#### Romans supergravity from five-dimensional holograms

HJE
Romans supergravity from ve-dimensional holograms
Chi-Ming Chang 0 1 4
Martin Fluder 0 1 2
Ying-Hsuan Lin 0 1 2
Yifan Wang 0 1 3
One Shields Avenue 0 1
Davis 0 1
U.S.A. 0 1
Supersymmetric Gauge Theory
0 Jadwin Hall Washington Road, Princeton, NJ 08544-0708 , U.S.A
1 1200 E California Blvd , Pasadena, CA 91125 , U.S.A
2 Walter Burke Institute for Theoretical Physics, California Institute of Technology , USA
3 Joseph Henry Laboratories, Department of Physics, Princeton University
4 Center for Quantum Mathematics and Physics (QMAP), University of California Davis , USA
We study ve-dimensional superconformal eld theories and their holographic dual, matter-coupled Romans supergravity. On the one hand, some recently derived formulae allow us to extract the central charges from deformations of the supersymmetric vesphere partition function, whose large N expansion can be computed using matrix model techniques. On the other hand, the conformal and avor central charges can be extracted from the six-dimensional supergravity action, by carefully analyzing its embedding into type I' string theory. The results match on the two sides of the holographic duality. Our results also provide analytic evidence for the symmetry enhancement in ve-dimensional superconformal eld theories.
AdS-CFT Correspondence; Conformal Field Theory; Supergravity Models
1 Introduction 2
Review of key ingredients
Supersymmetric ve-sphere partition function
Central charges from deformations of ve-sphere partition function
2.2.1
2.2.2
Conformal central charge from metric deformations
Flavor central charge from mass deformations
Chern-Simons-like counter-terms
Seiberg theories
3
The large N limit of Seiberg theories
Free energy and conformal central charge
Flavor central charges 3.3
Finite N numerics
4
Central charges from supergravity
2.1
2.2
2.3
2.4
3.1
3.2
4.1
4.2
4.3
4.4
5.1
5.2
5.3
6.1
6.2
3.2.1
3.2.2
3.2.3
4.4.1
4.4.2
4.4.3
5.3.1
5.3.2
5.3.3
5.3.4
5.3.5
Fundamental hypermultiplet symmetry
Instantonic symmetry
Mesonic symmetry
Central charges and couplings
Massive IIA supergravity solutions
Conformal central charge from supergravity
Flavor central charges from supergravity
Mesonic symmetry
Fundamental hypermultiplet symmetry
Instantonic symmetry
5
Orbifold theories and their large N central charges
Orbifold theories
Free energy and conformal central charge
Flavor central charges
Fundamental hypermultiplet symmetries
Bifundamental hypermultiplet symmetries
Antisymmetric hypermultiplet symmetries
Mesonic and baryonic symmetries
Instantonic symmetries
{ i {
6
7
Conclusions Central charges from the supergravity dual of orbifold theories
Central charges by comparison to Seiberg theories
Baryonic avor central charges
Numerical evaluation of central charges
1
Introduction
41
42
HJEP05(218)39
Since the discovery of AdS/CFT, the holographic equivalence between quantum gravity in
anti-de Sitter space and conformal eld theory has elucidated key aspects on both sides of
the duality [1]. Typically, strong coupling properties on one side are reincarnated with a
simple weak coupling description on the other side, where physical observables can be
computed in perturbation theory. However, there are also holographic dualities that do not have
genuine weak coupling corners in their parameter spaces. Such is the case for interacting
superconformal eld theories in
ve dimensions, which do not admit marginal
deformations, and have no known weak coupling limit [2]. On the gravity side, the dilaton coupling
of type I' string theory diverges towards the boundary of the internal space [3]. Therefore,
quantitative predictions of these holographic dualities are more di cult to come about.
First steps towards closing this gap have been taken in [4{15]. On the gravity side, it
has been realized that in an appropriate low energy regime, the type I' string theory admits
an e ective supergravity description, which upon reduction to six dimensions becomes
Romans F (4) supergravity [16, 17] plus additional matter [18, 19].1 It is expected that this
much simpler six-dimensional e ective description captures key aspects of ve-dimensional
superconformal eld theories. On the eld theory side, following the seminal work of [20],
new advances in supersymmetric localization have provided extremely powerful tools for
extracting supersymmetric observables even in these strongly interacting theories. Careful
analyses of the localized formulae for these observables in the large N limit have provided
checks of these holographic correspondences beyond kinematics [7, 9, 11{15].
The present paper aims at a more thorough study of an enlarged set of observables
in ve-dimensional superconformal eld theories, and their manifestations in the bulk
supergravity. The grander goal is to gain a clearer understanding of the strong coupling
phenomena on both sides of the holographic correspondence, and to elucidate the general
relationships connecting the rich landscape [21{24] of ve-dimensional superconformal eld
theories, such as possible F - or C-theorems concerning renormalization group ows [25{27]
(see also [28] and further references therein). Many of these theories exhibit avor
symmetry enhancement, where extra conserved currents carrying instanton charges emerge
when reaching the ultraviolet
xed point, and enhance the
avor symmetry [21{23]. In
the bulk, this enhancement is a non-perturbative phenomenon that involves D0-branes
1Romans F (4) (gauged) supergravity contains only the elds dual to the ve-dimensional stress tensor
dilaton diverges [3]. Thus, a deep understanding of avor symmetry enhancement may
shed light on the non-perturbative dynamics in string theory.
The physical observables that we shall pursue are the supersymmetric (squashed)
vesphere partition function, the conformal central charge, and the various
avor central
charges. These charges appear in the superconformal block decomposition of the BPS
fourpoint functions, and serve as inputs to the bootstrap analyses of these superconformal eld
theories [29]. The avor central charges also serve as indicators of symmetry enhancement,
including the aforementioned one.
They also signify special features of the particular
holographically dual pair. For instance, we shall see that the avor central charges for the
mesonic U(1)M symmetry and the SU(2)R R-symmetry have a simple relation, which is a
priori obscure from the eld theory description, but clear in the dual supergravity [6].
In previous work [29], the present authors used conformal perturbation theory to
establish a precise relation between the central charges and the squashed
ve-sphere partition
function with mass deformations, and computed the latter via supersymmetric
localization. The resulting central charges for the rank-one Seiberg theories led to a bootstrap
analysis that revealed information about the non-BPS spectra in these theories. We
further found that the quantities we computed have surprisingly small instanton corrections,
and the comparison of avor central charges provided strong numerical evidence for avor
symmetry enhancement.
The present paper concerns the large N regime, in the hope to understand the
holographic duality for ve-dimensional superconformal eld theories, and to what extent
Romans F (4) supergravity (plus additional matter) captures the hologram. In addition to
Seiberg theories, we extend our analysis to a larger class of orbifold theories proposed
by [6]. Let us rst review prior progress and results in this direction. By localization, the
perturbative (squashed) ve-sphere partition function can be reduced to an N -dimensional
Coulomb branch integral [30{35], which has been computed analytically in the large N
limit using matrix model techniques [7, 11]. It has further been argued that in this limit,
the instanton contributions are exponentially suppressed, and thus, the perturbative
results are exact [7]. The matrix model of the squashed
ve-sphere partition function was
studied in [11], and the result was matched holographically with the properly renormalized
on-shell action of Romans F (4) supergravity in the bulk.
The progress we make in the present paper is as follows.
1. Borrowing the results on the matrix model for the large N squashed
ve-sphere
partition function from [11], we compute the conformal central charge of the Seiberg
theories in the large N limit.
2. We further study the matrix model for the mass-deformed ve-sphere partition
function. In particular, we nd that the round sphere free energy, the conformal central
charge, and the mesonic and baryonic avor central charges scale to leading order as
N 5=2. Similarly, the hypermultiplet and instantonic avor central charges are found
to scale as N 3=2. The coe cients of the latter two exactly agree in the large N limit,
providing analytic evidence for avor symmetry enhancement in the Seiberg theories.
{ 2 {
conformal symmetry.
at large N .
3. A subtlety in the above analysis is the potential presence of Chern-Simons-like
counter-terms. We explicitly determine a scheme under which the one-point
function of the instanton number current vanishes in the ultraviolet, as is required by
4. Except the instantonic avor central charge, all the other central charges in the
eld theory exactly agree with the couplings in Romans F (4) supergravity (coupled
to additional vectors) obtained by a careful reduction from type I' supergravity.2
This match provides further evidence of the suppression of instanton contributions
5. Finally, we generalize the above considerations to a larger class of orbifold theories [6].
We match not only the conformal, hypermultiplet, and mesonic avor central charges,
but also a set of baryonic avor central charges (up to an overall constant), on the
two sides of the holographic duality.
The rest of this paper is organized as follows. In section 2, we review our previous
work [29], and highlight the main ingredients that are relevant to the present analysis.
In section 3, we employ supersymmetric localization and matrix model techniques to the
Seiberg theories, to compute the free energy and central charges in the large N limit. The
relevant properties of the triple sine function and some
nite N numerics are provided
in appendices A and B. In section 4, we look at the dual type I' string theory, examine
its reduction to matter-coupled Romans F (4) supergravity with additional vectors, and
thereby compute the free energy and central charges. In section 5, a similar analysis is
performed for a more general class of orbifold theories. section 7 ends with concluding
remarks and future directions.
2
Review of key ingredients
This section presents a review of the key ingredients for computing the central charges
in interacting ve-dimensional superconformal eld theories. First, we present the general
de nition of the (squashed) supersymmetric ve-sphere partition function, the key
localization results, and its relation to the conformal central charge and avor central charges. We
then discuss the admissible counter-terms that can appear (on a ve-sphere background).
Finally, we introduce the theories of primary interest | the Seiberg theories. For the
derivation and in-depth discussions, we refer the reader to an earlier paper by the present
authors [29].
2.1
Supersymmetric ve-sphere partition function
A large class of ve-dimensional superconformal eld theories has an infrared gauge theory
phase. This infrared Lagrangian description allows localization computations, the strong
coupling limit of which recovers quantities at the ultraviolet xed point. This section
2We do not calculate the instantonic
avor central charge from supergravity in this paper due to the
subtleties explained in section 4.4.3, the resolution of which is left to future work.
{ 3 {
HJEP05(218)39
is devoted to a review of the localization results of the ve-sphere partition function for
general gauge theories.
The present discussion largely omits instanton contributions, which are generally
crucial for the consistency (e.g., symmetry enhancement) of the partition function.
Nonetheless, as we shall argue in section 3 (see also [7]), in the large N limit which could be
compared to the corresponding (weakly coupled) supergravity dual, the instanton
contributions are exponentially suppressed. Consequently, we may simply deal with the
perturbative part.
The perturbative part of the squashed ve-sphere partition function (i.e. without
instantons) has been computed in [31, 32, 35]. More precisely, the localization formula was
derived for squashed supersymmetric backgrounds that retain U(1)
SU(3)
SO(6) isom
etry, and then conjectured for the most generic squashing with U(1)
U(1) U(1) isometry. The metric of a generically squashed (unit) ve-sphere is given by
3
i=1
ds2 =
X(dyi2 + yi2d i2) + 2
X aj yj2d j A ;
j=1
e
coordinates, and yi are constrained such that P3
j=1 yj2 = 1. The round sphere (in terms of
polar coordinates) is given by setting aj = 0. Thus, the latter part of the metric can be
viewed as a perturbation of order O(aj2) to the round ve-sphere metric.
For a general ve-dimensional gauge theory with a simple rank-N gauge group Gg and
Nf hypermultiplets in the real representation Rf
Rf of Gg, f = 1; : : : ; Nf , the perturbative
squashed ve-sphere partition function is given by3
Zpert =
where the products are taken over all the roots
of Gg, the avors f = 1; : : : ; Nf that run
over the hypermultiplets on which the avor symmetry group Gf acts, and the weights f
of the particular representations Rf
Rf . The (Roman font) subscripts g and f denote
gauge and
avor, respectively. We introduced the classical ( at space) prepotential F( ),
which is given by
F( ) =
Tr 2 +
1
2gY2M
k
24 2 Tr 3
;
with gYM the classical gauge coupling, k the Chern-Simons coupling, and Tr ( ) is the
Killing form de ned as (h_ is the dual Coxeter number)
Finally, S3 is the triple sine function, and
1
2h_
Tr ( )
tr adj( ) :
S30(0 j !~) = lim
x!0
S3(x j !~)
x
:
3We use
to collectively denote the Cartan generators. In the case of Gg = USp(2N ), in the de ning
representation is given by the diagonal matrix f 1; : : : ; N ;
1; : : : ;
N g.
{ 4 {
We refer the reader to appendix A for a de nition of S3, and some of its properties relevant
for the present paper.
There is another type of deformations. For a theory of given avor symmetry group Gf ,
we can introduce mass parameters into the partition function by coupling the
hypermultiplets to background vector multiplets. For an Hermitian mass matrix M 2 gf
Lie(Gf ),
the mass term is explicitly given by
Z
d5xpg
ij qiM qj + 2itij qiMqj
2
2
M
;
(2.6)
HJEP05(218)39
where qi and qi are the scalars, and
the fermion, in the hypermultiplet.4 These masses
arise from turning on vacuum expectation values for the scalars in the background vector
multiplets, akin to Coulomb branch masses from turning on scalars in dynamical vector
multiplets. Hence, these generic mass terms appear in the partition function (2.2) in the
same way as the Coulomb branch parameters. We defer the presentation of the explicit
formula for Zpert for Seiberg theories with a particular choice of mass deformations to
section 2.4.
2.2
Central charges from deformations of ve-sphere partition function
In [29], the present authors derived formulae for ve-dimensional superconformal eld
theories that relate the conformal central charge and
avor central charges to deformations
of the ve-sphere partition function. The proof proceeds by coupling the ve-sphere
background to the appropriate background supergravity multiplets. Here, we explain the
rationale and present the resulting formulae.
2.2.1
Conformal central charge from metric deformations
In order to extract the conformal central charge CT from a partition function, we study the
superconformal eld theory on a
ve-sphere background perturbed by coupling the stress
tensor multiplet to a background supergravity multiplet on a generically squashed
vesphere with metric given in (2.1). In order to preserve the full superconformal symmetry,
we couple the theory to the ve-dimensional N = 1 standard Weyl multiplet
g ; D; V ij ; v ;
i
;
i ;
(2.7)
which consists of the dilaton D, the metric g , an SU(2)R symmetry gauge eld V ij ,
a two-form
eld v , together with their fermionic partners | the gravitino
dilatino i. We deform the ve-sphere background by writing g
= gS5 + h . Upon doing
i and the
so, the (bosonic) linearized action is given by
4Although the physical mass matrix M in (2.6) is Hermitian, when performing localization, one has to
analytically continue M to an anti-Hermitian matrix for convergence [30, 31].
{ 5 {
where T , Jij , B
for the given metric.
and
are the (bosonic) components of the stress tensor multiplet.5
The linearized background elds of the standard Weyl multiplet for the generic squashed
ve-sphere can be explicitly determined by solving the relevant supersymmetry conditions
From the linearized perturbation (2.8), it follows that the ve-sphere free energy at
second order in the squashing parameters aj can be written in terms of the two-point
functions of the currents B
, J ij and
on the round sphere. These two-point functions
are related to the conformal central charge upon stereographic projection from
at space.
By evaluating them explicitly, and expanding the background values for the bosonic elds
in the standard Weyl multiplet to leading order in the squashing parameters, we nd that
the free energy is in fact related to the conformal central charge CT as
F a2 =
i
2CT
Flavor central charge from mass deformations
The same overall logic applies to extracting the
avor symmetry group Gf , by considering mass deformations of the ve-sphere free energy.
Working in conformal perturbation theory, we couple the avor current multiplet in the
ve-dimensional superconformal eld theory to a background vector multiplet
avor central charge CGf for a given
J
W a; M a;
a i; Y a ij ;
which contains the vector eld W a, the scalar M a, the gaugino
a i and the triplet of
auxiliary
elds Y a ij , all in the adjoint representation of Gf . The linearized perturbation
of the bosonic action is then given by5
where Liaj , J a , and N a are the bosonic components of the avor current multiplet. On
the
ve-sphere, the vector multiplet has to satisfy supersymmetry conditions, which can
be solved by W
=
i = 0 and Y ij =
M tij , for tij = 2i 3 and some M 2 gf
Lie(Gf ).
Thus, the leading order piece in the mass deformation of the ve-sphere free energy can be
expressed in terms of the two-point functions of Liaj and N a on the round sphere. Those
two-point functions are related to the corresponding avor central charge CJGf in at space
by stereographic projection, and we end up with
F M2 =
3 2CGf
J
256
abM aM b :
It remains to translate the mass parameters M a appearing in conformal perturbation
theory to the mass matrix appearing in the (infrared) gauge theory Lagrangian. We refer
5Notice that in the \rigid" limit, all fermionic elds of the background supergravity multiplets are set
to zero.
{ 6 {
the reader to [29] for a in-depth treatment of this. Here, we shall simply state the result,
F
M
where RKf is the representation of the hypermultiplets under Kf | the manifest avor
symmetry group of the infrared Lagrangian, IRKf is the Dynkin index associated with the
represenation RKf , and M is the mass matrix in the action deformation (2.6).
The global symmetry can be enhanced at the ultraviolet xed point, which is in fact a
common phenomenon in
ve dimensions. In that case, if G is simple, one can obtain CGf
from CJKf , by use of the embedding index Ik,!g, i.e.,
J
a physically relevant quantity, one must remove such divergences by introducing local
di eomorphism-invariant counter-terms, which would generically break conformal
invariance. For the case of ve-dimensional N = 1 superconformal eld theories, such
counterterms would break the superconformal group to su(4j1), but renders the nite part of the
ve-sphere partition function unambiguous and physical.
More explicitly, the counter-terms are supersymmetric completions of (mixed)
ChernSimons terms in Poincare supergravity, and have been classi ed in [29] by the present
authors. The three possible counter-terms involving a single scalar M in the background
vector multiplet (2.10) are given by
(2.13)
(2.14)
(2.15)
(2.16)
(1)
LT T J 3 i 1T T J
(2)
LT T J 3 i 2T T J
(3)
LT T J 3 i 3T T J
hpgM R2 +
i
;
hpgM
hpgM C
R
R
C
the theory restricts the constants jT T J to be real, and upon localization each of those terms
will contribute a real piece at linear order in the (real) parameters mf , where mf are the
demotion of Mf to constants, interpreted as mass parameters for the avor symmetry.6 If
more than one background vector multiplet are present, then there exists another
counterterm given by
LJJJ 3 i aJbJcJ hpgM aM bM cR +
i
:
As before, the constants aJbJcJ are real, and we conclude that such a counter-term
contributes a real piece to the localized
ve-sphere partition function, at cubic order in the
mass parameters mf for the avor symmetry. As we shall see in section 3.2.2, one has to
subtract such terms o the large N free energy to end up with a physically consistent result.
6Recall that localization requires M to be imaginary and here M is proportional to imf by a real number.
{ 7 {
In the next few sections, we focus on a particular class of ve-dimensional superconformal
eld theories, which were rst introduced by Seiberg [21].7 They can be constructed in
type IIA string theory (or rather type I' string theory) by a D4-D8/O8-brane setup. We
begin with two orientifold O8-planes located at x9 = 0 and x
coupling, the D0-branes become massless, and together with the massless string degrees of
freedom localized on the D4-branes yield the aforementioned Seiberg theories. The avor
symmetries for the Seiberg theories of arbitrary rank are given by
ENf +1
SU(2)M ;
(2.17)
where the rst factor is enhanced from the SO(2Nf ) symmetry of the Nf D8-branes and
the U(1)I instanton particle (D0-brane) symmetry [3, 21]. The latter factor is the mesonic
symmetry arising from rotations in the fx5; x6; x7; x8g directions.
A renormalization group ow connects a Seiberg theory of avor symmetry ENf +1 in
the ultraviolet to an infrared USp(2N ) gauge theory. The D4-D4 strings give rise to a vector
multiplet and a hypermultiplet in the antisymmetric representation of USp(2N ). The
D4D8 strings give rise to Nf
7 fundamental hypermultiplets. Starting from the ultraviolet,
the infrared gauge theory can be reached by turning on a deformation that becomes the
ve-dimensional Yang-Mills term towards the end of the ow. Although an in nite number
of irrelevant terms arise along this ow, they are believe to be Q-exact, and thus do not
contribute to the localized path integral [7, 32, 36]. Therefore, the supersymmetric partition
function of the ultraviolet superconformal Seiberg theory is fully captured by the infrared
USp(2N ) gauge theory.
The supersymmetric ve-sphere partition function has been computed by localization
using the infrared gauge theory description. The result is a sum over the contributions
from in nitely many saddle points. We presently review the result of the perturbative
saddle point, where all the hyper- and vector multiplets have trivial vacuum expectation
value [31, 32, 35]. For simplicity, we choose the mass matrix (2.6) for the hypermultiplet
SO(2Nf ) avor symmetry to be
and for the mesonic SU(2)M symmetry acting on antisymmetric hypermultiplet to be
M(SO(2Nf )) = mf (i 2
ve-sphere partition function for the infrared
USp(2N ) gauge theories with Nf fundamental and one antisymmetric hypermultiplets is
7In section 5, we consider a more general class of theories labeled by n 2 Z 1, of which n = 1 corresponds
to the Seiberg theories.
{ 8 {
given by8
where jWj = 2N N ! is the order of the Weyl group of USp(2N ). Note that since the
USp(2N ) gauge group has no cubic Casimir, the (gauge) Chern-Simons terms are absent,
and the prepotential only consists of the classical Yang-Mills piece.
The dependence of the supersymmetric ve-sphere partition function on the squashing
and mass parameters encodes the central charges of the superconformal eld theory. The
embedding indices, which by (2.14) relate the avor central charges of the infrared global
symmetry group to the ultraviolet enhanced global symmetry group, in the case of Seiberg
theories, are
Iso(2Nf ),!eNf +1 = 1;
Iu(1)I,!eNf +1 =
8
4
Nf
:
We refer to [29] for the explicit computations.
3
The large N limit of Seiberg theories
Motivated by holography, we proceed to the computation of the deformed ve-sphere
partition function for the Seiberg theories in the large N limit. As we shall see, the
supersymmetric partition function, which is hard to evaluate at nite N , simpli es in this limit.
To describe the partition function at the superconformal xed point, we are required
to send gYM ! 1 under which all saddles in the instanton expansion contribute equally
(without manifest suppression by a small parameter), and a perturbative analysis seems
to break down.
Nonetheless, it has been argued that the instanton contributions are
exponentially suppressed in the large N limit [7]. To understand this, we write the full
partition function schematically as
(2.20)
(2.21)
(3.1)
(3.2)
(3.3)
S3(x j !~)S3( x j !~).
1
X
n=0
Znqn ;
q = exp
{ 9 {
where Z
numbers, and
n are functions of the fugacities that are only explicitly known for small instanton
with gYM being the infrared gauge coupling. When moving out on the Coulomb branch,
the e ective Yang-Mills coupling receives a one-loop correction
where by
we collectively denote the Coulomb branch parameters of the theory. Since
the instanton particles are BPS, their masses are determined by the central charge (of the
supersymmetry algebra), which is a linear combination of the bare gauge coupling and
the Coulomb branch parameters. Accordingly, the e ective parameter for the instanton
expansion is governed by not the bare gauge coupling but an e ective coupling,
As we shall see below, in the large N limit, the Coulomb branch parameters
asymptotically as
Consequently, the contributions with nontrivial instanton numbers are exponentially
suppressed in the large N limit.
The upshot is that in the large N limit, the perturbative part of (squashed)
vesphere partition function is exact. We can therefore explicitly determine the conformal
central charge and the avor central charges for the ENf +1
SU(2)M
avor symmetry, via
the perturbative localization formula (2.20), as well as the prescription of section 2.2. The
enhancement of the avor symmetry from the manifest infrared SO(2Nf )
U(1)I to the
ultraviolet ENf +1 implies that the values of the exceptional avor central charge computed
from the SO(2Nf ) and U(1)I embeddings should agree. Whereas numerical evidence for
the agreement was found in [29] for the rank-one Seiberg theories, here we shall nd an
exact agreement in the large N limit. In section 4, the results of the present section
will be compared with the central charges extracted directly from the dual supergravity
description.
3.1
Free energy and conformal central charge
Let us rst recall the large N computation of the undeformed free energy in [7, 11, 12].
We start by rewriting the perturbative partition function (2.20) into the form
Z
where we have chosen to represent the exponent by the symbol F ( ) because when later
evaluated at the large N saddle point, it becomes the large N free energy F . In the
following, F ( ) will be referred to as the \localized action". For the Seiberg theories,9
N
i=1
N
+ X
i6=j
1
2
where GV and GH are the logarithms of the triple sine functions, i.e.,
GV (z j !~) =
log S3 (iz j !~) ;
GH (z j !~) = log S3 iz +
!tot
2 j !~ ;
with !tot = !1 + !2 + !3. By Weyl re ections of the USp(2N ) gauge group, we can restrict
the integration region to i
0, and compensate by a symmetry factor of 2N .
We now proceed to performing the large N saddle point approximation of the
integral (3.6). These saddle points can be studied numerically at large but nite N , and the
results suggest that the Coulomb branch parameters scale (to leading order) as i
in the asymptotic limit of large N and
nite x. This particular scaling of i with N can
N 1=2xi
be argued analytically, as follows. First, assume that
i = N xi ;
with
> 0 :
Then, note that the functions GV ( z j !~) and GH (z j !~) have the following asymptotic
expansions at jzj ! 1,
GV ( z j !~)
GH (z j !~)
3 !1!2!3 j j
which follow from the asymptotic formulae (A.9) and (A.10) for triple sine functions. As
reviewed in appendix A.1, the expansions (3.11) have no subleading power law correction,
a fact that will be important later. The leading order term in the rst line of (3.8) scales
as N 1+3 , whereas the leading order term in the second line of (3.8) scales as N 2+ . In
order to get a nontrivial saddle point, both terms must contribute to the same order, and
we thereby determine
In the large N limit, we introduce a density (x) for the rescaled Coulomb branch
parameters xi (we use the Weyl re ections of the USp(2N ) gauge group to restrict to xi
0),
(x)
Z
N
1 X
N i=1
dx (x) = 1 :
(x
xi) ;
Z x?
0
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
normalized such that
In the continuum limit, the localized action (3.8) becomes
!t2ot (x+y +jx yj)
(8
We are thus left with the simple variational problem of nding the function (x) that
extremizes F ( ). We add a Lagrange multiplier term for the constraint (3.13),
and solve F =
= 0 for (x) and x . The resulting saddle point con guration is
Evaluating (3.14) at this saddle gives the well-known result for the free energy [11, 12]
(x) =
2x
x2
?
;
x2? =
!t2ot
2(8
Nf )
:
F =
p
The leading corrections of order N 1=2 come from both the continuum approximation and
further terms in the series expansion in mf . Using
3.2
3.2.1
Flavor central charges
Fundamental hypermultiplet symmetry
To compute the avor central charge for SO(2Nf ), let us now deform the round-sphere
partition function (with squashing parameters !i = 1), by giving masses to the Nf
fundamental hypermultiplets. For simplicity, we choose the following mass matrix (see (2.6) and
the discussion in section 2.4),
M = mf (i 2
Nf identity matrix, and 2 is the second Pauli matrix. By adding
this particular mass term to the perturbative partition function, the localized action (3.8)
is modi ed by the replacement
We determine the round-sphere free energy to be
HJEP05(218)39
and extract the conformal central charge from the !i-dependence of the free energy by the
relation (2.9) to be
GH (
i j !~) ! GH (
i + mf j !~) :
Since the addition of this mass term only changes the saddle point to subleading order in
1=N , we may simply evaluate the mass term on the saddle point con guration (3.16) to
nd the modi cation to the free energy,
F =
=
mf2Nf N 3=2 Z x?
0
p
8
Nf
p
2 mf2Nf N 3=2 + O(N 1=2) :
dx (x)x + O(N 1=2)
tr vec(SO(2Nf ))(M2) = 2Nf mf2 ;
in the relation (2.13), we determine the avor central charge for SO(2Nf ) to be
J
CSO(2Nf ) =
256p2
3 p
8
Nf
In this case, the embedding index into the enhanced avor symmetry group at the
ultraviolet xed point is simply one, and therefore,
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
coupling by
piece reads
Next, we compute the avor central charge for the instantonic U(1)I symmetry. The avor
symmetry SO(2Nf ) U(1)I in the infrared is expected to enhance to ENf +1 at the ultraviolet
superconformal xed point. This implies that the value of CENf +1 predicted by the CJ of
J
the U(1)I subgroup and that of the SO(2Nf ) subgroup should agree (with the embedding
indices properly accounted for). For general N , this statement is di cult to prove, as it
requires re-summing all instanton contributions.10 However, we expect to obtain a precise
agreement in the large N limit, since such non-perturbative contributions are exponentially
suppressed as discussed before.
To compute the CJ for U(1)I, we simply keep track of the dependence of the
perturbative partition function on the instanton particle mass mI, which is related to the Yang-Mills
HJEP05(218)39
mI =
and only appears in the classical piece. The large N localized action including the classical
up to corrections that are non-perturbative in 1=N , since the asymptotic expansions (3.11)
of log S3 have no subleading power law correction.
If we naively proceed as in the previous section, by evaluating the additional instanton
mass term on the saddle point (3.16) without mass deformations, then we immediately run
into a problem, which is that the dependence on mI truncates at linear order. Thus, to nd
the CJ for U(1)I, we must include corrections to the saddle that are subleading in 1=N .
We assume that (x) has support on the interval [x1; x2], with xi
0. The saddle point
equation with inclusion of the instanton mass mI reads
0 =
+
9
4
(8
3
10In [29], the present authors obtained some numerical evidence for the agreement in the case of N = 1.
where as before
is the Lagrange multiplier for the normalization of (x). Taking the
derivative twice with respect to x shows that (x) is a linear function. The saddle point
equation can be straightforwardly solved to give
where11
2x
a
(x) =
+ b ;
xed point.
relation (2.12),
We can now compute the leading order large N free energy F at the saddle point,
F =
p
9 2 N 5=2
p
The leading correction of order N 1=2 comes from the continuum approximation of the
discrete Coulomb branch parameters i by
(x).
By regarding the discrete sum over
i = 1; 2; : : : ; N as a (midpoint) Riemann sum for the integral over x, we estimate the
error of the continuum approximation to be of order N 2 relative to the leading N 5=2. In
comparison, the correction coming from the Gaussian integral around the saddle point is
of order log N .
There appears to be an immediate problem. The term of order mI would imply a
non-vanishing sphere one-point function of the
avor currents, which violates conformal
invariance. However, as discovered in [29] and reviewed in section 2.3, there are ambiguities
in the free energy due to counter-terms. In particular, upon localization on the ve-sphere,
the mixed Chern-Simons counter-terms contribute real pieces at order mI and mI3 [29].
Thus, we may pick a regularization scheme to remove the terms with linear and cubic
dependence on mI in (3.31), to properly preserve conformal symmetry at the ultraviolet
We can now compute the avor central charge for the instantonic U(1)I current by the
J
The avor central charge of the enhanced ENf +1 avor symmetry is related to that of the
instantonic U(1)I by the embedding index (2.21), and we conclude that
J
CENf +1 =
8
4
Nf CU(1)I =
J
256p2
3 p
8
Nf
11To leading order at large N , we recover the previous saddle point (3.16).
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
This precisely agrees with the avor central charge for ENf +1 computed using the SO(2Nf )
mass deformation given in (3.24) and (3.25), and con rms the avor symmetry
enhancement in Seiberg theories.
Finally, we compute the avor central charge for the mesonic SU(2)M symmetry. We deform
the ve-sphere partition function by a mass for the antisymmetric hypermultiplet, which
transforms in the fundamental of SU(2)M. As before, we couple it to a background vector
multiplet with the following choice of mass matrix,
HJEP05(218)39
Then the localized action (3.8) is modi ed by the replacement
M = imas 3 2 su(2)M :
GH ( i
1
j j !~) ! 2
GH ( i
j
mas j !~) :
Again, we shall use the asymptotics for the triple sine functions as given in (3.11). The
leading order modi cation to the large N localized action (3.14), due to the addition of
the mass for the antisymmetric hypermultiplet, then reads
2
N 5=2m2as Z x?
0
Z x?
0
denotes the Heaviside theta function, and the leading corrections of order N 3=2
come from further terms in the series expansion in mas. This modi es the saddle point
con guration to
(3.34)
(3.35)
(3.36)
(3.37)
(3.38)
(3.39)
(3.40)
This large N formula for the mesonic avor central charge is in fact equal to that for the
R-symmetry avor central charge. The latter is related to the conformal central charge by
Consequently, the large N free energy is given by
As before, we apply the relation (2.13) with
(x) =
p
2
2x
x2
?
;
x2? =
2 (8
9 + 4m2as :
Nf )
F =
9 + 4m2as 3=2
15p8
Nf
J
CSU(2)M =
256p2
5 p
8
Nf
to compute the avor central charge of the mesonic SU(2)M
avor symmetry,
superconformal Ward identities, since the R-symmetry currents reside in the same
superconformal multiplet as the stress tensor. The agreement of these two avor central charges
at large N is expected from the dual supergravity perspective, where SU(2)M and SU(2)R
combine to form the SO(4) isometry of the internal four-hemisphere (see section 4), and are
exchanged under a frame rotation as described in section 4.4.1. However, this agreement
is obscure from eld theoretic considerations, and fails at nite N .
We have argued that the perturbative partition function is exact in the large N limit.
Away from large N , instanton corrections are no longer suppressed, and computing the
exact partition function or central charges becomes a di cult task. However, in [29], the
present authors observed that for Seiberg theories of rank one | which is the opposite
of large N
| the instanton contributions to the central charges as extracted from the
supersymmetric
ve-sphere partition function are also small. Combining these two facts,
it is natural to expect that instanton contributions are in fact small for all N .
Under this assumption, we can approximate the exact partition function by the
perturbative formula, and numerically compute the round-sphere free energy and the central
charges at nite values of N . For 1
N
3, the integral (2.20) can be computed
straightforwardly by direct numerical evaluation; for 1
N
40, we can estimate the integral by
a saddle point approximation. In the absence of a large parameter, the latter is a priori
illegal. However, after explicitly computing the round sphere free energy and the various
central charges, we nd that the saddle point approximation agrees with direct numerical
integration even for N = 3 to within 1% (these numerical results are presented in detail in
appendix B). Thus, we believe that this approximation can in fact be trusted for N
These results suggest that in situations where only approximate values of these
quantities are needed, such as for numerical bootstrap, the
nite N saddle point approximation
serves as an e cient method to perform the perturbative localization integral (2.20). In
gure 1, we juxtapose the results of direct numerical integration, nite N saddle point
approximation, and the large N formula, in the case of the Seiberg E8 theory.
4
Central charges from supergravity
We now move towards a study of the central charges in the holographic duals of the
Seiberg theories. As reviewed in section 2.4, the Seiberg theories can be constructed by
a system of D4-branes probing an orientifold singularity in type I' string theory [21], the
con guration of which is given in table 1. In summary, the con guration has Nf < 8
D8-branes coinciding with an O8, on top of which lie N D4-branes. The decoupling limit
suggests a holographic correspondence between Seiberg theories and type I' string theory
on a background of the form
M6
w HS4, a warped product of a four-hemisphere with
a asymptotically locally AdS6 space M6 [4{6]. In an appropriate low energy regime, the
gravity side is well approximated by type I' supergravity [3]; in particular, the region
between D8-branes is described by Romans massive type IIA supergravity [3, 37]. This
latter regime is what we mainly consider in the following.
106
105
104
1000
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50
CJSU (2)M
106
105
104
1000
100
■
■
■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
5
■
■
by numerically computing the perturbative partition function (squares), and by the large N formula
(dashed line). The physical quantities considered here are the round sphere free energy
conformal central charge CT , the mesonic avor central charge CSU(2)M , and the exceptional avor
J
central charge CJE8 . The numerical computations are done by direct integration up to N = 3, and
F0, the
by saddle point approximation up to N = 40.
0
1
2
3
4
5
6
7
8
9
This particular holographic correspondence has thus far been tested in various ways.
Initial checks include comparisons of the symmetries on both sides [4, 5], and more recent
checks include comparisons of the entanglement entropy [7] and the free energy/Wilson
lines in the case of M6 = AdS6 [9], as well as for more general asymptotically locally AdS6
spacetimes [11, 12, 15].
4.1
Central charges and couplings
The conformal central charge CT and the avor central charges CJ in a d-dimensional
superconformal eld theory with a weakly coupled AdSd+1 dual are related to speci c
coupling constants in the supergravity action. The leading order derivative expansion of
the weakly coupled bulk e ective action is schematically given by
Z
Sd+1 =
dd+1xp
gd+1
"
1
2 2d+1
(Rd+1 2 d+1)
1
2e2 Tr F (d+1)F (d+1)
#
+
; (4.1)
where Rd+1 is the (d + 1)-dimensional Ricci scalar, F (d+1) is the (d + 1)-dimensional eld
strength for the gauge eld which is sourced by the avor symmetry current at the
conformal boundary, Tr ( ) is the Killing form de ned in (2.4), and nally,
is the (d + 1)-dimensional cosmological constant, with ` the radius of AdSd+1. The
gravitational coupling
d+1 and the gauge coupling e are related to the central charge CT and
the avor central charge CJ by [38, 39]
HJEP05(218)39
These relations allow us to extract CT and CJ from the supergravity duals of the Seiberg
theories. The key is then to determine the precise values of the couplings d+1 and e that
properly embed the supergravity into the aforementioned type I' string theory background.
Massive IIA supergravity solutions
Let us rst review the AdS6
HS4 solutions in massive IIA supergravity [5]. In the string
frame, the bosonic part of the ten-dimensional massive IIA action reads
S1II0A =
2 210
d10x p
where g10 is the ten-dimensional metric, R10 the corresponding Ricci scalar, and
10 = 8 7=2`4
s
is the ten-dimensional gravitational coupling expressed in terms of the string length `s.
Furthermore, we denote by
the ten-dimensional dilaton, F6 the six-form
ux, and mIIA
the Romans mass. The string frame metric for the AdS6
HS4 background is explicitly
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
given by
hemisphere,
ds120 =
1
sin1=3
hL2ds2A\dS6
+ R
2 d 2 + cos2
ds2S3 i
;
where L is the AdS6 radius, R is the HS4 radius, and
the standard metric of the unit-radius AdS6, and by ds2S3 the unit S3-slices of the
four2 (0; =2]. We denote by ds2A\dS6
1 h
4
ds2S3 =
d 12 + sin2 1 d 22 + (d 3
cos 1 d 2)2i ;
with 1 2 [0; ], 2 2 [0; 2 ), and 3 2 [0; 4 ). Note that the isometry group of S3 is
SO(4) = SU(2)M
SU(2)R, where the former SU(2)M factor corresponds in the dual
vedimensional superconformal eld theory to the mesonic global symmetry, and the latter to
the R-symmetry.
SO 2Nf(1)
U N (2)
f
U N (k+1)
f
X12
USp(2N )
SU(2N )
X23
Xk(k+1)
SU(2N )
A
eld theories with odd n = 2k + 1.
Their charges are related by
The meson operator is A basis for the baryon operators is 1 2
Ba = det Xa(a+1) ;
a = 1; : : : ; k
1 ;
plus an additional one given by the Pfa an of the antisymmetric hypermultiplet A,
Bk = Pf(A) =
1 2N A 1 2
A 2N 1 2N ;
where
j are the USp(2N )-indices, raised and lowered by the corresponding invariant
tensor. Note that the alternative combination Pf(A0) is not independent of the chosen
basis fM; B1; : : : ; Bkg of gauge-invariant operators composed purely of hypermultiplets.
Odd orbifold ( gure 4)
This nal case is a hybrid between the two even cases: there is a USp(2N )
SU(2N )k
infrared gauge group, together with fundamental, bifundamental, and antisymmetric
hypermultiplets. The resulting overall infrared avor symmetry group is
G(n=2k+1) = U(1)I(1)
f
U(1)(b1)
SO(2Nf(1))
U(1)I(k+1)
U(1)A
U(Nf(k+1)) :
Yk hU(1)I(a)
a=2
U(1)(ba 1)
U(Nf(a))i
(5.10)
(5.6)
(5.7)
(5.8)
(5.9)
As before, we group the U(1) symmetries into mesonic U(1)M and baryonic U(1)(Ba) factors,
with a = 1;
; k. Their charges are related by
A basis for the baryon operators is
!
#
A :
Again, the Pfa an Pf(A) is not independent of the basis fM; B1; : : : ; Bkg of gauge-invariant
operators composed purely of hypermultiplets.
In all three cases, the total number of fundamental hypermultiplets equals the number
f
of D8-branes in the type I' string theory setup, i.e., Nf = P
be a UV
xed point, this number must fall into the range 0
N (a) must satisfy certain conditions.
Free energy and conformal central charge
a Nf(a). In order for there to
Nf < 8, and the distribution
We begin by writing down the localized action of the Coulomb branch integral
expression (3.6) for the perturbative
ve-sphere partition function. For ease of notation, it is
convenient to introduce a set of auxiliary Coulomb branch parameters i(a), where a labels
the gauge node, i = 1; : : : ; 2N , and then perform identi cations on
speci c gauge group. Schematically, the Coulomb branch integration measure is
i
(a) suitable for the
Z
[d ] e F ( ) (constraints) ;
up to some ambiguity in the Jacobian factor that only a ects the free energy at order N in
the large N limit. Since we only need to be accurate to order N 3=2 to extract the central
charges of interest, we shall be liberal in our de nition of this integration measure. The
forms of the localized action F ( ) for the three classes of orbifold theories are as follows.
Even orbifold with vector structure
(5.11)
(5.12)
(5.13)
(5.14)
(a+1)
#
(5.15)
Fv(s2k)( ) =
2N "
GV
1
2
+
GV
2N "
+X
i=1
1
2
k
a=2
(ja))
( i(k+1)
j
(k+1))
#
#
j
;
1
2
GV 2 i(1) +
GV 2 i(k+1) +X N (a)GH
f
with the constraints (1)
N+i =
i(1), (Nk++i1) =
(k+1), and Pi2=N1 i
i
(a) = 0 for 2
a
Fn(v2sk)( ) =
X GV
(ja)) +GH
i(1) + (j1) +GH
i(k) + (jk)
(a+1)
#
+X
X N (a)GH
f
i=1 a=1
(a)
i
#
;
with Pi2=N1 i(=a)1 = 0 for 1
(ja))
1 X2N "
2
2N "
+ X
i=1
1
2
1
2
+ GH
(k+1)
GV 2 i(1)
+ X N (a)GH
f
k+1
a=1
k+1
a=2
#
#
;
with
N+i =
(1) and Pi2=N1
i
(a) = 0.
Assuming that the Coulomb branch integral is convergent for these orbifold theories,
their large N limit can be computed by the saddle point approximation.21 In the large N
limit, provided that i(a) scale as N 1=2, the leading terms in the above F ( ) scale as N 7=2,
and are minimized by [7]
(a)
i
(a+1)
j
(5.17)
#
#
(5.16)
(5.18)
(5.19)
(a) = i ;
i
i =
N+i
i ;
1
1
i
i
2N ;
N ;
i = N 1=2xi ;
where i become the sole remaining Coulomb branch parameters. As before, de ne
and rewrite the localized action F ( ) as a functional F [ ] of the density (x). The
equations (5.18) actually render the SU(2N ) and the USp(2N ) gauge factors indistinguishable.
Now we can take the large N limit of the exponent, and nd that for all three cases, it is
simply given by
n !t2ot (x+y +jx yj)
(8
Nf )
3
x
The large N localized action F [ ] receives two types of corrections. First, there are N 3=2
terms coming from the asymptotic expansions (3.11) of F ( ). Second, there are order N 1
corrections to the saddle point con guration (5.18) due to the N 5=2 terms in F ( ), and
21While it requires extra work to understand the convergence of the Coulomb branch integral for general
quiver gauge theories (see [50] for a discussion), the existence of a holographic dual here provides us certain
con dence.
(x) =
2x2?
x2? =
9n
2(8
Nf )
;
F =
15 !1!2!3
p
8
2 !t3ot n3=2N 5=2
Nf
+ O(N 3=2) :
CT =
p
8
Nf
+ O(N 3=2) :
and the free energy
Invoking the general relation between the squashed free energy and the conformal central
charge (2.9), we nd that the conformal central charges of the family of orbifold theories
are given by
is given by
2N
i=1
1 Nf ) 2 so(2Nf(a))
F ( ) = N (a) X h
f
GH
i
(a) + m(a)
f
GH
(a) i
i
:
The leading order correction to the free energy is given by the large N asymptotics (3.11)
of
F ( ) evaluated on the leading order saddle (5.18) as well as (5.21) (from extremizing
the leading order N 7=2 piece of the exponent),
F =
p
2 N (a)(mf(a))2
f
p
8
Nf
tr vec(SO(2Nf ))(M2) = 2Nf(a)(mf(a))2;
in the relation (2.13), we determine the corresponding avor symmetry central charges to be
they also give rise to order N 7=2
(N 1)2 = N 3=2 corrections to F [ ].22 The saddle point
approximation to the Coulomb branch integral can be performed in complete analogy to
the case of Seiberg theories, leading to the saddle point con guration
Flavor central charges
Let us now turn to the computation of the avor central charges for the orbifold theories.
It su ces to consider the round sphere, so we set !i = 1 in the following.
Fundamental hypermultiplet symmetries
We rst compute the avor central charges for generic SO(2Nf(a)) factors | associated with
the gauge nodes a = 1 and a = k + 1 for even orbifold with vector structure, and a = 1 for
odd orbifold | in the avor symmetry groups of the orbifold theories. The modi cation
to the localized action F ( ) due to the introduction of a mass matrix
J
CSO(2Nf(a))
256p2
3 p
8
Nf
n1=2N 3=2 + O(N 1=2) :
22A correction
to the saddle point con guration (5.18) results in a correction of order
localized action F ( ), since by de nition, F ( )=
= 0 at the saddle point.
(5.21)
(5.22)
(5.23)
(5.24)
(5.25)
(5.26)
(5.27)
(5.28)
2 to the
The fact that the leading 1=N corrections to the avor central charges are of order
N 1=2 needs some explanation. Firstly, one may worry that the order N 3=2 corrections to
the undeformed free energy F in (5.22) would contaminate the avor central charges that
are also of order N 3=2. However, these corrections do not depend on the mass parameters,
and therefore do not a ect the avor central charges. Secondly, since the mass deformations
given by (5.25) scale as m2N 3=2 (N 2 relative to the leading N 7=2), there are corrections
to the saddle point con guration (5.18) that are of order m2N 2, in addition to the
mindependent corrections of order N 1 discussed below (5.22). The corresponding correction
to the free energy F in (5.22) that is quadratic in m is from the mixed term, of order
N 7=2
(m2N 2
N 1 = m2N 1=2.
We now consider the remaining gauge nodes that are associated with U(Nf(a)) avor
factors, by introducing the mass matrix,
into the Lagrangian. Proceeding as before, we nd that the leading order correction to the
F =
2 N (a)(mf(a))2
f
p
8
1
2
;
Nf
tr
J
CU(Nf(a))
=
256p2
3 p
8
Nf
in the relation (2.13), we determine the corresponding avor symmetry central charges to be
As will be discussed further in section 6.1, Hanany-Witten moves [51] relate an orbifold
theory considered here to one that has SO(2Nf ) avor acting on hypermultiplets charged
under the rst gauge node, and the
nite shifts in the ranks of the gauge nodes cannot
be detected in the large N limit. This large N
avor symmetry enhancement explains the
coincidence of the avor central charges for all gauge nodes in the original orbifold theory.
5.3.2
Bifundamental hypermultiplet symmetries
We now turn towards the avor central charges for the U(1)(ba)
Each bifundamental hypermultiplet Xa(a+1) has unit charge under the U(1)(ba)
metry associated to a mass parameter m(a) for Xa(a+1). The mass deformation of the
b
avor symmetry factors.
avor
symlocalized action is
i
F ( ) on the large N saddle (5.18), and taking the continuum
limit, the localized action becomes
Z x?
0
This term modi es the saddle to
which leads to the large N free energy,
(x) =
2
2x
x2
?
;
x2? =
9n + 32(m(ba))2
2(8
Nf )
;
F =
9n + 32(m(ba))2 3=2
15p8
Nf
J
Antisymmetric hypermultiplet symmetries
The antisymmetric hypermultiplet A in the odd orbifold has unit charge under the U(1)A
avor symmetry associated with a mass parameter mA. The mass deformation of the
F ( ) on the large N saddle (5.18), and taking the continuum limit, the
By the relation (2.12), we are led to the following U(1)(ba)
avor central charge
(5.35)
(5.36)
(5.37)
HJEP05(218)39
(5.40)
(5.41)
(5.42)
localized action becomes
This term modi es the saddle to
which gives the large N free energy,
(x) =
p
2
Z x?
0
Nf
Nf
2x
x2
?
;
x2? =
2(8
9n + 16m2A ;
Nf )
F =
15p8
9n + 16m2A 3=2
J
CU(1)A =
1024p2
5 p
8
n1=2N 5=2 + O(N 3=2) :
By the relation (2.12), we are led to the following U(1)(Aa)
avor central charge
An analogue analysis shows that the antisymmetric hypermultiplets A and A0 in the even
orbifold without vector structure have the same avor central charge as in (5.42).
CU(1)B = BB
J
CU(1)B = BB
J
CU(1)B = BB
J
0
2
B1
B
1
02
B1
B
B
B1
0
3
0
B2
B
2
1
2
1
2
3
2
1
1
.
.
.
2
2
.
.
.
1
2
1 1
C
C
C
A
2
1
1
2
0
0 1
0 C
C
C
C
C
1
k k
2 1
C
C
A
3
k k
J
J
J
As speci ed in (5.2), (5.6) and (5.11), the mesonic U(1)M and baryonic U(1)B symmetries
are linear combinations of the U(1) avor symmetry factors that act on the bifundamental
and antisymmetric hypermultiplets. The mesonic avor central charge is then given by
J
CU(1)M =
n
256p2
5 p
8
Nf
n3=2N 5=2 + O(N 3=2) :
(5.43)
The baryonic symmetries are not orthogonal. Their avor central charges are given by
matrices. These (symmetric) matrices can be computed by taking derivatives with respect
to the mass parameters of di erent baryonic U(1) factors in the overall avor symmetry.
We write these matrices in the bases (5.2), (5.6), and (5.11). In even orbifolds with vector
structure, the baryonic avor central charge matrix is
HJEP05(218)39
(5.44)
(5.45)
(5.46)
(5.47)
(5.48)
In even orbifolds without vector structure,
In odd orbifolds,
To compare with the results from gravity in section 6.2, it is convenient to perform a
change of basis for the charge lattice from the Q(Ba) de ned in (5.2), (5.6), and (5.11). For
even orbifold with vector structure, we de ne
For even orbifold without vector structure, we de ne
1
2
1
0
1
1
.
.
.
1
0
for a = 1;
; k
2 ;
; k
Q a = Q(Ba)
Q k 1 = Q(Bk 1) :
Q a = Q(Ba)
Q(Ba+1)
Q k 1 = Q(Bk 1) ;
Q k = Q(Bk) :
For odd orbifold without vector structure, we de ne
; k
(5.49)
In these bases, we nd
CJU;e(1v)eBn;vs = BB
CJU;e(1v)eBn;nvs = BBBB 0
CJU;o(1d)dB = BB
0 2
B
B 1
B
0
0 2
B 1
B
B
0
0 2
B
B 1
B
0
1
1
1
2
0
0
2
0
0
0
2
0
0
0
1
0
1
0
1
0
0
.
.
.
0
0
0
0
.
.
.
0
0
0
0
0
.
.
.
0
0
1
0
1
0
0
1
0
0
0
2
1
0
0
2
0
1
0
0
2
1
2
0
0
2
0
1
0 1
0 C
C
C
C
C
3
k k
k k
J
J
CU(1)b ;
J
CU(1)b ;
(5.50)
The above change of basis, (5.44), (5.45), and (5.46), from Q(Ba) to Q a is integral and
uni-modular, and therefore preserves the charge lattice. While such a change of basis is
convenient for the later holographic comparison, more fundamentally, we are matching the
charge lattice, which is basis-independent.
5.3.5
Instantonic symmetries
In order to extract the avor central charge for the U(1)I instantonic factor associated to
the a-th gauge node, we proceed as in section 3.2.2 for the Seiberg theories, by keeping the
contribution of the classical piece. The large N localized action is
a
Nf )
3
I
+
+ X m(a) 2 N 2
x1
x1
Z x2
x1
dx (x)x3 + O(N 3=2)
dx (x)x2 + O(N ) ;
(5.51)
for either the USp or SU gauge nodes. The saddle point solution to the relevant 1=N order is
(x) =
x1 = 0 ;
x2 =
4(8
Nf )x
9n
;
a mI(a) +
and gives rise to the large N free energy,
HJEP05(218)39
F =
p
9 2
p
5 8
Nf
X m(a)
I
n3=2N 5=2 + O(N 3=2) + X m(a)
I
Nf )
nN 2 + O(N )
!2 "
p
4 2
(8
Nf )3=2 n1=2N 3=2 + O(N 1=2) + O((mI(a))3) :
q
16(P
p
2 N (Nf
a mI(a))2 + 18 nN (8
Nf )
;
8)
.
.
.
9
#
1 1
1
As in the case of Seiberg theories, the linear m(a) piece would be inconsistent with
conformal symmetry, and must be removed by a counter-term. Using the relation (2.12), we nd
that the instanton symmetries are not orthogonal, and their avor central charge matrix is
J
whose dimensionality equals the number of gauge nodes. The reasoning for the leading 1=N
corrections to the avor central charges being of order N 1=2 proceeds as before. Since the
mass deformations in (5.51) scale as mN 2 (N 3=2 relative to the leading N 7=2), there are
corrections to the saddle point con guration (5.18) that are of order mN 3=2, in addition to
the m-independent corrections of order N 1 discussed below (5.22). The correction to the
free energy F in (5.22) that is quadratic in m is then of order N 7=2 (mN 3=2)2 = m2N 1=2.
At order N 3=2, the
avor central charge matrix is rank-one, meaning that we only
observe one independent combination of
avor central charges. In order to access the
other independent avor central charges that are of order N 1=2, we are required to carefully
study the subleading contributions to the matrix models (5.15), (5.16), and (5.17), and in
particular modify the leading order saddle (5.18).
6
Central charges from the supergravity dual of orbifold theories
Let us now study the massive IIA supergravity duals for the orbifold theories [6], and
subsequently match the various avor central charges associated to the global symmetry group.
M6
The holographic duals of the orbifold theories are the Zn orbifolds of the geometry
w HS4 discussed in section 4.2. More precisely, for each S3-slice at constant
of the
hemisphere, we impose the identi cation 3 !
3 + 4 =n on the coordinate 3 in (4.8).
(5.52)
(5.53)
(5.54)
This corresponds to replacing the three-sphere with the lens space L(n; 1) = S3=Zn. Notice
that this action reduces the volume of the four-hemisphere by a factor of n,
VolHS4=Zn =
VolHS4 ;
n
and a ects the e ective six-dimensional couplings upon compacti cation. Due to the
volume reduction, the F4
ux is multiplied by a factor of n to preserve the D4-brane charge
quantization condition (4.11). Hence, the supergravity solution of the orbifold theory is
given by the substitution N ! nN in the Seiberg theory background (4.7), (4.15), (4.10),
and (4.14). Together these considerations immediately give us many of the central charges.
Central charges by comparison to Seiberg theories
Compared to the holographic duals of Seiberg theories, the free energy receives an extra
factor of n 1
n5=2 = n3=2, where the n 1 comes from the reduction of the internal volume,
and the n5=2 comes from the N ! nN shift due to the modi ed charge quantization. This
precisely matches the large N conformal central charge (3.19) of the eld theory. By the
same argument, the SO(2Nf ) and U(Nf ) avor central charges each receives a factor of
n1=2, matching the eld theory results in (5.28) and (5.32), respectively.
In the large N limit, we are not able to distinguish between the distinct theories that
arise from di erent distributions of the
Nf =
X N (a)
f
a
(6.1)
(6.2)
(6.3)
fundamental hypermultiplets, for the following reason.23 On the gravity side, the orbifold
solutions in [6] describe the con gurations where all the Nf D8-branes sit at the rst gauge
node. However, starting with a di erent setup, in which the Nf D8-branes are distributed
among di erent gauge nodes, and performing consecutive Hanany-Witten moves [51], one
ends up with a eld theory description in which some of the ranks of the gauge nodes are
shifted N ! N + `, where `
Nf . But this e ect is not expected to be visible in the
leading order large N asymptotics. Thus, in the strict large N limit, we will only ever be
able to probe the SO(2Nf ) avor symmetry associated to the nal gauge node.24
The mesonic U(1)M symmetry of the orbifold theories corresponds to the U(1)M
factor in the isometry group of the orbifold hemisphere, SU(2)R
U(1)M. Compared to the
avor central charge of the U(1)M
SU(2)M subgroup in the Seiberg theories, the
orbifold theories receive the same extra n3=2 factor as discussed above for the conformal and
hypermultiplet avor central charges. Taking into account the embedding index
Iu(1)M,!su(2)M =
1
2
;
we nd accordance with the eld theory result (5.43).
that do.
charges at leading large N .
23Not all distributions give rise to ultraviolet superconformal xed points. Here, we restrict to the ones
24As a matter of fact, we cannot even distinguish between SO(2Nf ) and U(Nf ) by their avor central
Note that the Kaluza-Klein modes have minimal U(1)M charge n=2, which follows from
the periodicity 3
3 + 4n in the orbifold theory [6]. This holographic argument for the
normalization of U(1)M is in agreement with the mesonic charge de ned in
eld theory,
given by (5.2), (5.6), and (5.11), of the meson operator, given in (5.3), (5.7), and (5.12).
Let us brie y remark on the dual supergravity gauge elds for the U(1)I(a) instantonic
symmetries. In [6], it was argued that all but one of them arise from the reduction of the
R-R three-form C3 on the nontrivial two-cycles at the orbifold
xed point of HS4=Zn. The
remaining one is argued to arise by a reduction akin to the infrared U(1)I of the Seiberg
theories (see section 4.4.3). We leave the matching of the instantonic avor central charge
to future work, in light of the subtleties already present in the Seiberg theories, as discussed
The baryonic gauge elds are obtained from the reduction of the R-R three-form C3 in
ten-dimensional massive IIA on the internal two-cycles ea in HS4=Zn [6]. Thus, in order
to match the large N baryonic avor central charge matrix with the coupling matrix of the
gauge elds in the holographic dual, we are required to analyze the appropriate (baryonic)
two-cycles, and their intersection forms.
The orbifolded four-hemisphere, HS4=Zn, has an orbifold singularity at the north pole.
Thus, the usual Kaluza-Klein reduction from IIA supergravity needs to be complemented
by analyzing the twisted sectors of the string theory. We start by considering the covering
space S4=Zn (before the orientifold projection), or more precisely a mirror pair of C2=Zn
singularities, before the near horizon limit. We introduce two sets of vanishing
anti-selfdual two-cycles i
N;S, with i = 1; : : : ; n
1, for the two copies of C2=Zn, corresponding to
the twisted sector ground states of the orbifold CFT. Their intersection forms are given
by the An 1 Cartan matrices,
(6.4)
(6.5)
(6.6)
and
north and south copies of C2=Zn as follows
As we shall explain below, the twisted sector closed string states that survive the orientifold
projection give rise to six-dimensional gauge elds dual to the baryonic symmetries.
The orientifold action I, given by a composition of worldsheet parity and the
re ection I : x9
x9 (before the near-horizon limit), maps the vanishing cycles in the
N
i $
S
n i
;
N;S ) = BB
0 2
B
0 C
C
C
C
C
2
(n 1) (n 1)
for even n = 2k with i 6= k, and odd n = 2k + 1. For even n = 2k, the middle cycles k
are allowed to transform in two di erent ways corresponding to with and without vector
structure [52{55]
vs :
nvs :
N
N
k $
k $
S
k ;
S
k :
(again, we emphasize that this is prior to taking the near-horizon limit)25
To make contact with the notation in reference [6], we identify the cycles i and ei as
i <
i >
n
2
n
2
:
:
i = iN + iS ;
i =
N
i
S
i ;
ei = i
N
S
ei = i
S
i ;
iN :
k =
kN + k ;
S
ek =
N
S
k :
For even n = 2k, there is an additional middle cycle, which we de ne in terms of k
Note that while there are D2-branes wrapping combinations of i and sitting on the
O8plane, corresponding to (singular) instantons in the boundary theory, the baryonic cycles
ei only exist away from the O8-plane.
Then the six-dimensional baryonic gauge elds (coupled to the baryonic U(1)B currents
in eld theory), are given by reducing the R-R three-form C3 on linear combinations of the
two-cycles ei that are odd under the orientifold action
I, that we call ea,26
Ba =
C3 :
Z
ea
Thus, the coupling matrix for the baryonic gauge elds are proportional to the intersection
matrices ( ea ; eb ) which we compute for each case of the orbifolds below.
Even orbifold with vector structure
The odd two-cycles are
ea = ea + en a =
N
N
n a
S
S
n a ;
a = 1;
; k
1 :
(6.11)
(6.8)
(6.7)
N;S as
(6.9)
(6.10)
The intersection matrix is
( ea ; eb ) = 4
0 2
B
B
B
0
1
2
0
0
0
1
0
0
0 C
C
C
C
C
2
(k 1) (k 1)
:
(6.12)
25In [6], upon Kaluza-Klein reduction of the R-R three-form C3 from massive IIA, the di erent cycles
i and ei give rise to six-dimensional U(1) gauge elds sourced by instantonic and baryonic U(1) global
conserved currents, respectively.
26Similarly, there are cycles
a which are I-odd combinations of i. They give rise to six-dimensional
gauge elds dual to instanton symmetries. However due to the mixing with other R-R and NS-NS elds from
ten-dimensional supergravity as explained in section 4.4.3, the intersection matrix of a will not directly
produce the CJ matrix for U(1)I. We leave the complete analysis in this case for future.
The odd two-cycles are
and
Their intersection matrix is
( ea ; eb ) = 4
( ea ; ek ) = 0 ; ( ek ; ek ) = 4 ;
for a; b = 1;
; k
Odd orbifold
The odd cycles are
ea = ea + en a =
Their intersection matrix is
( ea ; eb ) = 4
0
0
2
1
0
1
C
C
C
C
2
(k 1) (k 1)
1
C
C
C
C
k k
N
N
n a
S
; k
ek = ek =
N
S
k :
0 2
B 1
B
B
B
B
0
1
2
0
0
N
0 2
B
B
B
B
0
0
1
N
n a
1
2
0
0
0
0
.
.
.
0
0
0
1
1
0
S
0
HJEP05(218)39
(6.13)
(6.14)
(6.15)
(6.17)
The intersection matrices (6.17), (6.15), and (6.12) are in agreement with the baryonic
avor central charge matrices (5.50), up to an overall normalization that can be xed.
7
In this paper, we study the central charges of a certain class of holographic ve-dimensional
superconformal eld theories that have constructions in type I' string theory, from both the
eld theory and supergravity perspective. On the eld theory side, we employ the formulae
discovered by the present authors in previous work [29], that relate particular deformations
of the ve-sphere partition function to the conformal and avor central charges. In order to
compare to the corresponding supergravity quantities, we take the large N limit of the eld
theory, where it is argued that the instanton contributions are exponentially suppressed.
Thus, we obtain exact large N results purely based on the perturbative part of the partition
function, and compute the large N conformal and
avor central charges for our class of
theories. The comparison of the large N
avor central charges for the manifest infrared
instantonic U(1)I and hypermultiplet SO(2Nf ) symmetries provides evidence for the avor
symmetry enhancement to ENf +1 in the ultraviolet. We further support our large N results
by juxtaposing the results from direct numerical integration, the saddle point method, and
the analytic large N formulae.
We then compare and explicitly match these large N
eld theory results against
their holographic duals. The central charges are related to certain couplings in the
sixdimensional e ective gravity dual that arise from the reduction of massive IIA
supergravity.
Although the corresponding ten-dimensional supergravity backgrounds have a
curvature singularity in the internal manifold, by explicitly reducing the relevant terms in
the ten-dimensional action to six dimensions, we obtain
nite values for the e ective
sixdimensional couplings, that precisely match with the conformal and avor central charges
in the eld theory.
The matching of the instantonic avor central charges is left for future work. On the
gravity side, as argued in section 4.4.3, in order to reproduce the correct e ective
sixdimensional kinetic term from which the corresponding central charge can be extracted,
we are required to take into account the reduction of all the
elds in ten-dimensional
supergravity. On the eld theory side, we found only one independent instantonic
avor
central charge to leading order in the large N limit, i.e., the avor central charge matrix
is rank-one. To capture the remaining (independent) central charges, we need to carry the
matrix model analysis for the orbifold theories to further subleading order in 1=N .
Finally, we are con dent that a similar large N analysis should provide evidence and
checks for some proposed large N dualities of ve-dimensional theories [48], as well as the
recently discovered type IIB AdS6 solutions [
56, 57
].27
Acknowledgments
We are grateful to Oren Bergman, Daniel L. Ja eris, Igor R. Klebanov, Silviu S. Pufu,
and Diego Rodr guez-Gomez for helpful discussions and correspondence. CC, YL, and
YW thank the Aspen Center for Physics, MF thanks the Simons Summer Workshop, and
YL thanks National Taiwan University for hospitality during the course of this work. CC
is supported in part by the U.S. Department of Energy grant DE-SC0009999.
MF is
supported by the David and Ellen Lee Postdoctoral Scholarship, YL is supported by the
Sherman Fairchild Foundation, and both MF and YL by the U.S. Department of Energy,
O ce of Science, O ce of High Energy Physics, under Award Number DE-SC0011632. YW
is supported in part by the US NSF under Grant No. PHY-1620059 and by the Simons
Foundation Grant No. 488653. This work was partially performed at the Aspen Center for
Physics, which is supported by National Science Foundation grant PHY-1607611.
A
Triple sine function
In this appendix, we introduce the triple sine function as well as its asymptotic limit, which
is important when taking the large N limit of the squashed
ve-sphere free energy.
The multiple sine function is de ned as
SN (z j !1; : : : ; !N ) =
N (z j !1; : : : ; !N ) 1
N (!tot
z j !1; : : : ; !N )( 1)N ;
(A.1)
27While AdS6 solutions in massive IIA supergravity are rather rare [58], there is a variety of solutions in
type IIB that contains a warped AdS6 factor. Some early works in this direction include [59, 60].
N is the multiplet gamma function de ned as
Here, we have introduced yet another special function
N (z j !1; : : : ; !N ) = exp [ N (z j !1; : : : ; !N )] :
N (z j !1; : : : ; !N ) =
N (s; z j !1; : : : ; !N ) ;
where the multiple zeta-function is given as follows
d
ds s=0
1
X
m1;:::;mN =0
N (s; z j !1; : : : ; !N ) =
(z + m1!1 +
mN !N ) s ;
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
where Re z > 0, Re s > N and !1; : : : ; !N > 0. The function N is a meromorphic function
with simple poles at s = 1; : : : ; N .
A.1
Asymptotics of the triple sine function
In order to compute the large N limit of the (squashed) ve-sphere free energy, we require
the asymptotic jzj !
1 expansion of log S3 (z) and we shall quickly mention how to
compute those. In [61], the author proved that there is an alternative de nition of
N (z)
N (z j !1; : : : ; !N ) =
BN;N (z) log z + ( 1)N X
N !
M
X
k=N+1
( 1)k
BN;k (0) zN k(k
1)! + RN;M (z) ;
k=0
N
N 1 BN;k(0)zN k N k
X 1
where Re z > 0, and M
was shown in [61] to behave as
N is an arbitrary integer. RN;M (z) is some remainder, which
RN;M (z)
O z
N M 1 ;
in the asymptotic limit jzj ! 1 as long as jarg zj <
. Similarly it is straightforward
to see that the third term in (A.5) is of order O z 1 in the asymptotic limit jzj ! 1.
Furthermore, we denoted by BN;N (z j !~) the generalized/multiple Bernoulli polynomials,
which can be explicitly computed by expanding and solving
tN ezt
QbN=1 (e!bt
1)
X1 t
n
BN;n (z)
order-by-order. For the case of interest in the present paper, i.e. N = 3, we have
B3;3 (z j !~) =
z
3
!1!2!3
2!1!2!3
3!tot z2 +
!t2ot + (!1!2 + !1!3 + !2!3) z
2!1!2!3
!tot (!1!2 + !1!3 + !2!3) :
4!1!2!3
Thus, it is now easy to compute (A.5) in the asymptotic limit and equivalently for the
triple sine function. Explicitly one obtains
log S3 (iz j !~) + log S3 ( iz j !~)
3 !1!2!3 jzj3 +
jzj + O(jzj2 M ) ;
log S3 iz +
z 3
3 jzj + O(jzj2 M ) ;
(A.10)
where z 2 R. In other words, for arbitrary M , the contributions from the third term in (A.5)
exactly cancel. Since the integer M can be chosen arbitrarily large, the expansions (A.9)
and (A.10) have no subleading power law corrections jzj n, for n
Numerical evaluation of central charges
We present tabulated values for the round sphere free energy
F0, the conformal central charge CT , the mesonic
J
avor central charge CSU(2)M , and the exceptional avor central
charges CJGf for Gf = E1; E2; : : : ; E8, for Seiberg theories up to rank three. Each quantity
is extracted from the perturbative partition function (2.20), computed via direct numerical
integration and an a priori illegal saddle point approximation. An important conclusion
we draw is that the nite N saddle point method actually produces good approximations
for the integrals.
Integral Saddle
Integral Saddle
Integral Saddle
Error
Gf
F0
E1
E2
E3
E4
E5
E6
E7
E8
5:0967
6:1401
7:3949
8:9590
11:007
13:898
18:538
28:473
1
5:2612
6:2817
7:5109
9:0441
11:052
13:886
18:440
28:215
Error
1:6%
1:1%
0:78%
0:47%
0:20%
0:041%
0:27%
0:46%
22:190
25:425
29:335
34:233
40:391
49:543
64:642
96:712
Error
0:067%
0:052%
0:16%
0:25%
0:016%
0:13%
0:49%
0:53%
55:114
61:896
70:122
80:430
93:965
113:63
143:41
214:24
3
54:960
61:645
69:760
79:965
93:454
112:70
144:08
212:68
0:14%
0:20%
0:26%
0:29%
0:27%
0:41%
0:23%
0:36%
2
22:220
25:398
29:243
34:061
40:404
49:413
64:010
95:699
F0 in the rank-one to rank-three
Seiberg theories, computed by numerical integration and by the saddle point approximation.
Gf
CT
E1
E2
E3
E4
E5
E6
E7
E8
333:39
422:94
529:78
662:00
834:00
1075:1
1459:5
2274:4
J
CSU(2)M
Gf
E1
E2
E3
E4
E5
E6
E7
E8
0:92%
0:75%
0:60%
0:43%
3673:0
4197:1
4829:4
5619:4
6653:6
8126:6
10504:
15651:
3
3741:8
4266:1
4898:5
5688:9
6727:3
8199:4
10580: 0:36%
15736: 0:27%
0:93%
0:82%
0:71%
0:61%
0:55%
0:45%
1
365:53
455:28
562:40
694:99
867:48
1109:2
1494:3
2309:8
66:277
70:839
76:440
83:556
93:064
106:81
129:60
180:73
4:6%
3:7%
3:0%
2:4%
2:0%
1:6%
1:2%
0:77%
1477:0
1737:1
2049:6
2438:4
2946:9
3663:9
4815:9
7287:9
2
66:218
70:733
76:268
83:300
92:708
106:34
129:06
180:35
2
1529:8
1790:2
2102:9
2492:2
3001:4
3719:6
4873:8
7351:0
avor central charge CSU(2)M in the rank-one
J
to rank-three Seiberg theories, computed by numerical integration and by the saddle point
approximation.
Seiberg theories, computed by numerical integration and by the saddle point approximation.
0:045%
0:074%
0:11%
0:15%
0:19%
0:22%
0:21%
0:11%
218:62
233:82
252:45
275:74
307:17
354:67
431:93
605:43
3
218:15
233:23
251:77
275:41
307:14
353:31
430:43
604:63
Error
0:11%
0:13%
0:13%
0:061%
0:0044%
0:19%
0:17%
0:067%
21:638
23:700
26:413
30:131
35:587
44:657
64:752
18:409
20:582
23:120
26:190
30:128
35:664
44:707
64:756
1
Saddle
22:275
24:400
27:158
30:895
36:324
45:256
64:766
1
17:966
19:956
22:342
25:307
29:192
34:722
43:783
63:825
J
CSO(2Nf )
Gf
CU(1)I
J
Iu(1)I,!gf
Gf
E2
E3
E4
E5
E6
E7
E8
E1
E2
E3
E4
E5
E6
E7
E8
Seiberg theories, obtained using the instanton particle mass, computed by numerical integration
and by the saddle point approximation.
0:79%
0:78%
0:75%
0:67%
0:54%
0:35%
0:019%
88:687
96:877
107:22
121:22
142:94
177:82
255:52
3
Saddle
89:681
97:981
108:66
123:07
144:01
178:69
255:60
0:56%
0:57%
0:67%
0:76%
0:37%
0:24%
0:015%
0:28%
0:051%
0:11%
0:19%
0:21%
0:18%
0:14%
0:10%
79:666
87:144
96:095
107:25
121:96
143:05
177:89
255:42
3
80:481
87:726
96:490
107:51
122:14
143:22
178:07
255:64
Error
0:51%
0:33%
0:21%
0:12%
0:075%
0:058%
0:051%
0:043%
Seiberg theories, obtained using the hypermultiplet masses, computed by numerical integration and
by the saddle point approximation.
Integral Saddle
Error
Integral Saddle
Integral Saddle
1:5%
1:5%
1:4%
1:3%
1:0%
0:67%
0:011%
Integral
51:739
56:609
62:941
71:551
84:126
105:01
151:41
1:2%
1:5%
1:7%
1:7%
1:6%
1:3%
1:0%
0:72%
45:661
50:318
55:843
62:655
71:546
84:218
105:07
151:42
2
Saddle
52:559
57:504
63:886
72:512
85:045
105:76
151:47
2
45:919
50:370
55:722
62:411
71:243
83:907
104:78
151:12
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HJEP05(218)39
JHEP 11 (2016) 135 [arXiv:1602.01217] [INSPIRE].
460 (1996) 525 [hep-th/9510169] [INSPIRE].
[1] J.M. Maldacena, The large N limit of superconformal eld theories and supergravity, Int. J.
[2] C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of Superconformal Theories,
[3] J. Polchinski and E. Witten, Evidence for heterotic | type-I string duality, Nucl. Phys. B
[4] S. Ferrara, A. Kehagias, H. Partouche and A. Za aroni, AdS6 interpretation of 5-D
superconformal eld theories, Phys. Lett. B 431 (1998) 57 [hep-th/9804006] [INSPIRE].
[5] A. Brandhuber and Y. Oz, The D4{D8 brane system and ve-dimensional xed points, Phys.
Lett. B 460 (1999) 307 [hep-th/9905148] [INSPIRE].
171 [arXiv:1206.3503] [INSPIRE].
[6] O. Bergman and D. Rodriguez-Gomez, 5d quivers and their AdS6 duals, JHEP 07 (2012)
[7] D.L. Ja eris and S.S. Pufu, Exact results for ve-dimensional superconformal eld theories
with gravity duals, JHEP 05 (2014) 032 [arXiv:1207.4359] [INSPIRE].
[8] O. Bergman and D. Rodriguez-Gomez, Probing the Higgs branch of 5d xed point theories
with dual giant gravitons in AdS6, JHEP 12 (2012) 047 [arXiv:1210.0589] [INSPIRE].
[9] B. Assel, J. Estes and M. Yamazaki, Wilson Loops in 5d N = 1 SCFTs and AdS/CFT,
Annales Henri Poincare 15 (2014) 589 [arXiv:1212.1202] [INSPIRE].
[10] O. Bergman, D. Rodr guez-Gomez and G. Zafrir, 5d superconformal indices at large N and
holography, JHEP 08 (2013) 081 [arXiv:1305.6870] [INSPIRE].
[11] L.F. Alday, M. Fluder, P. Richmond and J. Sparks, Gravity Dual of Supersymmetric Gauge
Theories on a Squashed Five-Sphere, Phys. Rev. Lett. 113 (2014) 141601 [arXiv:1404.1925]
[12] L.F. Alday, M. Fluder, C.M. Gregory, P. Richmond and J. Sparks, Supersymmetric gauge
theories on squashed
ve-spheres and their gravity duals, JHEP 09 (2014) 067
[arXiv:1405.7194] [INSPIRE].
[13] L.F. Alday, P. Richmond and J. Sparks, The holographic supersymmetric Renyi entropy in
ve dimensions, JHEP 02 (2015) 102 [arXiv:1410.0899] [INSPIRE].
[14] N. Hama, T. Nishioka and T. Ugajin, Supersymmetric Renyi entropy in ve dimensions,
JHEP 12 (2014) 048 [arXiv:1410.2206] [INSPIRE].
[15] L.F. Alday, M. Fluder, C.M. Gregory, P. Richmond and J. Sparks, Supersymmetric solutions
to Euclidean Romans supergravity, JHEP 02 (2016) 100 [arXiv:1505.04641] [INSPIRE].
[16] L.J. Romans, The F (4) Gauged Supergravity in Six-dimensions, Nucl. Phys. B 269 (1986)
691 [INSPIRE].
[17] M. Cvetic, H. Lu and C.N. Pope, Gauged six-dimensional supergravity from massive type
IIA, Phys. Rev. Lett. 83 (1999) 5226 [hep-th/9906221] [INSPIRE].
[18] R. D'Auria, S. Ferrara and S. Vaula, Matter coupled F(4) supergravity and the AdS6=CF T5
correspondence, JHEP 10 (2000) 013 [hep-th/0006107] [INSPIRE].
[19] L. Andrianopoli, R. D'Auria and S. Vaula, Matter coupled F(4) gauged supergravity
Lagrangian, JHEP 05 (2001) 065 [hep-th/0104155] [INSPIRE].
[20] V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops,
Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
[21] N. Seiberg, Five-dimensional SUSY
eld theories, nontrivial xed points and string
dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
[22] D.R. Morrison and N. Seiberg, Extremal transitions and ve-dimensional supersymmetric
eld theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
[23] K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge
theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56
[hep-th/9702198] [INSPIRE].
[24] P. Je erson, H.-C. Kim, C. Vafa and G. Zafrir, Towards Classi cation of 5d SCFTs: Single
Gauge Node, arXiv:1705.05836 [INSPIRE].
[25] D.L. Ja eris, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-Theorem: N = 2 Field
Theories on the Three-Sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
[26] I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP 10
[27] H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys.
(2011) 038 [arXiv:1105.4598] [INSPIRE].
Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
[arXiv:1608.02960] [INSPIRE].
[28] S.S. Pufu, The F-Theorem and F-Maximization, J. Phys. A 50 (2017) 443008
[29] C.-M. Chang, M. Fluder, Y.-H. Lin and Y. Wang, Spheres, Charges, Instantons and
Bootstrap: A Five-Dimensional Odyssey, JHEP 03 (2018) 123 [arXiv:1710.08418]
Five-Sphere, Nucl. Phys. B 865 (2012) 376 [arXiv:1203.0371] [INSPIRE].
[31] J. Kallen, J. Qiu and M. Zabzine, The perturbative partition function of supersymmetric 5D
Yang-Mills theory with matter on the ve-sphere, JHEP 08 (2012) 157 [arXiv:1206.6008]
[32] H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144
[33] Y. Imamura, Supersymmetric theories on squashed ve-sphere, PTEP 2013 (2013) 013B04
[arXiv:1206.6339] [INSPIRE].
[arXiv:1209.0561] [INSPIRE].
[arXiv:1210.6308] [INSPIRE].
arXiv:1211.0144 [INSPIRE].
[34] G. Lockhart and C. Vafa, Superconformal Partition Functions and Non-perturbative
Topological Strings, arXiv:1210.5909 [INSPIRE].
[35] Y. Imamura, Perturbative partition function for squashed S5, PTEP 2013 (2013) 073B01
[36] H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes,
[37] L.J. Romans, Massive N=2a Supergravity in Ten-Dimensions, Phys. Lett. B 169 (1986) 374
[38] D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the
[39] J. Penedones, TASI lectures on AdS/CFT, in Proceedings, Theoretical Advanced Study
Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015),
Boulder, CO, U.S.A., June 1{26, 2015, pp. 75{136 (2017) [DOI:10.1142/9789813149441 0002]
[arXiv:1608.04948] [INSPIRE].
[40] L.F. Alday, P. Benetti Genolini, M. Fluder, P. Richmond and J. Sparks, Supersymmetric
gauge theories on ve-manifolds, JHEP 08 (2015) 007 [arXiv:1503.09090] [INSPIRE].
[41] J. Polchinski, Dirichlet Branes and Ramond-Ramond charges, Phys. Rev. Lett. 75 (1995)
4724 [hep-th/9510017] [INSPIRE].
[42] O. Bergman, M.R. Gaberdiel and G. Lifschytz, String creation and heterotic type-I' duality,
Nucl. Phys. B 524 (1998) 524 [hep-th/9711098] [INSPIRE].
[43] D. Matalliotakis, H.-P. Nilles and S. Theisen, Matching the BPS spectra of heterotic Type I
and Type I-prime strings, Phys. Lett. B 421 (1998) 169 [hep-th/9710247] [INSPIRE].
[44] M.B. Green, C.M. Hull and P.K. Townsend, D-brane Wess-Zumino actions, t duality and the
cosmological constant, Phys. Lett. B 382 (1996) 65 [hep-th/9604119] [INSPIRE].
[45] M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167
[46] C.V. Johnson and R.C. Myers, Aspects of type IIB theory on ALE spaces, Phys. Rev. D 55
[47] O. Bergman and G. Zafrir, 5d xed points from brane webs and O7-planes, JHEP 12 (2015)
[53] J. Polchinski, Tensors from K3 orientifolds, Phys. Rev. D 55 (1997) 6423 [hep-th/9606165]
[54] K.A. Intriligator, RG xed points in six-dimensions via branes at orbifold singularities, Nucl.
Phys. B 496 (1997) 177 [hep-th/9702038] [INSPIRE].
163 [arXiv:1507.03860] [INSPIRE].
[INSPIRE].
[arXiv:1408.4040] [INSPIRE].
Gauge Node, arXiv:1705.05836 [INSPIRE].
[hep-th/9605184] [INSPIRE].
[INSPIRE].
[48] O. Bergman, D. Rodr guez-Gomez and G. Zafrir, 5-Brane Webs, Symmetry Enhancement
and Duality in 5d Supersymmetric Gauge Theory, JHEP 03 (2014) 112 [arXiv:1311.4199]
[49] G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories, JHEP 12 (2014) 116
[50] P. Je erson, H.-C. Kim, C. Vafa and G. Zafrir, Towards Classi cation of 5d SCFTs: Single
[51] A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional
gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
[52] M. Berkooz, R.G. Leigh, J. Polchinski, J.H. Schwarz, N. Seiberg and E. Witten, Anomalies,
dualities and topology of D = 6 N = 1 superstring vacua, Nucl. Phys. B 475 (1996) 115
S2 in Type IIB
supergravity I: Local solutions, JHEP 08 (2016) 046 [arXiv:1606.01254] [INSPIRE].
JHEP 01 (2013) 113 [arXiv:1209.3267] [INSPIRE].
Duality, Phys. Rev. Lett. 110 (2013) 231601 [arXiv:1212.1043] [INSPIRE].
supergravity, JHEP 05 (2013) 079 [arXiv:1302.2105] [INSPIRE].
[55] J.D. Blum and K.A. Intriligator , Consistency conditions for branes at orbifold singularities , Nucl. Phys. B 506 ( 1997 ) 223 [ hep -th/9705030] [INSPIRE].
[56] E. D'Hoker , M. Gutperle , A. Karch and C.F. Uhlemann , Warped AdS6 [57] E. D'Hoker , M. Gutperle and C.F. Uhlemann , Holographic duals for ve-dimensional eld theories , Phys. Rev. Lett . 118 ( 2017 ) 101601 [59] Y. Lozano , E. O Colgain , D. Rodr guez-Gomez and K. Sfetsos, Supersymmetric AdS6 via T [60] J. Jeong , O. Kelekci and E. O Colgain , An alternative IIB embedding of F (4 ) gauged [61] S.N.M. Ruijsenaars , On barnes' multiple zeta and gamma functions, Adv . Math. 156 ( 2000 )