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JHEP08(2017)111

Published for SISSA by Springer

Received: August 4, 2017

Accepted: August 4, 2017

Published: August 24, 2017

A holographic dual for string theory on

AdS

×S

Lorenz Eberhardt,

Matthias R. Gaberdiel

and Wei Li

Institut f¨ur Theoretische Physik, ETH Zurich,

CH-8093 Z¨urich, Switzerland

Institute of Theoretical Physics, Chinese Academy of Science,

100190 Beijing, P.R. China

E-mail: eberhardtl@itp.phys.ethz.ch, gaberdiel@itp.phys.ethz.ch,

weili@itp.ac.cn

Abstract: The CFT dual of string theory on AdS

× S

is conjectured to be the

symmetric orbifold of the S

theory, provided that one of the two Q

quantum numbers

is a multiple of the other. We determine the BPS spectrum of the symmetric orbifold in

detail, and show that it reproduces precisely the BPS spectrum that was recently calculated

in supergravity. We also determine the BPS spectrum of the world-sheet theory that is

formulated in terms of WZW models, and show that, apart from some gaps (which are

reminiscent of those that appear in the corresponding T

calculation), it also reproduces

this BPS spectrum. In fact, the matching seems to work as well as for the familiar T

case,

and thus our results give strong support for this proposal.

Keywords: AdS-CFT Correspondence, Extended Supersymmetry, Conformal Field The-

ory, D-branes

ArXiv ePrint: 1707.02705

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP08(2017)111

JHEP08(2017)111

D Character derivation of the twisted sector spectrum 28

D.1 Odd twist 29

D.2 Even twist 29

E BPS states in the spectrally ﬂowed NS sectors 30

F Non-renormalization of the chiral ring 32

1 Introduction

The AdS

/CFT

duality provides some unique insights into the AdS/CFT correspondence,

mainly because of the powerful CFT techniques that are available in two dimensions.

Indeed, the CFT dual of string theory on AdS

×S

×M

, where M

is a 4D hyper-K¨ahler

manifold (i.e. T

or K3), is one of the ﬁrst explicitly known AdS/CFT dualities constructed

from string theory. On the gravity side, the background arises from the near horizon limit

of Q

D1 branes smeared in Q

coincident D5 branes that wrap M. The dual CFT is

then realized as the decoupling limit of the world-volume theory of the D1-D5 system, i.e.

as a sigma model on the moduli space of Q

instantons in the U(Q

) gauge theory living

on M

As a consequence, the dual CFT lies on the same moduli space as the Q

-fold

symmetric orbifold of M

, which is essentially a free theory (at least for the case of T

Together with the large amount of supersymmetry (described by the small N = 4 super-

conformal algebra) this allows many aspects of the duality to be checked and matched

explicitly, see e.g. [1] for a review. In particular, the BPS spectrum on both sides agrees, as

do the three-point functions [2, 3]. Finally, for the case of K3, the matching of the elliptic

genus provides further non-trivial evidence [4].

For both T

and K3, the dual CFT has small N = 4 superconformal symmetry.

Replacing the hyperk¨ahler manifold M

by S

× S

(which is the smallest “hyperk¨ahler

manifold with torsion”), one obtains a close cousin to the above setup, i.e. string theory

on AdS

× S

. The dual CFT dual is then expected to have large N = 4

superconformal symmetry [5–7].

Despite its apparent similarity to the familiar T

or K3 case, the CFT dual of string

theory on AdS

× S

has proven much more diﬃcult to ﬁnd. This might be

surprising at ﬁrst sight, given that this background has slightly larger symmetry than the

small N = 4 algebra supported by the T

or K3 case. For example, the CFT dual of

the N = 4 Vasiliev higher-spin theory was actually ﬁrst identiﬁed for the AdS

× S

background and shown to be a relatively simple (Wolf) coset CFT [8], while the

identiﬁcation for the T

or K3 case was only subsequently found as a limiting case [9, 10].

One reason why ﬁnding the stringy CFT dual of AdS

× S

has turned

out to be diﬃcult, is that the corresponding brane constructions are rather non-trivial.

A simple D1-D5 system suﬃces for the T

or K3 case, and the large U-duality symmetry

of the system allows one to show that the CFT dual can only depend on the product

– 1 –

JHEP08(2017)111

N = Q

[11]. For the AdS

× S

case, on the other hand, there are, in addition

to the Q

quantum number, two diﬀerent Q

quantum numbers (corresponding to the

sizes of the two S

’s). We then either need two diﬀerent kinds of D5 branes, or we must

realize one set of charges in terms of some non-trivial ﬂux. In either case, the description

of the moduli space of instantons is much more complicated. Furthermore, the U-duality

group is much smaller in this case, as was emphasized already in [7]; in particular, one

should therefore not expect the answer to depend just on some simple combinations of the

diﬀerent brane charges.

The other main diﬃculty for ﬁnding the CFT dual had to do with the structure of the

BPS bounds for the large N = 4 superconformal algebra A

, and its relation to the BPS

bound of the corresponding supergravity algebra D(2, 1|α). In particular, as was stressed

in [6, 7], the BPS bound for A

is in general stronger than that for D(2, 1|α), with the bound

only coinciding for those BPS states whose spins with respect to the two su(2) algebras

(corresponding to the two S

’s) agree. Compounding the problem, it was long believed [6]

that the supergravity theory had lots of BPS states, including many states whose spins

with respect to the two su(2) algebra do not agree — and that need to acquire a magical

amount of quantum correction in order to just satisfy the A

BPS bound. Furthermore,

none of proposed CFT duals had a corresponding BPS spectrum, even for some special

choice of charges [7].

Recently, this problem was revisited in [12], where it was found that there are no

troublesome BPS states (i.e. states whose su(2)-spins do not agree) in supergravity. In-

deed, motivated by the suggestive results of a world-sheet analysis, we performed a ﬁrst

principle supergravity calculation [12], and found that the only BPS states that appear in

supergravity have the property that their su(2) spins agree — in the old analysis of [6], the

BPS spectrum of supergravity had only been guessed based on group theoretical methods.

Furthermore, all the states of supergravity satisfy also the A

BPS bound — in fact, this

is also true for the non-BPS states in supergravity whose spins diﬀer — and hence there is

no need for any miraculous quantum correction.

With this roadblock removed, we return in this paper to the search for the CFT dual

of string theory on AdS

× S

. The dual CFT will be motivated by largely

the same methods as those used for the small N = 4 case, i.e. we start with a D-brane

construction and invoke open/closed duality. (Proposals of this form were already discussed

before, in particular in [7], see also [5], but they were discarded because of their failure

to reproduce the ‘old’ BPS spectrum of supergravity as incorrectly predicted in [6].) The

most promising brane construction appears to be the one where we consider Q

D5 branes

wrapping S

× S

, where S

is the special Lagrangian sub-manifold, supported by Q

−

units

of ﬂux, that is wrapped by the Q

D5-branes — this is the “third” construction proposed

by [7]. In addition, we add an arbitrary number of Q

branes smeared on the D5 brane. We

argue that the dual CFT is then the symmetric orbifold of S

× S

, where the ﬂux through

the S

turns out to be (Q

−

) − 1. This can be fairly directly understood for the case

when Q

= 1 since we can then give a direct description of the instanton moduli space; the

result for Q

> 1 is somewhat more conjectural. Since the ﬂux has to be quantized, the

proposal only makes sense if Q

is a factor of Q

−

. We also give a microscopic argument

– 2 –

JHEP08(2017)111

(based on anomaly considerations, following [13]) for where this condition may come from.

We then subject this proposal to some consistency checks. In particular, we show that

the BPS spectra match from both sides. This requires us to determine the BPS spectrum

of the symmetric orbifold of S

in detail, completing the analysis of [7]. We also compute

the full perturbative BPS spectrum from the worldsheet perspective; in particular, we work

out the contribution from the spectrally ﬂowed sectors, extending the analysis of [12] where

only the unﬂowed sector was analysed. As it turns out, this analysis is quite intricate and

the resulting BPS spectrum has the same qualitative structure (including some gaps, see

the discussion in [11]) as for the familiar case of T

and K3.

This paper is organized as follows. In section 2 we discuss various brane scenarios and

discuss their implications on the dual CFT. For the case where only one class of D5-branes

is present, we can read oﬀ an explicit realization of the dual CFT from this picture as the

symmetric orbifold of the S

[7, 14] theory. This is only directly possible for Q

= 1, but

we also speculate how the construction should be generalized to Q

> 1 in section 2.4.

Finally, appendix C is then devoted to reviewing this theory.

Section 3 contains the calculation of the BPS spectrum of the proposed dual — the

symmetric orbifold of S

. This is done carefully and in detail, since there are a number

of subtleties (depending on whether the twist of the twisted sector is even or odd, see

sections 3.3.2 and 3.3.1, respectively) that have to be taken into account. Some of the

more technical arguments are explained in appendix D, but the main result is simple and

spelled out in (3.2).

The next section is concerned with explaining the BPS spectrum of string theory and

supergravity on the background AdS

× S

. Section 4.1 reviews the supergravity

calculation of [12], section 4.2 the corresponding worldsheet calculation. Section 4.3 clariﬁes

some issues relating to the missing chiral primaries in the case of T

, while section 4.4 ﬁnally

discusses the full perturbative BPS spectrum of string theory, described by (4.31). (Some

technical derivations are described in appendix E.)

Section 5 makes some comparisons between the two sides of the duality. First and

foremost, we concentrate on the BPS spectrum (section 5.1), but we oﬀer also some further

tests by employing the chiral ring (section 5.2) of an N = 2 subalgebra of the large N = 4

algebra. We also explain how our proposal leads to the symmetric orbifold of T

in the

inﬁnite radius limit of one of the two three-spheres. Finally, we conclude in section 6. For

the convenience of the reader we have also reviewed some of the algebras that appear in

appendices A and B. Furthermore, we have included the derivation of a non-renormalization

theorem we have used in the main text in appendix F.

2 Strings on AdS

× S

In this section we discuss a brane conﬁguration whose near horizon limit gives rise to

AdS

× S

. Since it is engineered from string theory, we can read oﬀ from it

various aspects of the holographic dual. It also guarantees that the duality is consistent

non-perturbatively.

– 3 –

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