The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere

Abstract

We present the gravity dual to a class of three-dimensional N=2
supersymmetric gauge theories on a biaxially squashed three-sphere, with a
non-trivial background gauge field. This is described by a 1/2 BPS Euclidean
solution of four-dimensional N=2 gauged supergravity, consisting of a
Taub-NUT-AdS metric with a non-trivial instanton for the graviphoton field. The
holographic free energy of this solution agrees precisely with the large N
limit of the free energy obtained from the localized partition function of a
class of Chern-Simons quiver gauge theories. We also discuss a different
supersymmetric solution, whose boundary is a biaxially squashed Lens space
S^3/Z_2 with a topologically non-trivial background gauge field. This metric is
of Eguchi-Hanson-AdS type, although it is not Einstein, and has a single unit
of gauge field flux through the S^2 cycle.

Full text

Page 1

arXiv:1111.6930v1 [hep-th] 29 Nov 2011
November 30, 2011
The nuts and bolts of supersymmetric gauge
theories on biaxially squashed three-spheres
Dario Martelli1and James Sparks2
1Department of Mathematics, King’s College, London,
The Strand, London WC2R 2LS, United Kingdom
2Mathematical Institute, University of Oxford,
24-29 St Giles’, Oxford OX1 3LB, United Kingdom
Abstract
We present the gravity dual to a class of three-dimensional N = 2 supersym-
metric gauge theories on a biaxially squashed three-sphere, with a non-trivial
background gauge field. This is described by a 1/2 BPS Euclidean solution of
four-dimensional N = 2 gauged supergravity, consisting of a Taub-NUT-AdS
metric with a non-trivial instanton for the graviphoton field. The holographic
free energy of this solution agrees precisely with the large N limit of the free
energy obtained from the localized partition function of a class of Chern-Simons
quiver gauge theories. We also discuss a different supersymmetric solution, whose
boundary is a biaxially squashed Lens space S3/Z2 with a topologically non-
trivial background gauge field. This metric is of Eguchi-Hanson-AdS type, al-
though it is not Einstein, and has a single unit of gauge field flux through the
S2cycle.

Page 2

1 Introduction
Supersymmetric gauge theories on compact curved backgrounds are interesting for var-
ious reasons. For example, supersymmetry may be combined with localization tech-
niques, allowing one to perform a variety of exact computations in strongly coupled field
theories. The authors of [1] presented a construction of N = 2 supersymmetric gauge
theories in three dimensions in the background of a U(1) × U(1)-invariant squashed
three-sphere and R-symmetry gauge field. The gravity dual of this construction was
recently given in [2]. It consists of a 1/4 BPS Euclidean solution of four-dimensional
N = 2 gauged supergravity, which in turn may be uplifted to a supersymmetric solu-
tion of eleven-dimensional supergravity. In particular, the bulk metric in [2] is simply
AdS4, and the graviphoton field is an instanton with (anti)-self-dual field strength. The
asymptotic metric and gauge field then reduce to the background considered in [1].
The purpose of this letter is to present the gravity dual to a different field theory
construction, obtained recently in [3]. In this reference the authors have constructed
three-dimensional N = 2 supersymmetric gauge theories in the background of the
SU(2)×U(1)-invariant squashed three-sphere (which we refer to as biaxially squashed)
and a non-trivial background U(1) gauge field, and have computed the corresponding
partition functions using localization. Differently from a superficially similar construc-
tion discussed briefly in [1], this partition function depends non-trivially on the squash-
ing parameter. As we will see, the gravity dual to this set-up will have some distinct
features with respect to the solution in [2]. In particular, the metric is not simply AdS4,
although it will again be an Einstein metric, and there is a self-dual graviphoton.
As an illustration of the generality of our constructions, in section 4 we discuss
yet another supersymmetric solution, consisting of a non-Einstein metric and a non-
instantonic graviphoton field. This has the topology of M4= T∗S2, with a single unit
of gauge field flux through the S2. We conjecture that this solution is dual to a class
of supersymmetric gauge theories on S3/Z2, with a background gauge field that is a
connection on the non-trivial torsion line bundle over S3/Z2.
1

Page 3

2 Supersymmetric gauge theories on the biaxially
squashed S3
In the construction of [3] the metric on the three-sphere is, up to an irrelevant overall
factor, given by
ds2
3
= σ2
1+ σ2
2+1
v2σ2
3, (2.1)
where σi are the standard SU(2) left-invariant one-forms on S3, defined as iσiτi =
−2g−1dg, where τidenote the Pauli matrices and g ∈ SU(2). The background U(1)
gauge field reads
√v2− 1
A(3)
=
2v2
σ3, (2.2)
and the spinors in the supersymmetry transformations obey the equation (setting the
radius r = 2 in [3])
∇(3)
αχ −
i
4vγαχ − A(3)
βγαβχ = 0 , (2.3)
where ∇(3)
(2.1), and γαgenerate Cliff(3,0). There are two linearly independent solutions to (2.3),
transforming as a doublet under SU(2), whose explicit form is given in [3]. This will
α , α = 1,2,3, is the spinor covariant derivative constructed from the metric
be important for identifying the gravity dual.
In [3] the authors constructed Chern-Simons, Yang-Mills, and matter Lagrangians for
the N = 2 vector multiplets V = (Aα,σ,λ,D) and chiral multiplets Φ = (φ,ψ,F), in
the background of the metric (2.1) and R-symmetry gauge field (2.2). These are invari-
ant under a set of supersymmetry transformations, provided the spinorial parameters
obey the equation (2.3). The supersymmetric completion of the Chern-Simons La-
grangian contains new terms, in addition to those appearing in flat space, proportional
to σ2and σA(3)∧ dA (cf. eq. (32) of [3]). The Yang-Mills and matter Lagrangians
are total supersymmetry variations (cf. eq. (31) of [3]) and therefore can be used to
compute the partition function using localization. In particular, the partition function
localizes on supersymmetric configurations obeying
Aα = D = 0 ,σ = u = constant , (2.4)
with the matter fields all being zero. Notice that although D = 0, the Chern-Simons
Lagrangian is non-zero because of the new term proportional to σ2, and therefore it
2

Page 4

contributes classically to the localized partition function, as in previous constructions.
The Yang-Mills and matter terms contribute one-loop determinants from the Gaussian
integration about the the classical solutions (2.4). The final partition function may be
expressed again in terms of double sine functions sb(z), and for a U(N) gauge theory
at Chern-Simons level k ∈ Z reads
Z=
?
?
Cartan
du exp
?iπk
v2Tru2
?
?
Rootsαsb
?
α(u)−i
v
?
?
?
Chirals,repRa
?
ρ∈Rasb
ρ(u)−i(1−∆a)
v
? , (2.5)
where b = (1 + i√v2− 1)/v. The exponential term is the classical contribution from
the Chern-Simons Lagrangian, evaluated on (2.4); the numerator is the one-loop vector
multiplet determinant and involves a product over the roots α of the gauge group G;
while the denominator is the one-loop matter determinant and involves a product over
chiral fields of R-charge ∆ain representations Ra, with ρ running over weights in the
weight-space decomposition of Ra. Following [4], one can easily extract the large N
behaviour of this partition function for a class of non-chiral N = 2 quiver Chern-
Simons-matter theories. The calculation was done in [3], and the result is that the
leading contribution to the free energy (defined as F = −logZ) is given by
Fv
=
1
v2Fv=1, (2.6)
and thus depends very simply on the squashing parameter v. In the next section we
will present the supergravity dual to this construction, in particular showing that the
holographic free energy precisely agrees with the field theory result (2.6).
3 The gravity dual
As anticipated in [2], we will show that the gravity dual to the set-up described in the
previous section is a supersymmetric solution of d = 4, N = 2 gauged supergravity. In
Lorentzian signature, the bosonic part of the action is given by
SLorentzian
=
1
16πG4
?
d4x?−detgµν
?R + 6g2− (FL)2?
. (3.1)
Here R denotes the Ricci scalar of the four-dimensional metric gµν, and the cosmo-
logical constant is given by Λ = −3g2. The graviphoton is an Abelian gauge field
ALwith field strength FL= dAL; here the superscript L emphasizes that this is a
3

Page 5

Lorentzian signature object. A solution to the equations of motion derived from (3.1)
is supersymmetric if there is a non-trivial spinor ǫ satisfying the Killing spinor equation
?∇µ+1
2gΓµ− igAL
µ+i
4FL
νρΓνρΓµ
?ǫ = 0 . (3.2)
Here Γµ, µ = 0,1,2,3, generate the Clifford algebra Cliff(1,3), so {Γµ,Γν} = 2gµν.
Since the background of [3] preserves half of the maximal supersymmetry in three
dimensions, we should seek a 1/2 BPS Euclidean solution of d = 4, N = 2 gauged
supergravity, whose metric has as conformal boundary the biaxially squashed metric
on S3(2.1), and whose background U(1) gauge field restricted to this asymptotic
boundary reduces to (2.2). This very strongly suggests that the appropriate solution is
a Euclideanized version of the 1/2 BPS Reissner-Nordstr¨ om-Taub-NUT-AdS solution
discussed in [5].
We will first present this Euclidean solution, and then discuss the Wick rotation that
leads to it. The metric reads
r2− s2
Ω(r)
ds2
4
=
dr2+ (r2− s2)(σ2
1+ σ2
2) +4s2Ω(r)
r2− s2σ2
3, (3.3)
where
Ω(r) = (s − r)2?1 + g2(r − s)(r + 3s)?
,(3.4)
and s is the NUT parameter.1The SU(2) left-invariant one-forms σimay be written
in terms of angular variables as
σ1+ iσ2 = e−iψ(dθ + isinθdϕ) ,σ3 = dψ + cosθdϕ . (3.5)
The graviphoton field is
A = sr − s
r + s
?
1 − 4g2s2σ3. (3.6)
In the orthonormal frame
e1 =
√r2− s2σ1,
?
e2 =
√r2− s2σ2,
?
Ω(r)
e3 = 2s
Ω(r)
r2− s2σ3,e4 =
r2− s2
dr ,(3.7)
the curvature may be written as
F = dA = −s?1 − 4g2s2
(r + s)2
(e12+ e34) .(3.8)
1This is denoted N in [5].
4