown in [40] (see also [109]), the coefficients of the perturbative series describing the solution of the BES equation at weak coupling admit an analytic continuation
to strong coupling, where they coincide with those predicted by string theory. The approach we adopt here can be considered as another, complementary way to analytically
continue from weak to strong coupling.
On the rapidity u-plane the leading fluctuation density σ̂ ` (u) appears to be constant
inside the unit interval |u| < 1. We could argue that this constant part of σ̂ ` (u) is an
artifact of the way the BES equation was derived: The non-vanishing constant part of
2
The leading term in the strong coupling asymptotic expansion of the fluctuation density σ̂(t) was
derived analytically also in [108], independently of the present work.
66
the leading density offsets the splitting of the weak-coupling density into a log-divergent
one-loop part and a regular higher loop piece carrying log S as a coefficient. Further, it is
not hard to show [97] that the subleading correction inside the unit interval is absent; in a
manner of speaking a gap opens between [−∞, −1] and [1, ∞]. This can be qualitatively
compared to the results obtained from string theory. Indeed, the solution of the integral
equation describing the classical spinning strings in AdS3 × S1 [110] in the limit S → ∞
with angular momentum J along S1 fixed has support only outside the interval |u| < 1.
The same behavior is expected for the GKP solution which is obtained in the limit
J → 0. For finite S the solution is elliptic and it exhibits logarithmic singularities in
the limit S → ∞. On the other hand, our strong coupling density (2.65) is an algebraic
function which carries log S as a normalization. Of course, this density leads to the
same energy as for the GKP string. It is desirable to understand the detailed matching
between the string density (higher conserved charges) and the density we found from
the strong coupling limit of the BES equation.
67
Chapter 3
On Normal Modes of a Warped
Throat
3.1
Introduction
Duality between the cascading SU(k(M + 1)) × SU(kM ) gauge theory and type IIB
strings on the warped deformed conifold [49] provides a rich yet tractable example of
gauge/string correspondence1 . This background demonstrates in a geometrical language
such features of the SU(M ) supersymmetric gluodynamics as color confinement and the
breaking of the Z2M chiral R-symmetry down to Z2 via gluino condensation [49]. In
fact, it has been argued [49] that by reducing the continuous parameter gs M one can
interpolate between the cascading theory solvable in the supergravity limit and N = 1
supersymmetric SU(M ) gauge theory.
The problem of finding the spectra of bound states at large gs M can be mapped
to finding normalizable fluctuations around the supergravity background. This problem
is complicated by the presence of 3-form and 5-form fluxes, but some results on the
spectra are already available in the literature [52, 53, 113, 114, 115, 116, 117, 118, 119].
1
For earlier work leading up to this duality, see [44, 51, 111, 112], and for reviews [10, 11, 12]. The
most pertinent facts were summarized above in Section 1.2.
68
A particularly impressive effort was made by Berg, Haack and Mück (BHM) who used
a generalized PT ansatz [54] to derive and numerically solve a system of seven coupled
scalar equations [116, 117]. Each of the resulting glueballs is even under the charge
conjugation Z2 symmetry preserved by the KS solution (this symmetry was called the
I-symmetry in [52, 53]), and therefore has S P C = 0++ . The present chapter will study
three other families of glueballs, which are odd under the I-symmetry. Two of them
originate from a pair of coupled scalar equations, generalizing the zero momentum case
studied in [52, 53], and have S P C = 0+− . The third, pseudoscalar family arises from a
decoupled fluctuation of the RR two-form C2 and has S P C = 0−− .
An important aspect of the low-energy dynamics is that the baryonic U (1)B symmetry is broken spontaneously by the condensates of baryonic operators A and B. This
phenomenon, anticipated in the cascading gauge theory in [49, 56], was later demonstrated on the supergravity side where the fluctuations corresponding to the pseudoscalar
Goldstone boson and its scalar superpartner [52, 53], as well as the fermionic superpartner [118], were identified. Furthermore, finite deformations along the scalar direction
give rise to a continuous family of supergravity solutions [58, 59] dual to the baryonic
branch, AB = const, of the gauge theory moduli space.
The main purpose of this chapter is to obtain a deeper understanding of the GHK
scalar fluctuations [52, 53] and their radial excitations. Our motivation is two-fold. On
the one hand, we seek an improved understanding of the glueball spectra and their
supermultiplet structure. On the other, we would like to shed new light on the normal
modes of the warped deformed conifold throat embedded into a string compactification,
which has played a role in models of moduli stabilization [120] and D-brane inflation
[121, 122, 123]. In such inflation models, the reheating of the universe involves emission
of modes localized near the bottom of the throat, which are dual to glueballs in the
gauge theory [124, 125, 126, 127].
This chapter is based on the paper [128] written in collaboration with A. Dymarsky,
69
I. R. Klebanov and A. Solovyov; it is structured as follows. In Section 3.2 we construct
a generalization of the ansatz for the NSNS 2-form and metric perturbations that allows
us to study radial excitations of the GHK scalar mode. We derive a system of coupled
radial equations and determine their spectrum (by numerically solving those differential
equations). In Section 3.3 we show that a similar ansatz for the RR 2-form perturbation
decouples from the metric giving rise to a single decoupled equation for pseudoscalar
glueballs. In Section 3.4 we argue that the scalar glueballs we find belong to massive axial
vector multiplets, and the pseudoscalar glueballs belong to massive vector multiplets.
Agreement of the corresponding equations is explicitly demonstrated in the large radius
(KT) limit. In Section 3.5 we give a perturbative treatment of the coupled equations
for small mass that allows us to study the scalar mass in models where the length of the
throat is finite.
3.2
Radial Excitations of the GHK scalar
The ansatz that produced a normalizable scalar mode independent of the 4-dimensional
coordinates xµ was [52, 53]
δB2 = χ(τ ) dg 5 ,
δG13 = δG24 = ψ(τ ) .
(3.1)
Our first goal is to find a generalization of this ansatz that will allow us to study the radial
excitations of this massless scalar, i.e. the series of modes that exist at non-vanishing
kµ2 = −m24 . Thus, we must include the dependence of all fields on xµ . Such an ansatz
70
that decouples from other fields at linear order is
δF3 = 0 ,
(3.2)
δ F̃5 = 0 ,
(3.3)
δB2 = χ(x, τ ) dg 5 + ∂µ σ(x, τ ) dxµ ∧ g 5 ,
(3.4)
δH3 ≡ dδB2 = χ0 dτ ∧ dg 5 + ∂µ (χ − σ) dxµ ∧ dg 5 + ∂µ σ 0 dτ ∧ dxµ ∧ g 5 ,
(3.5)
δG13 = δG24 = ψ(x, τ ) .
(3.6)
The ansatz for δB2 originates from the longitudinal component of a 5-dimensional vector:
δB2 = (Aτ dτ + Aµ dxµ ) ∧ g 5 .
(3.7)
Requiring the 4-dimensional field strength to vanish, Fµν = 0, restricts Aµ to be of the
form ∂µ acting on a function. Then, choosing
Aτ = −χ0 ,
Aµ = ∂µ (σ − χ) ,
(3.8)
we recover the ansatz (3.4) up to a gauge transformation.
Yet another gauge equivalent way of writing (3.4) is
δB2 = (χ − σ) dg 5 − σ 0 dτ ∧ g 5 .
(3.9)
The new feature of our ansatz compared to the generalized PT ansatz used in [116, 117]
is the presence of the second function in δB2 which multiplies dτ ∧ g 5 . Terms of this
type, which are allowed by the 4-dimensional Lorentz symmetry, turn out to be crucial
for studying the modes that are odd under the I-symmetry.
Using δ(G−1 ) = −G−1 δG G−1 , we find that δG13 = δG24 = −G11 G33 ψ. The unperturbed metric components (see Subsection 1.2.2 for a review of the KS solution)
71
are
G11 = G22 =
G33 = G44 =
2
ε4/3 K(τ ) sinh2 (τ /2)h1/2 (τ )
2
ε4/3 K(τ ) cosh2 (τ /2)h1/2 (τ )
,
(3.10)
,
(3.11)
6 K(τ )2
,
ε4/3 h1/2
(3.12)
Gµν = h1/2 η µν .
(3.13)
G55 = Gτ τ =
In order to find the dynamic equations for the functions ψ, χ and σ in (3.4)-(3.6) we
study the linearized supergravity equations below (the type IIB SUGRA equations are
reviewed in Appendix A.1).
3.2.1
Equations of Motion for NSNS- and RR-Forms
All the Bianchi identities are automatically satisfied with the ansatz (3.2)-(3.6). Indeed,
the relation dδH3 = 0 is obvious, and consistent with vanishing dδ F̃5 we find that
δH3 ∧ F3 = 0 (using (1.38) one can verify that dg 5 ∧ F3 = 0 and dτ ∧ g 5 ∧ F3 = 0).
The self-duality equation for F̃5 reads
δ ∗ F̃5 = 0 .
(3.14)
Given that F̃5 has components along g 1 ∧g 2 ∧g 3 ∧g 4 ∧g 5 and along d4 x∧dτ , our adopted
deformation of the metric does not affect ∗F̃5 to first order.
Even though the variations of the forms F3 and F̃5 are zero, the deformations of their
Hodge duals δ ∗ F3 and δ ∗ F̃5 will in general be non-zero because of the deformations of
metric components. In the equation for F3
dδ ∗ F3 = F̃5 ∧ δH3 ,
72
(3.15)
the product F̃5 ∧ δH3 on the right hand side vanishes identically. From the explicit
form of F3 given in (1.38) we see that the perturbation of its Hodge dual will be a
closed form δ ∗ F3 = A(x, τ ) d4 x ∧ dτ ∧ (. . .). Note that the term in (1.38) proportional
to dτ ∧ (g 1 ∧ g 3 + g 2 ∧ g 4 ) is not affected by the deformation of the metric, and thus
dδ ∗ F3 = 0 is satisfied identically.
The remaining equations are nontrivial. In particular
dδ ∗ H3 = 0 ,
(3.16)
turns out to be more complicated than the equation for F3 . The variation
δ ∗ H3 = ∗δH3 + δG ∗ H3
(3.17)
consists of two parts: ∗δH3 accounting for the deformation of the form H3 itself, and
δG ∗ H3 arising from the deformation of the Hodge star. Explicit calculation shows that
√
∗δH3 = − −G G11 G33 G55 χ0 d4 x ∧ dg 5 ∧ g 5
√
− −G G11 G33 |Gµµ | ∂µ (χ − σ) ∗4 dxµ ∧ dτ ∧ dg 5 ∧ g 5
√
+ 21 −G (G55 )2 |Gµµ | ∂µ σ 0 ∗4 dxµ ∧ dg 5 ∧ dg 5 ,
gs M α 0 √
δG ∗ H3 = −
−G G11 G33 G55 f 0 G11 + k 0 G33 ψ d4 x ∧ dg 5 ∧ g 5 ,
2
(3.18)
(3.19)
where in the last equation we have introduced the shorthands
f (τ ) ≡
τ coth τ − 1
(cosh τ − 1) ,
2 sinh τ
k(τ ) ≡
τ coth τ − 1
(cosh τ + 1) .
2 sinh τ
(3.20)
The four-dimensional Hodge star ∗4 is taken with respect to the standard Minkowski
metric. Differentiating the expression for δ ∗ H3 and equating to zero the coefficients
73
multiplying linearly independent forms gives three equations:
1
gs M α0 0 11
0 33
0
d x ∧ dg ∧ dg : 2 G G
f G + k G ψ + χ = G55 h 2 4 σ 0 , (3.21)
2
0
√
0
4
5
5
0 11
0 33
11 33 55 gs M α
d x ∧ dτ ∧ dg ∧ g : ∂τ
f G +kG ψ+χ
−G G G G
2
√
1
+ −G G11 G33 h 2 4 (χ − σ) = 0 ,
(3.22)
√
√
1
1
∗4 dxµ ∧ dτ ∧ dg 5 ∧ dg 5 : 2 −G G11 G33 h 2 ∂µ (χ − σ) + ∂τ
−G (G55 )2 h 2 ∂µ σ 0 = 0 ,
4
5
5
11
33
(3.23)
where we have substituted for the warp factor |Gµµ | = h1/2 (no summation over µ is
implied). Not all of these equations are independent. Indeed, using (3.21) equation
(3.22) simplifies to
∂τ
n√
55 2
1/2
−G (G ) h
4 σ
0
o
+2
√
−G G11 G33 h1/2 4 (χ − σ) = 0 .
(3.24)
This is exactly what we obtain by acting on (3.23) with ∂ µ and contracting indices. Thus
only (3.21) and (3.23) are independent. The coefficient functions in these equations are
√
given by (we have dropped some inessential constant factor in −G):
4 K(τ )2
2 (sinh 2τ − 2τ )
=
,
ε4/3 h1/2
ε4/3 h1/2 K(τ ) sinh3 τ
16
G11 G33 =
,
ε8/3 h K(τ )2 sinh2 τ
6 K(τ )2
G55 h1/2 =
,
ε4/3
√
4
−G G11 G33 h1/2 ∼
,
K(τ )2
√
−G (G55 )2 h1/2 ∼ 9 K(τ )4 sinh2 τ .
f 0 G11 + k 0 G33 =
74
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
Taking into account these expressions, equations (3.21) and (3.23) read
K(τ )2
3 4/3
p
ε h(τ ) K(τ )4 sinh2 τ 4 σ 0 ,(3.30)
ψ + χ0 =
4/3
16
ε
h(τ )
n
o
9
∂µ (χ − σ) + K(τ )2 ∂τ K 4 sinh2 τ ∂µ σ 0 = 0 .
(3.31)
8
2(gs M α0 )
3.2.2
Einstein Equations
The first order perturbation of the Ricci curvature tensor is given by
δRij =
1
−δGa a ;ij − δGij;a a + δGai;j a + δGaj;i a ,
2
(3.32)
where covariant derivatives and contractions of indices are performed using the unperturbed metric. The first term in this expression vanishes because the metric perturbation
is traceless. The remaining three terms combine to give the only non-zero perturbations
δR13 = δR24 :
K 00 1 h00 z 00 (K 0 )2 1 (h0 )2 K 0 h0
+
+
+
−
+
ε
K
2 h
z
K2
2 h2
K h
0
0 0
0
0
K z
4
h
K
z
1
1
+2
−
+ coth τ
+4
+2
+2−
K z
h
K
z
9 sinh2 τ K 6
sinh2 τ
δR13 = −
3
3
K sinh τ z
4/3
− 21 h(τ )K sinh τ 4 z
"
2
2 0 0
0 0
(K
sinh
τ
)
(ln
h)
(K
sinh
τ
)
z
3
1
= − 4/3 K 3 sinh τ z
+
2
2
ε
2
(K sinh τ )
(K sinh τ ) z
#
2
8
1
4 cosh τ
−
−
+
− 12 h(τ )K sinh τ 4 z ,
2
2
6
9 K sinh τ
3 K 3 sinh2 τ
sinh τ
(3.33)
where z(x, τ ) is defined by
ψ(x, τ ) = h1/2 K sinh τ z(x, τ ) = 2−1/3 (sinh 2τ − 2τ )1/3 h1/2 z(x, τ ) .
(3.34)
The source terms on the right hand side of the Einstein equation Rij = Tij , spelled
75
out in (A.3), are due to the deformations of the metric and B2 form. It turns out that the
only nontrivial deformations are those with indices 13 or 24, with δT13 = δT24 . E.g. for
the 13 component δT13 we have the following contributions:
1
1
δB (H1ab H3 ab ) = [H1ab δH3 ab + δH1ab H3 ab
4
4
1
= G11 H12τ δH32τ + G33 δH14τ H34τ G55
2
1
= − (gs M α0 ) G55 G11 f 0 + G33 k 0 χ0 ,
4
2
gs2
g
δG (F̃1abcd F̃3 abcd ) = s (G11 )2 (G33 )2 G55 (F̃12345 )2 ψ ,
96
4
1
1
δG (H1ab H3 ab ) = H135 H315 δG13 G55 + H12τ H34τ δG24 Gτ τ
4
2
1
= (H135 )2 − H12τ H34τ G11 G33 G55 ψ
2
1
1
0 2
2
0 0
(k − f ) − f k G11 G33 G55 ψ ,
= (gs M α )
8
4
gs2 gs2
ab
δG (F1ab F3 ) =
F125 F345 δG24 G55 + F13τ F31τ δG31 Gτ τ
4
2
gs2 (F13τ )2 − F125 F345 G11 G33 G55 ψ
=
2
1
= (gs M α0 )2 F 02 − F (1 − F ) G11 G33 G55 ψ ,
8
(3.35)
(3.36)
(3.37)
(3.38)
with F (τ ) ≡ (sinh τ − τ )/(2 sinh τ ) and
−
1
1
δG G13 (Habc H abc + gs2 Fabc F abc ) = − (H32 + gs2 F32 ) ψ
48
8
1
= − (gs M α0 )2 G55 (G11 )2 f 02 + (G33 )2 k 02
32
+ 12 G11 G33 (k − f )2 + (G11 )2 F 2
+(G33 )2 (1 − F )2 + 2 G11 G33 F 02 ψ .
(3.39)
Denoting
h
i
δT13 = A1 (τ ) + A2 (τ ) ψ(x, τ ) + B(τ ) χ0 (x, τ ) ,
76
(3.40)
where A1 stands for the contribution from F̃5 , we get
3(gs M α0 )4 (τ coth τ − 1)2 (sinh 2τ − 2τ )4/3
,
21/3 ε20/3 h5/2
sinh6 τ
i
3 (gs M α0 )2 h
2
2
3
cosh
4τ
−
8τ
sinh
2τ
−
8τ
cosh
2τ
−
8
cosh
2τ
+
16τ
+
5
,
A2 (τ ) = − 4 3/2
8ε h sinh6 τ
(sinh 2τ − 2τ ) K(τ )
B(τ ) = −3 (gs M α0 )
.
(3.41)
ε8/3 h sinh3 τ
A1 (τ ) =
Then eliminating χ0 with the help of (3.21) yields
3 (gs M α0 )4 (τ coth τ − 1)2 (sinh 2τ − 2τ )5/3
δT13 = 2/3 20/3 2
z(x, τ )
2
ε h
sinh6 τ
3
(gs M α0 )2 (sinh 2τ − 2τ )1/3
×
+
8 ·h21/3
ε4 h
sinh6 τ
i
× cosh 4τ + 8(1 + τ 2 ) cosh 2τ − 24τ sinh 2τ + 16τ 2 − 9 z(x, τ )
9 gs M α0 sinh 2τ − 2τ 5
K 4 σ 0 (x, τ )
16 ε4/3
sinh
τ
3
1 (h0 )2 1 h00 K 0 h0
h0
3
= − 4/3 K sinh τ −
+
+
+ coth τ
z
ε
2 h2
2 h
K h
h
9 gs M α0 sinh 2τ − 2τ 5
K 4 σ 0 .
−
16 ε4/3
sinh τ
−
(3.42)
Equating (3.33) and (3.42) we obtain the final form of the linearized Einstein equation.
3.2.3
Two Coupled Scalars
Combining the equations for the field strengths and the Einstein equations we have the
system
sinh 2τ − 2τ
3 4/3
0
z
+
χ
=
ε h(τ ) K(τ )4 sinh2 τ 4 σ 0 ,
(3.43)
2
16
ε4/3 sinh τ
n
o
9
∂µ (χ − σ) = − K(τ )2 ∂τ K 4 sinh2 τ ∂µ σ 0 ,
(3.44)
8
0
(K sinh τ )2 z 0
ε4/3 h
2
8
1
4 cosh τ
+
4 z =
+
−
z
(K sinh τ )2
6 K2
9 K 6 sinh2 τ
3 K 3 sinh2 τ
sinh2 τ
3
sinh 2τ − 2τ 2
+ (gs M α0 )
K 4 σ 0 .
(3.45)
2
16
sinh τ
(gs M α0 )
77
Note that χ can be eliminated between (3.43) and (3.44). Further, a change of variables
z̃ = zK sinh τ ,
(3.46)
4/3
w̃ =
ε
K 5 sinh2 τ σ 0 ,
gs M α0
(3.47)
leads to a more symmetric pair of equations
ε4/3 h
2
3(gs M α0 )2 sinh 2τ − 2τ
z̃
+
4 w̃ ,
z̃
=
4
6K 2
16ε4/3
sinh2 τ
K 2 sinh3 τ
cosh2 τ + 1
ε4/3 h
8 sinh 2τ − 2τ
w̃00 −
w̃
+
z̃ .
4 w̃ =
2
2
6K
9 K 2 sinh3 τ
sinh τ
z̃ 00 −
(3.48)
(3.49)
Introducing the dimensionless mass-squared m̃2 according to
m̃2 = m24
22/3 (gs M α0 )2
,
6 ε4/3
(3.50)
we can rewrite the equations for z̃ and w̃ as
2
I(τ )
9
z̃ + m̃2 2
z̃ = m̃2
K(τ ) w̃ ,
2
K (τ )
4 · 22/3
sinh τ
cosh2 τ + 1
16
I(τ )
w̃00 −
w̃ + m̃2 2
w̃ =
K(τ ) z̃ .
2
K (τ )
9
sinh τ
z̃ 00 −
(3.51)
(3.52)
This is a system of coupled equations which defines the mass spectrum of certain scalar
glueballs with positive 4-dimensional parity. The natural charge conjugation symmetry
of the KS background is the I-symmetry, under which these modes are odd. Therefore,
we assign S P C = 0+− to this family of glueballs.2
In the massless case these equations lead to the GHK solution [52, 53]. If we assume
that 4 = −kµ2 = m24 = 0, then there are two solutions [52, 53], z̃1 = coth τ and
2
For comparison, the glueballs found in [116, 117] are 0++ . The glueballs whose spectrum comes
from the minimal scalar equation [113, 114] resulting from the analysis of graviton fluctuations are 2++ .
The axial vector U (1)R fluctuations [119] give rise to 1++ glueballs whose masses are also determined
by the minimal scalar equation.
78
z̃2 = τ coth τ −1. The solution for z̃ which is non-singular at the origin is z̃ = τ coth τ −1.
Substituting it into the second equation, we find
cosh2 τ + 1
16
22/3 8 0
w̃ −
w̃ =
K(τ ) (τ coth τ − 1) ≡ −
I (τ ) sinh τ .
9
9
sinh2 τ
00
(3.53)
The two solutions of the homogeneous equation are given by w̃1 = 1/ sinh τ and by
w̃2 = (sinh 2τ − 2τ )/ sinh τ ; both of them are singular either at zero or at infinity. This
means that the regular solution of the inhomogeneous equation is uniquely fixed. With
the Wronskian W (w̃1 , w̃2 ) = w̃1 w̃20 − w̃10 w̃2 = 4, we can find a general solution
Z τ
w̃2 (x) 0
22/3 8 n
w̃1 (τ ) C1 −
dx
I (x) sinh x
w̃(τ ) = −
9
W (x)
Z τ
o
w̃1 (x) 0
dx
+w̃2 (τ ) C2 +
I (x) sinh x .
W (x)
(3.54)
Integrating by parts and choosing the particular homogeneous solution to make w̃ well
behaved at both zero and infinity we get
22/3 8 1
w̃(τ ) = −
9 sinh τ
Z
τ
dx I(x) sinh2 x .
(3.55)
0
Let us note that the non-vanishing w̃ in the zero momentum case kµ = 0 is not in
contradiction with the GHK solution. This is because w̃ enters (3.2)-(3.6) only through
∂µ σ, which is zero as long as the momentum vanishes.
3.2.4
Numerical Analysis
To determine the spectrum of glueballs in the field theory, we need to solve the eigenvalue
problem for m̃2 in the infinite throat limit. This system of equations (3.51)-(3.52) does
not seem amenable to analytical solution and we employ a numerical approach to find
the spectrum of normalizable solutions. It is convenient to use the determinant method,
which generalizes the usual shooting technique to a system of several coupled equations
79
(see [117]). Let us briefly describe the details of the numerical analysis as well as the
subtleties specific to the system (3.51)-(3.52).
A standard method of finding the spectrum of a single second-order differential equation is the shooting technique. For a system of several coupled linear equations the
shooting method has to be generalized. The idea underlying the calculation scheme
(called the determinant method [117]) is to set the initial conditions at infinity corresponding to the two solutions regular at infinity, (z̃1 (τ ), w̃1 (τ ))T and (z̃2 (τ ), w̃2 (τ ))T , and
extend them numerically to small τ . Then the matrix
z̃1 (0) z̃2 (0)
w̃1 (0) w̃2 (0)
(3.56)
becomes degenerate at the critical points (eigenvalues) in the spectral parameter space.
Let us find the asymptotic behavior of regular and singular solutions near both zero
and infinity. At small τ equations (3.51) and (3.52) decouple,
2
z̃ = 0 ,
τ2
2
w̃00 − 2 w̃ = 0 .
τ
z̃ 00 −
(3.57)
(3.58)
There are two regular solutions with z̃, w̃ ∼ τ 2 and also two singular solutions with
z̃, w̃ ∼ 1/τ . For large τ we have
9
e−τ /3 w̃ ,
4 · 21/3
16 · 21/3 −τ /3
w̃00 − w̃ =
e
z̃ .
9
z̃ 00 = m̃2
(3.59)
(3.60)
The asymptotic behavior of the two regular solutions is
z̃1
w̃1
=
1
4/3
−2 e−τ /3
,
80
z̃2
w̃2
=
81
64·21/3
m̃2 e−4τ /3
,
e−τ
(3.61)
and the singular solutions are
z̃3
w̃3
=
−24/3
τ
,
τ − 43 e−τ /3
z̃4
w̃4
=
81
16·21/3
m̃2 e2τ /3
.
eτ
(3.62)
A particular subtlety of this setup is that at large τ the two singular solutions don’t
diverge equally fast: one of them grows exponentially while the other is only linear in τ .
This makes it difficult to start shooting from zero, since imposing the regularity condition
at infinity would require vanishing of both linear and exponential terms. To cancel the
linear term in the presence of the exponential one is difficult to achieve numerically.
That is why for this particular system it is convenient to start shooting from large τ ,
since both singular solutions at zero share the same behavior (∼ 1/τ ).
The result is that the spectrum consists of two distinct series, each with a quadratic
growth of m̃2n for large n. These series are interpreted as the radial excitation spectra of
two different particles. The lowest eigenvalues (m̃2 < 100) for these spectra are shown
in Table 3.1. The quadratic fit for spectrum I is
m̃2In = 2.31 + 1.91 n + 0.294 n2 .
(3.63)
For spectrum II we drop the lowest eigenvalue when fitting, and find
m̃2IIn = 0.36 + 0.14 n + 0.279 n2 .
(3.64)
It is interesting to compare these results with those found for the 0++ modes by Berg,
Haack and Mück (BHM) [117]. Their conventions correspond to a particular choice of
the KS parameters, and the relation between the masses is
m2BHM = (3/2)2/3 I(0) m̃2 ≈ 0.9409 m̃2 .
81
(3.65)
Spectrum I:
m2
100
80
n
1
2
3
4
m̃2n
4.53
7.30
10.7
14.6
n
5
6
7
8
m̃2n
19.1
24.4
30.1
36.4
n
9
10
11
12
m̃2n
43.3
50.8
58.9
67.6
n m̃2n
13 76.9
14 86.7
15 97.1
60
40
20
5
10
15
n
Spectrum II:
m2
100
n
1
2
3
4
5
m̃2n
.129
.703
1.76
3.33
5.43
n
6
7
8
9
10
m̃2n
8.06
11.2
15.0
19.3
24.1
n
11
12
13
14
15
m̃2n
30.1
35.5
42.1
49.2
56.9
n
16
17
18
19
m̃2n
65.1
73.9
83.3
93.3
80
60
40
20
5
10
15
n
Table 3.1: Non-zero eigenvalues with m̃2 < 100. There are the two distinct spectra. Both
spectra can be fitted by quadratic polynomials in the eigenvalue number n (the red line in the
plots).
Using this relation one can convert the mass eigenvalues to the BHM normalization. We
note that the lightest glueball we find, the first entry of spectrum II in Table 3.1, has
m2BHM ≈ 0.121. For comparison, the lightest 0++ eigenvalue found in [117] has masssquared m2BHM ≈ 0.185. The fact that the 0+− sector has the lightest glueballs may
be qualitatively understood as follows. Roughly speaking, glueball masses increase with
the dimensions of the operators that create them. The lowest dimension operator from
the 0++ sector is the gluino bilinear Trλλ of dimension 3, but the 0+− sector contains
an operator of dimension 2, namely Tr(A† A − B † B).
82
Converting the asymptotics of the two spectra to BHM units, we find
m2I BHM ≈ 2.17 + 1.79 n + 0.277 n2 ,
(3.66)
m2II BHM ≈ 0.34 + 0.13 n + 0.262 n2 .
(3.67)
The coefficients of the quadratic terms are close to those found in [117]. The quadratic
dependence on n, which is characteristic of Kaluza-Klein theory, is a special feature
of strongly coupled gauge theories that have weakly curved gravity duals (see [129]
for a discussion). Note that m24 is obtained from m̃2 through multiplying by a factor
√
∼ Ts /(gs M ), where Ts is the confining string tension. Thus, for n gs M these
modes are much lighter than the string tension scale, and therefore much lighter than
all glueballs with spin > 2. Such anomalously light bound states appear to be typical
of gauge theories that stay very strongly coupled in the UV, such as the cascading
gauge theory; they do not appear in asymptotically free gauge theories. Therefore,
the anomalously light glueballs could perhaps be used as a ‘special signature’ of gauge
theories with gravity duals if they are realized in nature.
One may be puzzled why the spectrum in Table 3.1 does not include the GHK
massless mode. This is because in solving the coupled equations (3.51)-(3.52) we required
that both wave-functions z̃ and w̃ go to a constant as τ → ∞. This excludes the GHK
zero mode which grows as z̃ ∼ τ . On the other hand, this growth is a lot slower than
the exponential growth found for generic solutions. The meaning of the GHK mode
as the baryonic branch modulus seems to be well established since even the solutions
at finite distance along this modulus are available [58, 59]. Thus, the GHK scalar
zero-mode should be normalizable with a proper definition of norm. In fact, the GHK
pseudoscalar and its fermionic superpartner are normalizable [52, 53, 118]; therefore, the
supersymmetry of the problem implies that the GHK scalar is normalizable as well and
is part of the spectrum.
83
3.3
Pseudoscalar Modes from the RR Sector
The type of ansatz used in Section 3.2 works even more simply for the RR 2-form field:
δH3 = 0 ,
(3.68)
δ F̃5 = 0 ,
(3.69)
δC2 = χ(x, τ ) dg 5 + ∂µ σ(x, τ ) dxµ ∧ g 5 ,
(3.70)
δF3 ≡ dδC2 = χ0 dτ ∧ dg 5 + ∂µ (χ − σ) dxµ ∧ dg 5 + ∂µ σ 0 dτ ∧ dxµ ∧ g 5 .
(3.71)
This ansatz is similar to, but somewhat simpler than the GHK pseudoscalar ansatz
[52, 53] which involved mixing with δ F̃5 . Since δF3 ∧ H3 = 0, now it is consistent to set
δ F̃5 = 0. We also have F̃5 ∧ δF3 = 0, so it is consistent to take δH3 = 0. Finally, one
needs to study mixing with metric fluctuations. At first glance it seems that δG12 and
δG34 might need to be turned on, but a more detailed analysis shows that their sources
vanish:
δT12 = F13τ δF2 3τ + δF14τ F2 4τ =
M α0 33 55 0 0
G G F χ − F 0 χ0 = 0 ,
2
δT34 = F31τ δF4 1τ + δF32τ F4 2τ = 0 .
(3.72)
(3.73)
Thus, the perturbation (3.68)-(3.71) decouples from all other modes, and the only nontrivial linearized equation is
d ∗ δF3 = 0 .
(3.74)
The calculation we need to perform is the same as above, except we now set ψ = 0 and
find
3 4/3
ε h(τ ) K(τ )4 sinh2 τ 4 σ 0 ,
16
o
n
9
2
2
4
0
∂µ (χ − σ) = − K(τ ) ∂τ K sinh τ ∂µ σ .
8
χ0 =
84
(3.75)
(3.76)
Eliminating χ and changing variables as before,
w̃ =
ε4/3
K 5 sinh2 τ σ 0 ,
0
gs M α
(3.77)
we find
cosh2 τ + 1
ε4/3 h
w̃ −
w̃ +
4 w̃ = 0 .
6K 2
sinh2 τ
00
(3.78)
After introducing the dimensionless mass as in (3.50), we get a non-minimal scalar
equation
w̃00 −
I(τ )
cosh2 τ + 1
w̃ + m̃2
w̃ = 0 .
2
K(τ )2
sinh τ
(3.79)
Since the 4-dimensional parity operation includes sign reversal of RR fields, we identify
the family of glueballs coming from this eigenvalue problem as pseudoscalars whose S P C
quantum numbers are 0−− .
If we set m̃ = 0 the solution regular at small τ is (sinh 2τ − 2τ )/ sinh τ . Since this
blows up at large τ we conclude that this equation does not contain a massless glueball.
A simple numerical analysis using the shooting method allows one to find the spectrum.
The lowest eigenvalues (m̃2 < 100) are listed in Table 3.2. The quadratic fit is
m̃2IIIn = 0.996 + 1.15 n + 0.289 n2 ,
(3.80)
and in the BHM normalization it is given by
m2III BHM = 0.938 + 1.08 n + 0.272 n2 .
(3.81)
The spectrum can be reproduced with good accuracy using a semiclassical (WKB)
approximation. The effective potential in (3.79) is singular at τ = 0 which does not
allow us to use the conventional WKB approximation. Yet we can cast the equation
(3.79) in the form Q1 Q2 w̃ = m2 w̃, where Qi are first-order differential operators and
85
m2
100
80
n
1
2
3
4
m̃2n
2.41
4.47
7.08
10.3
n
5
6
7
8
m̃2n
14.0
18.0
23.2
28.7
n
9
10
11
12
m̃2n
m̃2n
34.8
41.4
48.6
56.4
n
13 64.7
14 73.7
15 83.2
16 93.3
60
40
20
5
10
15
n
Table 3.2: Non-zero eigenvalues with m̃2 < 100 in the RR sector. This spectrum can also be
fitted by a quadratic polynomial (red line).
˜ which must give rise to the same spectrum
then consider an equation Q2 Q1 w̃˜ = m2 w̃,
up to a zero mode. Namely, in our case this means that for A such that
A 2 + A0 =
cosh2 τ + 1
,
sinh2 τ
(3.82)
equation (3.79) shares the spectrum with an equation
I(τ ) ˜
w̃˜ 00 − (B 2 + B 0 )w̃˜ + m̃2
w̃ = 0 ,
K(τ )2
1 d
I(τ )
where B = −A −
log
.
2 dτ
K(τ )2
(3.83)
(3.84)
The general solution of (3.82) reads
2 sinh2 τ
.
A = − coth τ +
cosh τ sinh τ − τ + C
(3.85)
For (3.83) to be non-singular at the origin C has to be non-zero. For finite C the
potential is regular everywhere but not monotonic and (3.83) admits a zero mode. The
most convenient choice is to take infinite C, which leads to A = − coth τ . In this case
the WKB approximation is applicable in its simplest form (see [113, 114]) and yields the
same results as the shooting method up to the accuracy given above.
86
3.4
Organizing the Modes into Supermultiplets
The pseudoscalar Goldstone mode and the massless scalar found in [52, 53] belong to
a 4-dimensional chiral multiplet. These fields appear as the phase and the modulus of
the baryonic order parameters that vary along the baryonic branch. When a long KS
throat is embedded into a Calabi-Yau compactification with fluxes, the baryonic U(1)
symmetry becomes gauged and a supersymmetric version of the Higgs mechanism is
expected to take place. The axial vector U(1)B gauge field ‘eats’ the pseudoscalar mode
and acquires a mass degenerate with the mass of a scalar Higgs. These fields constitute
the bosonic content of a massive N = 1 axial vector supermultiplet.
Above we explicitly constructed the massive modes that are radial excitations of the
GHK scalar. It is, of course, interesting to find the supermultiplets they belong to. We
will argue that each of these scalar radial excitations is also a member of a massive
axial vector supermultiplet. Similarly, each pseudoscalar glueball found in Section 3.3
is a member of a massive vector multiplet. To prove these facts we would need to
demonstrate the existence of the S P C = 1+− glueballs degenerate with the 0+− glueballs
found in Section 3.2, as well as of 1−− glueballs degenerate with the 0−− glueballs found
in Section 3.3. Unfortunately, constructing decoupled equations for vector supergravity
fluctuations around the KS background is a difficult task. Instead, we will provide some
evidence for our claims by studying axial vector and vector fluctuation equations in the
large radius (KT) limit (setting α0 = gs = 1, N = 0 and M = 2; see Subsection 1.2.2).
First we reconsider the simple decoupled pseudoscalar equation from the RR sector
(3.78) and argue that its superpartner is given by the four-dimensional vector A1 in
δB2 = A1 ∧ g 5 ,
(3.86)
where we have chosen the ansatz so that the corresponding radial component ∼ dr ∧ g 5
vanishes. The equation for d ∗ H3 implies (with primes denoting derivatives with respect
87
to r)
h r3
6
∗4 A01
i0
−
4r
hr3
∗ 4 A1 −
d4 ∗4 d4 A1 = 0 ,
3
6
(3.87)
and d4 ∗4 A1 = 0, i.e. the vector is divergence-free. Since the Laplacian acting on such
a vector is 4 = − ∗4 d4 ∗4 d4 (note the Minkowski signature of the four dimensional
metric), we find
h r3
i0 4r
hr3
A01 −
A1 +
4 A1 = 0 .
6
3
6
(3.88)
Defining a new variable Ã1 = rA1 , it is easy to see that its equation of motion,
r h r 0 i0
hr2
Ã1 − Ã1 +
4 Ã1 = 0 ,
3 3
9
(3.89)
coincides with the KT limit of the equation for the decoupled pseudoscalar w̃, once
we identify r ∼ ε2/3 eτ /3 . In fact, if we make the same ansatz (3.86) in the full KS
background, the equation of motion for Ã1 = K 2 sinh τ A1 resulting from the terms
∼ d3 x ∧ dτ ∧ ω2 ∧ ω2 in d ∗ H3 = 0 is precisely as in (3.78):
ε4/3 h
d2
cosh2 τ + 1
Ã
+
Ã
−
4 Ã1 = 0 .
1
1
dτ 2
6K 2
sinh2 τ
(3.90)
However, this ansatz is not closed in the KS case. The Bianchi identity for F̃5 is not
satisfied, so this NSNS vector must mix with RR excitations of F3 and/or F̃5 in the KS
background.
Let us now turn to the massive axial vector superpartners of the coupled scalars
(3.48), (3.49) found above. We make the following ansatz, which is similar to the one
studied in [130],
δC4 = B1 ∧ ω3 + F2 ∧ ω2 + K1 ∧ dr ∧ ω2 ,
(3.91)
δC2 = C1 ∧ g 5 + D2 + E1 ∧ dr ,
(3.92)
δB2 = H2 + J1 ∧ dr ;
(3.93)
88
where B1 , C1 , E1 , J1 , K1 are axial vectors and D2 , F2 , H2 are two-forms in four dimensions. We choose to split the six degrees of freedom residing in the two-form into a
vector and a dual vector, e.g. D2 = d4 (. . .) + ∗4 d4 D1 . The degrees of freedom contained
in the first (exact) part are in fact the same as those in E1 , so we can simply write
D2 = ∗4 d4 D1 without loss of generality. Similarly, F2 = ∗4 d4 F1 and H2 = ∗4 d4 H1 , and
the corresponding exact parts can be absorbed into the vectors K1 and J1 , respectively.
The equations of motion then imply that B1 and C1 have to be divergence-free:
d4 ∗4 B1 = d4 ∗4 C1 = 0. If this were not the case their divergences would simply
couple to additional scalars δC4 ∼ dr ∧ ω3 and δC2 ∼ dr ∧ g 5 , respectively, but we
will not consider this here (i.e. as for A1 above we choose as gauge in which these radial
components vanish). In fact we will assume that all vectors in our ansatz are divergencefree, and that the terms appearing in the RR- and NSNS-potentials are eigenstates of
the Laplacian 4 = − ∗4 d4 ∗4 d4 with eigenvalue m2 .
Let us present some of the details of the derivation of the equations of motion. With
the ansatz (3.91)–(3.93), the deformations of the field strengths are
δH3 = − ∗4 4 H1 + ∗4 d4 H10 + d4 J1 ∧ dr ,
(3.94)
hr5
h2 r5
4 H1 ∧ ω2 ∧ ω3 ∧ dr +
d4 H10 − ∗4 d4 J1 ∧ ω2 ∧ ω3 ;
(3.95)
54
54
δF3 = d4 C1 ∧ g 5 − C10 ∧ dr ∧ g 5 − C1 ∧ dg 5 + ∗4 d4 D10 + d4 E1 ∧ dr + d4 ∗4 d4 D1 ,(3.96)
∗δH3 = −
∗δF3 =
r3
r
hr3
∗4 d4 C1 ∧ ω2 ∧ ω2 ∧ dr +
∗4 C10 ∧ ω2 ∧ ω2 + ∗4 C1 ∧ dg 5 ∧ g 5 ∧ dr
6
6
3
2 5
hr5
h
r
+
d4 D10 − ∗4 d4 E1 ∧ ω2 ∧ ω3 −
4 D1 ∧ ω2 ∧ ω3 ∧ dr ;
(3.97)
54
54
δF5 = δF5 + ∗δF5 ,
(3.98)
δF5 = d4 B1 − B10 ∧ dr ∧ ω3 + − ∗4 4 F1 + ∗4 d4 F10 ∧ dr ∧ ω2
+d4 K1 ∧ dr ∧ ω2 ,
∗δF5 =
(3.99)
3
3
hr
∗4 d4 B1 ∧ ω2 ∧ dr +
∗4 B10 ∧ ω2 −
4 F1 ∧ ω3 ∧ dr
r
hr
3
r
+ d4 F10 − ∗4 d4 K1 ∧ ω3 .
3
89
(3.100)
Substituting these expressions into the Bianchi identity for F̃5 , and the form equations
(A.2) for ∗F3 and ∗H3 , we find the equations of motions. Splitting them into exact and
coexact parts with respect to the 4-dimensional derivative operator d4 shows that the
vectors E1 , H1 and K1 decouple3 . The resulting equations for the remaining vectors
read
h3
i0 3
3
0
B1 + 4 B1 = − 4 D1 ,
hr
r
r
h r i0 hr
3
0
F1 +
4 F1 = J1 + C1 ,
3
3
r
h r3 i0 4r
3
hr
3
9
C10 −
C1 +
4 C1 = 4 F1 − 2 B10 ,
6
3
6
r
hr
h hr5
i0 h2 r5
3
r
D10 +
4 D1 = −F10 − B1 − 3 log J1 ,
54
54
r
r∗
h hr5 i0
r 0
3
J1 + 3 log D1 = F10 + B1 ,
54
r∗
r
5
3 0 hr
r
4 F1 −
B1 =
4 J1 + 3 log 4 D1 ,
hr
54
r∗
(3.101)
(3.102)
(3.103)
(3.104)
(3.105)
(3.106)
where (3.102), (3.104) and (3.105) hold modulo terms annihilated by d4 . It is easy to
see that (3.105) and (3.106) imply (3.101), so the latter is not independent. We thus
have the five coupled equations for the five vectors B1 , C1 , D1 , F1 and J1 .
In the massless case, our ansatz includes the pseudoscalar found in [52, 53]. Putting
C1 = −f2 (r) d4 a(x) ,
4 D1 = f1 d4 a(x) ,
B10 = −f1 h r log
F1 = J1 = 0 ,
(3.107)
(3.108)
r
d4 a(x) ,
r∗
(3.109)
(3.110)
for some constant f1 and a four-dimensional massless pseudoscalar a(x), all equations
5
r
r
More precisely, we set hr
54 E1 = −3 log r∗ H1 = r log r∗ K1 , and find a single second order differential
equation obeyed by these fields. Thus we have found another decoupled vector, but this is not the one
we are looking for. Given this relation between them, E1 , H1 and K1 do not mix with the other vectors.
3
90
of motion are satisfied provided
h r3
6
f20
i0
−
9
r
4r
f2 = − f1 log
;
3
r
r∗
(3.111)
in perfect agreement with the literature (see also Subsection 1.2.3).
Now we would like to consider massive excitations however, and find axial vectorlike solutions to the equations (3.101)-(3.106) which give rise to the superpartners of
the massive scalar excitations of (3.48) and (3.49). In particular, changing variables to
W1 = rC1 equation (3.103) becomes
r h r 0 i0
hr2
2
6
W1 − W1 +
4 W1 = 4 F1 − 2 B10 .
3 3
9
r
hr
(3.112)
Thus we can identify W1 with w̃ in (3.49), which suggests setting the right hand side of
this equation proportional to the counterpart of z̃/r. Hence we define
Z1 ≡ 4 F1 −
3 0
B .
hr 1
(3.113)
Using (3.101) and (3.102) one can deduce that this new field obeys
r h r 0 i0 hr2
1
r
Z1 +
4 Z1 = 4 W1 + 4 J1 + D10 .
3 3
9
r
3
(3.114)
Our reduced ansatz containing five axial vectors is still too general. In order to
match the spectrum of the scalar particles found above, we need to impose an additional
constraint to reduce the number of dynamical vectors obeying independent second order
differential equations to two. The correct constraint for our purposes is given by
3
4 J1 + D10 = 2 4 W1 .
r
(3.115)
In order to show that we can consistently impose this relation we need to examine the
91
remaining equations. First of all, with this constraint (3.106) reads
Z1 = −
hr5
hr3
r
4 D10 +
4 W1 + 3 log 4 D1 .
54
18
r∗
(3.116)
Adding (3.104) and (3.105), using the constraint and the fact that W1 is a mass eigenstate
we find
h hr3
18
W1
i0
h2 r5
9
r
4 D1 = 0 .
+ 2 log W1 +
r
r∗
54
(3.117)
Eliminating 4 D1 between the last two equations we obtain a second order differential
equation containing only W1 and Z1 . A non-trivial fact is that this equation is identical
to (3.112). This relies heavily on the precise expression for the warp factor (1.48), and
shows the consistency of the constraint equation with the equations of motion.
Finally, introducing a symbol for the other combination of the vectors F1 and B1
that appears in the equations of motion
Y1 ≡ F10 +
3
B1 ,
r
(3.118)
3
4 D1 .
r
(3.119)
equation (3.105) implies
4 Y1 = Z10 −
In summary, we have the two coupled dynamical equations
r h r 0 i0 hr2
2
Z1 +
4 Z1 = 4 W1 ,
3 3
9
r
h
i
2
0
r r 0
hr
2
W1 − W1 +
4 W1 = Z1 ,
3 3
9
r
(3.120)
(3.121)
which determine W1 and Z1 . In terms of these 4 D1 is determined by (3.117), J1 by
(3.115), and 4 Y1 by (3.119). Equations (3.120), (3.121) are precisely the KT limit of
the scalar equations (3.48), (3.49) up to a rescaling of the fields by a numerical factor.4
Looking at (3.48) one might have expected the term 2z̃/ sinh2 τ to give rise to a term proportional
to Z1 /r6 in (3.120), but in fact this is not the case because it is too small to be seen in the KT limit.
4
92
Since the KT limits of their equations of motion agree, we thus argue that the axial
vectors Z1 and W1 are the superpartners of the coupled scalars z̃ and w̃ found above,
and that their massive excitations combine into vector multiplets. Subsequently, this
was shown rigorously by explicit construction of the vector fluctuations of the full KS
solution in [131], where all I-odd SU(2) × SU(2) invariant perturbations of the warped
throat were found and categorized.
3.5
Effects of Compactification
Now we will embed the KS throat into a flux compactification, along the lines of [132],
and estimate the mass of the Higgs scalar. Generally, glueballs are dual to the normalizable modes localized near the bottom of the throat, and one does not expect them to
be strongly affected by the bulk of the Calabi-Yau. This is indeed the case for all the
massive radial excitations found in Sections 3.2 and 3.3. We will see, however, that the
case of the GHK scalar is more subtle and exhibits some UV sensitivity.
To model a compactification, we will introduce a UV cut-off on the radial coordinate,
τmax . We also need to include a deformation of the KS solution introduced by bulk effects.
On the field theory side this corresponds to perturbing the Lagrangian of the cascading
gauge theory by some irrelevant operators. Here we are not interested in classifying all
of them but rather model the compactification effects in the simplest way by considering
one perturbation which simulates the main features of the compactified solution. We
consider a shift of the warp factor δh = const., which corresponds to the dimension 8
operator on the field theory side [133, 134, 135]. This also has a simple geometrical
meaning: the warp factor of the compactified solution is a finite constant in the bulk of
the Calabi-Yau and therefore should not drop below a certain value along the throat.
In the KS background it arises from a subleading term in the variation of the Ricci tensor (3.33) but
such terms that are asymptotically suppressed by powers of r compared to the leading terms are not
taken into account in the KT metric. Indeed, if we write ansatz (3.2)-(3.6) in the KT background and
follow the same strategy as we did for the full KS background, the term proportional to z̃/r6 does not
appear in the Einstein equations.
93
Let us introduce a small parameter δ which shifts the rescaled warp factor, I(τ ) →
I(τ ) + δ, and consider the system (3.51)–(3.52) in perturbation theory near m̃2 = 0:
z̃ = z̃0 + m̃2 z̃1 ,
(3.122)
w̃ = w̃0 + m̃2 w̃1 ,
(3.123)
z̃0 = τ coth τ − 1 ,
w̃0 = −
(3.124)
22/3 8 1
9 sinh τ
Z
τ
dx I(x) sinh2 x .
(3.125)
0
At leading order in m̃2 we find
τ
Z
Z
τ
dx u(x) (x coth x − 1) ,
(3.126)
dx u(x) coth x − coth τ
z̃1 = (τ coth τ − 1)
0
Z τ 0
Z ∞
1
sinh 2x − 2x sinh 2τ − 2τ
1
w̃1 = −
dx v(x)
dx v(x)
−
, (3.127)
4 sinh τ 0
sinh x
4 sinh τ
sinh x
τ
where
9
δ
I(τ )
z̃0 +
K(τ ) w̃0 − 2 z̃0 ,
2
2/3
K (τ )
4·2
K
I(τ )
16
δ
v(τ ) = − 2
w̃0 +
K(τ ) z̃1 − 2 w̃0 .
K (τ )
9
K
u(τ ) = −
(3.128)
(3.129)
Keeping in mind that for large τ we have u ' −2−2/3 δ τ e2τ /3 one finds the asymptotic
behavior
−2/3
z̃1 (τ ) ' −2
Z
δ
τ
dx (τ − x) x e2x/3 ' −
0
9δ
τ e2τ /3 .
4 22/3
(3.130)
This yields v ' −22/3 δ τ eτ /3 and
Z τ
Z
1
sinh 2x − 2x sinh 2τ − 2τ ∞
1
w̃1 = −
dx v0 (x)
−
dx v0 (x)
4 sinh τ 0
sinh x
4 sinh τ
sinh x
τ
2/3
9 2 δ τ /3
'
τe .
(3.131)
8
94
log m
"3
"4
"5
"6
10
15
20
25
30
35
Τmax
Figure 3.1: The dependence of log m̃ on τmax is linear with the slope equal to -1/3. The three
lines shown correspond to δ = 1, δ = 0.01 and δ = 0.0001.
Finally, up to first order in the mass-squared and δ:
9 δ m̃2 2τ /3 e
z̃ ' τ 1 −
,
4 22/3
9 δ m̃2 2τ /3 e
w̃ ' −24/3 τ e−τ /3 1 −
.
8 22/3
(3.132)
This suggests that for generic boundary conditions the cut-off value
τmax ' − log (δ 3/2 m̃3 ) .
(3.133)
This prediction can be tested numerically. In order to do so one can specify some small
m̃ and plot the determinant
z̃1 (τ ) z̃2 (τ )
det
,
w̃1 (τ ) w̃2 (τ )
(3.134)
of the two linearly independent solutions regular at τ = 0 as a function of τ . The first zero
marks the point τmax such that there is a regular solution with z(τmax ) = w(τmax ) = 0.
Hence τmax is the corresponding cut-off value. As Figure 3.1 shows, the relation (3.133)
holds for τmax large enough that m̃2 is small, where
m̃2 ∼ δ −1 e−2τmax /3 .
(3.135)
Let us consider a simple model of compactification where the throat is embedded
95
into an asymptotically conical space that terminates at some large cut-off value τmax .
To calculate the mass from (3.135) we need to know δ as well as τmax . The former is the
asymptotic value of the (rescaled) warp factor. The point where the field theory warp
factor approaches δ marks the UV cutoff of the field theory
I(τU V ) ∼ τU V e−4τU V /3 ' δ .
(3.136)
Using this in (3.135) we find m̃2 ∼ e(4τU V −2τmax )/3 . This shows that the Higgs mass
becomes parametrically small only for τmax 2τU V . This is not satisfied in general;
the geometry requires only that τmax > τU V because τU V is the length of the throat
embedded into a CY space. With the ratio between the UV and IR scales of the field
theory around 4·103 [121] we estimate that τU V ' 25 [59]. The cut-off τmax can be related
to the warped volume of the Calabi-Yau which, in a singular conifold approximation, is
V6w = Vol(T 1,1 )
Z
s
rmax
dr h(r)
0
det g6
,
det gT 1,1
(3.137)
where r ∼ ε2/3 eτ /3 . The integral from zero to rU V is the warped volume of the throat,
and from rU V to rmax is the bulk volume. Assuming that the latter dominates,
V6w '
6
16π 3 4/3
ε (gs M α0 )2 rmax
− rU6 V rU−4V .
27
(3.138)
Requiring τmax 50 leads to an enormous V6w , far larger than, for example, V6w ' 56 α03
in [121]. Thus, while for τmax 2τU V the Higgs scalar becomes parametrically lighter
than the other normal modes, in compactifications with realistic parameters it may
actually be heavier. This is due to the special feature of its wave function z̃ which grows
linearly with τ in the throat. The only conclusion we can draw from our simplified model
of compactification is that this mode is rather UV sensitive, so to determine its mass we
need to know the details of the compactification.
96
Chapter 4
Baryonic Condensates on the
Conifold
4.1
Introduction
Consideration of a stack of N D3-branes leads to the duality of N = 4 super Yang-Mills
theory to type IIB string theory on AdS5 ×S5 [1, 2, 3]. A different, N = 1 supersymmetric
example of the AdS/CFT correspondence follows from placing the stack of D3-branes at
the tip of the conifold [44, 45]. This suggests a duality between a certain SU(N )×SU(N )
superconformal gauge theory and type IIB string theory on AdS5 × T 1,1 . Addition of M
D5-branes wrapped over the two-sphere near the tip of the conifold changes the gauge
group to SU(N + M ) × SU(N ) [111, 112]. This theory is non-conformal; it undergoes
a cascade of Seiberg dualities [50] SU(N + M ) × SU(N ) → SU(N − M ) × SU(N ) as it
flows from the UV to the IR [49, 51] (for reviews, see [10, 11, 12] and Section 1.2).
The gauge theory contains two doublets of bifundamental, chiral superfields Ai , Bj
(with i, j = 1, 2). In the conformal case, M = 0, it has continuous global symmetries
SU(2)A ×SU(2)B ×U(1)R ×U(1)B . The two SU(2) groups rotate the doublets Ai and Bj ,
while one U(1) is an R-symmetry. The remaining U(1) factor corresponds to the baryon
97
number symmetry which we will be most interested in. As argued in [49, 52, 53, 56, 59],
in the cascading theory where N is an integer multiple of M , N = kM , this symmetry
is spontaneously broken by condensates of baryonic operators. In this chapter we will
provide a quantitative verification of this effect.
For N = kM the last step of the cascade is an SU(2M )×SU(M ) theory which admits
two baryon operators (sometimes referred to as baryon and antibaryon)
α
α
α
α
A ∼ α1 α2 ...α2M (A1 )α1 1 (A1 )α2 2 . . . (A1 )αMM (A2 )1 M +1 (A2 )2 M +2 . . . (A1 )αM2M ,
B ∼ α1 α2 ...α2M (B1 )α1 1 (B1 )α2 2 . . . (B1 )αMM (B2 )1 M +1 (B2 )2 M +2 . . . (B1 )αM2M .
(4.1)
Baryon operators of the general SU(M (k + 1)) × SU(M k) theory have the schematic
form (A1 A2 )k(k+1)M/2 and (B1 B2 )k(k+1)M/2 , with appropriate contractions described in
[56]. Unlike the dibaryon operators of the conformal SU(N ) × SU(N ) theory [111], A
and B are singlets under the two global SU(2) symmetries. These operators acquire
expectation values that spontaneously break the U(1)B baryon number symmetry; this
is why the gauge theory is said to be on the baryonic branch of its moduli space [57].
Supersymmetric vacua on the one complex dimensional baryonic branch are subject to
the constraint AB = −Λ4M
2M , and thus we can parameterize it as follows
i
.
B = Λ2M
ζ 2M
A = iζΛ2M
2M ,
(4.2)
The non-singular supergravity dual of the theory with |ζ| = 1 is the warped deformed
conifold found in [49]. In [52, 53] the linearized scalar and pseudoscalar perturbations,
corresponding to small deviations of ζ from 1, were constructed, and in the previous
chapter we have generalized this construction to fluctuations with non-zero momentum.
The full set of first-order equations necessary to describe the entire moduli space of
supergravity backgrounds dual to the baryonic branch, called the resolved warped deformed conifolds, was derived and solved numerically in [58] (for further discussion of
98
these solutions, see [59] and Subsection 1.2.4).
The construction of this moduli space of supergravity backgrounds, which have just
the right symmetries to be identified with the baryonic branch in the cascading gauge
theory, provides an excellent check on the gauge/string duality in this intricate setting.
Yet, one question remains: how do we identify the baryonic expectation values on the
string side of this duality? Among other things, this is needed to construct a map
between the parameter U that labels the supergravity solutions, and the parameter |ζ|
in the gauge theory.
The dual string theory description of the baryon operators (4.1) was first considered
by Aharony [56]. He argued that the heavy particle dual to such an operator is described
at large r by a D5-brane wrapped over the T 1,1 , with some D3-branes dissolved in it
(to account for this, the world volume gauge field needs to be turned on). To calculate
the two-point function of baryon operators inserted at x1 and x2 we may use a semiclassical approach to the AdS/CFT correspondence. Then we need a (Euclidean) D5brane whose world volume has two T 1,1 boundaries at large r, located at x1 and x2 . Here
we will be interested in a simpler embedding of the D5-brane: as suggested by Witten
(unpublished, 2004), the object needed to calculate the baryonic expectation values
is the Euclidean D5-brane that has the appearance of a pointlike instanton from the
four-dimensional point of view, and wraps the remaining six (generalized Calabi-Yau)
directions of the ten-dimensional spacetime. This object has a single T 1,1 boundary
at large r, corresponding to insertion of just one baryon operator. As we will find,
supersymmetry requires that the world volume gauge field is also turned on, so there are
D3-branes dissolved in the D5. This identification will be corroborated by demonstrating
that the D5-brane couples correctly to the pseudoscalar zero-mode of the theory that
changes the phase of the baryon expectation value [52, 53].
Close to the boundary, a field ϕ dual to an operator of dimension ∆ in the AdS/CFT
99
correspondence behaves as
ϕ(x, r) = ϕ0 (x) r∆−4 + Aϕ (x) r−∆ ,
(4.3)
Here Aϕ is the operator expectation value [48], and ϕ0 is the source for it. In the
cascading theory, which is near-AdS in the UV, the same formulae hold modulo powers
of ln r [136, 137]. The field corresponding to a baryon will be identified, at a semiclassical level, with e−SD5 (r) , where SD5 (r) is the action of a D5-brane wrapping the
Calabi-Yau coordinates up to the radial coordinate cut-off r. The different baryon
operators A, A, B, B will be distinguished by the two possible D5-brane orientations,
and the two possible κ-symmetric choices for the world volume gauge field that has to
be turned on inside the D5-brane. In the cascading gauge theory there is no source
added for baryonic operators, hence we find that ϕ0 = 0. On the other hand, the term
scaling as r−∆ is indeed revealed by our calculation of e−SD5 (r) as a function of the radial
cut-off, allowing us to find the dimensions of the baryon operators, and the values of
their condensates.
This chapter relies heavily on material from the paper [138] coauthored with A. Dymarsky and I. R. Klebanov, and is subdivided as follows. In the remainder of Section 4.1
we discuss the Killing spinors of the warped supergravity backgrounds dual to the baryonic branch. We also review the κ-symmetry conditions for D-brane embeddings, and
briefly discuss a number of brane configurations that satisfy them. Section 4.2 is devoted to the derivation of the first-order equation for the gauge field. We first discuss a
Lorentzian D7-brane wrapping the warped deformed conifold directions, before presenting a parallel treatment for the more subtle case of the Euclidean D5-brane wrapping
the conifold. In Section 4.3 we investigate the physics of the D5-instanton in the KS
background. From the behavior of the D5-brane action as a function of the radial cut-off
we extract the dimension of the baryon operator, and show that it matches the expec-
100
tations from the dual cascading gauge theory. We also show that the D5-brane couples
to the baryonic branch complex modulus in the way consistent with our identification
of the condensates. In particular, we demonstrate that pseudoscalar perturbations of
the backgrounds shift the phase of the baryon expectation value. We generalize to the
complete baryonic branch in Section 4.4, where we compute the baryon expectation values as a function of the supergravity modulus U . The product of the expectation values
calculated from the D5-brane action is shown to be independent of U in agreement with
(4.2). Finally, we present an integral expression for their ratio and evaluate it numerically, which provides a relation between the baryonic branch modulus |ζ| in the gauge
theory and the modulus U in the dual supergravity description, and show that they
satisfy AB = const. We conclude briefly in Section 4.5.
4.1.1
D-Branes, κ-Symmetry and Killing Spinors of the Conifold
A Dirichlet p-brane (with p spatially extended dimensions) in string theory is described
by an action consisting of two terms [139, 140, 141]: the Dirac-Born-Infeld action, which
is essentially a minimal area action including non-linear electrodynamics, and the ChernSimons action, which describes the coupling to the RR background fields:
Z
S = SDBI + SCS = −
p+1
d
−φ
σe
W
Z
p
− det(G + F) +
eF ∧ C .
(4.4)
W
Here W is the worldvolume of the brane and we have set the brane tension to unity.
Further, G is the induced metric on the worldvolume, F = F2 +B2 is the sum of the gauge
P
field strength F2 = dA1 and the pullback of the NSNS two-form field, and C = i Ci is
the formal sum of the RR potentials. In superstring theory all these fields should really
be understood as superfields, but we shall ignore fermionic excitations here.
Wick rotation of this action to Euclidean space such that all p + 1 directions become
101
spatially extended (which leads to a Euclidean worldvolume D-instanton) effectively
multiplies the action by a factor of i. This cancels the minus sign under the square root
in the DBI term and leaves it real since the determinant is now positive. The CS term
however is purely imaginary now. Consequently the equations of motion that follow
from the DBI and CS terms now have be satisfied independently of each other if we
insist on the gauge field being real.
The action (4.4) is invariant on shell under the so-called κ-symmetry [142, 143, 144].
This allows us to find first-order equations for supersymmetric configurations which are
easier to solve than the second order equations of motion. The κ-symmetry condition
can be written as
Γκ ε = ε ,
(4.5)
where ε is a doublet of Majorana-Weyl spinors, and the operator Γκ is specified below.
Satisfying this equation guarantees worldvolume supersymmetry in the probe brane
approximation, and every solution for which ε is a Killing spinor corresponds to a supersymmetry compatible with those preserved by the background.
The decomposition of a Weyl spinor ε into a doublet of Majorana-Weyl spinors
ε1
ε=
ε2
(4.6)
is achieved by projecting onto the eigenstates of charge conjugation1 ε1 = (ε + ε∗ )/2 and
ε2 = (ε − ε∗ )/2i.
In IIB superstring theory on a (9, 1) signature spacetime, the κ-symmetry operator
Γκ for a Lorentzian D-brane extended along the time direction x0 and p spatial directions
Given any spinor ε we denote its charge conjugate by ε∗ , which of course is represented by complex
conjugation and left multiplication by a charge conjugation matrix B. We do not write B explicitly
here, though its presence is understood.
1
102
is given by
Γκ = p
√
− det G
∞
X
(−1)n F/ n Γ(p+1) ⊗ (σ3 )n+
p−3
2
− det(G + F) n=0
1
√
Γ(p+1) ≡
ε µ1 ...µp+1 Γµ1 ...µp+1 ,
(p + 1)! − det G
1
F/ n ≡ n Γν1 ...ν2n Fσ1 σ2 . . . Fσ2n−1 σ2n Gν1 σ1 . . . Gν2n σ2n .
2 n!
iσ2 ,
(4.7)
(4.8)
(4.9)
Here σi are the usual Pauli matrices. We use Greek labels for the worldvolume indices
of the D-brane and consequentially the Γµ are induced Dirac matrices. In what follows
we denote the Minkowski spacetime coordinates by x0 . . . x3 and label the tangent space
of the internal manifold M by 1, 2 . . . 6 in reference to the basis one-forms (1.66). The
expression for Γκ can be significantly simplified for an embedding covering all six directions of the deformed conifold, in which case we simply align the worldvolume tangent
space with that of M.
The Killing spinor Ψ of the supergravity backgrounds dual to the baryonic branch
is built out of a six-dimensional pure spinor η − and an arbitrary spinor ζ − of negative
four-dimensional chirality,
Ψ = α ζ − ⊗ η − + iβ ζ + ⊗ η + ,
(Γ1 − iΓ2 )ζ − ⊗ η − = (Γ3 − iΓ4 )ζ − ⊗ η − = (Γ5 − iΓ6 )ζ − ⊗ η − = 0 ,
(4.10)
(4.11)
where η + = (η − )∗ and ζ + = (ζ − )∗ . The functions α and β are real [58, 59] and given by
eφ/4 (1 + eφ )3/8
α=
,
(1 − eφ )1/8
eφ/4 (1 − eφ )3/8
β=
.
(1 + eφ )1/8
(4.12)
(this expression for β is for U > 0; β changes sign when U does). The corresponding
103
Majorana-Weyl spinors Ψ1 and Ψ2 are
1
(α − iβ)ζ − ⊗ η − + (α + iβ)ζ + ⊗ η + ,
2
1
Ψ2 =
(α + iβ)ζ − ⊗ η − − (α − iβ)ζ + ⊗ η + .
2i
Ψ1 =
4.1.2
(4.13)
(4.14)
Branes Wrapping the Angular Directions
In the context of the conifold, the closest analogue to the baryon vertex in AdS5 × S5
that was discussed in [145, 146, 147], would be a D5-brane wrapping the five angular
directions of the internal space, with worldvolume coordinates σ µ = (x0 , θ1 , φ1 , θ2 , φ2 , ψ).
The brane describing the baryon vertex in AdS5 × S5 has “BI-on” spikes corresponding to fundamental strings attached to the brane and ending on the boundary of AdS,
indicating that it is not a gauge-invariant object. Here however, we are interested in
gauge-invariant, supersymmetric objects, that are candidate duals to chiral operators
in the gauge theory, so we might try to consider a smooth embedding at constant radial coordinate (the difference between a “baryon” and a “baryon vertex” was already
stressed in [145]).
To avoid having the BI-on spikes, it was proposed [56] that we should use an appropriate combination of D5-branes wrapping all the angular coordinates, and of D3-branes
wrapping the S3 . This is equivalent to turning on a particular gauge field on the wrapped
D5-brane. Unfortunately, it is not clear how to maintain the supersymmetry of such an
object. It is not hard to see, for example from the appropriate κ-symmetry equations,
that a (Lorentzian) D5-brane wrapping the five angular direction of the conifold and
embedded at constant r cannot be a supersymmetric object. The κ-symmetry equation
seems to call for an additional constraint of the form Γx0 ψ ε∗ = −iε on the Killing spinors,
which would imply also Γx0 r ε∗ = −ε, i.e. precisely what we would expect for strings
stretched in the radial direction. However, such a projection does not commute with
the other conditions that the Killing spinors have to satisfy and thus is not consistent.
104
This was pointed out in [148] for the case of the singular conifold [44], and the argument carries over to the deformed conifold. Even with a worldvolume gauge field such
a D5-brane cannot be a BPS object.
The same conclusion also follows from the equation of motion for the radial component of the embedding X M (σ µ ). The leading term (as r → ∞) in the D-brane Lagrangian
arises from the B2 -field contribution to the DBI term and is proportional to r (ln r)2 , so
this brane is bound to contract and move to smaller r, until eventually it reaches the
tip of the conifold, where the two-cycle collapses.
On the other hand, as suggested by Aharony [56], the D5-branes with D3-branes
dissolved within them are the particles dual to the baryon operators. As proposed by
Witten (unpublished, 2004), to find the baryonic condensates we need to consider a
Euclidean D5-brane wrapping the deformed conifold directions, with a certain gauge
field turned on. While there are no non-trivial two-cycles in this case, the worldvolume
gauge field does modify the coupling of this D-instanton to the RR potential C4 . We will
show that such a configuration can be made κ-symmetric and then yields the baryonic
condensates consistent with the gauge theory expectations.
As a first example of a supersymmetric brane wrapping all the angular directions,
we shall discuss a D7-brane wrapping the warped deformed conifold, with the remaining
one space and one time directions extended in R3,1 . The supersymmetry conditions for
general D-branes in N = 1 backgrounds were derived in [149, 150, 151], and our results
will be consistent with theirs. We will show that the Lorentzian D7-brane configuration
on the KS background is supersymmetric in the absence of a worldvolume gauge-field,
though the κ-symmetry analysis will also reveal supersymmetric configurations with nonzero gauge field. The fact that switching on this field is not required for supersymmetry
might have been guessed from a naive counting argument. This embedding of the D7brane should be mutually supersymmetric with the D3-branes filling the R3,1 , since the
number of Neumann-Dirichlet directions for strings stretched between them equals eight.
105
The object we are most interested in is the Euclidean D5-brane completely wrapped
on the conifold. In contrast to the case of the D7-brane, we will find that supersymmetry
requires a non-trivial gauge field on the worldvolume. Again this is consistent with the
naive count of Neumann-Dirichlet directions with the D3-branes, which gives ten in this
case and thus indicates that these branes cannot be mutually supersymmetric if F2 = 0.
4.2
Derivation of the First-Order Equation for the
Worldvolume Gauge Bundle
In this section we derive the first-order equation of motion that the U(1) gauge field
has to satisfy to obtain a supersymmetric configuration. Because the κ-symmetry of the
Euclidean D5-brane is subtle, we will first discuss the closely related case of a Lorentzian
D7-brane wrapping the six-dimensional deformed conifold, with non-zero gauge bundle
only in these directions. This object is extended as a string in the R3,1 but in the case
of a non-compact space dual to the cascading gauge theory the tension of such a string
diverges with the cut-off as e2τ /3 . Therefore, this string is not part of the gauge theory
spectrum.
4.2.1
κ-Symmetry of the Lorentzian D7-Brane
The explicit form of the κ-symmetry equation for the D7-brane with non-trivial U(1)
bundle on the six-dimensional internal space is given by
ε1
ε1
ε1
3
2
= Γκ
∼ −(F/ + F/ ) σ3 + (1 + F/ ) iσ2 Γx0 x1 123456
,
ε2
ε2
ε2
(4.15)
For the case of Euclidean D-branes wrapping certain cycles in Calabi-Yau manifolds,
it was shown in [149] that the κ-symmetry condition (4.15) can be rewritten in more
106
geometrical terms. This results in the conditions that F 2,0 = 0, and that
1
1
J ∧J ∧F − F ∧F ∧F =g
2!
3!
1
1
J ∧J ∧J − J ∧F ∧F
3!
2!
.
(4.16)
The constant g was found [149] to encode some information about the geometry, namely
a relative phase between coefficients of the covariantly constant spinors in the expansion
of the εi [149]. As we shall see below, the same equation holds in our case of a generalized
Calabi-Yau with fluxes, except that g becomes coordinate dependent.
With the SU(2) × SU(2) invariant ansatz for the gauge potential
A1 = ξ(τ )g5 ,
(4.17)
we find that the gauge-invariant two-form field strength is given by
F=
ie−x
×
(4.18)
2 sinh(τ )
h
i
˜
e−g ξ(cosh(τ
) + 2a + a2 cosh(τ )) + h2 sinh2 (τ )(1 − a2 ) (G1 + iG2 ) ∧ (G1 − iG2 )
i
2
˜
+ e ξ cosh(τ ) − h2 sinh (τ ) (G3 + iG4 ) ∧ (G3 − iG4 )
i
h
˜ + a cosh(τ )) − h2 a sinh2 (τ )
+ ξ 0 v sinh(τ )(G5 + iG6 ) ∧ (G5 − iG6 ) + ξ(1
(G1 + iG2 ) ∧ (G3 − iG4 ) + (G3 + iG4 ) ∧ (G1 − iG2 ) ,
g
h
where ξ˜ = ξ + χ. This explicitly shows that F is a (1, 1) form, which is one of the
κ-symmetry conditions [149, 150, 151]. Now it is convenient to define
a(ξ, τ ) ≡ e−2x [e2x + h22 sinh2 (τ ) − (ξ + χ)2 ] ,
b(ξ, τ ) ≡ 2e−x−g sinh(τ )[a(ξ + χ) − h2 (1 + a cosh(τ ))] .
107
(4.19)
In terms of these expressions we find that
1
1
J ∧ J ∧ J − J ∧ F ∧ F = (a + ve−x b ξ 0 ) vol6 ,
3!
2!
1
1
J ∧ J ∧ F − F ∧ F ∧ F = (−b + ve−x a ξ 0 ) vol6 ,
2!
3!
(4.20)
where vol6 = (J ∧ J ∧ J)/3!. Thus (4.16) would lead to a differential equation of the
form
ξ0 =
ex (ga + b)
,
v(a − gb)
(4.21)
for some as yet undetermined g. In order to confirm the validity of this equation and
determine the function g we return to the full κ-symmetry equation (4.15) with the
Majorana-Weyl spinors ε1 = (Ψ + Ψ∗ )/2 and ε2 = (Ψ − Ψ∗ )/2i constructed from the
Killing spinor. The analysis of this equation is much simplified by noting that Γ1..6 η ± =
∓iη ± and that the spinors η ± are in fact eigenspinors2 of F/ n
/ ± = ±iη ± (F12 + F34 + F56 ) ,
Fη
(4.22)
F/ 2 η ± = −η ± (F12 F34 + F14 F23 + F12 F56 + F34 F56 ) ,
(4.23)
F/ 3 η ± = ∓iη ± (F12 F34 F56 + F14 F23 F56 ) ,
(4.24)
where the indices refer the basis one-forms (1.66). Then it follows from (4.20) that the
two terms in the κ-symmetry equation act on the spinors in a rather simple fashion:
1 + F/ 2 η ± = a + ve−x bξ 0 η ± ,
F/ + F/ 3 η ± = ±i −b + ve−x aξ 0 η ± .
(4.25)
Using these relations it is easy to see that the Killing spinor (4.10) indeed solves (4.15)
provided we impose the conditions that its four-dimensional parts ζ ± obey the condition
2
For simplicity we drop the four-dimensional spinors ζ ± in ζ ± ⊗ η ± .
108
Γx0 x1 ζ ± ⊗ η ± = ζ ± ⊗ η ± , and that the gauge field ξ(τ ) satisfies (4.21) with
g(τ ) = g7 (τ ) ≡ −
p
2αβ
−φ
=
−e
1 − e2φ .
α2 − β 2
(4.26)
Thus indeed (4.16) holds and (4.21) is the correct first order differential equation given
this function g(τ ).
The fact that the κ-symmetry condition (4.15) is satisfied implies worldvolume supersymmetry in the probe brane approximation. However, we also ask for the worldvolume
supersymmetries to be compatible with those of the background. In order to check how
many supersymmetries of the background are preserved by the brane we need to enumerate the solutions of (4.15) for which ε1 + iε2 is not just any spinor, but a Killing
spinor. For the particular case of the D7-brane with U(1) gauge bundle determined by
the first-order equation (4.21) we saw that Killing spinors of the form (4.10) solve the
κ-symmetry equation if Γx0 x1 ζ ± ⊗ η ± = ζ ± ⊗ η ± , and thus half of the supersymmetries
of the background are preserved.
4.2.2
An Equivalent Derivation Starting from the Equation of
Motion
Here we present an alternative derivation of the first-order equations for the gauge field
ξ(τ ), starting from the second-order equation of motion. This method has the advantage
that it applies equally well to Lorentzian D7 and Euclidean D5-branes wrapping the
conifold. The κ-symmetry argument we employed in the previous section for the D7brane is somewhat complicated in the case of the D5-instanton by the fact that we are
forced to Wick rotate to Euclidean spacetime signature where there are no MajoranaWeyl spinors. However, knowing that a first-order differential equation for the gauge
field exists, as well as its general features, it is not hard to derive it directly from the
second-order equation of motion.
109
Since with Euclidean signature the DBI action is real and the CS action pure imaginary, two sets of equations of motion have to be satisfied simultaneously if we insist
on the gauge field being real. With the ansatz (4.17) for the gauge potential, the CS
equations are automatically satisfied, as are five of the DBI equations; only the one for
the g5 component of the gauge field (or equivalently its ψ component) is non-trivial.
In terms of the (implicitly U -dependent) functions defined in [58] the determinant
that appears in the DBI action is given by
"
detM (G + F) = v −2 e6x (1 + (ξ 0 )2 v 2 e−2x ) 1 + e−4x (ξ + χ)2 − sinh2 (τ )h22
−2e−2x (ξ + χ)2 + sinh2 (τ )h22
1 − 2e−2g a2 sinh2 (τ )
#
−8e−2x−2g sinh2 (τ ) a h2 (ξ + χ)(1 + a cosh(τ )) ,
2
(4.27)
where we have omitted the angular dependence ∼ sin2 θ1 sin2 θ2 . Here we have only taken
into account the six-dimensional internal manifold M. If the brane is also extended in
the Minkowski directions (but carries zero gauge bundle in these directions) there are
additional ξ-independent factors multiplying the DBI determinant that appears in the
action (4.4). E.g. for the Lorentzian D7-brane this factor is equal to e4A . Using the
definitions (4.19), the term in square brackets in (4.27) can be written as a sum of
squares a2 + b2 .
We know from the form of the κ-symmetry equation that the first-order differential
equation we are looking for must
i) be polynomial (of at most third order) in ξ and its first derivative,
ii) contain ξ 0 only at linear order (i.e. no (ξ 0 )2 terms),
iii) be such that the determinant factorizes.
In particular the last condition means that when we eliminate ξ 0 from the action,
p
the ξ-dependent term must be a perfect square, else the factor of detM (G + F) in the
denominator of (4.7) cannot be cancelled by the numerator to give unit eigenvalue. This
110
implies that we must have
a2 + b2
,
f2 (ξ, τ )
(4.28)
a2 + b2 − f2 (ξ, τ )
.
v f(ξ, τ )
(4.29)
(1 + (ξ 0 )2 v 2 e−2x ) =
for some f(ξ, τ ), so that
0
ξ =
ex
p
Because we expect the equation to be polynomial in ξ one must be able to explicitly
take the square root, and thus f(ξ, τ ) can be written as
a − g(τ )b
f(ξ, τ ) = p
,
1 + g2 (τ )
(4.30)
for some function g(τ ), where all the ξ-dependence is now implicit in a and b. With this
ansatz we have
ξ0 =
ex (ga + b)
,
v(a − gb)
(4.31)
which is of the same form as the first order differential equation we derived for the D7brane in the previous section. The function g follows by varying the action with respect
to ξ and substituting for ξ 0 using (4.31). It is not difficult to check that the equations
p
R
of motion that follow from the DBI action of the D7-brane e2A−φ detM (G + F) are
indeed implied by the first order equation (4.31) with
g = g7 =
p
ex−g (1 + a cosh(τ ))
= −e−φ 1 − e2φ ,
h2 sinh(τ )
(4.32)
as we found above using a κ-symmetry argument.
Using the same method, we can now find the first-order equation for the gauge field
on the Euclidean D5-brane. Having constrained the equation we are looking for to the
p
R
form (4.31) we vary the DBI action e−φ detM (G + F) using (4.27) and eliminate ξ 0
111
to obtain
i
δ h −φ p
e
det(G + F + B) = 0 =
δξ
√
d e−φ e2x (ga + b)
2e−φ e2x 1 + g2 −x
−g
√
−(ξ + χ)e a + e a sinh(τ )b −
. (4.33)
v(a − gb)
dτ
1 + g2
Collecting powers of ξ and equating their coefficients to zero we find differential equations
for g(τ ) which are solved simultaneously by
g = g5 ≡ −
e−x+g h2 sinh(τ )
eφ
=√
.
(1 + a cosh(τ ))
1 − e2φ
(4.34)
Substituting this into (4.31), the first-order equation we were looking for, written out in
full detail, is
ξ0 =
h
− h2 sinh(τ )e2g [e2x + h22 sinh2 (τ ) − (ξ + χ)2 ]
i
+2e2x sinh(τ )(1 + a cosh(τ ))[a(ξ + χ) − h2 (1 + a cosh(τ ))] ×
h veg (1 + a cosh(τ ))[e2x + h22 sinh2 (τ ) − (ξ + χ)2 ]
i−1
2
.
+2h2 sinh (τ )[a(ξ + χ) − h2 (1 + a cosh(τ ))]
(4.35)
In spite of its complicated appearance, this equation can be integrated and can in fact
be solved fairly explicitly. In the KS limit it reduces to a simpler equation (4.40) that
will be discussed in Section 4.3.
Let us note here the interesting fact that the Euclidean D5-brane and the Lorentzian
D7-brane are related by g5 = −1/g7 . For the D7-brane we find g7 = 0 for the KS
background (since there 1 + a cosh(τ ) = 0), while g7 diverges far along the baryonic
branch where h2 → 0, and correspondingly for g5 the situation is the other way around3 .
The first order equation for the gauge bundle we have derived is in fact more general
3
As a p
curious aside, note that taking g = 0 in (4.31) leads to an equation consistent with the action
e
detM (G + F). This coincides with the D7 brane case for the KS solution (since here φ = 0),
but in general it is not clear what (if anything) this corresponds to.
R
2A−2φ
112
than we have made explicit, and when written in the form (4.35) applies to the whole
two-parameter (η, U ) family of SU(3) structure backgrounds discussed in [58]. The
baryonic branch in particular corresponds to the choice of boundary condition η = 1
at τ = ∞ in the notation of [58], but the above family of solutions also includes the
CVMN background [60, 61, 62], which has the linear dilation boundary condition η = 0
at infinity.
4.2.3
κ-Symmetry of the Euclidean D5-Brane
Let us now reconsider the Euclidean D5-brane using the κ-symmetry approach. The
κ-symmetry projection operator in [142, 144] was derived using the superspace formalism for Lorentzian worldvolume branes in (9,1) signature spacetimes, and thus it is not
immediately clear if it is applicable to the case of a Euclidean worldvolume instanton
which necessarily has to reside in a (10,0) signature spacetime. For now we shall nevertheless proceed by performing just a naive Wick-rotation of the κ-symmetry projector,
which simply introduces a factor −i in (4.7) such that Γ2κ = 1 still holds.
The analog of the κ-symmetry condition (4.15) for the Euclidean D5-brane is then
given by
ε1
ε1
ε1
3
2
= Γκ
∼ −(F/ + F/ ) + (1 + F/ ) σ3 σ2 Γ123456
.
ε2
ε2
ε2
(4.36)
Re-expressing this in geometrical terms leads to an equation of the same form as (4.16),
but now we expect g(τ ) to be equal to g5 (τ ). Using the same ansatz A1 = ξ(τ ) g5 as
above it is clear that equations (4.20) and thus (4.21) still hold, and of course F is still
a (1,1) form.
However, with the gauge bundle we derived in the previous subsection (i.e. with g =
g5 = (α2 − β 2 )/(2αβ)) the κ-symmetry equation (4.36) does not have solutions for
ε1 + iε2 being equal to the Killing spinor (4.10). We can find solutions for other spinors
113
by expanding the εi in terms of pure spinors:
εi = xi (τ ) ζ − ⊗ η − + yi (τ ) ζ + ⊗ η + ,
(4.37)
where i = 1, 2. We find that with this ansatz (4.36) is solved if the coefficients satisfy
x1
(α − iβ)2
=i 2
,
x2
α + β2
(α + iβ)2
y1
=i 2
.
y2
α + β2
(4.38)
Thus we have obtained a family of spinors (4.38) that solves the κ-symmetry equation
with the correct gauge bundle, but this family does not seem to contain the Killing
spinor (which differs by a sign in y1 /y2 ). This would imply that even though for the
gauge field configuration we have found there is worldvolume supersymmetry in the
probe brane approximation, these supersymmetries would not be compatible with those
of the background.
We believe that this difficulty is just an artefact of applying the κ-symmetry operator
in a Euclidean spacetime to a Euclidean worldvolume brane without properly taking into
account the subtleties of Wick-rotating the spinors and the projector itself, and that
the D5-instanton does preserve the background supersymmetries. In fact it is known
that for a Euclidean D5-brane wrapping six internal dimensions the correct κ-symmetry
equations are not the ones obtained by the naive Wick rotation we performed above, but
instead are identical to those for a Lorentzian D9-brane4 . The κ-symmetry conditions
for the Lorentzian D9-brane lead to equations identical to (4.38) except for a change of
sign on the right hand side of the equation for y1 /y2 , so that they are now satisfied by the
Killing spinor. This shows that the worldvolume gauge field found above is consistent
with properly defined κ-symmetry.
In either case we consider the independent derivation of the first-order equation
(4.35) in the previous subsection a compelling argument that this gauge bundle is in
4
We would like to thank L. Martucci for pointing this out to us.
114
fact the correct one for our purposes, which will be corroborated below by the successful
extraction of the baryon operator dimension from its large τ behaviour.
4.3
Euclidean D5-Brane on the KS Background
We will now specialize the discussion of the previous section to the case of a Euclidean
D5-brane wrapping the deformed conifold in the KS background. Since this background
is known analytically, the formulae are more explicit in this case. We interpret the
Euclidean D5-brane (which has the appearance of a pointlike instanton in Minkowski
space) as the dual of the baryon in the field theory, in the sense that its action captures
information about the (scale-dependent) anomalous dimension of the baryon operator,
as well as its expectation value.
4.3.1
The Gauge Field and the Integrated Form of the Action
For the KS background (where for simplicity we set M α0 = 2 and gs = ε = 1), we have
a = −1/ cosh(τ ) and χ = 0, so the first-order differential equation (4.35) simplifies to
ξ0 =
e2x + h22 sinh2 (τ ) − ξ 2
,
2vξ
(4.39)
or more explicitly, substituting in the KS expressions for x, h2 and v:
3
(sinh(τ ) cosh(τ ) − τ )2/3 h 1
sinh(τ ) cosh(τ ) − τ 0
2
ξ
ξ
+
ξ
=
+ (τ coth(τ ) − 1)2 . (4.40)
2
16
4
sinh (τ )
Note that there is no ξ 0 ξ 2 term. Thus we can multiply the equation by an integrating
factor to turn the left hand side into the total derivative [(sinh(τ ) cosh(τ ) − τ )1/3 ξ 2 ]0 and
reduce the equation to the integral
ξ 2 = (sinh(τ ) cosh(τ ) − τ )−1/3 J(τ ) ,
115
(4.41)
where5
Z
J(τ ) =
0
τ
sinh2 (x) h(x)
sinh2 (x)(x coth(x) − 1)2
+
24
6 (sinh(x) cosh(x) − x)2/3
dx .
(4.43)
We have set the integration constant to zero by requiring regularity at τ = 0.
Now consider the DBI action of the Euclidean D5-brane with this worldvolume gauge
field. Neglecting the five angular integrals for the time being, and focussing on the radial
integral, we see that the Lagrangian is in fact a total derivative, and thus the action is
given by
Z
SDBI ∼
√
dτ e−φ det G + F
1
J 3/2
(4.44)
1/2
3(sinh(τ ) cosh(τ ) − τ )
(τ coth(τ ) − 1)2
(sinh(τ ) cosh(τ ) − τ )1/2 h
+
+
J 1/2 .
16
4(sinh(τ ) cosh(τ ) − τ )1/6
=−
We are particularly interested in the UV behaviour of these quantities. From (4.41)
it is easy to find the asymptotic expansion of the gauge field as τ → ∞:
1
7
47
+ O(e−2τ /3 ) .
ξ2 → τ 2 − τ +
4
8
32
(4.45)
Note that to leading order this approximates h22 sinh2 (τ ), so for large τ the coefficients of
the F2 and B2 fields become equal and cancellations occur in the action. This is essential
for obtaining the τ 3 behaviour of the action for large cut-off τ , which as we will see gives
the correct τ 2 scaling of the baryon operator dimensions.
To extract the asymptotic behaviour of the action we will use the integrated form
5
The integral looks “almost” like the explicitly computable one
Z τ
sinh2 (x)(x coth(x) − 1)2
sinh2 (x) h(x)
+
dx
24
18 (sinh(x) cosh(x) − x)2/3
0
1
1
(sinh(τ ) cosh(τ ) − τ ) h(τ ) +
(τ coth(τ ) − 1)2 (sinh(τ ) cosh(τ ) − τ )1/3 ,
=
48
12
(4.42)
but a relative factor of 3 in the second term of (4.43) prevents us from performing it in closed form.
116
(4.44). The leading terms in the expansion are easily found analytically, with the result
Z
SDBI =
−φ
dτ e
p
1/2
1 2 7
47
1 2
τ − τ+
det(G + F) → (τ + τ − 2)
+ O(e−2τ /3 )
6
4
8
32
1 3
1 2
25
943
→ τ − τ −
τ+
+ O(1/τ ) .
(4.46)
12
16
128
1536
Below we will argue that the O(1) term in this expansion determines the expectation
value of the baryon operator. Of particular interest is the variation of this expectation
value along the baryonic branch; we will investigate it in the next section. First, however,
we will give a field theoretic interpretation to the terms that increase with τ . As we will
see, the coefficients of these divergent terms are universal for all backgrounds along the
baryonic branch.
4.3.2
Scaling Dimension of Baryon Operator
We have seen that for large cut-off r (i.e. large τ ), the DBI action of the Euclidean
D5-brane will behave as S(r) ∼ (ln(r))3 . Since this object corresponds to the baryon
in the field theory, we expect that exp(−S) is related to r−∆ , where ∆ is the scaling
dimension of the baryon operator.
To make this statement more precise we consider the RG flow equation relating the
operator dimension ∆ to the boundary behavior of the dual field ϕ(r):
−r
dϕ(r)
= ∆(r)ϕ(r) .
dr
(4.47)
This equation obviously holds in the usual AdS/CFT case where all operator dimensions
have a limit as the UV cut-off is removed. The case of cascading theories is more subtle,
since there exist operators, such as the baryons, whose dimensions grow in the UV. As
we will see, in these cases (4.47) is still applicable. Identifying the field dual to a baryon
117
operator as
ϕ(r) ∼ exp(−S(r)) ,
(4.48)
we find
∆(r) = r
dS(r)
dS(r)
=
.
dr
d ln(r)
(4.49)
To calculate the scaling dimension of the baryon in the gauge theory, we simply count
the number of constituent fields required to build a baryon operator for a given gauge
group SU(kM ) × SU((k + 1)M ) and multiply by the dimension of the chiral superfield
A or B; the latter approaches 3/4 in the UV where the theory is quasi-conformal. This
gives
27gs2 M 3
3
(ln(r))2 + O(ln(r)) ,
∆(r) = M k(k + 1) =
4
16π 2
(4.50)
where k labels the cascade steps and we have used the asymptotic expression for the
radius (energy scale) at which the kth Seiberg duality is performed:
rk = r0 exp
2πk
3gs M
.
(4.51)
Here and in the remainder of this subsection we keep factors of gs , M, ε and α0 explicit.
Let us now compare this to the scaling dimension we obtain from the action of the D5instanton according to equation (4.49). The leading term in the action is τ 3 /12, which
is multiplied by a factor (gs M α0 /2)3 that we had previously set to one, a factor 64π 3
from the previously neglected five angular integrals and a factor of T5 = (2π)−5 α0−3 gs−1 .
Therefore, using (1.29) we have
τ3
S=
12
gs M α0
2
3
64π 3
9gs2 M 3
2
+
O(τ
)
=
(ln(r))3 + O((ln(r))2 ) .
(2π)5 α03 gs
16π 2
118
(4.52)
From (4.49) we find that this string theoretic calculation gives
∆(r) =
27gs2 M 3
(ln(r))2 + O(ln(r)) .
2
16π
(4.53)
The term of leading order in ln(r) is in perfect agreement with the gauge theory result (4.50). We consider this a strong argument that the relation (4.48) between the
Euclidean D5-brane action and the field dual to the baryon is indeed correct.
4.3.3
The Pseudoscalar Mode and the Phase of the Baryonic
Condensate
Let us now turn to a discussion of the Chern-Simons terms in the D-brane action. Given
our conventions (1.39) for the gauge-invariant and self-dual five-form field strength F̃5 ,
there is a slight subtlety in the CS term of the action (4.4). Its standard form, given
above, is valid with the choice of conventions where F̃5 = F5 + H3 ∧ C2 = dC4 + dB2 ∧ C2 .
In these conventions dC4 is invariant under B2 gauge transformations B2 → B2 + dλ1 ,
but transforms under C2 gauge transformations C2 → C2 + dΛ1 such as to leave F˜5
invariant. However, we work in different conventions where F̃5 = dC4 + B2 ∧ F3 ; here
dC4 changes under B2 gauge transformations. This choice also alters the form of the CS
term in the action. The new RR fields are obtained by C4 → C4 +B2 ∧ C2 combined with
C2 → −C2 everywhere else, which modifies some of the terms in the CS action that will
be relevant for us:
1
2
Z
Z
C2 ∧ F ∧ F +
1
C4 ∧ F → −
2
Z
1
C2 ∧ F ∧ F +
2
Z
Z
C2 ∧ B ∧ B +
C4 ∧ F . (4.54)
For the KS background the CS action simply vanishes. However, it is interesting to
consider small perturbations around it. The pseudoscalar glueball discovered in [52, 53]
is the Goldstone boson of the broken U (1) baryon number symmetry; it is associated
119
with the phase of the baryon expectation value. This massless mode is a deformation of
the RR fields (which is generated for example by a D1-string extended in R3,1 ) given by
δF3 = ∗4 da + f2 (τ ) da ∧ dg 5 + f20 (τ ) da ∧ dτ ∧ g 5 ,
h(τ )
5
δ F̃5 = (1 + ∗)δF3 ∧ B2 = ∗4 da −
da ∧ dτ ∧ g ∧ B2 ,
6K 2 (τ )
(4.55)
where a(x0 , x1 , x2 , x3 ) is a pseudoscalar field in four dimensions that satisfies d ∗4 da =
0 and would experience monodromy around a D-string. This deformation solves the
supergravity equations with
1
f2 (τ ) =
6 K 2 (τ ) sinh2 τ
Z
τ
dxh(x) sinh2 (x) .
(4.56)
0
If we wish to identify the exponential exp(−S) = exp(−SDBI − SCS ) of the brane action
(or more precisely the constant term in its asymptotic expansion as τ → ∞) with the
baryon expectation value, then the pseudoscalar massless mode has to shift the phase
of this quantity, contained in the imaginary Chern-Simons term. The DBI action is
obviously unaffected by this deformation of the background since the NSNS fields are
unchanged. This is consistent with the magnitudes of the baryon expectation values
being unaffected by the pseudoscalar mode; these magnitudes depend only on the scalar
modulus U in supergravity, corresponding to |ζ| in the gauge theory.
The phase exp(−SCS ) by itself is not gauge invariant and thus not physical. Because our brane configuration has a boundary at τ = ∞, only the difference in phase
exp(−∆SCS ) = exp(−i∆φ) between two Euclidean D5-branes displaced slightly in one
of the transverse directions (i.e. between two instantons at different points in Minkowski
space) is gauge-invariant. Taking into account the anomalous Bianchi identity for F̃5
and the RR gauge transformations we see that this gauge-invariant phase difference is
120
given by
∆φ = ∆φB + ∆φF ,
(4.57)
where
Z 1
∆φB =
δF3 ∧ B ∧ B + δF5 ∧ B + ∗10 δF3
2
Z 1
∆φF =
− δF3 ∧ F ∧ F + δF5 ∧ F .
2
,
(4.58)
(4.59)
The integrals are performed over the six internal dimensions as well as a line in Minkowski
space. Note that here F5 ≡ dC4 = F˜5 − B2 ∧ F3 . For small perturbations around KS the
contribution ∆φF from the coupling to the gauge field vanishes (the first term in (4.59)
is a total derivative with vanishing boundary terms, while the second term doesn’t have
the right angular structure to give a non-zero result). Substituting the explicit form of
the RR deformations from (4.55) we find that the phase difference is
1
∆φB = −
2
Z h
0
+ f2 (τ coth(τ ) − 1)2 da ∧ dτ ∧ g 1 ∧ g 2 ∧ g 3 ∧ g 4 ∧ g 5 .
6K 2
(4.60)
We can interpret ∆φ as ∆a times a baryon number. It is satisfying to see that the
pseudoscalar Goldstone mode indeed shifts the phase of the baryon expectation value
and not its magnitude. A more stringent test of our interpretation would be to carry
out this computation for the whole baryonic branch and check whether the numerical
value of the baryon number computed this way is independent of the modulus U . This
is rather difficult, since the pseudoscalar mode at a general point along the baryonic
branch is not explicitly known at present.
121
4.4
Euclidean D5-Brane on the Baryonic Branch
In this section we extend the discussion of the previous section from the KS solution
to the entire baryonic branch. In particular we are interested in the dependence of the
baryon expectation value on the modulus U of the supergravity solutions. All supergravity backgrounds dual to the baryonic branch have the same asymptotics [59] and we will
see that the leading terms (cubic, quadratic and linear in τ ) in the asymptotic expansion
of the action (4.46) are universal. This implies that the leading scaling dimensions of the
baryon operators do not depend on U , consistent with field theory expectations. However, the finite term in the asymptotic expansion of the brane action does depend on U .
This provides a map from the one-parameter family of supergravity solutions labelled
by U to the family of field theory vacua with different baryon expectation values (4.2),
parameterized by ζ.
4.4.1
Solving for the Gauge Field and Integrating the Action
Having derived the differential equation that determines the gauge field in full generality
in Section 4.2, let us now turn to a more detailed investigation of the first order equation
(4.35). First of all we note that it can be rewritten as
d
dτ
"
1
− ξ3 +
3
#
2
ah2 sinh2 (τ )
2ah
sinh
(τ
)
2
− χ ξ 2 + e2x − h22 sinh2 (τ ) − χ2 +
χ ξ
1 + a cosh(τ )
1 + a cosh(τ )
h2 sinh(τ )eg
[e2x + h22 sinh2 (τ ) − χ2 ]
v(1 + a cosh(τ ))
2e2x sinh(τ )
[aχ − h2 (1 + a cosh(τ ))] .
+
veg
=−
(4.61)
For notational convenience we define ξ˜ ≡ ξ + χ,
A(τ ) ≡
ah2 sinh2 (τ )
,
1 + a cosh(τ )
B(τ ) ≡ e2x − h22 sinh2 (τ ) ,
122
(4.62)
(4.63)
and
ρ(τ ) ≡
τ
h2 sinh(τ )eg
[e2x + h22 sinh2 (τ )]
v(1 + a cosh(τ ))
0
2x
2e h2 sinh(τ )(1 + a cosh(τ ))
2
2x
2
0
+
− [e − h2 sinh (τ )]χ dτ ,
veg
Z
(4.64)
which allows us to write (4.61) more compactly as
i
d h 1 ˜3
2
˜
˜
− ξ + A(τ )ξ + B(τ )ξ + ρ(τ ) = 0 .
dτ
3
(4.65)
Thus the solutions for the shifted field ξ˜ are given by the roots of the third order
polynomial
1
− ξ˜3 + A(τ )ξ˜2 + B(τ )ξ˜ + ρ(τ ) = C ,
3
(4.66)
where C is the integration constant.6 To fix it, we consider the small τ expansion, which
is valid for any U
A ∼ τ + O(τ 3 ) ,
(4.67)
B ∼ τ 2 + O(τ 4 ) ,
(4.68)
ρ ∼ τ 3 + O(τ 4 ) .
(4.69)
Note that at τ = 0 all coefficients in (4.66) vanish, except the first one; therefore, the
integration constant C has to be zero for this cubic to admit more than one real solution.
Then we find that ξ˜ = 0 at τ = 0 for any solution on the baryonic branch.
Let us examine the cubic equation (4.66) more closely in the KS limit (U → 0) to
see how our earlier result (4.41) is recovered. In the U → 0 limit a → −1/ cosh(τ ) and
6
This equation is quite general; it does not assume boundary conditions η = 1 that characterize the
baryonic branch [59]. In particular this result is also valid for a brane embedded in the CVMN solution
[60, 61, 62].
123
therefore (1 + a cosh(τ )) vanishes. For small U [52, 53, 58, 59]
(1 + a cosh(τ )) = 2−5/3 U Z(τ ) + O(U 2 ) ,
Z(τ ) ≡
(τ − tanh(τ ))
.
(sinh(τ ) cosh(τ ) − 1)1/3
(4.70)
(4.71)
In this case A and the first term in ρ diverge as U −1 . All other terms can be dropped
and we have instead of (4.65)
ah2 sinh2 (τ )
+
ξ˜2
Z(τ )
Z
τ
dτ
0
h2 sinh(τ )eg 2x
[e + h22 sinh(τ )2 ] = 0 .
vZ(τ )
(4.72)
After substituting the KS values for a, v, h2 , x we recover (4.41).
While it would be desirable to obtain a closed form expression for the integral ρ(τ )
in order to evaluate ξ explicitly, this appears to be impossible, since even in the KS case
we cannot perform the corresponding integral J(τ ).
Evaluating the DBI Lagrangian on-shell using (4.31) we find
−φ
e
p
√
e−φ e3x 1 + g2 (a2 + b2 )
det(G + F) =
,
v|a − gb|
(4.73)
where we have taken the absolute value since the sign of a − gb will turn out to depend
on which root of equation (4.66) we pick.
For the baryonic branch backgrounds we can show that the action is a total derivative.
First note that the DBI Lagrangian (4.73) can be rewritten in the form
e−φ
p
e−φ e3x (ga + b)2 + (a − gb)2
det(G + F) = √
|a − gb|
v 1 + g2
4x
e (1 + a cosh(τ )) −x 0
,
=
[ve
ξ
(ga
+
b)
+
(a
−
gb)]
vh2 sinh(τ )eg
(4.74)
where the right hand side is now cubic in ξ (and its derivative) much like the differential
equation (4.31). In fact, substituting for a, b and g = g5 this equation can be integrated
124
in the same manner, which results in the action
1
3
2
˜
˜
˜
S = − ξ + C(τ )ξ + D(τ )ξ + σ(τ ) ,
3
(4.75)
with C, D, σ defined as
C=−
e2x a (1 + a cosh(τ ))
,
h2 e2g
(4.76)
D = [e2x + h22 sinh2 (τ ) + 2e2x (1 + a cosh(τ ))2 e−2g ] ,
Z τ 2x
e (1 + a cosh(τ )) 2x
[e − h22 sinh2 (τ )] +
σ=−
g
vh
sinh(τ
)e
2
0
[e
2x
+
h22
2
2x
2 −2g
sinh (τ ) + 2e (1 + a cosh(τ )) e
(4.77)
(4.78)
]χ dτ .
0
(4.79)
Again the ξ-independent term is an integral, that we denoted by σ(τ ). Thus we have a
fairly explicit expression for the action involving two integrals: ρ(τ ), which appears in
˜ and σ(τ ).
the equation for ξ,
To conclude this subsection we will demonstrate that the third solution of (4.65),
which is absent (formally divergent for all τ ) in the KS case (4.41), produces a badly
divergent action and is therefore unacceptable for any point on the branch. Restoring
the −ξ˜3 /3 term in (4.72) we see that in the GHK region U → 0 the third solution is
simply
ξ=−
22/3 3
(cosh(τ ) sinh(τ ) − τ )1/3 + O(U ) .
U
(4.80)
The value of the Lagrangian in this case is
p
36
det(G + F) = 3 sinh2 (τ ) + O(U −2 ) .
U
(4.81)
This expression can be used to extract the leading UV asymptotics of the Lagrangian
125
for any U as the UV behavior is universal for all U :
p
det(G + F) →
9 2τ
e .
U3
(4.82)
Since the action for the third solution diverges exponentially at large τ it does not seem
possible to interpret this solution as the dual of an operator in the same sense as we do
for the other two solutions.
4.4.2
Baryonic Condensates
We shall now study the D5-brane action (4.75) in more detail. First we develop an
asymptotic expansion of the action (4.75) as a function of the cut-off. This expansion
is useful because the divergent terms give the scaling dimension of the baryon operator,
while the finite term encodes its expectation value.7 Then we present a perturbative
treatment of small U region followed by a numerical analysis of the whole baryonic
branch. The main result of this section will be an expression for the expectation value
as a function of U which can be evaluated numerically. This leads to an explicit relation
between the field theory modulus |ζ| and the string theory modulus U .
To calculate the baryonic condensates we need asymptotic the behavior of A, B, ρ
and C, D for large τ . Notice that since for any U the solution approaches the KS solution
at large τ , the terms divergent at U = 0 are UV divergent as well
A→
e2τ /3
+ O(e−2τ /3 ) ,
U
B → O(τ 2 ) ,
e2τ /3 1 2 7
47
ρ→−
τ − τ+
+ O(1) ,
U
4
8
32
7
(4.83)
(4.84)
(4.85)
A systematic procedure for isolating the finite terms is holographic renormalization [152, 153], but
here we limit ourselves to a more heuristic approach.
126
and similarly
C → O(e−2τ /3 ) ,
1 2 1
5
D→
τ − τ+
+ O(e−4τ /3 ) .
4
8
32
(4.86)
(4.87)
From the expansion for A, B, ρ we find that at large τ the gauge field ξ˜ grows linearly
with τ and approaches the KS value with exponential precision
˜ U) → ±
ξ(τ,
1 2 7
47
τ − τ+
4
8
32
1/2
+ O(e−2τ /3 ) .
(4.88)
It is crucial that the dependence on U in (4.88) is exponentially suppressed.
Since C is exponentially small and the leading term in D is U -independent we can
explicitly express the action (4.75) in terms of σ:
S± (U, τ ) = Sdiv (τ ) ± σ(U, τ ) + O(e−2τ /3 ) ,
(4.89)
where the U -independent divergent part of the action is given by
1
Sdiv (τ ) = (τ 2 + τ − 2)
6
1 2 7
47
τ − τ+
4
8
32
1/2
,
(4.90)
Note that
1 3
− ξ˜ + D(τ )ξ˜ = Sdiv (τ ) + O(e−2τ /3 ) .
3
(4.91)
The two signs stand for the two well-behaved solutions ξ(τ ) corresponding to the two
baryons A and B. As we argued in Section 4.1, the I-symmetry which exchanges the A
and B baryons is equivalent to changing the sign of U . Our explicit expression (4.89)
127
confirms that
S+ (U, τ ) = S− (−U, τ ) ,
(4.92)
S− (U, τ ) = S+ (−U, τ ) ,
(4.93)
since σ(U, τ ) is antisymmetric in U according to the arguments presented around (1.80).
In order to find the expectation value of the baryons we evaluate the action (4.75) on
these solutions and remove the divergence by subtracting the KS value. The expectation
values hence are given by exp[− limτ →∞ S0 (ξ1,2 )], where by S0 we denote the finite part
of the action. It is simplest to work with the product (normalized to the KS value) and
ratio of the expectation values. The former is given by
hAihBi
= lim exp [S+ (U, τ ) + S− (U, τ ) − 2S(0, τ )] ,
hAiKS hBiKS τ →∞
(4.94)
where we have used the fact that the two solutions coincide in the KS case, where σ = 0.
It follows from (4.94) that
hAihBi = hAiKS hBiKS ,
(4.95)
which corresponds to the constraint AB = −Λ4M
2M in the gauge theory. The ratio of the
baryon condensates is given by
hAi
= lim exp [S+ (U, τ ) − S− (U, τ )] = lim e2σ ,
τ →∞
hBi τ →∞
(4.96)
loghAi ' lim σ(τ ) .
(4.97)
or
τ →∞
Unfortunately we were not able to calculate σ analytically, since the U -dependent
128
terms of order O(τ n ) exp(−2τ /3) in the integrand are significant. However, we can
evaluate the integral8 to first order in U for small U :
σ ' 3.3773 U + O(U 3 ) ,
(4.98)
and thus obtain the slope of the expectation values in the vicinity of KS. Even though
we lack analytical arguments that would fix the behavior of the expectation values for
large U , we can compute the integral σ(τ ) numerically. Our results for the expectation
value as a function of the modulus are shown in Figure 4.1. Since hAi ∼ ζ this plot
provides a mapping from the supergravity modulus U to the field theory modulus ζ (as
we remarked before, careful holographic renormalization is needed to check this relation).
4.5
Conclusions
In previous work, increasingly convincing evidence has been emerging [49, 52, 53, 56, 59]
that the warped deformed conifold background of [49] is dual to the cascading gauge
theory with condensates of the baryon operators A and B. Furthermore, a one-parameter
family of more general warped deformed conifold backgrounds was constructed [58, 59]
and argued to be dual to the entire baryonic branch of the moduli space, AB = const.
In this chapter we have presented additional, and more direct, evidence for this identification by calculating the baryonic condensates on the string theory side of the duality.
Following [56], we have identified the Euclidean D5-branes wrapped over the deformed
conifold, with appropriate gauge fields turned on, with the fields dual to the baryonic
8
The coefficient of the term linear in U is given by
Z ∞ "
(τ coth(τ ) − 1)2
h sinh2 (τ )
h(sinh(τ ) cosh(τ ) − τ )2/3
−5/3
−
2
dτ
16
4
12(sinh(τ ) cosh(τ ) − τ )2/3
0
#
(τ coth(τ ) − 1)(sinh(τ ) cosh(τ ) − τ )2/3 h(sinh(τ ) cosh(τ ) − τ )2/3
(τ coth(τ ) − 1)2
−
+
.
16
4
sinh2 (τ )
129
150
100
!
50
0
!50
!100
!150
!150
!100
!50
0
50
100
150
U
Figure 4.1: Plot of numerical results for the O(τ 0 ) term in the asymptotic expansion of
the action versus U . The slope at U = 0 matches the value from (4.98). The baryon
expectation value hAi ∼ hBi−1 in units of Λ2M
2M is given by the exponential of this
function.
operators in the sense of gauge/string dualities. We derived the first order equations for
the gauge fields and solved them explicitly. The solutions were subjected to a number of
tests. From the behavior of the D5-brane action at large radial cut-off r we have deduced
the r-dependence of the baryon operator dimensions and matched it with that in the
cascading gauge theory. Furthermore, we used the D5-brane action to calculate the condensates as functions of the modulus U that is explicit in the supergravity backgrounds.
We found that the product of the A and B condensates indeed does not depend on U .
This calculation also establishes a map between the parameterizations of the baryonic
branch on the string theory and on the gauge theory sides of the duality, which should
be useful for comparing other physical quantities along the baryonic branch.
130
Chapter 5
Charges of Monopole Operators in
Chern-Simons Yang-Mills Theory
5.1
Introduction
Superconformal Chern-Simons gauge theories are excellent candidates for describing the
dynamics of coincident M2-branes [68]. Bagger and Lambert [63, 64, 65], and Gustavsson
[66] succeeded in constructing the first N = 8 supersymmetric classical actions for
Chern-Simons gauge fields coupled to matter. Requiring manifest unitarity restricts the
gauge group to SO(4) [154, 155]; this model may be reformulated as SU(2) × SU(2)
gauge theory with conventional Chern-Simons terms having opposite levels k and −k
[71, 73]. Aharony, Bergman, Jafferis, and Maldacena (ABJM) [72] proposed that a
similar U(N ) × U(N ) Chern-Simons gauge theory with levels k and −k arises on the
world volume of N M2-branes placed at the singularity of R8 /Zk , where Zk acts by
simultaneous rotation in the four planes. Therefore, the ABJM theory was conjectured to
be dual, in the sense of AdS/CFT correspondence [1, 2, 3], to M-theory on AdS4 ×S 7 /Zk .
For k > 2 this orbifold preserves only N = 6 supersymmetry, and so does the ABJM
theory [5, 72, 156]. The conjectured duality predicts that for k = 1, 2 the supersymmetry
131
of the gauge theory must be enhanced to N = 8.
The mechanism for this symmetry enhancement in the quantum theory was suggested
in [72]; it relies on the existence of certain monopole operators in 3-dimensional gauge
theories [157, 158, 159, 160, 161, 162, 163]1 . Insertion of such an operator at some
point creates quantized flux in a U(1) subgroup of the gauge group through a sphere
surrounding this point. For example, in a U(1) gauge theory on R3 , a monopole operator
placed at the origin creates the Dirac monopole field
A=
H ±1 − cos θ
dϕ ,
2
r
(5.1)
where the upper sign is for the northern hemisphere and the lower sign for the southern
one. In addition, some scalar fields may need to be turned on as well; they are required
for BPS monopoles that preserve supersymmetry. The fluctuations of fermionic matter
fields can shift the dimension of such a monopole operator. These effects were studied in
some simple models, mostly with U(1) gauge group, in [160, 161, 162]. In this chapter
we will generalize these calculations to more complicated models. Our primary goal is
to calculate R-charges and dimensions of the monopole operators in ABJM theory for
any level k. Since for small values of k we cannot use perturbation theory in 1/k, we
will actually study the N = 3 supersymmetric Yang-Mills Chern-Simons theory that
provides a weakly coupled UV completion of the ABJM theory.2
In gauge theories with U(N ) gauge group there exists a rich set of monopole operators labeled by the generator H that specifies the embedding of U(1) into U(N )
[165] (see Appendix D of [166] for a brief discussion). The generalized Dirac quantization condition restricts, up to gauge equivalence, the background gauge field to be
proportional to the Cartan generator H = diag(q1 , . . . , qN ) where the integers qi satisfy
1
A more appropriate name may be “instanton operators” since they create instantons of a Euclidean
3-dimensional theory whose spacetime dependence resembles the spatial profile of monopoles in 3+1
dimensions.
2
We are grateful to Juan Maldacena for this suggestion. A similar trick was used in [164].
132
q1 ≥ q2 . . . ≥ qN . If the action of the gauge theory includes a Chern-Simons term with
level k, then the monopole operators are expected to transform non-trivially under the
U(N ) gauge group, in U(N ) representations given by the Young tableaux with rows of
length kq1 , kq2 , . . . , kqN [165].
The monopole operators in the U(N ) × U(N ) ABJM theory have been a subject
of several recent investigations [164, 166, 167, 168, 169]. In general the monopoles are
described by two different generators, H and Ĥ, which specify the form of the two
P
P
gauge potentials, A and Â, subject to the constraint i qi = i q̂i [164]. However, the
monopoles are believed to be BPS and thus not suffer renormalization of their dimensions
for H = Ĥ [164]. A proposal for the U(1) R-charge of the monopole operator in gauge
theories with N = 3 supersymmetry was made in [170, 171] based on the results of
[161] and group theoretic arguments. It states that the R-charge induced by fermionic
fluctuations is
Qmon
R
1
=
2
!
X
|hi | −
i
X
|vi |
,
(5.2)
j
where hi and vj are, respectively, the R-charges of the fermions in hyper and vector
multiplets weighted by the effective monopole charges appropriate for their gauge representations. We establish this formula through an explicit calculation in Section 5.4,
and derive its non-abelian generalization in Section 5.5. In the ABJM theory, if we consider the diagonal U(N ), we find two hyper multiplets and two vector multiplets with
charges such that there is a desired cancellation of the anomalous dimension. Using the
monopole operators (M−2 )âabb̂ with kq1 = k q̂1 = 2 which exist for k = 1, 2, we can form
twelve real (or six complex) conserved currents of dimension 2,
JµAB
h
i
A
B
A B
†A µ †B
= M Y Dµ Y − Dµ Y Y + iψ γ ψ
,
−2
which are responsible for the symmetry enhancement from SU(4)R to SO(8)R .
133
(5.3)
Monopoles are crucial not only for the supersymmetry enhancement at small level
k = 1, 2 but also for matching the spectrum of the dual gravity theory at any level k.
In fact, supergravity modes with momentum along the M-theory direction are dual to
gauge invariant operators involving monopoles. The spectra match if there are monopole
operators which can render a gauge theory operator gauge invariant without altering the
global charges and dimensions of the matter fields. These monopole operator themselves
have to be singlets under all global symmetries and have to have vanishing dimension.
In U(N ) × U(N ) gauge theories coupled to Nf bifundamental hyper multiplets, this
dimension is proportional to Nf − 2. Thus, the requisite monopole operators exist in
the ABJM theory, which has Nf = 2.
The outline of this chapter, which is based on joint work with I. R. Klebanov and
T. Klose [172], is as follows. Section 5.2 summarizes our reasoning and the results, and
ties together the subsequent more technical sections. Section 5.3 contains the details
of the theories under consideration. In Section 5.4 and Section 5.5 we present the
computation of the U(1)R and SU(2)R charges of the monopole operators, respectively.
Brief conclusions and an outlook are given in Section 5.6.
5.2
Summary
In this section we go through the logic of our arguments and present the results of our
computations while omitting the technical details.
A monopole operator is defined by specifying the singular behavior of the gauge field,
as in (5.1), and appropriate matter fields close to the insertion point. As such it cannot
be written as a polynomial in the fundamental fields appearing in the Lagrangian of
the theory. However, often the bare monopole operator has to be supplemented by a
number of fundamental fields in order to construct a gauge invariant operator.
Our aim is to find the conformal dimensions of such monopole operators in the ABJM
134
model. In particular we are interested in small Chern-Simons level k, as for k = 1, 2
we expect supersymmetry enhancement in this theory. For small k, however, ABJM is
strongly coupled and thus we cannot employ perturbation theory since there is no small
parameter. The key idea to circumvent this obstacle is to add a Yang-Mills term for the
gauge fields in the action which introduces another coupling3 g as a second parameter
besides k. This also requires adding dynamical fields in the adjoint representation in
order to preserve N = 3 supersymmetry and an SU(2)R subgroup of the SU(4)R group
of ABJM (which has N = 6 supersymmetry).
The coupling g is dimensionful and not a parameter we can dial. Since the Yang-Mills
term is irrelevant in three dimensions, there is a renormalization group (RG) flow from
the UV, where the theory is free (vanishing g) to a conformal IR fixed point (divergent
g). This RG flow appears naturally in the brane construction of the ABJM model and
in fact is essential for understanding how a pure Chern-Simons theory such as ABJM
can arise from D-branes that support Yang-Mills theories.
In the IR, the Yang-Mills terms as well as the kinetic terms for the adjoint matter
fields drop out of the action and the equations of motion for the latter degenerate into
constraint equations, allowing us to integrate them out and recover the ABJM theory.
Now the idea is to perform all relevant computations in the far UV where g is
small and the theory weakly coupled, then flow to the IR. The scaling dimensions of
monopole operators are of course only well-defined at the IR fixed point where the
theory is conformal, so they cannot be determined directly in this fashion. However,
we can instead compute a quantity that is preserved along the RG flow and related by
supersymmetry to the scaling dimensions in the IR: the non-abelian R-charges of the
monopole operators. Their one-loop value is exact and preserved along the RG flow
because non-abelian representations cannot change continuously and therefore cannot
depend (non-trivially) on g.
3
The reader is cautioned that this is not the same as the coupling g used in Chapter 2. There the
Yang-Mills coupling was denoted as gYM but here we drop the subscript YM for notational convenience.
135
Let us describe the computation of the non-abelian R-charges. In the UV there is
a separation of scales between the BPS background that inserts a flux at a spacetime
point in accordance with (5.1) – recall that its magnitude is constrained by the Dirac
quantization condition – and the typical size of quantum fluctuations of fields. Therefore
we can treat the monopole operator as a classical background. For this background to
satisfy the BPS condition the scalar fields φi and φ̂i , which are in the same N = 3 vector
multiplets as the gauge fields Aµ and µ , respectively, need to be turned on. A possible
choice of scalar background in radial quantization on R × S2 is
φi = −φ̂i = −
H
δi3 .
2
(5.4)
These scalar fields transform in the 3 of SU(2)R and therefore a non-zero expectation
value breaks SU(2)R to U(1)R .
As a first step we will find the one-loop U(1)R charge of the monopole operator
described by this fixed background. This is done by computing the normal ordering
constant for the U(1)R charge operator. Such a calculation has been performed in a
different context in [160, 161]. We will present the argument as applicable to our case
in Section 5.4. The result will turn out to be expression (5.2) above or more concretely
(5.69) for N = 3 U(N )×U(N ) gauge theory with hyper multiplets in the bifundamental.
These two formulas are related as follows. From Table 5.2, or equivalently from the
R-current (5.38), one can read off the R-charges y(ζ A ) = y(ωA ) = − 12 (for A = 1, . . . , Nf )
for the hyper multiplet fermions and y(χσ ) = y(χ̂σ ) = 1, y(χφ ) = y(χ̂φ ) = 0 for those
in the vector multiplets. In the expression for the U(1)R charge of the BPS monopole
P
(5.2), these R-charges are weighted by r,s |qr − qs | and the result is given by
Qmon
R
X
X
1
1
Nf
=
|qr − qs | ,
2Nf · − 2 − 2 · | + 1|
|qr − qs | =
−1
2
2
r,s
r,s
as found in (5.69). For the ABJM model we have Nf = 2 and hence Qmon
= 0.
R
136
(5.5)
These U(1)R charges might in principle be renormalized as we flow to the IR, since
abelian charges can vary continuously. To exclude this possibility, we need to find the
non-abelian SU(2)R charge of the monopole operator by taking into account the bosonic
zero modes of the background (5.4). These zero modes are described by a unit vector ~n
on the two-sphere SU(2)R /U(1)R . In other words, we will treat the SU(2)R orientation
~n of the scalar field as a collective coordinate of the BPS background and quantize its
motion. To this end we consider a more general background than (5.4), which is allowed
to depend on (Euclidean) time τ
φi = −φ̂i = −
H
ni (τ ) .
2
(5.6)
The rotation of the background is assumed to be adiabatic such that it cannot excite finite energy quantum fluctuations around the background. In Section 5.5 we will
compute the quantum mechanical effective action for the collective coordinate ~n(τ ) and
find the allowed SU(2)R representations. These representations are precisely the SU(2)R
spectrum of BPS monopole operators.
If there were no interactions between the bosonic zero modes and other fields in
the Lagrangian, the collective coordinate would simply be described by a free particle
on a sphere. Such a particle – and therefore the monopole operator – could be in any
representation of SU(2)R . The crucial point is, however, that the collective coordinate ~n
is not free, but subject to interactions with the fermions of the theory. It turns out that
the induced interaction term is a coupling of ~n to a Dirac monopole of magnetic charge
h ∈ Z on the collective coordinate moduli space (not to be confused with the spacetime
monopole background). The charge h depends on the background flux H and the field
content of the theory. In Section 5.5 we derive
h = (Nf − 2) qtot = (Nf − 2)
X
r,s
137
|qr − qs | ,
(5.7)
see (5.103), which is nothing but h = 2Qmon
R .
The equation of motion for the collective coordinates ~n has the structure
¨ + M (~n˙ )2 ~n + h ~n × ~n˙ = 0 ,
M ~n
2
(5.8)
which allows for a simple particle interpretation. The first term is the usual kinetic
term while the second term represents a constraining force that keeps the particle on
the sphere ~n2 = 1. If we assign unit electric charge to the particle, then the third term
~ due to a monopole field B
~ = h ~n. The SU(2)R charge is given
is the Lorentz force ~n˙ × B
2
by the conserved angular momentum
~ = M ~n × ~n˙ − h ~n .
L
2
(5.9)
The presence of the second term forces the quantized angular momentum to have an
orbital quantum number of at least l =
with l =
|h| |h|
, 2
2
|h|
,
2
~ 2 = l(l+1)
the possible representations being L
+ 1, |h|
+ 2, . . .. Hence, a non-zero coupling h will necessarily give the
2
monopole a non-trivial SU(2)R charge. The ABJM theory is special (as compared to
theories with different field content) in that all contributions to the effective h cancel
each other, and thus it allows SU(2)R singlet monopoles which do not contribute to the
IR dimensions of gauge invariant operators.
5.3
N = 3 Chern-Simons Yang-Mills Theory
We will study N = 3 supersymmetric U(N ) × Û(N ) 3-dimensional gauge theory coupled
to bifundamental matter fields and SU(Nf )fl flavor symmetry. For Nf = 1 and Nf = 2
this model is the UV completion of the N = 4 model of GW [74] and the N = 6 model
of ABJM [72], respectively.
138
Manifest
symmetry
Component fields
Super
fields
dynamical in the IR
V
A
V̂
Â
χ̂σ , χ̂†σ
D̂
χφ
Fφ
φ†
χ†φ
Fφ†
φ̂
χ̂φ
F̂φ
φ̂†
χ̂†φ
F̂φ†
ζA
FA
Z̄A
ZA†
ζA†
FA†
WA
WA
ωA
GA
W̄ A
SU(2)R × SU(Nf )fl
D
φ
Φ̂
ˆ
Φ̄
ZA
χσ , χ†σ
σ̂
Φ̄
ZA
aux.
σ
Φ
U(1)R × SU(Nf )fl
auxiliary in the IR
W †A
X Aa
ω †A
†
XAa
ξ Aa
G†A
†
ξAa
φi
φ̂i
λab
λ̂ab
Table 5.1: Field content of N = 3 Chern-Simons Yang-Mills theory. Each row (except
the last one) shows the components of one N = 2 superfield ordered into columns
according to whether they are dynamical or auxiliary. Within each of these columns
they have been arranged such that one can read off from the last row which components
join to form SU(2)R multiplets. A = 1, . . . , Nf is an SU(Nf ) flavor index, a = 1, 2 is an
SU(2)R spinor index, and i = 1, 2, 3 is an SU(2)R vector index.
5.3.1
Action and Supersymmetry Transformations
The complete field content is listed in Table 5.1. We use N = 2 superfields and our notation is explained in Appendix A.2. In the gauge sector we have two vector superfields
V = (σ, Aµ , χσ , D) and V̂ = (σ̂, µ , χ̂σ , D̂) and two chiral superfields Φ = (φ, χφ , Fφ )
and Φ̂ = (φ̂, χ̂φ , F̂φ ) in the adjoint of the two gauge groups U(N ) and Û(N ), respectively, which together comprise two N = 3 gauge multiplets. In the matter sector we
have Nf chiral superfields Z A = (Z A , ζ A , F A ) and WA = (WA , ωA , GA ) in the gauge
representations (N, N̄) and (N̄, N), respectively.
We write down the N = 3 action on R1,2 with signature (−, +, +). It consists of five
139
parts:
S = SCS + SYM + Sadj + Smat + Spot .
(5.10)
First of all there are Chern-Simons terms
SCS
k
= −i
8π
Z
3
Z
4
d xd θ
1
h
i
ds tr V D̄α esV Dα e−sV − V̂ D̄α esV̂ Dα e−sV̂ , (5.11)
0
with opposite levels k and −k for the two gauge group factors. The s integral is nothing
but a convenient way of writing the non-abelian Chern-Simons action. Secondly, there
is a Yang-Mills term
SYM
1
= 2
4g
Z
h
i
d3 x d2 θ tr U α Uα + Û α Ûα ,
(5.12)
which introduces a coupling g of mass dimension 21 . The super field strength is given by
Uα = 14 D̄2 eV Dα e−V and similarly for Û. The last term in the gauge sector is given by
the kinetic terms for the adjoint scalar fields
Sadj
1
= 2
g
Z
h
i
ˆ e−V̂ Φ̂eV̂ .
d3 x d4 θ tr −Φ̄e−V ΦeV − Φ̄
(5.13)
In the matter sector we have the minimally coupled action for the bifundamental fields
Z
h
i
d3 x d4 θ tr −Z̄A e−V Z A eV̂ − W̄ A e−V̂ WA eV
Smat =
(5.14)
and a super potential term
Z
Spot =
3
Z
2
d xd θ W −
140
d3 xd2 θ̄ W̄
(5.15)
with
k
W = tr ΦZ A WA + Φ̂WA Z A +
tr ΦΦ − Φ̂Φ̂ ,
8π
ˆ Z̄ W̄ A + k tr Φ̄Φ̄ − Φ̄
ˆ Φ̄
ˆ .
W̄ = tr Φ̄W̄ A Z̄A + Φ̄
A
8π
(5.16)
(5.17)
This theory is not conformal and will flow to an IR fixed point which is strongly coupled
unless k is large. At the fixed point g diverges, which renders the gauge fields and
the adjoint scalars non-dynamical. If we integrate out Φ and Φ̂, we recover the ABJM
superpotential for Nf = 2. This theory has enhanced flavor symmetry, SU(2)fl × SU(2)fl ,
under which Z A and WA transform separately.
For our computation below we need the action in terms of component fields. To this
end we perform the Grassmann integrals in the action and integrate out the auxiliary
fields D, D̂, F A , GA , Fφ and F̂φ . The remaining component fields can be arranged into
SU(2)R multiplets as follows.
The adjoint matter fields constitute two scalars (where the lower/upper index is the
row/column index)




†
†
 −σ φ 

φab = φi (σi )ab = 


φ σ
 σ̂ φ̂ 
 ,
φ̂ab = φ̂i (σi )ab = 


φ̂ −σ̂
,
(5.18)
transforming in the 3 of SU(2)R , and two fermions


†
 χσ e−iπ/4 χφ e−iπ/4 


λ =

χφ e+iπ/4 −χ†σ e+iπ/4
ab

,

†
 χ̂σ e−iπ/4 −χ̂φ e−iπ/4 

 ,
λ̂ = 

−χ̂φ e+iπ/4 −χ̂†σ e+iπ/4
ab
(5.19)
transforming in the reducible representation 2×2 = 3+1 of SU(2)R , which in particular
implies that λab is neither symmetric nor anti-symmetric in its indices. These fields
141
satisfy
(λab )∗ = −λab = −ac bd λcd
,
(φab )∗ = φba = ac bd φcd
(5.20)
and the same for the hatted fields.
The bifundamental matter fields can be grouped into Nf doublets of the SU(2)R
symmetry group in the following way:

X
Aa



†
 ZA 

=


W †A
†
XAa
,
 ZA 
 ,
=


WA
(5.21)
and

ξ
Aa

 ω †A eiπ/4 

=


ζ A e−iπ/4

,
†
ξAa

 ωA e−iπ/4 
 .
=


ζA† eiπ/4
(5.22)
The component action with manifest SU(2)R symmetry and the corresponding supersymmetry transformations are given in Appendix A.3. One observes that in the IR,
where g becomes large, the Yang-Mills terms disappears together with the kinetic terms
for φ, φ̂, λ and λ̂, and we can integrate out these fields. Doing this for Nf = 2 we end
up with ABJM theory. The ABJM Lagrangian with manifest SU(4)R symmetry [5] is
recovered when one defines Y A = {X 11 , X 21 , X 12 , X 22 } and ψA = {−ξ 22 , ξ 12 , ξ 21 , −ξ 11 }.
As outlined in the overview, Section 5.2, we will carry out our computations in the
far UV region of the radially quantized theory. The point of going to the far UV is
that the theory becomes perturbative in g and we can find the quantum R-charges
from a one-loop computation. This one-loop result is in fact the exact answer because
SU(2)R is preserved along the flow from small to large g and non-abelian representations
cannot change continuously. We will use radial quantization because we are interested
142
in the spectrum of conformal dimensions in the IR theory, which is related to the energy
spectrum by the operator state correspondence.
The steps required for deriving the action relevant for radial quantization are as
follows. First we perform a Wick rotation from R1,2 to R3 by defining Euclidean coordinates (x1 , x2 , x3 ) = (x1 , x2 , ix0 ), then we change to polar coordinates (r, θ, ϕ) and finally
we introduce a new radial variable τ by setting r = eτ . The result is a theory on R × S2
described by the coordinates (τ, θ, ϕ), where τ ∈ R is the “Euclidean time”. The change
of the coordinates is accompanied by a Weyl rescaling of the fields according to
A = e− dim(A)τ Ã ,
(5.23)
where A is a generic field of dimension dim(A) on R3 and à is the field we use on
R × S2 . After the transformation we will drop the tildes in order to avoid cluttered
notation. Since the theory is not conformal, the action will change under these rescaling.
In addition to the mass terms which are generated even in the conformal case, every
factor of the coupling g will turn into
g̃ = eτ /2 g .
(5.24)
This relation makes the RG-flow explicit and relates it to the “time” on R × S2 . In the
infinite past the effective Yang-Mills coupling g̃ vanishes, and it grows without bound
toward future infinity. To compute in the UV therefore means to work at τ → −∞ and
to flow to the IR means to send τ → +∞.
Now we are ready to give the complete action of N = 3 Chern-Simons Yang-Mills
theory on R × S2 . In addition to the manipulations just described we have rescaled
λ → gλ and λ̂ → g λ̂ (before the Weyl rescaling) which is the appropriate scaling for
fluctuations in the UV. We do not perform a similar rescaling of the other fields in
the gauge sector, A, Â, φ, or φ̂, since these fields will later provide a large classical
143
background. Introducing finally the rescaled Chern-Simons coupling κ =
k
,
4π
the kinetic
part of the action reads4
E
Skin
Z
=
h
dτ dΩ tr + 2g̃12 F mn Fmn − κ imnk Am ∂n Ak +
2i
A A A
3 m n k
+ 2g̃12 F̂ mn F̂mn + κ imnk Âm ∂n Âk +
2i
  Â
3 m n k
(5.25)
/
+ Dm X † Dm X + 41 X † X − iξ † Dξ
+ 2g̃12 Dm φab Dm φba +
+
i ab /
λ Dλab
2
+
κg̃ 2
2
κ2 g̃ 2
2
ab
φab φba +
iλ λba +
1
D φ̂a Dm φ̂ba
2g̃ 2 m b
i ab /
λ̂ Dλ̂ab
2
−
κg̃ 2
2
+
ab
κ2 g̃ 2
2
iλ̂ λ̂ba
φ̂ab φ̂ba
i
and the interaction terms are given by
E
Sint
Z
=
h
dτ dΩ tr + κg̃ 2 Xa† φab X b − κg̃ 2 X a φ̂ba Xb† + iξa† φab ξ b + iξ a φ̂ba ξb†
(5.26)
− g̃ ac λcb X a ξb† + g̃ ac λcb ξ b Xa† + g̃ ac λ̂cb ξb† X a − g̃ ac λ̂cb Xa† ξ b
− κ6 φab [φbc , φca ] − κ6 φ̂ab [φ̂bc , φ̂ca ] + 2i λab [φbc , λac ] − 2i λ̂ab [φ̂bc , λ̂ac ]
2
+ g̃4 (Xσi X † )(Xσi X † ) +
g̃ 2
(X † σi X)(X † σi X)
4
†
φbc X Aa φ̂cb
+ 12 (XX † )φab φba + 12 (X † X)φ̂ab φ̂ba + XAa
i
− 8g̃12 [φab , φcd ][φba , φdc ] − 8g̃12 [φ̂ab , φ̂cd ][φ̂ba , φ̂dc ] .
The covariant derivatives are given by Dm X = ∇m X +iAm X −iX Âm etc. We also translate the supersymmetry variations from flat Lorentzian space as given in Appendix A.3
to Euclidean R × S2 . They are conveniently expressed in terms of a rescaled parameter
ε̃ab (τ ) = εab e−τ /2 .
(5.27)
In the following we will use the τ dependent parameter; however, we will drop the tilde
4
Suppressed indices are assumed to be in the standard positions as defined in (5.21) and (5.22).
†
The indices of the Pauli matrices are placed accordingly, e.g. Xσi X † ≡ X Aa (σi )a b XAb
or X † σi X ≡
†
XAa
(σi )a b X Ab . And by definition we have (σi )a b = σi and (σi )a b = σiT .
144
for notational simplicity. This parameter satisfies the Killing spinor equation
∇m ε = − 21 γm γ τ ε ,
(5.28)
which is the curved spacetime generalization of the usual condition of (covariant) constancy that the supersymmetry variation parameter obeys in flat space. The N = 3
supersymmetry transformations read
δAm = − ig̃2 εab γm λab ,
(5.29)
/ bc εac −
δλab = 2g̃i mnk Fmn γk εab − g̃i Dφ
+g̃ iX a Xc† εcb −
ig̃
(XX † )εab
2
2i b / ac
φ ∇ε
3g̃ c
+
i
[φb , φcd ]εad
2g̃ c
+ κg̃ iφbc εac (5.30)
,
δφab = −g̃εcb λca + g̃2 δba εcd λcd ,
(5.31)
δ Âm = − ig̃2 εab γm λ̂ab ,
(5.32)
/ φ̂bc εac +
δ λ̂ab = 2g̃i mnk F̂mn γk εab + g̃i D
−g̃ iεbc Xc† X a +
ig̃
(X † X)εab
2
2i b / ac
φ̂ ∇ε
3g̃ c
+
i
[φ̂b , φ̂cd ]εad
2g̃ c
+ κg̃ iφ̂bc εac (5.33)
,
δ φ̂ab = −g̃εcb λ̂ca + g̃2 δba εcd λ̂cd ,
(5.34)
δX Aa = −iεab ξ Ab ,
/ Ab εab + 13 X Ab ∇ε
/ ab + φab εbc X Ac + X Ac εbc φ̂ab ,
δξ Aa = DX
†
† b
δXAa
= −iξAb
εa ,
†
† b
†
†
† c b
/ Ab
/ ba + φ̂ba εcb XAc
= DX
εa + 13 XAb
+ XAc
ε b φa .
δξAa
∇ε
(5.35)
SU(2)R and U(1)R charge. Fundamental and anti-fundamental SU(2)R indices a, b
transform under infinitesimal rotations as
δAa = iεab Ab
,
δAa = −iεba A†b ,
145
(5.36)
yi
−1
field χ†σ , χ̂†σ
− 21
0
+ 12
+1
ζ A , ωA ,
χφ , χ̂φ , χ†φ , χ̂†φ
ζA† , ω †A
χσ , χ̂σ
Table 5.2: U(1)R charges of the fermion fields. These numbers show what the sum
(chiral only) of R-charges vanishes precisely for two flavors, A = 1, 2.
where A represents a generic field. The Noether current is
h
←
→
←
→
→
†
† ←
γ µ ξ Ab
D µ X Ab + iφca D µ φbc + iφ̂ca D µ φ̂bc − ξAa
Jaµb ∼ tr iXAa
(5.37)
i
+ 21 λac γ µ λbc + 21 λca γ µ λcb + 12 λ̂ac γ µ λ̂bc + 12 λ̂ca γ µ λ̂cb .
In Section 5.4 we will be dealing with the U(1)R component of this current which is
related to the transformation (5.36) with εab ∼ (σ3 )ab . Hence the current is the contraction
of (5.37) with (σ3 )ab . The part due to the fermions is given by
i
h
J µ ∼ tr − 21 ζA† γ µ ζ A − 21 ω †A γ µ ωA + χ†σ γ µ χσ + χ̂†σ γ µ χ̂σ ,
(5.38)
where we have reverted back to the U(1)R × SU(Nf )fl fields, see Table 5.1. From this
expression we read off the U(1)R charges of the fermions as given in Table 5.2.
5.3.2
Classical Monopole Solution
Since we are interested in BPS monopoles, our first task is to find a classical BPS
solution with flux emanating from a point in spacetime. Starting from the gauge field
configuration of a Dirac monopole in R3 given in (5.1) and performing the Weyl rescaling
appropriate for fields on R × S2 as described above, we find that the dependence on the
radial coordinate disappears from the gauge potential. In fact the Hodge dual of the
corresponding field strength, mnk Fmn , is constant in magnitude and purely radial (i.e.
only its τ component is non-vanishing).
In order to show that there is indeed a BPS solution with such a Dirac monopole
146
potential, let us examine in detail the supersymmetry variations of λab . It contains
terms of order g̃ and ones of order g̃ −1 , but since we are looking for a solution that is
supersymmetric along the whole RG flow they should cancel separately. We will also
assume that all background fields (in our choice of gauge) are valued in the Cartan
subalgebras of the gauge group factors, so that all commutators vanish.
Focusing on the terms of order g̃ −1 for now, we thus want the sum of the first three
terms in δλab , as given in (5.29), to vanish for an appropriate, non-trivial choice of
supersymmetry variation parameter. The fact that mnk Fmn is constant suggests that
φi should also be constant, in which case the second term vanishes by itself. Hence we
simply have to balance the first term and the third one, which simplifies upon using the
Killing spinor equation (5.28).
Recalling that εab = εi (σi )ab is traceless Hermitian, we can take the SU(2)R trace of
δλab which implies that εi φi = 0, i.e. the non-trivial supersymmetry variation parameter
has to be orthogonal to the background scalar in the R-symmetry directions.
Given this restriction, we can now contract δλab with (σi )ab and find
i mnl
Fmn γl
2
εi + ijk φj γ τ εk = 0 .
(5.39)
Thus the magnitudes of the scalar background and gauge fields are related, and picking
an SU(2)R orientation we can choose e.g. 21 mnτ Fmn = −η φ3 , where η = ±1 distinguishes
BPS from anti-BPS monopoles. In this case ε3 = 0 and the remaining supersymmetry
parameters have to satisfy5 ε1 − iη ε2 = 0.
Let us now turn to the terms of order g̃. Following the same lines of reasoning, they
imply
φba = − κ1 X b Xa† − 12 δab XX †
5
⇔
1
φi = − 2κ
Xσi X † .
In Euclidean space we treat ε1 ± iε2 as two independent supersymmetry parameters.
147
(5.40)
It is evident that if we choose  = A and φ̂i = −φi the variations δ λ̂ab (5.32) will vanish
also in the same manner, provided that
φ̂ba =
1
κ
Xa† X b − 12 δab X † X
⇔
φ̂i =
1
X † σi X
2κ
.
(5.41)
This leaves the supersymmetry transformations of the remaining fermions, δξ Aa and its
complex conjugate (5.35), to be verified. Given that φ̂i = −φi causes the last two terms
to cancel, they simply fix the functional dependence of the bifundamental scalars to be
†
X A1 ∼ exp (−η τ /2) and X A2 ∼ exp (η τ /2).6 Then all of the δξ Aa and δξAa
vanish
either by virtue of this particular τ -dependence, which is consistent with the equation
of motion for X
D2 X Aa − 14 X Aa = 0 ,
(5.42)
or because of a vanishing variation parameter.
Fixing the coefficients of the X fields such that (5.40) and (5.41) are satisfied, the
above BPS conditions are of course also consistent with the remaining equations of
motion. In particular, those for the gauge fields, given that the Dirac monopole potential
satisfies Dn g̃12 F mn = Dn g̃12 F̂ mn = 0, reduce to
κ mnk Fnk = XDm X † − Dm XX † ,
(5.43)
κ mnk F̂nk = Dm X † X − X † Dm X .
In summary, a convenient choice of classical (anti-)BPS solution is given by
A = Â =
H
(±1 − cos θ)dϕ ,
2
φi = −φ̂i = −η
H
δi3 ,
2
(5.44)
(where the upper sign holds on the northern and the lower one on the southern hemi†
†
And similarly XA1
∼ exp (η τ /2) and XA2
∼ exp (−η τ /2). While it may appear unusual that X
†
and X are not complex conjugates of each other, this is simply an artefact of the Euclidean signature
of spacetime. After a suitable Wick rotation they would evidently be conjugate in the usual sense.
6
148
sphere), supplemented by an appropriate X expectation value chosen to satisfy (5.40),
(5.41) and (5.43), e.g. for positive semi-definite H
X 12 = W 1†
√
Hκ e−τ /2 ,
√
= Hκ e−τ /2 ,
X 11 = Z 1 =
†
X11
= Z1† =
†
X12
√
Hκ eτ /2 for η = 1 or
√
= W1 = Hκ eτ /2 for η = −1 ,
(5.45)
with all other X’s vanishing. Note that the choice of X expectation value breaks the
flavor symmetry and here we have arbitrarily used the first flavor. This background is
invariant under supersymmetry transformations with the parameter ε1 + iηε2 , and can
be generalized to any preferred SU(2)R orientation.
Our classical solution is BPS along the full RG flow for any value of g̃, which is an
important prerequisite for our arguments. However, the actual calculation we wish to
carry out will be performed in the far UV, and here the role of the X expectation value
is quite different from the expectation values of the adjoint scalars and gauge fields.
If we were to do perturbation theory in g around this background, we would be led to
rescale the (quantum) fields A and φ (and their hatted analogues) by a factor of g (before
carrying out the Weyl transformation), and thus g sets the scale of quantum fluctuations.
Since the background values of A and φ are of order unity, they are parametrically larger
than these fluctuations in the UV, and thus can be treated classically.
The fluctuations of the bifundamental field X do not suffer such a rescaling however,
and both quantum excitations as well as the expectation value in our classical solution
are of the same order of magnitude. Therefore, we shall not treat the X fields as a
classical background in the UV theory. They are to be though of as quantum excitations
which dress up the monopole background, and thus we shall drop them for the purpose of
describing the bare monopole operator. We keep in mind however, that a bare monopole
operator is not gauge invariant, and will eventually have to be contracted with a number
of basic fields appearing in the Lagrangian in order to form a gauge invariant operator.
149
In this context it is interesting to note that the number of X excitations corresponding
to our classical solution is of order Hk, which is related to the number of fundamental
fields we expect to contract the bare monopole operator with.
Finally, we will need to generalize the monopole background to arbitrary, possibly
τ -dependent SU(2)R orientation ni and thus, to conclude this discussion, we collect the
basic properties of the semi-classical BPS monopoles background we will make use of
below.
BPS monopole background. For the background to be BPS, we have to essentially
identify the two gauge groups U(N ) and Û(N ), and turn on expectation values for the
gauge fields and adjoint scalars φi and φ̂i given by
A = Â =
H
(±1 − cos θ)dϕ ,
2
φi = −φ̂i = −
H
ni (τ ) .
2
(5.46)
The monopole background is diagonal in the U(N ) gauge indices H = diag(q1 , . . . , qN )
where qr ∈ Z are the U(1)N gauge charges. They determine how the monopole transforms under gauge transformations. Furthermore, the background is labeled by a unit
SU(2)R vector ni which gives the direction of the scalar fields. This vector is a collective
coordinate of the background (5.46) and spans the moduli space S2 .
5.4
U(1)R Charges from Normal Ordering
In this section we compute the quantum corrections to the U(1)R charge which is preserved by the static background (5.46) with ni = δi3 . These quantum corrections are
due to fermions fluctuations and are encoded in the normal ordering constant of the
U(1)R charge operator. Before going into the specifics of ABJM theory, we will discuss
the computation for a toy example which is then simple to generalize.
150
5.4.1
Prototype
Let us consider a single fermion ψ(τ, Ω) in an abelian gauge theory subject to the equation of motion
η
/ + qψ = 0 ,
Dψ
2
(η = ±1)
(5.47)
and compute the oscillator expansion of the charge
Z
Q = −i
dΩ ψ † γ τ ψ .
(5.48)
/ =∇
/ + iA
/ in (5.47) includes a monopole background of the kind
The Dirac operator D
(5.46) with magnetic charge H → q. The mass term in (5.47) whose magnitude is
proportional to q plays the role of the coupling to the background scalar. The two signs,
η = ±1, correspond to a BPS and an anti-BPS background, respectively.
The easiest way to solve (5.47) is to expand ψ in monopole spinor harmonics which
/ S . This operator is
are eigenfunctions of the monopole Dirac operator on the sphere, D
/ = γ τ ∂τ + D
/ S , since the monopole
contained in the full operator in (5.47) simply as D
potential does not have any component along τ . We note the explicit form of the
monopole spinor harmonics in Appendix A.4. Here we only need their eigenvalues and
multiplicities. For given magnetic charge q, the quantum numbers of the total angular
momentum are given by
j=
|q| − 1
+p
2
with
m = −j, −j + 1, . . . , j ,
where the state j =
|q|−1
2
is absent for q = 0.
151
p = 0, 1, 2, . . . ,
(5.49)
We denote the eigenfunctions by
/ S Υ0qm = 0
D
± ±
/ S Υ±
D
qjm = i∆jq Υqjm
for j =
|q|−1
2
(5.50)
for j =
|q|+1 |q|+3
, 2 ,...
2
(5.51)
with eigenvalues
1
∆±
jq = ± 2
p
(2j + 1)2 − q 2 .
(5.52)
This spectrum is plotted in Figure 5.1(a). The |q| zero modes which exist for non-zero
magnetic charge will be responsible for a shift of the quantized version of the charge
(5.48).
Now we are ready to write the harmonic expansion of the wave function as
ψ(τ, Ω) =
X
ψm (τ )Υ0qm (Ω) +
m
X
ε
ψjm
(τ )Υεqjm (Ω) ,
(5.53)
jmε
where ε = ±1. Plugging this expansion into the equation of motion (5.47) and using
the properties (A.36),(A.38) and the orthogonality (A.39) one finds

ψ̇m = −η
|q|
ψm
2
,



+
 ψ̇jm 

=
 
−
−i∆ −
0

+
η 2q   ψjm

 .

(5.54)
i
i
Xh
|q|
|q|
cjm uεj e−Ej τ + d†jm vjε eEj τ Υεjm ,
cm u0 e− 2 τ + d†m v 0 e 2 τ Υ0m +
(5.55)


−
ψ̇jm
−i∆+ −
η 2q
0


−
ψjm
The solution is
ψ=
Xh
m
jmε
where the energies are given by Ej = j + 21 . The wavefunctions for the BPS case (η = +1)
152
are given by
0
0
u =1, v =0,
u+
j
=
vj+
=1,
u−
j
=
−vj−
=
√1
2
q
2j+1
+i
q
1−
2
q
2j+1
,
(5.56)
and the ones for the anti-BPS case (η = −1) are obtained from this by exchanging
∓
u0 ↔ v 0 and u±
j ↔ vj . The normalization of the wave functions are such that we have
{ψα (τ, Ω), Πβ (t, Ω0 )} = δαβ δ (2) (Ω − Ω0 ) ,
(5.57)
{cjm , c†j 0 m0 } = δjj 0 δmm0 ,
(5.58)
{djm , d†j 0 m0 } = δjj 0 δmm0 ,
(5.59)
where Π = −iψ † γ τ is the canonically conjugate momentum.
The energy spectra are plotted in Figure 5.1(b) and Figure 5.1(c). One observes
that the zero modes of the Dirac operator have turned into “unpaired states”, i.e. states
for which there are no states with the opposite energy in the spectrum. There are
, which is positive for the BPS and negative
2j + 1 = |q| unpaired states with energy η |q|
2
for the anti-BPS case.
Now we compute the U(1)R charge using point splitting regularization as in [161]:
i
Q(β) = −
2
Z
h
τ
τ †
i
β
β
β
β
†
dΩ ψ τ + 2 γ ψ τ − 2 − ψ τ + 2 γ ψ τ − 2 .
(5.60)
where β > 0 will be taken to zero in the end. Inserting the oscillator expansion7 and
ordering the terms, we find the normal ordered piece
Q1 (β = 0) =
Xh
c†jm cjm − d†jm djm
i
(5.61)
jm
The oscillator expansion of ψ † is the complex conjugate of that of ψ where in addition the sign of
τ is reversed.
7
153
5
4
3
2
1
∆±
jq
9
8
7
6
5
4
3
!2
!3
!4
!5
6
4
9
7
5
10
8
6
9
7
10
8
11
5
12
4
9
10
3
2
2
1
1
0
!1
8
2
3
4
5
6
7
8
E
2
3
4
5
6
4
6
7
8
8
9
10
0
10
111 2
5
7
9
11
3
6
8
10
4
q
7
9
11
5
8
10
12
6
!2
9
11
10
!3
!4
12
!5
713 8
5
9
8
4
7
7
7
6
6
3
5
5
4
4
2
3
2
0
!1
8
8
6
9
9
9
8
1
E
1
2
2
3
4
4
5
6
5
6
7
8
8
10
(a) Dirac operator
1
7
9
2
6
8
10
9
4
q
8
9
10
5
!3
7
8
10
3
!2
5
6
7
9
0
4
6
10
7
8
5
5
5
4
4
4
8
7
6
6
6
9
8
7
7
3
3
2
2
1
0
!1
3
8
8
7
6
9
9
9
8
!4
!5
(b) BPS
2
3
4
4
5
6
5
6
7
8
8
9
10
0
6
7
9
10
1
7
8
2
8
9
10
3
4
q
9
10
5
6
10
7
8
(c) anti-BPS
Figure 5.1: Eigenvalues of the Dirac operator on thep
S2 and energy spectra for BPS
±
1
and anti-BPS backgrounds. The eigenvalues ∆jq = ± 2 (2j + 1)2 − q 2 and E = ±Ej =
±(j + 12 ) are parametrized by j = |q|−1
+p for p = 0, 1, 2, . . .. The dashed lines correspond
2
to a fixed value p. The numbers next to the points denote the multiplicities of the
corresponding eigenvalues. Note in particular that there are |q| zero modes of the Dirac
operator for monopole charge q, which are lifted to non-zero unpaired modes when the
background scalar is turned on.
and a normal ordering constant
Q0 (β) = −
i
1 Xh ε† ε
uj uj − vjε† vjε e−βEj ,
2 jmε
where in both sums we understand the zero modes with j =
(5.62)
|q|−1
2
to be included. For
that value of j there is no sum over ε = ±1 and Ej = |q|
. The normal ordering constant
2
P
ε
can be written in a concise form by noticing that ε uε†
j uj = 1 gives a contribution for
P
every positive energy state and ε vjε† vjε = 1 one for every negative energy state. Hence
we can write
Q0 (β) = −
1 X
sign(E) e−β|E| ,
2 states
154
(5.63)
where the sum extends over all states in the spectrum8 and vanishes if the spectrum
is symmetric with respect to E = 0. This is not the case due to the unpaired states.
Instead we find
Q0 = −η
|q|
,
2
(5.64)
where the factor |q| is the number of unpaired states (the degeneracy of the mode
j=
|q|−1
)
2
and η the sign of their energy. This normal ordering constant gives the U(1)R
charge of the BPS or anti-BPS monopole background, in agreement with [161]. As it
was also shown in [161], the bosonic fields do not contribute the induced charge because
their spectrum is symmetric.
5.4.2
Application to N = 3 Gauge Theory
The above discussion is readily applied to the N = 3 gauge theories with Nf hyper
multiplets. In place of the charge (5.48) we now have
Z
Q = −i
i
h
dΩ tr − 21 ζA† γ τ ζ A − 21 ω †A γ τ ωA + χ†σ γ τ χσ + χ̂†σ γ τ χ̂σ ,
(5.65)
see (5.38), and the equation of motion (5.47) is replaced by
/ A + η2 [H, ζ A ] = 0 ,
Dζ
/ σ + η2 [H, χσ ] = 0 ,
Dχ
/ χ̂σ + η2 [H, χ̂σ ] = 0 ,
D
/ A + η2 [H, ωA ] = 0 ,
Dω
/ φ + η2 [H, χφ ] = 0 ,
Dχ
/ χ̂φ + η2 [H, χ̂φ ] = 0 ,
D
(5.66)
which hold in the static background (5.44) and in the far UV where g̃ → 0. The
only qualitative difference is that we are now dealing with a non-abelian theory. Since
the background fields have the same U(N ) dependence for both gauge group factors,
8
This formulae looks superficially different from that in [162] because we sum over all states including
their degeneracy, not just over different energies levels.
155
namely H = diag(q1 , . . . , qN ), all fields effectively transform in the adjoint of the diagonal
U(N )d ⊂ U(N )× Û(N ). This is apparent from the fact that we could write commutators
in (5.66), and also the gauge field inside the Dirac operator acts via a commutator. Now
the key observation is
[H, ψ]rs = qr δrt ψts − ψrt qt δts = (qr − qs )ψrs ,
(5.67)
where ψrs is one of the N × N matrix elements. This allows us to consider all matrix
elements separately, if we use
q → qrs ≡ qr − qs
(5.68)
as the effective monopole charge. This immediately implies that the U(1)R charge of the
vacuum, i.e. the monopole background, is
Qmon
=
R
i
Xh
− 12 · Nf · −η |q2rs | − 21 · Nf · −η |q2rs | + 1 · −η |q2rs | + 1 · −η |q2rs |
r,s
=η
X
N
Nf
−1
|qr − qs | .
2
r,s=1
(5.69)
The sum produces an answer at least of order N unless all qr are equal. For example, for
the simplest monopole whose only non-vanishing label is q1 = 1, and therefore transforms
in the bifundamental of U(N ) × U(N ) for k = 1, we find that the induced R-charge is
(Nf − 2)(N − 1).
For the BPS background where η = +1, the R-charge of the monopole is positive if
Nf > 2. For Nf = 1 the monopole R-charges are negative and the theory may not flow
to a conformal limit [170, 171]. The BPS monopoles in ABJM theory have vanishing
R-charge since there are two flavors, Nf = 2. However, this does not mean that there are
gauge invariant operators with vanishing R-charge because such operators necessarily
156
involve matter fields in addition to the monopoles. In fact, in a strongly coupled theory it
is generally impossible to separate the monopole part from the matter part; nevertheless,
this is possible in the weakly coupled UV limit where the monopole part is semi-classical.
We finally remark that from a similar computation, where we normal order the flavor
charge operator instead of the R-charge operator, one can see that the monopole does
not carry any induced flavor representation. Using the mode expansion in the flavor
charge
Qnfl
Z
∼
dΩ
tr ξA† (T n )A B γ τ ξ B
Z
=
h
i
dΩ tr ζA† (T n )A B γ τ ζ B − ω †B (T n )A B γ τ ωA (5.70)
yields a vanishing normal ordering constant simply because the generators (T i )A B of
SU(Nf ) are traceless. Note that considering the static background (5.44) which preserves
only U(1)R ⊂ SU(2)R is general enough for this argument, because flavor and R-charge
are completely independent.
5.5
SU(2)R Charges and Collective Coordinate Quantization
In this section we compute the SU(2)R charges of the BPS monopole operators by quantizing the collective coordinate ~n(τ ) of the corresponding class of classical monopole
backgrounds (5.46). The dynamics of the collective coordinate is governed by the interaction with the other fields of the theory. We take the influence of these interactions
into account by computing the effective action Γ(~n). As explained in Section 5.2 it is
sufficient to carry out this computation in the UV limit of the theory where the YangMills coupling g̃ goes to zero. In this limit precisely the interactions with the fermions
157
survive. Thus the effective action is obtained by integrating out the fermions:
−Γ(~
n)
e
Z
=
[dξ † ][dξ][dλ][dλ̂] e−S ,
(5.71)
where the relevant part of the action is
Z
S=
h
†
†
/ Aa − 2i ni ξAa
dτ dΩ tr − iξAa
Dξ
(σi )a b [H, ξ Ab ]
(5.72)
/ ab − 2i ni λab (σi )b c [H, λac ]
+ 2i λab Dλ
(5.73)
i
/ λ̂ab − 2i ni λ̂ab (σi )b c [H, λ̂ac ] .
+ 2i λ̂ab D
(5.74)
Since the action is quadratic the path integral is “simply” a determinant, albeit a determinant of a matrix operator with very many indices. We have displayed explicitly
the SU(2)fl index (A) and the SU(2)R indices (a, b, c). Besides those, there are implicit
U(N ) gauge indices which we denote by ξrs , λrs , etc. below. Then there is the spatial
dependence of the fermions which we will trade for a set of mode indices by expanding
the fields into harmonics on the sphere.
The good news is that the operator is fairly diagonal and couples only very few
components together. For instance it is completely diagonal in the flavor and gauge
indices, and couples at most two modes of the harmonic expansion. Thus we can perform
the computation for a generic component and take the sum over the indices into account
later.
5.5.1
Prototype
We start the discussion with a quantum mechanical model where there is only a (Euclidean) time coordinate τ . This example already exposes the essential point of the
whole argument. The dependence of the fields on the angular coordinates on the sphere
will be taken into account in the next paragraph.
158
Quantum mechanics. Let us consider one fermion ψ a (τ ) in the fundamental representation of SU(2)R , as indicated by the index a, with the action9
Z
S=
dτ
h
−iψa† ∂τ ψ a
−
i
2
qni (τ ) ψa† (σi )a b ψ b
i
.
(5.75)
Since we have not included any spatial dependence in this example, there is no monopole
gauge field either. The coupling of ψ to the collective coordinate has been written as q
in anticipation of it becoming the monopole charge later.
Formally we find for the effective action10
Γ(~n) = − ln det i∂τ − i 2q ni (τ )σi .
(5.76)
Due to the unspecified τ -dependence of ni , we cannot evaluate this determinant exactly. However, since the collective coordinate is considered as being quasi-static, it is
legitimate to do a derivative expansion. The general form of the effective action then is
Z
Γ(~n) =
i
h
dτ −Veff (~n) + i ṅi Ai (~n) + 21 ṅi ṅj Bij (~n) + . . . .
(5.77)
Our aim is to find the function Ai (~n) by expanding (5.76). However, it is not immediately
possible to expand (5.76) in ṅi , as it does not contain ṅi explicitly. The trick due to
[174] is to write
ni (τ ) = n̊i + ñi (τ ) ,
(5.78)
where n̊i is constant with n̊2 = 1 and ñi (τ ) a small “fluctuation”. Then the form of the
9
A similar model with a fermion in the 3 of SU(2) was studied in [173].
The relative sign between the two terms may superficially appear to have changed since (σi )a b in
(5.75) are transposed Pauli matrices, while in (5.76) we use non-transposed ones, (σi )a b .
10
159
effective action to second order in fluctuations reads
Z
Γ(~n) =
h
dτ −Veff (˚
~n) − ñi ∂i Veff (˚
~n) − 12 ñi ñj ∂i ∂j Veff (˚
~n)
(5.79)
i
~n) + . . . .
+i ñ˙ i Ai (˚
~n) + i ñ˙ i ñj ∂j Ai (˚
~n) + 21 ñ˙ i ñ˙ j Bij (˚
The point of (5.78) is that we can now expand (5.76) in powers of ñi , some of which will
come with τ -derivatives, and compare the result with the general expression (5.79). We
cannot determine Ai from the term ñ˙ i Ai (˚
~n) as this is a total derivative and hence will
not show up in the expansion of (5.76). Therefore we will focus on the next term, the
unique term with two powers of ñ and one τ -derivative
Z
Γ(2,1) (~n) = i
i
dτ ñ˙ i ñj ∂j Ai (˚
~n) = −
2
Z
dτ ñ˙ i ñj ∂i Aj (˚
~n) − ∂j Ai (˚
~n) .
(5.80)
Now we start from (5.76) using (5.78)
Γ(~n) = − tr ln i∂τ − im̊
\ − im̃
\ = − tr ln i∂τ − im̊
\ − tr ln 1 −
1
m̃
\
∂τ −m̊
\
, (5.81)
where we have introduced the shorthands m̊i = 2q n̊i , m̃i = 2q ñi , and m
\ ≡ mi σi . We
isolate the term with two powers of ñi ∼ m̃i by expanding the logarithm:
1
∂τ + m̊
\
∂τ + m̊
\
Γ(2) (~n) = tr 2
m̃
\
m̃
\ .
2
∂τ − m̊2 ∂τ2 − m̊2
(5.82)
Next we move all derivatives to the right using
∞
X
1
1
φ
=
(−1)k [∂ 2 , [∂ 2 , . . . , [∂ 2 , φ] . . .]] 2
,
2
2
|
{z
}
∂ − m̊
(∂ − m̊2 )k+1
k=0
(5.83)
k
and perform the trace over SU(2)R indices using
tr σi σj = 2δij
,
tr σi σj σk = 2iijk
,
tr σi σj σk σl = 2δij δkl − 2δik δjl + 2δil δjk . (5.84)
160
We find for the terms which contain exactly one derivative
2
2
1
˙ i m̃i ∂τ (∂τ + m̊ ) − i tr ijk m̃
˙ i m̃j m̊k
Γ(2,1) (~n) = − 3 tr m̃
2
2
3
2
(∂τ − m̊ )
(∂τ − m̊2 )2
∂τ
˙ i m̃j m̊i m̊j − m̃
˙ j m̃j m̊i m̊i
− 6 tr 2m̃
.
(∂τ2 − m̊2 )3
(5.85)
Now that the coordinate dependent part is separated from the derivatives it is easy to
evaluate the functional trace: it leads to one integral over τ and one over the energy
ω, where ∂τ → −iω. All but the second term will lead to an integral over a total τ derivative and therefore can be dropped. We are left with only the second term which
becomes
Z
Γ(2,1) (~n) = −i
˙ i m̃j m̊k
dτ ijk m̃
Z
1
i
dω
= − sign(q)
2
2
2
2π (ω + m̊ )
4
Z
dτ ijk ñ˙ i ñj
n̊k
. (5.86)
|˚
~n|3
Comparing this to (5.80), we read off
∂i Aj (~n) − ∂j Ai (~n) =
sign(q)
nk
ijk 3 .
2
|~n|
(5.87)
This expression is recognized as the field strength for a magnetic monopole with charge
sign(q)
2
and hence Ai (~n) is the corresponding gauge potential. We should stress that this
monopole has nothing to do with the original monopole background of ABJM theory
which we set out to study. In fact, the action (5.75) does not contain any monopole
background for ψ. The monopole potential we are finding here lives on the space spanned
by ~n, which just happens to be the moduli space of supersymmetric monopoles in N = 3
gauge theory.
At any rate what we have found is that the effect of the fermion ψ is to induce a
Wess-Zumino term in the effective action for the collective coordinate. In other words,
the dynamics of the collective coordinate ~n is the same as that of a point particle on a
sphere with a magnetic monopole at its center. The coefficient of the Wess-Zumino term
161
is the product of the electric charge of the point particle and the magnetic charge of the
monopole. This analogy immediately implies that the allowed SU(2)R representations
for the quantized collective coordinate, and hence the ABJM monopoles, are bounded
from below by the Wess-Zumino coefficient. Thus, if we want to have monopole operators
in the singlet representation, we will have to find that the Wess-Zumino term cancels
from the effective action when all contributions are included. This is what will indeed
happen for ABJM theory, but not in general.
Spatial dependence. We reinstate the dependence of ψ on the coordinates of the
sphere and consider the case
Z
S=
i
h
†
a
b
†/ a
i
dτ dΩ −iψa Dψ − 2 qni (τ ) ψa (σi ) b ψ ,
(5.88)
which contains a monopole background with magnetic charge q but is still abelian. The
easiest way to deal with the spatial dependence is to expand ψ(τ, Ω) into monopole
spinor harmonics and perform the S2 -integration in the action. All relevant properties
of these harmonics have already been discussed in Subsection 5.4.1, see also Appendix
A.4.
The expansion of ψ is the same as in Section 5.4, eq. (5.53). Since ~n(τ ) is constant
on the sphere, the orthogonality of the monopole harmonics (A.39) implies that different
(jm)-modes of ψ do not couple to each other. The only coupling is between the ±-modes
which is due to the property (A.36). Indeed the action for the modes becomes
S=
XZ
h
i
†
†
dτ −i sign(q) ψm
∂τ ψm − 2i qni ψm
σi ψm
(5.89)
m
+
XZ
i
h
−ε† ε
ε†
−ε†
ε
ε
dτ −iψjm
∂τ ψjm
+ ∆εjq ψjm
ψjm − 2i qni ψjm
σi ψjm
.
jmε
This decoupling means that we can compute the contribution to the effective action
for each pair (jm) individually. One such term in the zero-mode sector, j =
162
|q|−1
,
2
is
essentially the previously considered case (5.75). The additional sign(q) in (5.89), which
originates from (A.38), removes the sign(q) from the result (5.86). The sum over m
introduces a factor of 2j + 1 = |q|. This is how the Wess-Zumino term acquires the
dependence on the monopole charge. Thus the Wess-Zumino potential is
∂i Aj (~n) − ∂j Ai (~n) =
|q|
nk
ijk 3 .
2
|~n|
(5.90)
Now we turn to the non-zero-mode sector. It will turn out that there is no contribution to the effective action from this sector, and that (5.90) is the final result. In the
following we think of j and m as fixed to some values and suppress these labels. Due
to the coupling between the ±-modes, we now have a 2 × 2 matrix in “mode space” on
top of the matrix structure that mixes the two components of the SU(2)R doublet:


Γ(~n) = − ln det 


−∆− − im
\
 .

+
−∆ − im
\
i∂τ
i∂τ
(5.91)
The generalization of (5.82) is

2

\
  −∂τ −i∆ − m̊




1 

\ −∂τ
Γ(2) (~n) = tr  i∆ − m̊
2 

∂τ2 − ∆2 − m̊2



\ 
 0 m̃


,



m̃
\ 0 


(5.92)
where ∆ ≡ ∆+ = −∆− . Evaluating this expression analogously to the previous case
yields that the term Γ(2,1) (~n) is zero (up to surface terms), i.e. the non-zero modes do
not contribute to the Wess-Zumino term.
Recall that in the computation of the U(1)R charge of the monopole operator in
Section 5.4, we also found that the non-zero modes did not contribute. There the cancellation occurred between states of equal but opposite energy. In order to demonstrate
163
explicitly that the same mechanism is at work here, too, we pretend that ∆+ and ∆−
are unrelated for the time being and repeat the computation. Without presenting any
of the lengthy intermediate steps, we arrive at
Z
Γ(2,1) =
Z
dτ
˙ i m̃j m̊k (∆+ + ∆− )(ω 2 + ∆+ ∆− ) ω
−4iijk m̃
dω
.
2π ω 4 + 2(∆+ ∆− − m̊2 )ω 2 + (∆+ ∆− )2 + (∆+ )2 + (∆− )2 m̊2 + m̊4 2
(5.93)
Indeed we see that the vanishing is due to the pairing of eigenvalues, ∆+ = −∆− .
5.5.2
Application to N = 3 Gauge Theory
Having obtained the result (5.90) for the prototype action (5.88) it is a simple matter
to specialize to ABJM theory and its N = 3 UV completion. All that needs to be done
is to include the gauge indices and sum over the field content.
Gauge structure. The non-abelian nature of the theory is taken care of just as in
Section 5.4.2. From (5.67) it is clear that the action is diagonal in gauge indices and
therefore every matrix element contributes independently from the others to the effective
action. The effective monopole charge that the matrix element ψrs experiences is given
by
qrs ≡ qr − qs .
(5.94)
Hence the non-abelian result is obtained from the abelian one by the replacement
Γq (~n) →
N
X
Γqrs (~n) .
(5.95)
r,s=1
We will see that the sum over r, s will factorize, because all fermions transform effectively
in the same representation.
164
Therefore, it will be convenient to define the total charge
qtot =
N
X
|qrs | = 2
r,s=1
X
|qr − qs | .
(5.96)
r>s
Hyper multiplet fermions. Upon using (5.67), the action for ξ A is just Nf copies of
the prototype (5.88). Therefore we can immediately write down the total contribution
from the hyper multiplet fermions
∂i Aj (~n) − ∂j Ai (~n) = Nf
qtot
nk
ijk 3 .
2
|~n|
(5.97)
Vector multiplet fermions. The computation for λ and λ̂ reduces to (5.88) as well,
but we have to be careful not to over-count the degrees of freedom. Due to the relations
(5.20) there are only two independent (complex) components. We choose λ11 ∼ χσ and
λ12 ∼ χ†φ as the independent components and denote their complex conjugates by
λ†11 ≡ (λ11 )∗
,
λ†12 ≡ (λ12 )∗ ,
(5.98)
and similarly for λ̂. When expressed in terms of these fields, the action reads
Z
S=
h
/ 1a + 2i ni λ†1a (σi )a b [H, λ1b ]
dτ dΩ tr − iλ†1a Dλ
(5.99)
i
/ λ̂1a + 2i ni λ̂†1a (σi )a b [H, λ̂1b ] .
− iλ̂†1a D
This is nothing but the action for ξ with reversed sign of the interaction, cf. (5.72). Thus
the contribution from the vector multiplet fermions is
∂i Aj (~n) − ∂j Ai (~n) = −2 ×
qtot
nk
ijk 3 ,
2
|~n|
where the factor of 2 arises because we have λ and λ̂.
165
(5.100)
Collective coordinate quantization. Adding all contributions together, the total
effective action for the collective coordinate is given by
Z
Γ(~n) =
dτ
h
1
M ~n˙ 2
2
i
2
˙
~
+ i A(~n) · ~n + λ(~n − 1) ,
(5.101)
where the kinetic term is simply the sum of the kinetic terms for φ and φ̂. The last
term is a Lagrange multiplier term which enforces the constraint that the modulus of ~n
is fixed to one. This actions describes a particle with unit electric charge and large mass
(as g̃ → 0 in the UV)
M = M (τ ) =
X
1
2
−2 −2τ
tr
H
=
g
e
qr2
g̃ 2
r
(5.102)
on a sphere surrounding a magnetic monopole with charge
h = (Nf − 2) qtot = (Nf − 2)
X
|qr − qs | ,
(5.103)
r,s
~ =∇
~ ×A
~ = h ~n.
and field strength B
2
This is the Euclidean version of the system discussed in Section 5.2 with the modification that now M depends on time. Still, the conserved angular momentum is given
by
~ = M ~n × ~n˙ − h ~n ,
L
2
(5.104)
and its quantized values are the SU(2)R charges of the monopole operator described by
the BPS background (5.46). Due to the second term in (5.104), the smallest possible
SU(2)R representation has spin
X
|h| Nf
l=
=
− 1
|qr − qs |
2
2
r,s
166
(5.105)
and dimension 2l + 1 = |h| + 1.
A monopole in the singlet representation and hence of vanishing IR dimension, is
only possible if h = 0. One way of achieving this is to set all fluxes qr equal. This
corresponds to a monopole in the diagonal U(1)d ⊂ U(N ) × U(N ) of the gauge group
which decouples from the matter fields. Another way to have singlet monopoles is to
consider Nf = 2 hyper multiplets, which is the field content of ABJM theory. This is
true regardless of how the fluxes qr are distributed inside the total U(N )d flux H.
5.6
Conclusions and Outlook
In this chapter we have calculated the global charges and dimensions of monopole operators in certain three-dimensional N = 3 supersymmetric Yang-Mills Chern-Simons
theories. This is the smallest amount of supersymmetry leading to a non-abelian Rsymmetry which was crucial for our argument, because the SU(2)R spin of a monopole
operator cannot change along an RG flow. This allowed us to find the exact charges
from a one-loop calculation in the weakly coupled UV limit of the gauge theory.
In the far UV the monopole operator was adequately described by a classical Dirac
monopole background for the gauge fields. For the description of BPS monopoles the
background needs to be supersymmetric which required us to also turn on a classical
background for the adjoint scalar fields.
In Section 5.4 we considered a static scalar background. Since such a background
breaks the R-symmetry from SU(2)R to U(1)R , we could only determine the abelian
charge of the monopole in this case. Using the methods developed in [161], we found that
fermionic fluctuations around the background induce a U(1)R charge of the monopole
proportional to the R-charges of the fermions times their magnetic coupling to the
background. Our complete formula (5.69) for the R-charge in U(N ) × U(N ) gauge
theory coupled to Nf hyper multiplets in the bifundamental representation is consistent
167
with the proposal made in [170, 171].
However, knowing the U(1)R charges at small Yang-Mills coupling is in general not
enough as this quantity is not protected and one cannot make any reliable statement
about their IR values. In [161] the computation was performed directly in the IR limit
(of SQED) which was possible by assuming a large number of flavors. Here we could not
resort to this trick because we wanted to keep the number of flavors arbitrary. However,
we can make use of the N = 3 supersymmetry and compute non-abelian R-charges which
are protected. They follow from quantization of the SU(2)/U(1) collective coordinate of
the background. By calculating the fermionic determinants which induce a Wess-Zumino
term in the effective action of the collective coordinate, we demonstrated in Section 5.5
that the smallest allowed SU(2)R representation is given by (5.105). The largest U(1)R
charge within this representation coincides with our findings in Section 5.4.
Note that the induced R-charge of the monopole is entirely due to the fermions
of the theory. In the normal ordering computation, Section 5.4, the reason why the
bosons do not contribute is because their spectrum is symmetric with respect to zero
and therefore the states with positive energy cancel the effect of those with negative
energy. In the collective coordinate computation, Section 5.5, this follows from the fact
that the coupling between the bosons and the collective coordinate goes to zero in the
UV.
After the theory has flown to the superconformal Chern-Simons fixed point, we can
use these results to argue that the contribution of the monopole operator to gaugeinvariant operator dimensions is given by (5.69), as long as it is non-negative (when it is
negative, a conventional fixed point does not exist). For ABJM theory, where Nf = 2,
this contribution vanishes which is crucial for matching the spectrum with supergravity
on AdS4 × S7 /Zk and for the supersymmetry enhancement to N = 8.
Since the gauginos make a crucial negative contribution to the R-charge, and they
are not even dynamical in the IR Chern-Simons theory, it is not clear how to carry out
168
this calculation reliably without appealing to the UV theory containing the Yang-Mills
term. Luckily, there are various other theories to which the method of starting with a
weakly coupled UV theory can be applied. One obvious example is to consider quiver
theories with more than two U(N ) gauge groups and bifundamental hyper multiplets.
These Yang-Mills Chern-Simons theories can flow in the IR to N = 3 superconformal
fixed points; some of them have M-theory AdS4 duals found by Jafferis and Tomasiello
[175].
Determination of monopole operator dimensions poses more of a challenge in N =
2 superconformal Chern-Simons theories, since we cannot rely on a non-abelian Rsymmetry. Nevertheless, it should again be possible to define some of these theories
via flow from weakly coupled Yang-Mills Chern-Simons theories, where monopole operator R-charge can be computed semiclassically. It is conceivable that the U(1)R charge
does not change under the RG flow, which would then determine it in the superconformal
theory.
As we have noted, in some quiver theories the induced R-charge of the monopole in
the UV is simply proportional to the sum over the R-charges of all the fermions. If a
“parent” quiver Yang-Mills gauge theory can be written down in four dimensions with
the same superpotential, then all the fermion R-charges in it are the same as in three
dimensions. In this “parent theory” the sum over all fermion R-charges determines the
U(1)R anomaly. In particular, if this quantity vanishes, then the 4-dimensional gauge
theory is superconformal. Thus, it is tempting to conjecture a relation between U(1)R
anomaly in a parent 4-dimensional gauge theory and the induced monopole operator
R-charge in a descendant 3-dimensional gauge theory. For example, in the class of
U(N ) × U(N ) theories we have considered in this paper, the induced monopole R-charge
is ∼ (1 − Nf /2). In its parent 4-dimensional gauge theory, the U(1)R anomaly coefficient
is (1 − Nf /2)N , with the first term due to an adjoint gluino of R-charge 1, and the
second due to Nf bifundamentals of R-charge −1/2. The anomaly cancellation for
169
Nf = 2 singles out the 4-dimensional superconformal gauge theory describing D3-branes
on the conifold [44].
This discussion suggests that, if a quiver gauge theory is superconformal in four
dimensions, then at least some monopole operators in its 3-dimensional descendant
(presumably the ones that correspond to turning on monopoles in all gauge groups)
have vanishing monopole operator dimensions. Clearly, the possibility of a connection
between monopole R-charges in three dimensions and anomaly coefficients in four dimensions requires a more detailed study.
170
Appendix: Useful Formulae,
Notations and Conventions
A.1 The Type IIB Supergravity Equations
Here we succinctly list the equations of motion of IIB supergravity. They are used
heavily in Chapter 3 to study 2-form potential and metric perturbations. For simplicity
we have set the dilaton and RR scalar to zero.
Bianchi identities:
dF3 = 0 ,
dH3 = 0 ,
(A.1)
dF̃5 = H3 ∧ F3 .
Dynamic equations:
d ∗ H3 = −gs2 F̃5 ∧ F3 ,
d ∗ F3 = F̃5 ∧ H3 ,
(A.2)
F̃5 = ∗F̃5 .
Einstein equation:
Rij = Tij =
1
g2
gs2
1
g2
F̃iabcd F̃j abcd + Hiab Hj ab − Gij Habc H abc + s Fiab Fj ab − s Gij Fabc F abc .
96
4
48
4
48
(A.3)
171
A.2 Chern-Simons Field Theory Notations and Conventions
Lorentzian ABJM model.
The world-volume metric is g µν = diag(−1, +1, +1)
with index range µ = 0, 1, 2. We use Dirac matrices (γ µ )α β = (iσ 2 , σ 1 , σ 3 ) satisfying
γ µ γ ν = g µν +µνρ γρ . The fermionic coordinate of superspace is a complex two-component
spinor θ. Indices are raised, θα = αβ θβ , and lowered, θα = αβ θβ , with 12 = −12 = 1.
Note that lowering the spinor indices of the Dirac matrices makes them symmetric
µ
γαβ
= (−1, −σ 3 , σ 1 ). In products like θα θα ≡ θ2 , θα θ̄α ≡ θθ̄ etc and θα γ µαβ θ̄β ≡ θγ µ θ̄ we
suppress the indices. We have
θα θβ = 12 αβ θ2
,
θα θβ = 12 αβ θ2
(A.4)
and likewise for θ̄ and derivatives. The Fierz identities are11
(ψ1 ψ2 )(ψ3 ψ4 ) = − 12 (ψ1 ψ4 )(ψ3 ψ2 ) − 12 (ψ1 γ µ ψ4 )(ψ3 γµ ψ2 ) ,
(A.5)
(ψ1 ψ2 )(ψ3 γ µ ψ4 ) = − 21 (ψ1 γ µ ψ4 )(ψ3 ψ2 ) − 21 (ψ1 ψ4 )(ψ3 γ µ ψ2 ) − 21 µνρ (ψ1 γν ψ4 )(ψ3 γρ ψ2 ) ,
(ψ1 γ µ ψ2 )(ψ3 γ ν ψ4 ) = − 21 g µν (ψ1 ψ4 )(ψ3 ψ2 ) + 21 g µν (ψ1 γ ρ ψ4 )(ψ3 γρ ψ2 ) − (ψ1 γ (µ ψ4 )(ψ3 γ ν) ψ2 )
− 21 µνρ (ψ1 γρ ψ4 )(ψ3 ψ2 ) − (ψ1 ψ4 )(ψ3 γρ ψ2 ) ,
which imply in particular
(θθ̄)2 = − 21 θ2 θ̄2 ,
(θθ̄)(θγ ν θ̄) = 0 ,
(θγ µ θ̄)(θγ ν θ̄) = 12 g µν θ2 θ̄2 .
11
1
2
(A.6)
(A.7)
(A.8)
Here and everywhere
we use symmetrization
and anti-symmetrization with weight one X[a Yb] =
Xa Yb − Xb Ya and X(a Yb) = 12 Xa Yb + Xb Ya .
172
The supercovariant derivatives and supersymmetry generators are
µ β
θ̄ ∂µ ,
Dα = ∂α + iγαβ
µ β
θ̄ ∂µ ,
Qα = ∂α − iγαβ
µ
∂µ ,
D̄α = −∂¯α − iθβ γαβ
µ
∂µ ,
Q̄α = −∂¯α + iθβ γαβ
(A.9)
(A.10)
where the only non-trivial anti-commutators are
µ
∂µ
{Dα , D̄β } = −2iγαβ
,
µ
∂µ .
{Qα , Q̄β } = 2iγαβ
(A.11)
We use the following conventions for integration
d2 θ ≡ − 41 dθα dθα
,
d2 θ̄ ≡ − 41 dθ̄α dθ̄α
,
d4 θ ≡ d2 θ d2 θ̄ ,
(A.12)
such that
Z
2
Z
2
d θθ = 1 ,
2
2
d θ̄ θ̄ = 1 ,
Z
d4 θ θ2 θ̄2 = 1 .
(A.13)
It is useful to note that up to a total derivative
Z
d4 θ . . . =
1
D2 D̄2 . . . |θ=θ̄=0 .
16
(A.14)
We use the N × N hermitian matrix generators T n (n = 0, . . . , N 2 − 1) and tn
(n = 1, . . . , N 2 − 1) for U(N ) and SU(N ) respectively. We have T n = (T 0 , tn ) with
√
T 0 = 1/ N .
The generators are normalized as tr T n T m = δ nm . Their completeness implies that
tr AT n tr BT n = tr AB and tr AT n BT n = tr A tr B for U(N ). Similarly for SU(N ) we
have tr Atn tr Btn = tr AB −
1
N
tr A tr B and tr Atn Btn = tr A tr B −
173
1
N
tr AB.
Euclidean Chern-Simons Yang Mills theory in radial quantization. The main
part of our computations in Chapter 5 are performed on R × S2 with the metric
ds2 = gmn dxm dxn = dτ 2 + dθ2 + sin2 θ dϕ2 . As Dirac matrices in the tangent frame
we use (γ a )α β = (−σ 2 , σ 1 , σ 3 ), which satisfy γ a γ b = δ ab + iabc γ c . As above all spinor
indices are raised and lowered from the left, ψ α = αβ ψβ and ψα = αβ ψ β , with
12 = −12 = 1. Again, (γ a )αβ = (−i1, −σ 3 , σ 1 ) are symmetric and we also have
(γ a )α β = (γ a )β α . For contracting spinor indices we use the NW-SE convention, e.g
ψγ m γ n χ ≡ ψ α (γ m )α β (γ n )β γ χγ etc. The spin connection is
∇m ψ = (∂m + ωm )ψ
,
ωm = 41 ωmab γ ab
,
γ ab = 12 [γ a , γ b ]
(A.15)
with the only non-zero component being ωϕ21 = −ωϕ12 = cos θ. For raising and lowering
SU(2)R indices (also called a, b, . . .) we use the same conventions as for spinor indices.
The standard index position for Pauli matrices is (σi )a b .
N = 2 superfields.
The component expansion of the N = 2 superfields are as
follows. The vector superfield V(x, θ, θ̄) in Wess-Zumino gauge contains the U(N )×Û(N )
gauge field Aµ , a complex two-component fermion χσ , a real scalar σ and an auxiliary
scalar D, such that
√
√ 2 †
2i θ θ̄χσ (x) − 2i θ̄2 θχσ (x) + θ2 θ̄2 D(x) , (A.16)
√
√
V̂ = 2i θθ̄ σ̂(x) − 2 θγ m θ̄ Âm (x) + 2i θ2 θ̄χ̂†σ (x) − 2i θ̄2 θχ̂σ (x) + θ2 θ̄2 D̂(x) , (A.17)
V = 2i θθ̄ σ(x) − 2 θγ m θ̄ Am (x) +
and similarly the chiral superfields in the adjoints of the two gauge group factors are
Φ = φ(xL ) +
Φ̂ = φ̂(xL ) +
√
√
2 θχφ (xL ) + θ2 Fφ (xL ) ,
2 θχ̂φ (xL ) + θ2 F̂φ (xL ) ,
√
2 θ̄χ†φ (xR ) − θ̄2 Fφ† (xR ) , (A.18)
ˆ = φ̂† (x ) − √2 θ̄χ̂† (x ) − θ̄2 F̂ † (x ) . (A.19)
Φ̄
R
R
φ R
φ
Φ̄ = φ† (xR ) −
174
The components of the chiral and anti-chiral superfields, Z(xL , θ) and Z̄(xR , θ̄), are a
complex boson Z, a complex two-component fermion ζ as well as a complex auxiliary
scalar F , and similarly for W and W̄. The expansions of these bifundamental matter
fields are given by
√
2 θζ(xL ) + θ2 F (xL ) ,
√
Z̄ = Z † (xR ) − 2 θ̄ζ † (xR ) − θ̄2 F † (xR ) ,
√
W = W (xL ) + 2 θω(xL ) + θ2 G(xL ) ,
√
W̄ = W † (xR ) − 2 θ̄ω † (xR ) − θ̄2 G† (xR ) ,
Z = Z(xL ) +
(A.20)
(A.21)
(A.22)
(A.23)
m
m
m
m
m
where xm
L = x − iθγ θ̄ and xR = x + iθγ θ̄.
A.3 N = 3 Chern-Simons Yang-Mills on
R1,2
Here we present the action of the N = 3 Chern-Simons Yang-Mills theory on R1,2 with
signature (−, +, +). For further explanations we refer the reader to Section 5.3. The
kinetic and mass terms are
Z
Skin =
h
d3 x tr − 2g12 F µν Fµν + κ µνλ Aµ ∂ν Aλ +
2i
A A A
3 µ ν λ
− 2g12 F̂ µν F̂µν − κ µνλ µ ∂ν Âλ +
2i
  Â
3 µ ν λ
(A.24)
/
− Dµ X † Dµ X + iξ † Dξ
− 2g12 Dµ φab Dµ φba − 12 κ2 g 2 φab φba −
1
D φ̂a Dµ φ̂ba
2g 2 µ b
− 12 κ2 g 2 φ̂ab φ̂ba
i
/ ab − κ2 iλab λba − 2gi 2 λ̂ab D
/ λ̂ab + κ2 iλ̂ab λ̂ba ,
− 2gi 2 λab Dλ
175
and the interactions, which involve cubic and quartic scalar vertices, as well as Yukawa
terms, are given by
Z
Sint =
h
d x tr − κg 2 Xa† φab X b + κg 2 X a φ̂ba Xb† − iξa† φab ξ b − iξ a φ̂ba ξb†
3
(A.25)
+ ac λcb X a ξb† − ac λcb ξ b Xa† − ac λ̂cb ξb† X a + ac λ̂cb Xa† ξ b
+ κ6 φab [φbc , φca ] + κ6 φ̂ab [φ̂bc , φ̂ca ] −
2
− g4 (Xσi X † )(Xσi X † ) −
1
iλab [φbc , λac ]
2g 2
1
iλ̂ab [φ̂bc , λ̂ac ]
2g 2
+
g2
(X † σi X)(X † σi X)
4
†
φbc X Aa φ̂cb
− 12 (XX † )φab φba − 12 (X † X)φ̂ab φ̂ba − XAa
i
+ 8g12 [φab , φcd ][φba , φdc ] + 8g12 [φ̂ab , φ̂cd ][φ̂ba , φ̂dc ] .
The supersymmetry variations with parameter εab = εi (σi )ab in the 3 of SU(2)R read
δAµ = − 2i εab γµ λab ,
(A.26)
/ bc εac + 2i [φbc , φcd ]εad
δλab = 12 µνλ Fµν γλ εab − iDφ
+κg 2 iφbc εac + g 2 iX a Xc† εcb −
ig 2
(XX † )εab
2
,
(A.27)
δφab = −εcb λca + 12 δba εcd λcd ,
(A.28)
δ µ = − 2i εab γµ λ̂ab ,
(A.29)
/ φ̂bc εac + 2i [φ̂bc , φ̂cd ]εad
δ λ̂ab = 12 µνλ F̂µν γλ εab + iD
+κg 2 iφ̂bc εac − g 2 iεbc Xc† X a +
ig 2
(X † X)εab
2
,
δ φ̂ab = −εcb λ̂ca + 12 δba εcd λ̂cd ,
(A.30)
(A.31)
δX Aa = −iεab ξ Ab ,
/ Ab εab + φab εbc X Ac + X Ac εbc φ̂ab ,
δξ Aa = DX
(A.32)
†
† b
δXAa
= −iξAb
εa ,
†
† b
†
† c b
/ Ab
δξAa
= DX
εa + φ̂ba εcb XAc
+ XAc
ε b φa .
(A.33)
176
A.4 Monopole Spinor Harmonics
We define monopole spinor harmonics as eigenspinors of the Dirac operator on the sphere
in a monopole background with magnetic charge q:
± ±
/ S Υ±
−iD
qjm = ∆jq Υqjm
(A.34)
with eigenvalues
1
∆±
jq = ± 2
for j =
|q|−1 |q|+1
, 2 ,...
2
p
(2j + 1)2 − q 2
(A.35)
and m = −j, −j + 1, . . . , j. The spectrum is drawn in Figure
5.1(a) on page 154. These spinors also satisfy
∓
γ τ Υ±
qjm = Υqjm ,
(A.36)
which couples modes with positive and negative eigenvalue. The lowest modes, j =
|q|−1
,
2
which only exists for q 6= 0 are zero-modes and the corresponding Υ± -spinors are not
independent. We introduce a special notation for them
1 −
+
sign(q)
Υ
.
Υ0qm ≡ √ Υ+
qjm
qjm
|q|−1
j= 2
2
(A.37)
γ τ Υ0qm = sign(q) Υ0qm .
(A.38)
Then (A.36) implies
Further properties are the orthogonality
Z
dΩ
Υ0†
qm
γ
τ
Υ0qm0
Z
= iδmm0
,
0
0
εε
τ
ε
dΩ Υε†
δjj 0 δmm0 ,
qjm γ Υqj 0 m0 = iδ
177
(A.39)
and completeness relations
X
0
Υ0qm (Ω)Υ0†
qm (Ω ) +
m
X
0
τ 2
0
Υεqjm (Ω)Υε†
qjm (Ω ) = iγ δ (Ω − Ω ) .
(A.40)
jmε
We also note the explicit expressions. The generalization of the spinor harmonics in
[176] to non-zero monopole background is
q
1+rqj
2
q
1−r
qj
+ i sign(q)
Ω−
qjm ,
2
q
q
1−rqj
1+rqj
Υ̃−
Ω+
Ω−
qjm = sign(q)
qjm + i
qjm ,
2
2
Υ̃+
qjm
with rqj =
q
1−
q2
(2j+1)2
j−m
Ω±
qjm = q
(−)
=
Ω+
qjm
(A.41)
(A.42)
and
1
i j+ 2
2
j+
1
2
s
q
q
(j − m)!
(j + m)!
“
q”
i m+ ϕ
2
e
Γ j + 32 − 2 Γ j + 32 + 2

j+m
√
m−+ /2
m+− /2 d
∓
∓i
(1
−
x)
(1
+
x)

dxj+m

×
 √
dj+m
± ±i (1 − x)m++ /2 (1 + x)m−− /2 j+m
dx
√
2π
×
(A.43)
(1 − x)
(1 − x)j−− (1 + x)j++
j+−
j−+
(1 + x)





where
jε1 ε2 ≡ j + ε1 21 + ε2 2q
,
mε1 ε2 ≡ m + ε1 21 + ε2 2q .
We rotate to our basis of Dirac matrices by defining Υ±
qjm =
178
√1 (1
2
− iσ 1 )Υ̃±
qjm .
(A.44)
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