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JHEP05(2017)053

Published for SISSA by Springer

Received: March 17, 2017

Accepted: May 7, 2017

Published: May 10, 2017

Partition function of free conformal ﬁelds in 3-plet

representation

Matteo Beccaria

and Arkady A. Tseytlin

b,1

Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Universit`a del Salento & INFN,

Via Arnesano, 73100 Lecce, Italy

The Blackett Laboratory, Imperial College,

London SW7 2AZ, U.K.

E-mail: matteo.beccaria@le.infn.it, tseytlin@imperial.ac.uk

Abstract: Simplest examples of AdS/CFT duality correspond to free CFTs in d dimen-

sions with ﬁelds in vector or adjoint representation of an internal symmetry group dual in

the large N limit to a theory of massless or massless plus massive higher spins in AdS

d+1

One may also study generalizations when conformal ﬁelds belong to higher dimensional

representations, i.e. carry more than two internal symmetry indices. Here we consider the

case of the 3-fundamental (“3-plet”) representation. One motivation is a conjectured con-

nection to multiple M5-brane theory: heuristic arguments suggest that it may be related

to an (interacting) CFT of 6d (2,0) tensor multiplets in 3-plet representation of large N

symmetry group that has an AdS

dual. We compute the singlet partition function Z on

× S

d−1

for a free ﬁeld in 3-plet representation of U(N ) and analyse its novel large N

behaviour. The large N limit of the low temperature expansion of Z which is convergent

in the vector and adjoint cases here is only asymptotic, reﬂecting the much faster growth

of the number of singlet operators with dimension, indicating a phase transition at very

low temperature. Indeed, while the critical temperatures in the vector (T

∼ N

, γ > 0)

and adjoint (T

∼ 1) cases are ﬁnite, we ﬁnd that in the 3-plet case T

∼ (log N)

−1

, i.e. it

approaches zero at large N. We discuss some details of large N solution for the eigenvalue

distribution. Similar conclusions apply to higher p-plet representations of U(N ) or O(N)

and also to the free p-tensor theories invariant under [U(N)]

or [O(N )]

with p ≥ 3.

Keywords: AdS-CFT Correspondence, Conformal Field Theory

ArXiv ePrint: 1703.04460

Also at Lebedev Institute, Moscow, Russia.

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP05(2017)053

JHEP05(2017)053

1 Introduction

Examples of unitary conformal ﬁeld theories are free theories containing scalar, spinor

or (

− 1)-form ﬁelds in d dimensions. Assuming that these ﬁelds transform in some

representation of a global U(N) or O(N) symmetry group one expects that in the large N

limit this theory should be dual to a theory in AdS

d+1

. The latter should contain a tower

of massless higher spins dual to bilinear conserved currents as well as an inﬁnite collection

of massive higher spins dual to primary operators containing irreducible (“single-trace”)

contractions of more than two ﬁelds.

The simplest case is that of a fundamental (vectorial) representation [1–3] where the

spectrum is particularly simple, containing conserved currents in CFT

or massless higher

spins in AdS

d+1

The dual AdS theory is then a massless higher spin theory with N as

the coeﬃcient in front of the classical action.

Another well-known example is when the free ﬁelds (e.g. spin-1 vectors in 4d) belong to

the adjoint representation [5–9]. Here the dual AdS theory should be string-like: in addition

to massless higher spin ﬁelds it should contain also families of massive ﬁelds which together

may have an interpretation of a spectrum of a “tensionless” string theory in AdS

d+1

The

coupling constant on the AdS side is then 1/N , i.e. the coeﬃcient in front of the AdS ﬁeld

theory action is N

Here we would like to study the next in complexity case when the CFT ﬁeld belongs to

a 3-fundamental or “3-plet” representation, i.e. to a general, symmetric or antisymmetric

3-index tensor representation of a global symmetry group. Already in the simplest case of

the scalar ﬁeld the spectrum of “single-trace” operators with more than two ﬁelds (dual

to massive ﬁelds in AdS) here is much more intricate than in the adjoint case, suggestive

of a “tensionless membrane” interpretation. The coeﬃcient in front of the dual AdS ﬁeld

theory action should now be N

The spectrum of CFT operators on R

(or states in R × S

d−1

) is naturally encoded

in the small temperature T = β

−1

or small x = e

−β

expansion of the partition function

Z(x) on S

×S

d−1

. The singlet-state large N partition function was previously computed

in the U(N) [10] and the O(N ) [11, 12] vector representation case and was matched to the

corresponding massless higher spin partition function in AdS. The partition function in the

adjoint representation case was computed in [13–15] (see also [16–18]) and its matching to

the AdS partition function was discussed in [7–9]. Once the temperature becomes large

enough (of order N

with γ > 0 in the vector case and γ = 0 in the adjoint case) these

partition functions exhibit phase transitions [10, 13, 15] that may have some dual AdS

interpretation (cf. [19]).

Here we will compute the singlet partition function Z(x) in the case of a free CFT in a

3-plet (i.e. 3-index tensor) representation of internal symmetry group.

We shall analyse its

In the (

− 1)-form (e.g. 4d vector ﬁeld) case there are also few additional massive ﬁelds [3, 4].

In the maximally supersymmetric case this interpretation is of course suggested by the canonical ex-

ample of duality between the N = 4 SYM theory and the superstring theory in AdS

×S

where taking the

’t Hooft coupling to zero corresponds to a “tensionless” limit in the quantum string theory.

We shall consider the general (reducible) rank 3 tensor U(N) representation as well as totally symmetric

and antisymmetric cases. We shall also discuss the case of the 3-plet representation of O(N). The large N

results will be similar in all of these cases.

– 2 –

JHEP05(2017)053

small temperature expansion matching the direct operator count and also present the large

N solution of the corresponding matrix model implying the presence of a phase transition

at small x

= e

−β

of order of N

−a

with a > 0, i.e. at T

log N

1.1 Heuristic motivation: 3-plet (2,0) tensor multiplet as M5-brane theory

Before turning to the details of analysis of the large N limit of the 3-plet partition function

let us ﬁrst make some speculative remarks providing a motivation behind this work which is

related to attempts to understand a 6d CFT describing N  1 coinciding M5-branes [20–

32].

The idea that N

scaling of observables in the multiple M5-brane case may be explained

in terms of M2-branes with three holes attached to three diﬀerent M5-branes thus leading

to 3-index world volume ﬁelds was originally suggested in [21]. Then a natural proposal

that a 6d superconformal theory describing multiple M5-branes should involve (2, 0) tensor

multiplets in 3-tensor representation of O(N ) was made in [27, 30].

To recall, the structure of a world-volume theory of a single M5-brane follows from

consideration of symmetries and collective coordinates corresponding to the classical M5-

brane solution in 11d supergravity [40–42]. It is represented by a free (2, 0) tensor multiplet

containing (anti)selfdual 3-form H

µνλ

= 3∂

[µ

νλ]

, 5 scalars φ

and 2 Weyl fermions ψ

This is an example of a free 6d CFT with conformal anomalies and correlators that can be

directly computed [27, 28].

In the case of multiple D-branes the low energy theory is the SYM theory, i.e. one

gets N

vector multiplets at weak coupling and that matches the leading N

scaling pre-

dicted by the dual supergravity for BPS observables. By formal analogy, one needs N

free tensor multiplets to match the leading large N scaling of protected 11d supergravity

observables [20, 21, 24, 27, 28]. This suggests that the (2, 0) multiplet should carry a 3-

index representation of an internal O(N) symmetry group that has dimension ∝ N

large N.

Thus, if the conjectured 6d theory were to have a weak coupling limit its ﬁeld

content would be (B

ijk

µν

, φ

ijk

, ψ

ijk

) with i, j, k = 1, 2, . . . , N.

The reason why the M5-brane world-volume 2-form ﬁeld may carry 3 internal indices

can be heuristically argued for as follows [20, 21, 27]. Replacing the standard picture of

multiple D-branes connected by open strings by M5-branes connected by M2-branes one

may attempt to explain the N

scaling of multiple M5-brane entropy [20] by assuming that

Let us note that a possible connection of multiple M5-brane theory to interacting tensor models was

mentioned in footnote 1 in [33]. Interacting tensor models [34–36] in large N limit in one dimension where

recently investigated in [33, 37] in connection with SYK model. Some properties of scalar ﬁeld tensor models

in d > 1 were studied in [33].

Note that the suggestion to consider tensor multiplets in a 3-plet representation is diﬀerent from at-

tempts to construct an interacting theory of tensor multiplets assigned to adjoint representation of an

internal symmetry group [38, 39].

Explicitly, the number of components of an antisymmetric 3-tensor is

N(N − 1)(N − 2) and of a

symmetric traceless 3-tensor is

N(N − 1)(N + 4).

– 3 –

JHEP05(2017)053

triple M5-brane connections by “pants-like” membrane surfaces are providing the dominant

contribution (pair-wise “cylinder” connections should give subleading N

contributions).

Alternatively, one may think of an interacting (2,0) tensor multiplet theory as a low-

energy limit of a tensionless 6d string theory with closed strings carrying internal 3-plet

indices which originate from virtual membranes connecting three parallel M5-branes: when

the M5-branes get close, the membranes with 3 holes may eﬀectively reduce to strings that

then have 3 internal labels and thus B

ijk

µν

massless modes. The correlation between the 3

Lorentz indices and the 3 internal indices of the corresponding ﬁeld strength H

ijk

µνλ

= dB

ijk

µν

is thus a direct analog of the fact that the YM ﬁeld strength which is a rank 2 Lorentz

tensor carries also 2 internal indices (takes values in adjoint representation).

Ignoring the self-duality constraint on H

µνλ

one may start with a free theory

= H

ijk

µνλ

ijk

µνλ

, δB

ijk

µν

= ∂

[µ



ijk

ν]

, (1.1)

where 

ijk

is a gauge parameter. A speculative idea of how to generalize this to an inter-

acting theory is to assume that the role of the would-be structure constants here should

be played by the generalized ﬁeld strength

ijk

µνλ

itself, i.e. that a non-linear generalization

of the transformation rule for B in (1.1) should have the following structure

δB

ijk

µν

= ∂

[µ



ijk

ν]

µρλ

νρ



+ . . . ,

ijk

µνλ

= ∂

[µ

ijk

νλ]

µρκ

νρ

λκ

+ . . . , (1.2)

where dots stand for terms with other possible contractions of indices. The full non-linear

ﬁeld strength

H should then be non-polynomial in B. Such couplings required to contract

3-plet internal indices may have a generalization to self-dual case and may correspond

to a “soft” gauge algebra structure (thus possibly avoiding no-go arguments against the

existence of an interacting chiral antisymmetric tensor theory in [43, 44]).

One consequence of this 3-index assumption is that the leading interaction between

the B

ijk

µν

-ﬁelds should be quartic [30] rather than cubic as in the adjoint-representation YM

theory

L =

ijk

µνλ

ijk

µνλ

→ ∂B

ijk

∂B

ijk

∂B

+. . . . = G(B)∂B∂B+. . . . (1.3)

Adding scalar ﬁelds of tensor multiplets one may expect to get similar non-linear interac-

tions, e.g., through covariant derivative D

ijk

= ∂

ijk

µρκ

ρκ

+ . . .. Super-

The relevance of triple M5-brane connections by membranes with 3 boundaries was suggested in [21] in

order to explain the

√

scaling of the entropy of the extremal 4-d black hole described by the 2555

intersecting M-brane conﬁgurations. Similar virtual triple connections are not dominant in the case of open

strings ending on D-branes as 3-string interactions are subleading in string coupling. 3-hole “pants-like”

conﬁgurations may be viewed as basic building blocks of 2d membrane surfaces: any membrane surface

ending on several M5-branes may be cut into “pants” and thus surfaces with more than 3 holes should give

subleading contributions in membrane interaction strength. An indication that 3-plet M2-brane interactions

may be relevant is that the C

form of 11d supergravity naturally couples to M2-brane world volume while

the 11d supergravity action contains the cubic interaction term C

∧dC

. In the limit when M5-branes

coincide and thus the M2-brane conﬁguration connecting them becomes small with string-like boundary,

the membrane coupling

should lead to

coupling at the boundary of a membrane and this 2-form

should then carry (ijk) indices. This is analogous to how the B

-ﬁeld coupling to world volume of a string

translates into vector

couplings at the two boundary points of an open string connecting two D-branes.

– 4 –

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