A matrix model for WZW

Abstract

We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal field theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large N limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework.

Published for SISSA by Springer Received: June 7, 2016 Accepted: July 27, 2016

Published: August 1, 2016

A matrix model for WZW

Nick Dorey,a David Tonga,b,c and Carl Turnera

aDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 OWA, U.K.

bDepartment of Theoretical Physics, TIFR,

Homi Bhabha Road, Mumbai 400 005, India

cStanford Institute for Theoretical Physics,Via Pueblo, Stanford, CA 94305, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal eld theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large N limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework.

Keywords: Chern-Simons Theories, Conformal and W Symmetry, Matrix Models

ArXiv ePrint: 1604.05711

JHEP08(2016)007

Open Access, c

The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP08(2016)007

Web End =10.1007/JHEP08(2016)007

Contents

1 Introduction 1

2 The current algebra 42.1 The currents 52.2 Deriving the Kac-Moody algebra 6

3 The partition function 83.1 A digression on symmetric functions 103.2 Back to the partition function 153.3 The continuum limit 19

A Proofs of two classical identities 23

B Kostka polynomials 26

C (A ne) Lie algebra conventions 28

1 Introduction

The purpose of this paper is to describe how a simple quantum mechanical matrix model is related to the chiral WZW conformal eld theory in d = 1 + 1 dimensions.

The matrix model consists of a U(N) gauge eld, which we denote as , coupled to a complex adjoint scalar Z and p fundamental scalars i, i = 1, . . . , p. The action is rst order in time derivatives and given by

S = Z

su(p) a ne Lie algebra at level k can be constructed from the quan

tum mechanical operators Z and i.

1

JHEP08(2016)007

Xi=1iDti (k + p) tr tr ZZ#(1.1)

The covariant derivatives are DtZ = tZ i[, Z] and Dti = ti ii and tr denotes

the trace over U(N) gauge indices. Here and in the following k is a positive integer.

In addition to the U(N) gauge symmetry, the quantum mechanics has an SU(p) global symmetry. We will show that, in the large N limit, this matrix model captures the physics of the SU(p)k WZW conformal eld theory. Specically, we demonstrate the following two results:

The left-moving

dt

"i tr

ZDtZ + ip

[

The partition function of the matrix model can be computed exactly, for all N, as a

function of both temperature and chemical potentials for the SU(p) global symmetry. The result (3.19) is an expansion in Schur polynomials and Kostka polynomials (both of which will be dened later in the paper). In the large N limit, the partition function is proportional to the partition function of the chiral SU(p)k WZW model.

This second property requires some elaboration as the matrix model partition function depends in a rather delicate way on how we take the large N limit. To recover the chiral WZW partition function also known as the vacuum character one should set N divisible by p and subsequently take the large N limit.

One can also ask what happens if we take the large N limit when N = M mod p. In this case, we show that the quantum mechanics partition function is equal to the character of the WZW model associated to a primary in a representation which is perhaps best described as the kth-fold symmetrisation of the Mth antisymmetric representation of SU(p). In terms of Young diagrams, this representation is

M

(

k

z }| { (1.2)

Relationship to Chern-Simons theory. The connection between the matrix model (1.1) and the WZW model is not coincidental; both are related to Chern-Simons theories. Before we derive the results above, we will rst explain why the results described in this paper are not unexpected.

The matrix model (1.1) describes the dynamics of vortices in a d = 2 + 1 dimensional Chern-Simons theory, coupled to non-relativistic matter. This Chern-Simons theory has gauge group and levels

U(p)k,k = U(1)k

JHEP08(2016)007

SU(p)k

Zp (1.3)

where k = (k + p)p.

The relationship between the matrix model and non-Abelian vortex dynamics was explained in [1], following earlier work on vortices in the p = 1 Abelian Chern-Simons theory [2, 3]. The vortices sit in a harmonic trap which forces them to cluster around the origin, where they form a droplet of size N. Outside this region, the gauge group U(p)

is broken; inside it is unbroken.

The upshot is that the solitonic vortex provides a way to engineer Chern-Simons theory on a manifold with boundary, where the role of the boundary is played by the edge of the vortex. It is well known that the gapless excitations of the Chern-Simons theory are chiral edge modes, described by a WZW model with algebra U(p)k,k [4, 5]. The advantage of the present set-up is that we can identify the microscopic origin of these edge modes as the excitations of the vortices. These excitations are captured by the matrix model (1.1).

The vortex perspective also provides a way to understand the delicate manner in which we should take the large N limit. It was shown in [1] that the vortices have a

2

unique, SU(p) singlet, ground state only when N is divisible by p. As we described above, with this restriction in place, the large N limit of the partition function coincides with the partition function of the WZW model.

In contrast, when N = M mod p, the ground state of the vortices is not unique; rather, it transforms in the representation (1.2). This explains why taking the large N limit keeping N = M mod p results in the character of the Kac-Moody algebra associated to this representation.

Relationship to the quantum Hall e ect. Our original interest in the matrix model (1.1) was through its connection to the quantum Hall e ect. When p = 1, it reduces to the matrix model introduced by Polychronakos to describe Laughlin states at lling fraction = 1/(k + 1) [6]. (This matrix model was inspired by an earlier approach using non-commutative geometry [7].) With p 2, the matrix model describes a class of

non-Abelian quantum Hall states with lling fraction = p/(k + p) [1]. These lie in the same universality class as states previously introduced by Blok and Wen [8].

There is a deep connection between the bulk properties of quantum Hall states and the d = 1 + 1 conformal eld theory which describes the dynamics of the edge modes. This connection was rst highlighted in [9] where it was shown that the bulk wavefunction can be reconstructed as a CFT correlation function. This relationship was subsequently used to derive several interesting non-Abelian quantum Hall states [911].

However, one can also go the other way. Starting from a quantum Hall wavefunction, one can enumerate its full set of excitations. These can then be matched to the excitations of the boundary conformal eld theory. This was rst done by Wen for Abelian quantum Hall states [12, 13] and later extended to a number of paired, non-Abelian quantum Hall states in [14].

The connection between the matrix model (1.1) and the WZW model highlighted in this paper falls naturally into this larger quantum Hall narrative. Indeed, it is known that the Blok-Wen states which are the ground states of our matrix model can be reconstructed from correlation functions in the U(p)k,k WZW model [1, 8]. The results of this paper can be thought of as a derivation of the converse story: the excitations of the matrix model coincide with those of the boundary CFT.

The excitations arising from the p = 1 matrix model were previously shown to coincide with those of a chiral boson [15]. For p 2, the story is much richer as the partition function

now depends on both temperature and chemical potentials for the SU(p) avour symmetry. Nonetheless, our results show that the excitations above the quantum Hall state do coincide with those of the boundary conformal eld theory.

The plan of the paper. This paper contains two main results. In section 2 we construct the Kac-Moody current algebra from the quantum mechanics. In section 3 we compute the partition function of the matrix model and explain how to take the large N limit.

The computation of the partition function involves a number of results from the theory of symmetric functions. In an attempt to make this paper self-contained, we have included in section 3.1 a review of the properties of Schur, Hall-Littlewood and Kostka polynomials,

3

JHEP08(2016)007

which are the lead characters in our story. Appendix B contains further details about Kostka polynomials. Other appendices describe our conventions for a ne Lie algebras and the details of the current algebra computation.

2 The current algebra

The purpose of this paper is to explain how the N limit of the matrix model (1.1) is

related to the d = 1 + 1 WZW conformal eld theory. The smoking gun for the emergence of a WZW model is, of course, a current algebra. In this section we will show how to construct such an algebra from the matrix model degrees of freedom Z and i.

The key point is that the U(N) gauge symmetry ensures that Z and are not independent. In particular, Gauss law of the matrix model (1.1) constrains the degrees of freedom to obey

[Z, Z] +

p

Xi=1ii = (k + p)1N (2.1)

Well see that the current algebra arises, in part, due to these constraints.

Both the classical and quantum matrix models exhibit the Kac-Moody algebra. The di erence between the two appears only to be a shift of the level. We will prove that the classical matrix model has an [

su(p) algebra at level k + p. In the quantum theory we nd level k. However, the extra complications in the quantum theory mean that we have been unable to complete the proof of the existence of the algebra; we rely on two conjectured identities which we present below.

This shift of the level can already be seen in the quantum version of the constraint equation (2.1). In the quantum theory, the individual matrix and vector entries Zab and

ia become operators, obeying the canonical commutation relations

[Zab, Zcd] = adbc and [ia, j b] = abij (2.2)

We choose a reference state |0i obeying Zab|0i = ia|0i = 0 and construct a Hilbert space by acting with Zab and ia. The quantum version of the Gauss law constraint (2.1) is interpreted as the requirement that physical states are SU(N) singlets; this can be written in normal ordered form as

: [Z, Z] : +

p

JHEP08(2016)007

Xi=1ii = (k + p)1N (2.3)

Here the : : determines the order in which operators appear with Z moved to the right but not the way that U(N) group indices are contracted; this is determined by the matrix commutator [ , ]. Meanwhile, the level determines the charge under U(1) U(N) that

physical states must carry. Taking the trace of Gauss law, and using the commutation relations (2.2), gives

N

Xa=1

p

p

Xi=1iaia = kN (2.4)

We will see below that a similar normal ordering issue shifts the level of the Kac-Moody algebra.

4

Xi=1iaia = (k + p)N

N

Xa=1

2.1 The currents

It is straightforward to construct generators of the positive graded current algebra in the matrix model. The problem factorises into U(1) and SU(p) parts. The U(1) currents are simply

m = tr Zm while the SU(p) adjoint-valued currents are

mij = i

1pij kZmk

Here i, j, k = 1, . . . , p are avour indices, while m 0 denotes the grading.

It is simple to show that the commutators (2.2) imply that these currents give a representation of half of the Kac-Moody algebra,

[mij,

nkl] = i

ilm+nkj kj

m, J nij] = 0. This holds for any N. This same expression holds in

both the quantum theory and the classical theory where, in the latter, the commutation relations (2.2) should be replaced by classical Poisson brackets.

While the result (2.5) is heartening, our interest really lies in the full Kac-Moody algebra and, in particular, the central extension term. Here we will see the di erence between classical and quantum theories.

The central charge of the U(1) current is harder to pin down due to a possible rescaling. For this reason, we focus on the SU(p) currents. Here too there is a normalisation issue, but one that will turn out to be uniquely xed. To this end, we rescale the positive-graded currents

J mij =

m/2mij m 0

Note that these still obey the algebra (2.5) since the overall scaling is a power of m. We will see that only these rescaled currents will give rise to the full Kac-Moody algebra. We then dene the negative graded currents as

J mij = J |m|ji m < 0and similarly for. These too obey the graded Lie algebra (2.5) if we restrict to m, n < 0.

Of course, the central term only arises when we consider mixed commutators of the form [J mij, J nkl] with m, n > 0. These are trickier to compute because now the con

straint (2.1) comes into play. However, things simplify somewhat in the N limit. We

will show that the currents obey the Kac-Moody algebra

[J mij, J nkl] i(ilJ m+nkj kjJ m+nil) + km m+n,0

(2.6)

Here means up to 1/N corrections. Moreover, the operators in this equation should

act on states that are constructed from the vacuum |0i by acting with fewer than O(N)

creation operators.

The rest of this section is devoted to the derivation of (2.6). (We also show this structure arises perturbatively, in a sense made clear in the appendix, in the Poisson brackets of the classical theory. In that setting we obtain an algebra at the unshifted level k + p.)

5

iZmj

JHEP08(2016)007

m+nil

(2.5)

while [m,

n] = [

(k + p)N p

jkil

1 pijkl

2.2 Deriving the Kac-Moody algebra

The novelty in deriving (2.6) arises from the commutator [Z, Z] terms between currents travelling in opposite directions. We take m, n > 0 and look at

[mij,

nkl] = [iZmj, kZnl]= jkiZmZnl ilkZnZmj + iakb[Zmac, Znbd]jcld = jki[Zm, Zn]l + jkiZnZml

ilkZnZmj + iakb[Zmac, Znbd]jcld (2.7)

All U(N) group indices are contracted in the obvious manner apart from in the nal term where weve written them explicitly.

Our rst goal is to simplify the two commutators in this expression. We deal with them in turn. For the rst, we write

[Zm, Zn] =

m1

Xr=0

JHEP08(2016)007

n1

Xs=0ZrZs[Z, Z]Zn1sZm1r (2.8)

Were going to replace the factor [Z, Z] appearing here with some combination of ii using Gauss law (2.3). However, Gauss law is not an identity between operators; instead it holds only when evaluated on physical states |physi. If we dont include the normal

ordering in (2.3), then the constraint is written as

[Z, Z] + ii

|

physi = (k + p + N)|physi (2.9)

To this end, we consider the operator i[Zm, Zn]l acting on a physical state. Then, after some manipulation, we can use (2.9) to write

i[Zm, Zn]l|physi =

m1

Xr=0

n1

Xs=0iZrZs[Z, Z]Zn1sZm1rl|physi

=

m1

Xr=0

n1

Xs=0iZrZs(k + p i i)Zn1sZm1rl|physi

=

m1

Xr=0

n1

Xs=0iZrZsi iZn1sZm1rl|physi (2.10)

+(k + p)n

m1

Xr=0 iZrZn1Zm1rl|physi

This term above proportional to (k + p)n is key: it will become the central term in the algebra. Well come back to this shortly. Meanwhile, the rst term combines nicely with the second commutator in (2.7). Using the expansion (2.8), it can be written as

iakb[Zmac, Znbd]jcld = iakb

m1

Xr=0

n1

Xs=0 (ZrZs)ad(Zn1sZm1r)bc

!jcld

n1

Xs=0 (iZrZsl)(kZn1sZm1rj) + kl

=

m1

Xr=0

6

where the term proportional to kl arises from commuting kb past ld. It can be neglected simply because we are ultimately interested in the kl-traceless part of this expression.

The four- term above is very close to that appearing in (2.10); it di ers only in its

U(p) indices and overall sign. Adding the two together gives

=

m1

Xr=0

n1

Xs=0

h(iZrZsl)(kZn1sZm1rj)

jk (iZrZsi )(iZn1sZm1rl)

i

We can manipulate the index structure to exploit this similarity: we separate the two double sums into their trace and traceless parts with respect to il, jk, ii and il. Doing

this we nd that the products of traces cancel between the two pairs, as do half of the trace-traceless terms, leaving only traceless-traceless terms which we neglect on the grounds that they are subleading in the large N limit. Were left with

1 p

m1

Xr=0

n1

Xs=0

hil(iZrZsi )(kZn1sZm1rj)

jk(iZrZsi

)(iZn1sZm1rl)

i

JHEP08(2016)007

The rst term above and the third term in (2.7) are both proportional to il; similarly, the second term above and the second term in (2.7) are both proportional to jk. In each

case, the two terms combine together in the large N limit. This follows from the following identity:

Identity 1: for m n,

iZnZml

1 p

m1

Xr=0

n1

Xs=0(iZrZsi )(iZn1sZm1rl) (k + p)N p

n iZmnl

where again means up to 1/N corrections. Further, we are neglecting a trace, propor

tional to il on both sides. A similar expression holds when n > m.

The proof of this identity in the classical theory is already somewhat involved, so we relegate it to appendix A. We have not been able to extend the proof to the quantum case where the additional commutators (2.2) make it much more challenging, though we have checked it for small n and m. In what follows, we will make the natural assumption that this identity generalises directly to the quantum case.

It remains only to discuss the second term in (2.10); this is our central term. We again decompose it into the trace and traceless components with respect to the i, l indices. At large N, the traceless component is subleading; we have

(k + p)njk

m1

Xr=0iZrZn1Zr1rl (k + p)np jkil

m1

Xr=0 iZrZn1Zm1ri

To proceed, we need a second large N identity. This time the identity takes a di erent form in the classical and quantum theories. Evaluated on classical matrices, the identity reads

7

Identity 2 (Classical Version): for m n,

m1

Xr=0iZrZn1Zm1ri p (k + p)N p

We present a proof of this identity in appendix A.

Meanwhile, in the quantum theory there is an extra term which arises due to the shift k + p k seen in (2.4). The corresponding large N identity now reads

Identity 2 (Quantum Version): for m n,

m1

Xr=0iZrZn1Zm1ri p (k + p)N p

We do not have a general proof of this result in the quantum theory. Nonetheless, as before the existence of the new factor can be checked numerically in a number of simple examples.

Putting all of this together, we arrive at our nal result. In the large N limit, up to terms proportional to ij and kl, we have

[iZmj, kZnl]

(k + p)N p

n mn

JHEP08(2016)007

n

1 pk + p

mn

"jkiZmnl ilkZmnj + kn mn jkil#

Written in terms of currents, this is equivalent to the Kac-Moody algebra (2.6).

3 The partition function

In this section, we compute the partition function of the matrix model. In the limit of large particle number, N , we will show that this partition function is proportional

to a character of the chiralp1 current algebra at level k (2.6).

There is a well-established machinery for solving matrix models in the N limit;

the usual route is through the path integral which, at large N, can typically be evaluated by nding an appropriate saddle point for the Wilson lines arising from the gauge eld . Here we will do better and compute the partition function exactly for all values of N. The resulting formula for the partition function, given in equation (3.19), can then be analysed directly in the large N limit. However, the nature of this limit is subtle; in particular, it depends on the value of N modulo p, and does not seem to have a direct interpretation in terms of a saddle point of the original matrix integral.1

In fact, our formula for the partition function of the matrix model can be related [17 19] to the partition function of a certain integrable lattice model in two dimensions which gives rise to conformal eld theory with a ne Lie algebra symmetry in the continuum

1If one tries to take the standard large N approach, the hurdle is to nd a correct way to implement the level constraint (2.4). One can show that integrating out the fundamental matter i results in a Wilson line for the SU(N) gauge eld which sits in the kNth symmetric representation [16]. Because this representation scales with N, it shifts the saddle point in a complicated manner. We have not been able to evaluate the partition function in this approach.

8

n

limit. This limit has been studied in detail in [17], and the results therein lead to a closed formula for the large N limit of the matrix model partition function as an a ne character.

Our partition function will depend on both the (inverse) temperature and the chemical potentials i for the U(1) Cartan elements of the SU(p) global symmetry. Including these chemical potentials, the Hamiltonian for the matrix model (1.1) is

H = trZZ

p

Xi=1iii (3.1)

The Hamiltonian is trivial: it simply counts the number of Z and i excitations, weighted by and i respectively. Evaluated on any physical state, the Hamiltonian gives

H|physi =

p

Xi=1 iJi

!|physi

where the quantum numbers and Ji are integers labelling each state.Our interest is in the partition function

Z(q, xi) = Tr eH = Tr q

p

Yi=1 xJii

JHEP08(2016)007

where Tr is the trace over all states, q = e and xi = ei.

All the complexity in the problem lies not in the Hamiltonian, but instead in the nontrivial structure of the physical Hilbert space originating in the constraints imposed by the U(N) gauge symmetry. Our strategy is to rst enumerate all gauge non-invariant states and only later project onto the gauge invariant subset. With this in mind, we introduce further fugacities for each Cartan element of the gauge symmetry, U(1)N U(N). We call

these fugacities a with a = 1, . . . , N.

If we ignore the restrictions of gauge invariance, then the Hilbert space is simple to dene: it consists of any number of Zab or ai operators acting on |0i. Lets deal with

each species of operator in turn. The Z operators lie in the adjoint representation of U(N) and are singlets under SU(p). They carry quantum numbers of +1a1b (for some a 6= b)

and = 1. Taking the trace over states of the form Zr|0i for all possible r gives the

contribution to the partition function of the form

ZZ =

N

Ya,b=111 qa/b

(3.2)

Meanwhile, the operators transform in the fundamental of both U(N) and SU(p). This means that they come with a factor +1ax+1i for some a and i. They have = 0. Taking the trace over states of the form r|0i gives the contribution to the partition function

Z =

N

Ya=1

Yi=111 axi

p

(3.3)

9

We now impose the requirements of gauge invariance. The physical states must be SU(N) singlets. Further, the level constraint (2.4) requires that physical states carry charge k under the U(1) centre of U(N) but are singlets under SU(N) U(N). This can be

imposed by contour integration, giving us the expression

Z(q, xi) = 1 N!

N

Ya=11 2i

I

da k+1a

! Y

ZZ Z

(3.4)

where the contour of integration is the unit circle in the complex plane for each integration variable. Here the product factor arises from the Haar measure on the group manifold of U(N). The factor of k+1 in the denominator ensures that the only contributions we pick up in the contour integral are those with correct overall charge.

Our strategy for evaluating the partition function will be to expand the integrand of (3.4) in a suitable basis of polynomials. The integration variables wa and the fugacities xi are invariant under permutations corresponding to the Weyl groups of U(N) and SU(p) respectively. This means that the partition function can be expanded in terms of symmetric polynomials. Before proceeding we pause to review some elementary facts about these functions.

3.1 A digression on symmetric functions

In this subsection we review some standard facts about symmetric functions. For further details and proofs of the statements reviewed below see [21]. As symmetric functions are labelled by partitions we will begin by reviewing basic features of the latter.

A partition is a non-increasing sequence of non-negative integers,

1 2 3 . . . () > ()+1 = 0The number () of non-zero elements in the sequence is called the length of the partition.

The sum of all the elements, || =

b6=c

1 b c

JHEP08(2016)007

Pi1 i, is called the weight of the partition. We will write P for the set of all partitions.

The multiplicity mj() of the positive integer j is the number of times that j appears in the partition ; i.e.

mj() = |{i 1 : i = j}|

We can specify a partition either by listing its non-zero parts, = (1, 2, . . . , ()) or by specifying multiplicities. For example the partition (7, 5, 5, 3, 3) can alternatively be written as (71, 52, 32) where the exponent of each entry indicates its multiplicity. We will use this notation extensively below.

A partition can be represented graphically by a Young diagram Y(). This is an array of boxes where the ith row contains i boxes. Each row is aligned so that the left-most boxes sit under each other. For example, the Young diagram for the partition (71, 52, 32)

looks like this:

10

Concretely the set Y() contains boxes x = (r, s) labelled by their coordinate r and s specifying the row and column respectively of the diagram relative to the top left hand corner of the diagram. The Young diagram Y() therefore contains boxes x = (r, s) with r = 1, . . . , () and, for each value of r, s = 1, . . . , r.

The transpose T of the partition is obtained by interchanging the rows and columns of the Young diagram Y(). Explicitly the non-zero parts of T are

Ti = |{j 1 : j i}|for i = 1, . . . , (T ) = 1. For example, (71, 52, 32)T = (53, 32, 12). Finally, we also dene the function n : P

Z0 by

n[] =

JHEP08(2016)007

Xi1(i 1)i

We now turn to symmetric functions. Let X = {x1, . . . , xn} denote a set of n variables.

A symmetric function f(X) = f(x1, x2, . . . , xn) is any polynomial of the xi invariant under the action of the permutation group Sn acting on the variables X, so

f x(1), x(2), . . . , x(n)

= f(x1, x2, . . . , xn) Sn

We will frequently use the shorthand notation {g(X)} = g(x(1), x(2), . . . , x(n)) for the action of a permutation Sn on an arbitrary function g(X).

The set of all symmetric polynomials forms a vector space. Though innite dimensional, it is naturally written as a direct sum of vector subspaces of nite dimension corresponding to symmetric polynomials of xed degree. The basis vectors are labelled by partitions P with at most n parts: () n. In particular, one possible choice of basis

vectors are monomial symmetric functions, given by

m(X) =

XSn/Sn {x11x22 . . . xnn} (3.5)

Here Sn denotes the stabiliser of the monomial X = x11x22 . . . xnn in Sn and thus the sum is taken over distinct permutations {X} = x1(1)x2(2) . . . xn(n) of the monomial X. The

degree of the monomial symmetric function m(X) corresponds to the weight || of the

permutation . One can easily dene an inner product on the space of symmetric functions with respect to which the monomial symmetric functions form an orthonormal basis:

hm, mi

1 n!

Yi=11 2i

ICdxi xi

!

m (X) m X1

= ,

n

Here the contour of integration is the unit circle C in the complex xi-plane for i = 1, 2, . . . , n and X1 denotes the n variables {x11, x12, . . . , x1n}.

Another possible set of basis vectors for the space of symmetric functions of n variables is provided by the Schur functions. For each partition P, we dene the Schur function

s(X) =

XSn

x11x22 . . . xnn

Yi>j1

1 xixj

11

Although not immediately apparent from this denition, the Schur function s(X),

like the monomial symmetric function m(X), is a polynomial in the variables X of degree

||. The signicance of the Schur functions for our problem lies in their close relation to the

representation theory of the Lie algebra u(n). The nite dimensional, irreducible representations of u(n) are inherited from those of its complexication gl(n, C). Each such representation is labelled by a partition of length () n. Equivalently, the representation is

labelled by the Young diagram Y(). As discussed in more detail below, the Schur function s(X), evaluated on the n variables X = {x1, . . . , xn} is essentially the character of the cor

responding representation R. This correspondence is a consequence of the famous Schur-Weyl duality between the representation theory of u(n) and that of the permutation group.

Like the monomial symmetric functions discussed above, the Schur functions provide a basis for the vector space of symmetric functions. Indeed one can construct a matrix K giving the explicit linear transformation between these two bases by writing

s (X) =

X

K,m (X) (3.6)

Here K, is zero unless || = ||. The non-vanishing entries of K, are all positive inte

gers, known as Kostka numbers. Thinking of s as the character of R, each monomial in the Schur polynomial corresponds to a weight of the representation and the corresponding coe cient is simply the multiplicity of this weight. More precisely, each monomial symmetric function m appearing on the r.h.s. of (3.6) corresponds to a family of gl(n, C)

weights related by the action of the Weyl group. The Kostka number K, is the precisely

the common multiplicity of these weights in R.

The Kostka numbers also have a second interpretation in the representation theory of u(n) which will be important below. Let nj denote the jth symmetric power of the fundamental representation. Then K, is the multiplicity of the irreducible representation

R in the decomposition of the tensor product

T () = n1 n2 . . . n() (3.7)

The Schur functions form a complete basis for the symmetric functions in n variables. They are orthonormal with respect to a modied inner product h , iS dened by

hs, siS

1 n!

JHEP08(2016)007

Yi=11 2i

ICdxi xi

! Y

i6=j

1 xi xj

s (X) s X1

n

= , (3.8)

In the group theoretic context described above, this relation is just the familiar orthogonality of U(n) characters with respect to integration over the group manifold with the Haar measure. The completeness of the Schur functions as a basis is expressed by the Cauchy identity. For any two sets of variables X = {x1, . . . , xn} and Y = {y1, . . . , ym} we have

n

Yi=1

m

Yj=111 xiyj=

Xs(X)s(Y ) (3.9)

The sum on the right-hand side can be taken over all partitions as the product of Schur functions in the summand will vanish identically for () > min{n, m}.

12

As stated above, our goal will be to evaluate the matrix model partition function by expanding the integrand of (3.4) in terms of symmetric functions. Because of the presence in this integrand of the Haar measure factor, together with the adjoint partition function ZZ, it will be convenient to introduce yet another inner product h , iP on the

space of symmetric functions depending on an arbitrary complex parameter q. For any

two symmetric functions f(X) and g(X) we dene

hf, giP

1 n!

Yi=11 2i

ICdxi xi

! Q

i6=j

n

1 xixj

Qi6=j

(3.10)

Note that our new inner product reduces to h , iS in the special case q = 0. Can we nd

a new set of basis functions, generalising the Schur functions, which are orthogonal with respect to new measure with q 6= 0? In fact the Hall-Littlewood polynomials have exactly

this property. Moreover, many of the properties of the Schur polynomials discussed above are generalised in a nice way. For each partition P we dene

P(X; q) = 1

NXSn

1 q xixj f (X) g X1

x11x22 . . . xnn

Yi>j

1 q xixj

1 xixj

JHEP08(2016)007

(3.11)

The normalisation factor is given by

N =

n()

Qj1 mj() (1 q)n

where

m =

m

Yj=1

1 qj

and mj() denotes the multiplicity of the positive integer j in the partition as dened above. As before, P(X; q), is a homogeneous polynomial in the variables X of degree ||.

It is useful to rewrite the denition (3.11) as

P(X; q) =

XSn/Sn

x11x22 . . . xnn

Yi<j

1 q xixj

1 xixj

(3.12)

where the sum is over distinct permutations of the monomial X = x11x22 . . . xnn.

As already mentioned, we have the orthogonality property

hP, PiP =

1

N , (3.13)

An even more striking fact is that, with the given normalisation, each term in P(X; q) is

itself a polynomial in the parameter q with integer coe cients. One might instead choose to normalise these functions to achieve orthonormality with respect to the inner product

h , iP ; however then the basis functions would no longer be polynomial in q.

13

The completeness of the resulting basis is expressed in a generalisation of the Cauchy identity. As before we consider two sets of variables X = {x1, . . . , xn} and

Y = {y1, . . . , ym}. We now have

n

Yi=1

where b(q) =

Qj1 mj()(q). From the denition (3.11), the Hall-Littlewood polynomial P(X; q) reduces to the Schur function s(X) for q = 0. On setting q = 1 in the denition, we also nd P(X, 1) = m(X) where m is the monomial symmetric function dened in (3.5) above. Again we can nd a matrix describing the change of basis from Schur to Hall-Littlewood. Relation (3.6) is now generalised to

s (X) =

X

K,(q)P (X; q) (3.15)

For each choice of , P, the matrix elements K,(q) are polynomials in the parameter

q. They are known as Kostka polynomials (see e.g. chapter III.6 of [21]) and they will play a central role in our evaluation of the partition function. An explicit combinatoric formula for the Kostka polynomials due to Kirillov and Reshetikhin is given in appendix B. Here we will list some of their main features:2

They are polynomials in q of degree n[]n[] with leading coe cient equal to unity.

All non-zero coe cients are positive integers.

K,(q) = 0 unless || = ||.

They reduce to the Kostka numbers for q = 1: K,(1) = K, for all partitions and .

K,(0) = ,.

These properties ensure that the Kostka polynomials can be regarded as a graded generalisation of the Kostka numbers. As the Kostka numbers K, count the number of occurrences of the representation R in the tensor product T () dened in (3.7), the cor

responding Kostka polynomial K,(q) receives a contribution q for some

Z0, for

each such occurrence. Hence as we vary the partition , the Kostka polynomial assigns an integer-valued energy (, ) to each irreducible component of the tensor product T ().

It is useful to think of the representation space of T () as the Hilbert space of a spin chain

with () sites with a u(n) spin in the representation ni at the ith site. Remarkably, the energy precisely corresponds to one of the Hamiltonians of the Heisenberg spin chain with these spins. In fact is the essentially lattice momentum for a spin chain with periodic boundary conditions. The Bethe ansatz solution of this system provides an e cient

2The last two listed properties follow easily from the denition (3.15) and the third follows from the invertibility of the change of basis proven in [21] (see eq. (2.6) in chapter III of this reference). The rst two properties are highly non-trivial and were rst proven in [22].

14

Yj=11 qxiyj1 xiyj=

Xb(q)P(X; q)P(Y ; q) (3.14)

m

JHEP08(2016)007

combinatoric description of the corresponding Kostka polynomials and leads directly to the explicit formulae given in appendix B.

To evaluate the partition function we will need one more class of symmetric functions known as3 Modied Hall-Littlewood polynomials [23, 24] Q(X; q). For our purposes, this polynomial is dened by the formula

Q(X; q) =

X

K,(q)s (X) (3.16)

Importantly, this denition yields a non-zero answer for partitions of any length4 Thus, unlike the other symmetric functions dened above, Q(X; q) does not vanish identically5 for partitions with () > n.

As discussed above, each Schur function s is the character of a u(n) representation R. Meanwhile, the Kostka polynomial K,(q) is non-zero only for irreducible representations R occurring in the tensor product T () dened in (3.7). Further, for each , K,(q)

receives a contribution q for each occurrence of the irrep R in T () where is the

appropriate spin chain Hamiltonian. Putting these facts together we learn that Q(X; q)

has a natural interpretation as the partition function of a spin chain dened on the tensor product space T ().

Using the properties of the Schur functions and Kostka polynomials, we see that Q(X; q) is a homogeneous polynomial in the variables X = {x1, . . . , xn} of degree ||.

Moreover the coe cients are themselves polynomials in q with positive integer coe cients. The polynomial Q has the following key property: it is adjoint to the ordinary Hall-

Littlewood polynomials P with respect to the inner product h , iS for Schur functions [23].

For any two sets of variables X = {x1, . . . , xn} and Y = {y1, . . . , ym} we have

n

Yi=1

m

X

Q(X; q)P(Y ; q) (3.17)

using (3.15) and (3.16). The nal sum on the r.h.s. can be taken over all partitions but the summand will vanish unless () m.

3.2 Back to the partition function

We are now ready to compute the partition function Z dened in (3.4). The partition

function is symmetric in the u(p) fugacities X = {x1, x2, . . . , xp} so we can expand it

3These are also sometimes referred to as Milne polynomials in the mathematical literature.

4This follows because, although the r.h.s. of (3.16) vanishes identically for partitions of length greater than n, there is no such constraint for the partition appearing in the Kostka polynomial K,(t). Indeed

we evaluate several examples of this type in the following using the combinatorial algorithm of appendix B.

5Note that in some references the denition of the modied Hall-Littlewood polynomial in n variables is nevertheless restricted to the case () n.

15

JHEP08(2016)007

Yj=111 xiyj=

=

=

Xs(X)s(Y )

X,

K,(q) P(Y ; q) s(X)

in terms of Schur functions. As each Schur function corresponds to the character of a nite-dimensional irreducible representation of U(p), the resulting expansion determines the multiplets of the SU(p) U(p) global symmetry present in the matrix model spectrum.

The integrand of the partition function is also a symmetric function of the U(N) fugacities = {1 . . . , N} and we may thus expand it in terms of a suitable set of basis functions.

To proceed to the answer by the shortest path, we will use the Cauchy identity in the form (3.17) to expand the factor of the integrand corresponding to the fundamental-valued elds as

Z =

N

Ya=1

Yi=111 axi=

p

X

Q(X; q)P( ; q)

In contrast, the corresponding factor ZZ for the adjoint-valued eld will be left unexpanded

as part of the integration measure. For the next step, we use the denition of the Hall polynomials in its second form (3.12) to write

1

QNa=1 ka

= P(kN ) 1; q

where, as above, (kN) denotes the partition with N non-zero parts each equal to k. The resulting integral over the variables can then be written, using (3.13), as an inner product

Z =X

Q(X; q) 1 N!

N

JHEP08(2016)007

Ya=11 2i

ICda a

! Q

a6=b

1 ab

Qa,b

1 q ab

P ( ; q) P(kN ) 1; q

X

Q(X; q) 1(1 q)N hP, P(kN )iP (3.18)

= 1

N(q) Q(kN)(X; q)

Thus our nal result for the partition function of the matrix model is

Z =

N

=

Yj=11(1 qj)

X

K,(kN )(q) s(X) (3.19)

where the sum on the right-hand side runs over all partitions P. However, as explained

above the summand vanishes unless we have || = |(kN)| = kN and () p.

Ground state energy. Our result for the partition function (3.19) holds for all positive integral values of the level, k, the rank p of the global symmetry and the particle number N. In the remainder of this section we will extract the ground state energy E0(k, p, N), which simply corresponds to the leading power of q appearing in the expansion of the partition function for |q| 1, and compare it with our expectations based on the analysis of the

ground state given in [1].

We begin with the abelian case p = 1. In this case there is only one partition = (kN) which satises the conditions || = kN, and () 1. In this special case, the formulae

16

for the Kostka polynomial given in appendix B simplify, giving6

K(kN),(kN )(q) = qn[

where, as above, n[] =

where x = x1 acts as a fugacity for the U(1) charge which is xed by the D-term constraint. The ground state energy,

E0(k, 1, N) = k2 N(N 1)agrees with the identication of the ground state given in [1]. The remaining factor in (3.20) is a plethystic exponential accounting for excitations corresponding to all possible products of the N independent single-trace operators tr(Zl) with l = 1, 2, . . . , N. This partition function for the p = 1 matrix model was previously computed in [15].

The partition function for general p 1 is somewhat richer; it also depends on the

fugacities xi for the SU(p) Cartan elements. To understand the form of this partition function, we start by recalling that the Kostka polynomial K,(kN )(q) specialises for q = 1 to the Kostka number K,(kN ). This in turn coincides with the multiplicity of the irreducible representation of u(p) specied by the partition in the tensor product

TN = pk pk . . . pkof N copies of the kth symmetric power of the fundamental representation p. The corresponding Kostka polynomial is a gradation of the Kostka number where each power of q appears with a non-negative integer coe cient. As the Kostka polynomial K,(kN )(q)

appears in (3.19) multiplied by the corresponding Schur function s(X), we deduce that the full spectrum of the matrix model transforms in the reducible u(p) representation TN.

More precisely, the overall prefactor of

QNj=1(1 qj)1 means we actually have an innite number of copies of this representation. The additional information contained in the partition function is the energy of each irreducible component in the tensor product.

To go further we will need to use the combinatoric description of the Kostka polynomials given in appendix B. We will start with the easiest case k = 1 where an explicit formula is available. Here we have

K,(1N )(q) = qn[

where H(q) is the hook-length polynomial given by

H(q) =

YxY()

17

(kN )]

Pi1(i 1)i. Thus the partition function for p = 1 reads

Zp=1 = xkNq

k

2 N(N1)

N

Yj=11(1 qj)

(3.20)

JHEP08(2016)007

T ]

QNj=1 (1 qj)H(q)

1 qh(x)

6Here we see explicitly that K,(t) can be nonzero when () > () as mentioned above.

Here the product is over the boxes x = (r, s) of the Young diagram Y() corresponding to the partition and h(x) = r + Ts r s + 1 > 0 is the length of the hook passing

through box x.

As we described above, T denotes the transpose of the partition , obtained by interchanging the rows and columns of the Young diagram Y(). Explicitly the non-zero parts of T are

Ti = |{j 1 : j i}|for i = 1, . . . , (T ) = 1. To nd the ground state energy of the model we must therefore minimise the quantity

n

(pL, M) = 12L(L 1)p + LM

for N = Lp + M. The Schur polynomial corresponds to the representation of U(p)

U(1) SU(p) with U(1) charge N, which coincides with the Mth antisymmetric power of

the fundamental. Again, this yields complete agreement with the properties of the ground state discussed in [1].

Although the formulae for the Kostka polynomials are more complicated, the generalisation of this analysis to k > 1 is straightforward. As we discuss in appendix B, the minimum energy is obtained for the partition

0 = (kL + k)M, (kL)pM

and takes the value

E0(k, p, N) = k2 L(L 1)p + kLMfor N = Lp + M. The resulting ground state has U(1) charge kN and transforms in an irreducible representation of SU(p) corresponding to a k-fold symmetrisation of the Mth

antisymmetric power of the fundamental representation. This is the representation (1.2) that we mentioned in the introduction; it is in agreement with the results of [1].

18

JHEP08(2016)007

T =

Xi1(i 1)Ti

as we vary over partitions with || = N and () p. These restrictions correspond to

demanding that |T | = || = N and that Ti p for all i 1. Writing N = Lp + M for

non-negative integers L and M < p, the minimum occurs for the partition T0 = (pL, M) corresponding to

0 = (L + 1)M, LpM

Thus the leading term in the partition sum for |q| 1 is

Z qE0(N) s((L+1)

with vacuum energy

M ,LpM ) (x1, . . . , xp)

E0(1, p, N) = n

3.3 The continuum limit

In this section we will investigate the N limit of the partition function (3.19). As

we saw in the previous section, the ground state energy and its quantum numbers under the global U(p) symmetry depend sensitively on the value of N mod p. This means that in order to get a sensible limit, we must hold this value xed as N . Setting

N = Lp + M

for non-negative integers M < p and L, we therefore take the limit L with M and p

held xed.It is also convenient to factorise the partition function as

Z = qE0wkN/p

Z

where

E0 = E0(k, p, N) = k2 L(L 1)p + kLMis the ground state energy and w = x1x2 . . . xp is the fugacity for the u(1) centre of u(p).

The reduced partition function

Z =

N

JHEP08(2016)007

Yj=11(1 qj)qE0

X

K,(kN )(q) wkN/ps(X) (3.21)

thus encodes the energies and u(1) charges of the states in the spectrum relative to those of the ground state. As we will see, it is Z rather than the original partition function

which has a non-singular N limit. The main result of this section is (3.30) which

says (roughly) that

lim

N Z =

Yj=11(1 qj)Rk,M (q; X) (3.22)

Here the roughly refers to a slight notational subtlety regarding the di erence between the U(p) fugacities labelled by X and the SU(p) fugacities; this will be explained below. The key part of the result is that Rk,M denotes the character of thep a ne Lie algebra at level k associated to the representation Rk,M of SU(p). When M = 0, so N is divisible by p,

this is the vacuum character which coincides with the partition function of the WZW model. Meanwhile, for M 6= 0, Rk,M is the k-fold symmetrisation of the Mth antisymmetric repre

sentation, namely (1.2). The remainder of this section is devoted to the derivation of (3.22).

As discussed above, each Schur function s(X) appearing in the sum on the r.h.s. of (3.21) is the character of the irreducible representation of u(p) corresponding to the partition . In the following it will be convenient to decompose the global symmetry as u(p) u(1) su(p). Recall that the nite-dimensional, irreducible representations of

su(p) are in one-to-one correspondence with Young diagrams having at most p 1 rows

or, equivalently with partitions ~

having (~

) < p. In contrast, representations of u(p) correspond to diagrams with at most p rows or to partitions with () p. Given an

irreducible representation of u(p), we obtain a unique irreducible representation of su(p)

19

by removing all columns of height p from the corresponding Young diagram. Similarly, for any partition with () p we may nd a unique partition

~

with (~

) < p such that

i = ~

i + Q with i = 1, . . . , p for some non-negative integer Q. In the following we will abbreviate this relation as

= ~

+ (Qp) (3.23)

The Kostka polynomial K,(kN )(q) is only non-zero if || = |(kN)| = kLp + kM. Given

any partition ~

with (~

) < p and |

~

| kN, we may nd a unique partition =

~

+ (Qp)

obeying this constraint if and only if |

~

| kM is divisible by p, in which case we set

Q = Q(~

) = kL

1 p

|~| kM

(3.24)

As the Schur function s(X) is a homogeneous polynomial of degree || in the variables

X = {x1, x2, . . . , xp}, we may write

s(X) = w||/ps~( ~X)

where

~X = X/w1/p = {x1w1/p, . . . , xpw1/p} (3.25)

In particular note that, by construction, ~x1~x2 . . . ~xp = 1, which implies s( ~X) = s~( ~X).

Using the above results, we can trade the sum over all partitions appearing in (3.21) for a sum over ~

of length (~

) < p. This gives

Z =

N

JHEP08(2016)007

Yj=11(1 qj)qE0

X~, (~)<pK~+(Q(~)p),(kN)(q) s~( ~X) (3.26)

where Q(~

) is a non-negative integer given by (3.24) when |

~

| = kM mod p and |

~

| kN,

and is set to zero otherwise.

It will also be useful to make the relation between Schur functions and the characters of the simple Lie algebra su(p) and its complexication Ap1 = sl(p, C) more explicit (see appendix C for our Lie algebra conventions). The nite-dimensional irreducible representations of Ap1 are labelled by dominant integral weights L+W . Each such has an

expansion in terms of the fundamental weights { (1), . . . , (p1)}:

=

p1

Xj=1 j (j)

with coe cients j

Z0 known as Dynkin labels. Let R denote the corresponding Ap1 representation with representation space V . The character of R is a function of variables

Z = {z1, . . . , zp1} which encodes the weights of this representation or, equivalently, the

eigenvalues of the matrices R (hi) representing the Cartan subalgebra generators hi with i = 1, . . . , p 1 in the Chevalley basis. Explicitly we dene

(Z) = TrV

p1

Yj=1zR (hj ) j

20

As mentioned above irreducible representations of su(p) can also be labelled by partitions ~

with (~

) < p. For each dominant integral weight L+W , the corresponding partition

~

( ) has parts ~

i =

Pp1j=i j. The character of R can then be related to the Schur function of the partition ~( ):

(Z) = s~( )( ~X)

where the variables ~X = {~x1, . . . , ~xp}, obeying ~x1~x2 . . . ~xp = 1, are related to Z = {z1, . . . , zp1} by

z1 = ~x1

z2 = ~x1~x2...

zp1 = ~x1~x2 . . . ~xp1 (3.27)

Irreducible representations R of Ap1 are further classied by the congruence class of the corresponding highest weight , given in terms of the Dynkin labels by the value P( ) of 1 + 22 + . . . (p 1)p1 modulo p. Equivalently

P( ) is equal modulo p to the weight |

~

|

of the partition ~

corresponding to or to the number of boxes in the corresponding Young diagram. We denote by L+W (

P) the subset of the positive weight lattice L+W corresponding

to positive weights in Pth congruence class.The above results mean that we can rewrite the sum over partitions ~

appearing in the reduced partition function (3.26) as a sum over dominant integral weights of Ap1 in the

congruence class P( ) = kM mod p. Our nal rewriting of the reduced partition function is

Z =

Here

and

where

JHEP08(2016)007

Yj=11(1 qj)

X L+W (kM)

K (q) (Z) (3.28)

N

K (q) = qE0(k,p,N)K( ),(kN )(q)

( ) = ~

( ) + (Q( )p)

|~( )| kM

We will now switch gears and consider something seemingly quite unrelated to the above discussion; the representation theory of the a ne Lie algebrap1. (Again, see appendix C for conventions.) We work in a Chevalley basis with generators {hi, ei, fi}

where the index i now runs from zero to p1. A complete basis also includes the derivation

or grading operator L0 associated with the imaginary root.

The weights of any representation ofp1 lie in the a ne weight lattice, whose basis vectors are the fundamental weights

(j) with j = 0, 1, . . . , p 1. The integrable represen

tations ofp1 are labelled by a highest weight

=

X

Q( ) = max

0, kL 1 p

21

(j)

whose p Dynkin labels {

j} are non-negative integers. Each integrable representation has

a denite level which is a non-negative integer given by the sum of the Dynkin indices,

k =

0 +

1 + . . . +

p1

The resulting representations R are the a ne analogs of the nite-dimensional irreducible representations R of the simple Lie algebra Ap1 discussed above. We denote the corresponding representation space V . The character (q; Z) of the representation

R is a function of the variables q and Z = {z1, . . . , zp1} which encodes the weights of

the representation or, equivalently, the eigenvalues of the representatives of the Cartan generators hi, for i = 1, . . . p 1 of the global subalgebra Ap1

p1 together with those

of the derivation L0 acting in V . Explicitly we dene

(q; Z) = TrV

qR (L0)

p1

Yj=1zR (hj ) j

Any representation ofp1 must also provide a representation of the global subalgebra Ap1. Thus the a ne character must have an expansion in terms of Ap1 characters of the form

(q; Z) =

X L+Wb (q) (Z)

The coe cients b (q) are polynomials in q with non-negative integral coe cients. They are known as the branching functions for the embedding of Ap1 inp1. Another way to characterise them is to pick out only those vectors in the representation space V which are

highest weight with respect to the global generators. Thus we dene, for each dominant

integral weight of Ap1 with =

Pp1i=1 i (i), the following subspace:

V = {|

JHEP08(2016)007

i V : hi|

i = i|

i, ei|

i = 0 i = 1, . . . , p 1}

Then we have

b (q) = TrV

hqR (L0)

i

Remarkably a relation between the large N = Lp + M limit (with xed M and p) of the object K (q) dened in (3.28) above and a particular a ne branching function ofp1 is obtained in [17], proving an earlier conjecture of [20]. In particular, we must consider the integrable representation R with

= k (M). The primary states in the representation (i.e. those with the lowest L0 eigenvalue) transform in the Ap1 representation with 0 = k (M);

this corresponds to the k-fold symmetrisation of the Mth antisymmetric power of the fundamental representation. This is indeed the expected representation (1.2) for the ground state of the model. This representation has congruence class P( 0) = kM mod p and the remaining dominant integral weights for which b (q) is non-zero necessarily lie in the

same congruence class. Corollary 4.8 of [17] states that, for all L+W (kM), we have lim

N K (q) = b k (M)(q) (3.29)

22

This result has its origin [18] in the relation between the Kostka polynomials and the partition function of an integrable Ap1 spin chain to which we alluded above. Under favourable conditions, the relevant spin chain is believed to go over to the SU(p) Wess-Zumino-Witten model in the continuum limit [26, 27]. Kostka polynomials also appear [19] in the partition function of the so-called RSOS models, which yield coset conformal eld theories with a ne Lie algebra symmetry in the continuum limit.

Incorporating the above limit in the reduced partition function as given in (3.28) we reach our nal result

lim

N Z =

Yj=11(1 qj)k (M) (q; Z) (3.30)

where the variables Z = {z1, . . . , zp1} are related to the su(p) fugacities of the matrix

model by equations (3.25) and (3.27). The prefactor encoding the excitation spectrum of the u(1) sector of the model precisely corresponds to the partition function of a chiral boson.

Acknowledgments

We are grateful to Sean Hartnoll, Gautam Mandal and Shiraz Minwalla for many useful conversations. DT and CT give thanks to the theory group in TIFR for their very kind hospitality while this work was undertaken. DT is also grateful to the Stanford Institute for Theoretical Physics for hospitality while this work was written up. We are supported by STFC and by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013), ERC grant agreement STG 279943, Strongly Coupled Systems.

A Proofs of two classical identities

In this appendix we prove the two classical identities that we used to exhibit the existence of a Kac-Moody algebra. Assuming m n, they are

Identity 1:

iZnZml

1 p

JHEP08(2016)007

m1

Xr=0

n1

Xs=0(iZrZsi )(iZn1sZm1rl) il( )

kN p

niZmnl il( ) (A.1)

Identity 2 (Classical Version):

Xr=0iZrZn1Zm1ri p (k + p)N p

nmn (A.2)

m1

where means up to 1/N corrections and, in the rst identity, we subtract o the il trace

on both sides.

The phrase up to order 1/N corrections implicitly includes a restriction on the kind of classical solutions on which we should evaluate these expressions. Roughly speaking,

23

the solutions shouldnt deviate by O(N) from the ground state. We start by describing in

more detail what this means.For the p = 1 matrix model, the ground state was given in [6]

Z = Z0(N)

k

and = 0(N)

k

0 0

...0 N

0 1

0 2

...

0 N 1

0

together with = (2/B) diag(N 1, N 2, . . . , 2, 1, 0).

It is simple to embed these solutions in the more general matrix model. The number of ground states now depends on the relative values of N mod p. It is simplest when N is divisible by p. In this case there is a unique ground state which takes the block diagonal form

Z = Z0(N/p) 1p , = 0(N/p) 1p (A.3) where weve written the i (with i = 1, . . . , p) as an N p matrix, denoted by .

If N is not divisible by p then there are multiple classical ground states, transforming in the representation (1.2) [1]. For example, if N = 1 mod p then each of the blocks has Z0((N 1)/p), except for one which has Z0((N +p1)/N). There are p such choices; these

ground states transform in the p of the SU(p) global symmetry. Similarly, if N = q mod p then there are p

q

ground states, transforming in the qth antisymmetric representation of

SU(p).

In what follows, we will assume that N is divisible by p. Now we can make our statement about O(1/N) corrections more precise. We should treat O(N1/2) and

Z O(N1/2), since in the ground state the largest components of either scale like the

square root of N, and even when contracting indices there is only one non-zero entry per row or column. (This is important to check because there are O(N) components, which could upset our counting.) We will evaluate the identities on states which di er from the ground state by O(1) when measured naturally by the norm squared of Z and . It is

important that these states still satisfy the Gauss law constraint (2.1).

These restrictions make it fairly straightforward to prove the classical version of Identity 2. Consider a linear expansion of the left-hand side around the ground state in powers of N1/2; we obtain the zeroth order term plus something we can bound by Nn1/2. If we decide to neglect terms of this order, we can simply substitute the expression for the ground state into the left hand side. It is trivial to check that Zi = 0, and hence the only contribution is from iZn1Zm1i.

Next, observe that

ZmZmi = (k + p)m (N/p 1)!

(N/p 1 m)!

i

JHEP08(2016)007

(k + p)N p

mi

Upon using Zi = 0 once more, the nm factor in (A.2) follows. The nal ingredient is to observe ii = kN, completing the proof of (A.2).

24

Identity 1 is a little harder to prove. Let us start by rewriting it slightly:

iZnZn+ml

1

p

n+m1

Xr=0n1

Xs=0(iZrZsi )(iZn1sZn+m1rl) (k + p)N p

n iZml

where for brevity the prime denotes (asymptotic) equality of the il-traceless parts. We will proceed by rstly showing that only one term in the double sum contributes at leading order, namely that obtained at r = s = 0, reducing the problem to proving

iZnZn+ml

kNp (iZn1Zn+m1l)

(k + p)N p

n iZml

JHEP08(2016)007

Then we will inductively demonstrate that

iZnZn+ml (n + 1)

(k + p)N p

niZml (A.4)

from which the original identity follows immediately.

So to begin, let us estimate the size of the terms we wish to keep. The traceless part of the right-hand side vanishes in the ground state, so we must sacrice at least one term for something of order ; this is then generically non-vanishing. Therefore, the right-hand side is of order O(Nn+(m+1)/2).

We can now consider a single term of the double sum at general (r, s). The traceless part of the second bracket, (iZn1sZn+m1rl), vanishes in the ground state, and hence is at most order O(Nn+(mrs1)/2). Thus the rst bracket must be at least of order

O(N1+(r+s)/2). But this N-scaling is only possible if all terms in the rst bracket come from the ground state, when this term vanishes by the observations above except for r = s = 0.

This leaves us only with deriving (A.4). We will induct on n to establish this; note that the case n = 0 is trivial. Write

iZnZn+ml = iZmZnZnl i[Zm, Zn]Znl

The rst term is simple to handle. Since at leading order iZm = 0, we can safely make the approximation ZnZni ((k + p)N/p)ni.

The second term can be expanded into a double sum, and simplied slightly using the asymptotic version of Gauss constraint, [Z, Z] . Then almost all terms can be

shown to be subleading, using the ideas above, except for the one where we have the appearing at the far left. Hence

iZnZn+ml

(k + p)N p

niZml + ijjZn1Zm1Znl

(k + p)N p

niZml +

(k + p)N p

iZn1Zn1+ml

where we have also used the trick of separating the ij and jl traces out, discarding more irrelevant terms. Finally, applying the inductive hypothesis to the second term, we establish (A.4), and hence identity 1.

25

B Kostka polynomials

In this appendix we give an explicit description of the Kostka polynomial K,(q) due to

Kirillov and Reshetikhin [25].Given , P, we dene a sequence of partitions (K) with K = 0, 1, 2, . . . , () 1

with (0) = and

|(K)| =XjK+1j (B.1)

For each such sequence we dene the vacancy numbers

P(K)n =

Xj1

hmin{n, (K+1)j} 2min{n, (K)j} + min{n, (K1)j}i

for all positive integers n and K = 0, 1, 2, . . . , ()1 with the understanding that (()) 0. An admissible conguration, {} is any such sequence of partitions with non-negative

vacancy numbers, i.e.P(K)n 0

for all values of n and K. The charge c({}) of an admissible conguration is dened as

c ({}) = n[] +

()1

XK=1

M

h(K), (K)i

M

h(K), (K1)

i

JHEP08(2016)007

where, for any two partitions , P, we dene the function

M : P P

Z0 by

Xi,j1min{i, j}

Finally the Kostka polynomial can be dened as a sum over all admissible congurations; explicitly,

K,(q) =

X{} qc[{}]

()1

YK=1

M [, ] =

Yn1

"

P(K)n + mn (K)

m n (K)

#q

where we dene the q-binomial coe cient

"

m

n

#q

= m(q)

n(q)mn(q)

Qnj=1(1 qj).

Ground state energy. As in the text we set = (kN) and look for the ground state by searching for the partition , satisfying || = kN and () p, such that K,(k)

and, as in the text, we have n(q) =

N (q) yields

the leading term in the q expansion for |q| 1. The general formula given above can be

simplied [25] in the case k = 1 where = (1N). We nd

K,(1N )(q) = qn[

T ]

QNj=1 (1 qj) H(q)

26

where H(q) is the hook-length polynomial given by

H(q) =

YxY()

1 qh(x)

Here the product is over the boxes x = (r, s) of the Young diagram Y() corresponding to the partition and h(x) = r + Ts r s + 1 > 0 is the length of the hook passing

through box x.

As explained in the text, the minumum for the case k = 1 is attained for the partition 0 = ((L + 1)M, LpM) where N = Lp + M for non-negative integers L and M < p which gives a ground state energy

E0(1, p, N) = 1

2L(L 1)p + LM

It is instructive to reproduce this result from the general formula given above in the case = (1N) and = 0. According to the recipe we must nd a sequence of partitions (K)

for K = 0, 1, 2, . . . , (0) 1 = p 1 with (0) = = (1N) and

|(K)| =XjK+1(0)j =

((M K)(L + 1) + L(P M) 0 K M L(p K) M + 1 K p 1

with non-negative occupation numbers which minimises the charge c({}). It not hard to

see that this is achieved by maximising the number of parts in each partition (K). Thus

we set

(K) =( 1((MK)(L+1)+L(P

M))

JHEP08(2016)007

0 K M

1(L(pK))

M + 1 K p 1One may then check that the corresponding occupation numbers are non-negative and that

c ({}) = E0(1, p, N) =

1

2L(L 1)p + LM

The above conguration has a straightforward generalisation to k 1. As in the text we

set0 = ((kL + k)M, (kL)pM)

and one simply scales each partition (K) in the conguration by a factor of k setting

(K) =( k((MK)(L+1)+L(P

M))

0 K M

k(L(pK))

M + 1 K p 1The vacancy numbers are remain non-negative and the charge of the conguration scales linearly with k. Thus the new ground state energy is

c ({}) = E0(k, p, N) =

k

2 L(L 1)p + kLM

which is the result stated in the main text.

27

C (A ne) Lie algebra conventions

Here we give our conventions for the simple Lie algebra Ap1 = sl(p, C) and its A ne counterpartp1.

For Ap1, we work in a Chevalley basis with generators {hi, ei, fi; i = 1, . . . p 1} with

brackets

[hi, hj] = 0 , [hi, ej] = Ajiej , [hi, fj] = Ajifj , [ei, fj] = ijhi

where Aij is the Ap1 Cartan matrix. Weights of each irreducible representation lie in the weight lattice

LW = SpanZ{ (i), i = 1, . . . , p 1}whose basis vectors are the fundamental weights (i). Finite-dimensional, irreducible representations R are labelled by a highest weight , lying in the positive weight lattice

L+W = SpanZ0{ (i), i = 1, . . . , p 1}

We denote the corresponding representation space V . For each weight of R there is an

element

=

of LW with Dynkin labels i

Z. We then have a basis vector | i of V which is a

simultaneous eigenvector of the Cartan generators satisfying

R hi

jiej , [ei, fj] = ijhi

whereij is the a ne Cartan matrix. The basis elements with i > 0 generate an Ap1 subalgebra. Weights of an integrable representation have an expansion

=

for integers

i and n where is the imaginary root. The fundamental weights ofp1 can be written as

(i) =

(0) + (i)

for i > 0, where (i) are fundamental weights of the global Ap1 subalgebra.

The integrable representations R ofp1 are characterised by a highest weight

with non-negative Dynkin labels, and have the representation space V . Each weight of

28

p1

JHEP08(2016)007

Xi=1i (i) (C.1)

| i = i| i

for i = 1, . . . , p 1.

For the a ne Lie algebrap1 we have Chevalley generators {hi, ei, fi; i = 0, . . . p1}

with brackets

[hi, hj] = 0 , [hi, ej] =jiej , [hi, fj] =

p1

Xi=0i (i) + n (C.2)

R has an expansion of the form (C.2). The corresponding basis vector |

simultaneous eigenvector of the Cartan generators, with

R hi

for i = 1, . . . , p 1, and the derivation or grading operator, with

R (L0) |

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] N. Dorey, D. Tong and C. Turner, A matrix model for non-Abelian quantum Hall states,

arXiv:1603.09688 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1603.09688

Web End =INSPIRE ].[2] D. Tong, A quantum Hall uid of vortices, http://dx.doi.org/10.1088/1126-6708/2004/02/046

Web End =JHEP 02 (2004) 046 [http://arxiv.org/abs/hep-th/0306266

Web End =hep-th/0306266 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0306266

Web End =INSPIRE ].[3] D. Tong and C. Turner, Quantum Hall e ect in supersymmetric Chern-Simons theories,

http://dx.doi.org/10.1103/PhysRevB.92.235125

Web End =Phys. Rev. B 92 (2015) 235125 [arXiv:1508.00580] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1508.00580

Web End =INSPIRE ].[4] E. Witten, Quantum eld theory and the Jones polynomial,

http://dx.doi.org/10.1007/BF01217730

Web End =Commun. Math. Phys. 121 (1989) 351 [http://inspirehep.net/search?p=find+J+%22Comm.Math.Phys.,121,351%22

Web End =INSPIRE ].[5] S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical

quantization of the Chern-Simons-Witten theory, http://dx.doi.org/10.1016/0550-3213(89)90436-7

Web End =Nucl. Phys. B 326 (1989) 108 [http://inspirehep.net/search?p=find+J+%22Nucl.Phys.,B326,108%22

Web End =INSPIRE ].[6] A.P. Polychronakos, Quantum Hall states as matrix Chern-Simons theory,

http://dx.doi.org/10.1088/1126-6708/2001/04/011

Web End =JHEP 04 (2001) 011 [http://arxiv.org/abs/hep-th/0103013

Web End =hep-th/0103013 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0103013

Web End =INSPIRE ].[7] L. Susskind, The quantum Hall uid and noncommutative Chern-Simons theory,

http://arxiv.org/abs/hep-th/0101029

Web End =hep-th/0101029 [http://inspirehep.net/search?p=find+EPRINT+hep-th/0101029

Web End =INSPIRE ].[8] B. Blok and X.G. Wen, Many body systems with non-Abelian statistics,

http://dx.doi.org/10.1016/0550-3213(92)90402-W

Web End =Nucl. Phys. B 374 (1992) 615 [http://inspirehep.net/search?p=find+J+%22Nucl.Phys.,B374,615%22

Web End =INSPIRE ].[9] G.W. Moore and N. Read, Nonabelions in the fractional quantum Hall e ect,

http://dx.doi.org/10.1016/0550-3213(91)90407-O

Web End =Nucl. Phys. B 360 (1991) 362 [http://inspirehep.net/search?p=find+J+%22Nucl.Phys.,B360,362%22

Web End =INSPIRE ].[10] N. Read and E. Rezayi, Beyond paired quantum Hall states: parafermions and incompressible

states in the rst excited Landau level, http://dx.doi.org/10.1103/PhysRevB.59.8084

Web End =Phys. Rev. B 59 (1999) 8084 [http://arxiv.org/abs/cond-mat/9809384

Web End =cond-mat/9809384 ]

[http://inspirehep.net/search?p=find+EPRINT+cond-mat/9809384

Web End =INSPIRE ].

[11] E. Ardonnes and K. Schoutens, New class of non-Abelian spin-singlet quantum Hall states,

http://dx.doi.org/10.1103/PhysRevLett.82.5096

Web End =Phys. Rev. Lett. 82 (1999) 5096 [http://arxiv.org/abs/cond-mat/9811352

Web End =cond-mat/9811352 ].[12] X.G. Wen, Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall

states, http://dx.doi.org/10.1103/PhysRevB.41.12838

Web End =Phys. Rev. B 41 (1990) 12838 [http://inspirehep.net/search?p=find+J+%22Phys.Rev.,B41,12838%22

Web End =INSPIRE ].[13] X.-G. Wen, Theory of the edge states in fractional quantum Hall e ects,

http://dx.doi.org/10.1142/S0217979292000840

Web End =Int. J. Mod. Phys. B 6 (1992) 1711 [http://inspirehep.net/search?p=find+J+%22Int.J.Mod.Phys.,B6,1711%22

Web End =INSPIRE ].[14] M. Milovanovic and N. Read, Edge excitations of paired fractional quantum Hall states,

http://dx.doi.org/10.1103/PhysRevB.53.13559

Web End =Phys. Rev. B 53 (1996) 13559 [http://arxiv.org/abs/cond-mat/9602113

Web End =cond-mat/9602113 ] [http://inspirehep.net/search?p=find+EPRINT+cond-mat/9602113

Web End =INSPIRE ].

29

i of V is a

|

i =

i|

i

i = n|

i

JHEP08(2016)007

[15] I.D. Rodriguez, Edge excitations of the Chern Simons matrix theory for the FQHE,

http://dx.doi.org/10.1088/1126-6708/2009/07/100

Web End =JHEP 07 (2009) 100 [arXiv:0812.4531] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0812.4531

Web End =INSPIRE ].[16] B. Morariu and A.P. Polychronakos, Finite noncommutative Chern-Simons with a Wilson

line and the quantum Hall e ect, http://dx.doi.org/10.1088/1126-6708/2001/07/006

Web End =JHEP 07 (2001) 006 [http://arxiv.org/abs/hep-th/0106072

Web End =hep-th/0106072 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0106072

Web End =INSPIRE ].[17] A. Nakayashiki and Y. Yamada, Kostka polynomials and energy functions in solvable lattice

models, http://dx.doi.org/10.1007/s000290050020

Web End =Sel. Math. 3 (1997) 547 [http://arxiv.org/abs/q-alg/9512027

Web End =q-alg/9512027 ] [http://inspirehep.net/search?p=find+EPRINT+q-alg/9512027

Web End =INSPIRE ].[18] A. Nakayashiki and Y. Yamada, Crystalizing the spinon basis,

http://dx.doi.org/10.1007/BF02104914

Web End =Commun. Math. Phys. 178 (1996) 179 [http://arxiv.org/abs/hep-th/9504052

Web End =hep-th/9504052 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9504052

Web End =INSPIRE ].[19] A. Nakayashiki and Y. Yamada, Crystalline spinon basis for RSOS models,

http://dx.doi.org/10.1142/S0217751X96000201

Web End =Int. J. Mod. Phys. A 11 (1996) 395 [http://arxiv.org/abs/hep-th/9505083

Web End =hep-th/9505083 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9505083

Web End =INSPIRE ].[20] A.N. Kirillov, Dilogarithm identities, http://dx.doi.org/10.1143/PTPS.118.61

Web End =Prog. Theor. Phys. Suppl. 118 (1995) 61

[http://arxiv.org/abs/hep-th/9408113

Web End =hep-th/9408113 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9408113

Web End =INSPIRE ].[21] I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical

Monographs, Oxford University Press, Oxford U.K. (1995).[22] A. Lascoux and M.P. Schutzenberger, Sur un conjecture de H.O. Foulkes (in French), C. R.

Acad. Sci. Paris 288A (1978) 323.[23] A.N. Kirillov, New combinatorial formula for modied Hall-Littlewood polynomials, in

q-series from a contemporary perspective, Contemp. Math. 254, Amer. Math. Soc.,

Providence RI U.S.A. (2000) [http://arxiv.org/abs/math/9803006

Web End =math/9803006 ].[24] J. Dsarmnien, B. Leclerc and J.-Y. Thibon, Hall-Littlewood functions and Kostka-Foulkes

polynomials in representation theory, http://www.emis.de/journals/SLC/opapers/s32leclerc.html

Web End =Sm. Lotharingien Combin. 32 (1994) 38 .[25] A.N. Kirillov and N.Yu. Reshetikhin, The Bethe ansatz and the combinatorics of Young

tableaux, http://dx.doi.org/10.1007/BF01247088

Web End =J. Sov. Math. 41 (1988) 925 .[26] I. A eck, Universal term in the free energy at a critical point and the conformal anomaly,

http://dx.doi.org/10.1103/PhysRevLett.56.746

Web End =Phys. Rev. Lett. 56 (1986) 746 [http://inspirehep.net/search?p=find+J+%22Phys.Rev.Lett.,56,746%22

Web End =INSPIRE ].[27] I. A eck and F.D.M. Haldane, Critical theory of quantum spin chains,

http://dx.doi.org/10.1103/PhysRevB.36.5291

Web End =Phys. Rev. B 36 (1987) 5291 [http://inspirehep.net/search?p=find+J+%22Phys.Rev.,B36,5291%22

Web End =INSPIRE ].

30

JHEP08(2016)007