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JHEP05(2019)106

Published for SISSA by Springer

Received: March 14, 2019

Accepted: May 15, 2019

Published: May 20, 2019

A supersymmetric color superconductor from

holography

Ant´on F. Faedo,

David Mateos,

a,b

Christiana Pantelidou

and Javier Tarr´ıo

Departament de F´ısica Qu´antica i Astrof´ısica and Institut de Ci`encies del Cosmos (ICC),

Universitat de Barcelona,

Mart´ı i Franqu`es 1, ES-08028, Barcelona, Spain

Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA),

Passeig Llu´ıs Companys 23, ES-08010, Barcelona, Spain

Centre for Particle Theory and Department of Mathematical Sciences, Durham University,

Durham, DH1 3LE, U.K.

Physique Th´eorique et Math´ematique, Universit´e Libre de Bruxelles (ULB),

and International Solvay Institutes,

Campus de la Plaine CP 231, B-1050, Brussels, Belgium

E-mail: afaedo@ffn.ub.es, dmateos@icrea.cat,

christiana.pantelidou@durham.ac.uk, tarrio@gmail.com

Abstract: We use holography to study d = 4, N = 4, SU(N

) super Yang-Mills cou-

pled to N

 N

quark ﬂavors. We place the theory at ﬁnite isospin density n

by turning

on an isospin chemical potential µ

= M

, with M

the quark mass. We also turn on two

R-symmetry charge densities n

= n

. We show that the ground state is a supersymmetric,

superﬂuid, color superconductor, namely a ﬁnite-density state that preserves a fraction of

supersymmetry in which part of the global symmetries and part of the gauge symmetries

are spontaneously broken. The holographic description consists of N

D7-brane probes in

AdS

× S

. The symmetry breaking is due to the dissolution of some D3-branes inside

the D7-branes triggered by the electric ﬁeld associated to the isospin charge. The mass-

less spectrum contains Goldstone bosons and their fermionic superpartners. The massive

spectrum contains long-lived, mesonic quasi-particles if n

 µ

, and no quasi-particles

otherwise. We discuss the possibility that, despite the presence of mass scales and charge

densities in the theory, conformal and relativistic invariance arise as emergent symmetries

in the infrared.

Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence

ArXiv ePrint: 1807.09712

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP05(2019)106

JHEP05(2019)106

Contents

1 Introduction 1

2 Model 2

3 Higgs branch 3

4 Solution 4

5 Physical interpretation 5

6 Symmetry breaking 6

7 Spectrum 7

8 Discussion 7

1 Introduction

Quantum Chromodynamics (QCD) at non-zero baryon density n

is notoriously diﬃcult

to analyze. Because of asymptotic freedom, the preferred phase at asymptotically high

density can be shown to be a color-ﬂavor locked (CFL) conﬁguration [1, 2] (for a review

see [3]). The ground state in this regime is a color superconductor, namely a ﬁnite-density

state in which the color symmetry is Higgsed. Following a common abuse of language,

we will refer to this as the spontaneous breaking of the color symmetry. In addition, the

CFL ground state is also a superﬂuid, since the baryon number symmetry is spontaneously

broken too. In the regime of high but ﬁnite density, such as at the core of neutron stars,

no ﬁrst-principle calculations are possible. The only non-perturbative tool, namely lattice

QCD, is of limited applicability due to the so-called sign problem [4].

This situation provides one motivation to study the physics of QCD as some other

conserved charge is taken to be large, for example the isospin charge. In this case the sign

problem is absent and the theory can be simulated on the lattice (see e.g. [5]). Analytical

methods can also be used [6, 7]. The emergent picture is that the ground state is a superﬂuid

with superﬂuidity driven by a pion condensate at low density and by a quark-antiquark

condensate at high density. No color superconductivity was found in these analysis.

In this paper we give a step towards the holographic description of color supercon-

ducting phases. In this context the goal is not to do precision physics but to perform

ﬁrst-principle calculations that may lead to interesting insights [8]. In the case of QCD

at non-zero temperature, the insights obtained through this program range from static

properties to far-from-equilibrium dynamics of strongly coupled plasmas (see e.g. [9] and

references therein).

– 1 –

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