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JHEP10(2016)057

Published for SISSA by Springer

Received: March 2, 2016

Revised: August 8, 2016

Accepted: September 6, 2016

Published: October 12, 2016

Resolved gravity duals of N = 4 quiver ﬁeld theories

in 2 + 1 dimensions

William Cottrell and Akikazu Hashimoto

Department of Physics, University of Wisconsin-Madison,

1150 University Avenue, Madison, WI, 53706 U.S.A.

E-mail: wcottrell@wisc.edu, aki@physics.wisc.edu

Abstract: We generalize the construction by Aharony, Hashimoto, Hirano, and Ouyang

of N = 4 quiver gauge theory with gauge group U(N + M) × U(N), k fundamentals

charged under U(N) and bi-fundamentals, to the case with gauge group

i=1

U(N

) with

fundamentals charged under U(N

). This construction is facilitated by considering the

resolved ALE

× T N

background in M-theory including non-trivial ﬂuxes through the

resolved 4-cycles in the geometry. We also describe the M-theory lift of the IIA Page

charge quantization condition. Finally, we clarify the role of string corrections in various

regimes of parameter space.

Keywords: AdS-CFT Correspondence, Brane Dynamics in Gauge Theories, Gauge-

gravity correspondence

ArXiv ePrint: 1602.04765

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP10(2016)057

JHEP10(2016)057

Contents

1 Introduction 1

2 Review of AHHO and CHH 6

3 Brane construction 8

3.1 Summary of brane construction 14

4 Gravity dual description 16

5 IIA description 21

6 Conclusions 26

1 Introduction

N = 4 gauge ﬁeld theories in 2 + 1 dimensions is a rich dynamical system, exhibiting

features such as quantum corrected moduli spaces and enhancons in the holographic dual.

The number of supersymmetries is the same as theories in 3+1 dimensions with N = 2

supersymmetry, whose vacuum structure can be analyzed exactly using the techniques of

Seiberg and Witten [1, 2]. The relation between N = 2 theories in 3 + 1 dimensions and

N = 4 theories in 2 +1 dimensions is a rich subject on its own [3]. One complication stems

from the fact that N = 4 vector multiplets in 2 + 1 dimensions have twice as many scalar

components compared to N = 2 vector multipletes in 3+1 dimensions. There are, however,

other powerful tools at our disposal to explore the quantum dynamics of N = 4 theories

in 2+1 dimensions, such as mirror symmetry [4], localization, and holography. Recently,

the structure of Coulomb branch of N = 4 theories in 2+1 dimensions was mapped out

formally in [5].

The fact that full quantum dynamics is accessible on the ﬁeld theory side provides an

opportunity to explore subtle issues in gauge/gravity duality where string and quantum

corrections are expected on the gravity side of the correspondence. Knowing the existence,

and in some instance, the analytic form, of certain physical quantities on the ﬁeld theory

side provides a concrete target that one can aim to reproduce on gravity side.

The aim of this paper is to lay the foundation for such analysis. For a particular

class of N = 4 theories in 2 + 1 dimensions with gauge and matter content illustrated in

ﬁgure 1a, a gravity dual in type IIA was constructed explicitly and was analyzed in some

detail by Aharony, Hashimoto, Hirano, and Ouyang (AHHO) in [6] and more recently

by Cottrell, Hanson, and Hashimoto (CHH) in [7]. In ﬁgure 1b, we illustrate the brane

construction for these models in type IIB string theory. In the brane construction, one ﬁrst

constructs a defect ﬁeld theory in 3+1 dimensions on R

1,2

× S

. One recovers the theory

– 1 –

JHEP10(2016)057

N N+Mk

(a)

N+M

(b)

Figure 1. (a) Quiver diagram of U(N + M) × U(N) with k ﬂavors. (b) Brane construction for

the theory studied in AHHO and CHH. We have M fractional D3 branes suspended between two

NS5 branes in addition to N integer branes. The horizontal direction is periodically identiﬁed at

the dotted lines.

in 2+1 dimensions by taking the radius of S

to zero keeping the gauge coupling in 2+1

dimensions ﬁxed.

These theories are formulated as U(N + M) × U(N) gauge ﬁeld theories with k ﬂavors

charged under U(N) and two bifundamental matter ﬁelds in the ultra-violet. Such a system

will then undergo a renormalization group ﬂow. The structure in the IR will depend on the

gauge group and the matter content. In [8] Gaiotto and Witten classiﬁed the IR dynamics

of broad class of N = 4 theories in 2+1 dimensions into categories “good”, “ugly”, and

“bad”. The “good” theories ﬂows in the IR to interacting superconformal ﬁxed point. The

“bad” theories appear to exhibit a more more complicated ﬂow that depends sensitively on

details of the model [9, 10]. For the class of models illustrated in ﬁgure 1a, the condition

for the theory to be “good” is

0 ≤ − min(M, 0) ≤ N (1.1)

and

0 < −2M < k . (1.2)

The ﬁrst condition (1.1) is simply the requiment that the rank of each gauge group in

U(N + M) × U(N) is positive. The second is the condition for the gauge coupling not to

diverge in the RG ﬂow.

It would be interesting and desirable to identify the manifestation of conditions (1.1)

and (1.2) in the gravity dual. This very issue was studied recently in [7], which provided

the following general picture. A supergravity ansatz can be set up, and solved, for any

choice of N, M, and k satisfying

N −

≥ 0 . (1.3)

Violating this bound will give rise to a repulson singularities. For the set of repulson

free solutions satisfying (1.3), one can explore the possibility of various brane probes be-

coming tensionless. If that happens, we say that “the background suﬀers from enhancon

– 2 –

JHEP10(2016)057

N+M

Figure 3. Quiver diagram for a theory with ﬂavors charged under both gauge groups.

Now that we have a better understanding of the relationship between (1.1) and (1.3),

there are a number of interesting directions one can explore. For the “good” theories

satisfying (1.2), one can use localization techniques [13, 14] to compute the free energy

exactly. It would be interesting to recover (1.3) as the leading large N/k approximation as

well as subleading terms and compare the ﬁrst few corrections to curvature corrections on

the gravity side.

For theories not in the range (1.2), the task of resolving the enhancon seems quite

daunting. On the other hand, we know from previous studies on related systems in 3+1

dimensions [15] that these enhancons are closely associated with the locus of enhanced

gauge symmetry on the Coulomb branch as well as being the baryonic root, a point on

the Coulomb branch from which the Higgs branch eminates [16]. More detailed analysis

of string correction in gauge gravity correspondence of N = 2 systems in 3+1 dimensions

were carried out in [17, 18]. It would be very interesting to understand how the version

of this story in 2+1 dimensions is manifested on the gravity side of the gauge/gravity

correspondence.

One way in which one might imagine approaching the baryonic root is to consider

turning on FI parameters. In gauge theory FI parameters generically smooth out the

origin of Higgs branch and lift the Coulomb branch. One can then approach the baryonic

root by studying the limit in which FI parameter is taken to zero.

In order to fully explore the relationship between the ﬁeld theory and the gravity

formulation of these N = 4 systems, it would also be useful to have access to more general

construction than the class of models covered in ﬁgure 1a. One obvious generalization is

to allow matter to be charged under U(N + M) as well as U(N), i.e. to consider a quiver

of the type illustrated in ﬁgure 3.

It is not too diﬃcult, it turns out, to generalize the construction of the gravity solution

to the one corresponding to the brane construction illustrated in ﬁgure 4 with generic FI

and mass parameters turned on.

The goal of the remainder of this note is to describe such a construction. As we

will describe in detail below, considering generic FI and quark mass naturally leads to

the generalization of the ansatz to include the possibility of adding matter charged under

diﬀerent components of the product gauge group. One can then consider the limit of

vanishing FI and mass parameters and obtain a candidate gravity dual for the ﬁeld theory

with vanishing FI and mass parameters at least for the particular point on the Coulomb

branch that one reaches in this limiting procedure.

– 4 –

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