JHEP08(2016)007 N Springer dels June 7, 2016 July 27, 2016 : August 1, 2016 : : as recently shown to Received Accepted Published , 10.1007/JHEP08(2016)007 limit, it coincides with the eory. This manifests itself lgebra from matrix degrees doi: he matrix model in terms of N hip to the WZW model can be , University of Cambridge, Published for SISSA by a [email protected] , and Carl Turner ) gauged matrix quantum mechanics which, in the large N a,b,c . 3 1604.05711 The Authors. David Tong c Chern-Simons Theories, Conformal and W Symmetry, Matrix Mo

We study a U( a , [email protected] [email protected] Department of Theoretical Physics, TIFR, Homi Bhabha Road, Mumbai 400Stanford 005, Institute India for Theoretical Physics, Via Pueblo, Stanford, CA 94305,E-mail: U.S.A. Department of Applied MathematicsWilberforce and Road, Theoretical Cambridge, Physics CB3 OWA, U.K. b c a ArXiv ePrint: Schur and Kostka polynomials and show that, in the large limit, is closely relatedin two to ways. the First, chiralof we WZW construct freedom. conformal the Secondly, left-moving field we Kac-Moody compute th a the partition function of t Keywords: partition function of thedescribe WZW non-Abelian quantum model. Hall statesunderstood This and in the same this relations matrix framework. model w A matrix model for WZW Open Access Article funded by SCOAP Abstract: Nick Dorey, JHEP08(2016)007 5 6 1 4 8 10 15 19 23 26 28 (1.1) ) global p , coupled to following two # α and tr denotes Z † i Z . The action is first iαϕ tr ω − i echanical matrix model − ϕ t is a positive integer. , . . . , p α ∂ k = ) tr = 1 i p i ϕ , = 1 + 1 dimensions. t + i can be constructed from the quan- d D k ϕ ( k − i ] and ϕ t D † i limit, this matrix model captures the physics α, Z [ ϕ i N – 1 – p ) gauge field, which we denote as . =1 − i i X i N ϕ Z t + ∂  and = fundamental scalars Z t Z Z p t D † D Z and ) gauge symmetry, the quantum mechanics has an SU(  ) affine Lie algebra at level p Z N tr ( i [ " su dt ) gauge indices. Here and in the following N Z = WZW conformal field theory. Specifically, we demonstrate the k S ) p tum mechanical operators The left-moving In addition to the U( The matrix model consists of a U( • 3.2 Back to3.3 the partition function The continuum limit 2.1 The currents 2.2 Deriving the Kac-Moody algebra 3.1 A digression on symmetric functions results: of the SU( symmetry. We will show that, in the large the trace over U( The covariant derivatives are order in time derivatives and given by a complex adjoint scalar 1 Introduction The purpose of thisis paper related is to to the describe chiral how WZW a conformal simple field quantum theory m in C (Affine) Lie algebra conventions A Proofs of two classical identities B Kostka polynomials 3 The partition function Contents 1 Introduction 2 The current algebra JHEP08(2016)007 ) p N mod (1.2) (1.3) , as a (both ). N M 1.1 = ern-Simons N ster around the one should set ) global symmetry. delicate manner in p = 2 + 1 dimensional actly, for all ortex dynamics was ]. The advantage of 5 d WZW model. , unction is equal to the 4 k [ e SU( played by the edge of the entation which is perhaps Kostka polynomials ) neer Chern-Simons theory limit when ] that the vortices have a p limit, the partition function x model partition function limit. To recover the chiral ,k 1 = 1 Abelian Chern-Simons ′ k hern-Simons theory are chiral N ) N and p N igin of these edge modes as the plain why the results described p This Chern-Simons theory has antisymmetric representation of k d by the matrix model ( ) th p M { SU( p The connection between the matrix limit. × Z ′ k k . Outside this region, the gauge group U( N }| N √ – 2 – U(1) Schur polynomials z ∼ = ( ,k ′ limit. It was shown in [ M k ) p N U( ) describes the dynamics of vortices in a -fold symmetrisation of the 1.1 th k ) is an expansion in . p ) 3.19 p ], following earlier work on vortices in the + and subsequently take the large 1 k p ]. The vortices sit in a harmonic trap which forces them to clu ) and the WZW model is not coincidental; both are related to Ch 3 = ( , ′ 1.1 2 k The result ( function of both temperature and chemical potentials for th of which will be defined later in the paper). In the large The partition function of the matrix model can be computed ex is proportional to the partition function of the chiral SU( )”. In terms of Young diagrams, this representation is The matrix model ( The relationship between the matrix model and non-Abelian v The upshot is that the solitonic vortex provides a way to engi The vortex perspective also provides a way to understand the One can also ask what happens if we take the large p • . In this case, we show that the quantum mechanics partition f explained in [ Chern-Simons theory, coupled togauge non-relativistic group matter. and levels edge modes, described by a WZW model with algebra U( on a manifold with boundary,vortex. where It the is role well of known the that the boundary gapless is excitations of the C the present set-up is thatexcitations we can of identify the the vortices. microscopic These or excitations are capture which we should take the large origin, where they form a droplet of size theories. Before we derive thein results this above, paper we are will not first unexpected. ex Relationship tomodel ( Chern-Simons theory. where is broken; inside it is unbroken. depends in a rather delicate way on how we take the large This second property requires some elaboration as the matri p WZW partition function — also known as the vacuum character — character of the WZW modelbest associated described to as a the primary “ in a repres SU( divisible by theory [ JHEP08(2016)007 d N = 1, it um Hall limit. p we compute N we construct 3 2 ]. 8 ]. The results of tka polynomials, 8 ]. These lie in the , ) flavour symmetry. 1 1 p he edge modes. This m Hall wavefunction, ntum Hall states and )[ ]. atrix model — can be p en for Abelian quantum n section 11 . As we described above, of results from the theory + – p escribe Laughlin states at ory: the excitations of the take the large 9 k the SU( ( rative. Indeed, it is known -contained, we have included be matched to the excitations onship was subsequently used p/ tes [ Blok and Wen [ e quantum Hall state do coincide = WZW model [ ν ,k ′ Our original interest in the matrix k ) and the WZW model highlighted in ) p 2, the matrix model describes a class of is divisible by 1.1 ). This explains why taking the large ≥ N p 1.2 – 3 – limit of the partition function coincides with the 2, the story is much richer as the partition function , the ground state of the vortices is not unique; N p = 1 matrix model were previously shown to coincide ≥ ] where it was shown that the bulk wavefunction can p p ].) With 9 7 mod ]. (This matrix model was inspired by an earlier approach 6 ]. For M results in the character of the Kac-Moody algebra associate 15 This paper contains two main results. In section p = + 1) [ N k mod ( / M ] and later extended to a number of paired, non-Abelian quant = 1 = 13 ν , a review of the properties of Schur, Hall-Littlewood and Kos N ) singlet, ground state only when 12 ]. p ) was through its connection to the quantum Hall effect. When 3.1 14 1.1 = 1+1 conformal field theory which describes the dynamics of t The computation of the partition function involves a number The excitations arising from the The connection between the matrix model ( However, one can also go the other way. Starting from a quantu There is a deep connection between the bulk properties of qua In contrast, when d of symmetric functions. In anin attempt section to make this paper self Nonetheless, our results show that thewith excitations those above th of the boundary conformal field theory. The plan of the paper. now depends on both temperature and chemical potentials for the Kac-Moody current algebrathe from partition function the of quantum the mechanics. matrix model I and explain how to that the Blok-Wen statesreconstructed — from which correlation are functions the inthis ground the paper U( states can of bematrix our thought model m of coincide as with a those derivation of of the the boundary converse CFT. st states in [ this paper falls naturally into this larger quantum Hall nar be reconstructed as ato CFT derive correlation several function. interesting non-Abelian This quantum relati Hall sta using non-commutative geometry [ same universality class as states previously introduced by limit keeping unique, SU( to this representation. Relationship tomodel the ( quantum Hall effect. non-Abelian quantum Hall states with filling fraction with those of a chiral boson [ rather, it transforms in the representation ( with this restriction in place, the large connection was first highlighted in [ partition function of the WZW model. reduces to the matrixfilling model fraction introduced by Polychronakos to d of the boundary conformalHall field states theory. [ This was first done by W one can enumerate its full set of excitations. These can then the JHEP08(2016)007 rix ) is ) is and ence (2.2) (2.1) (2.3) (2.4) 1.1 ) that 2.1 ab Z N U( are not in- . i tries ϕ ⊂ ϕ and and n that we have been kN Z Z ion of the constraint moved to the right — -Moody algebra. The = Z n we will show how to ij i a δ ions el. We will prove that the ) constrains the degrees of ith nd using the commutation ϕ we rely on two conjectured ab eedom the level of the Kac-Moody δ † i a ) singlets; this can be written 1.1 N se constraints. ions for affine Lie algebras and ϕ contains further details about 1 N N ) ] = . In the quantum theory we find 1 p limit of the matrix model ( p B =1 ) p = 0 and construct a Hilbert space i X † j b p + i + 0 , ϕ =1 + N k | k a X i a k → ∞ i a ϕ = ( ϕ N † i = ( ⇒ = ϕ † i i i ϕ ϕ 0 i | and [ ϕ – 4 – N p ab =1 ) i X Z p p =1 bc i X δ + ad ) gauge symmetry ensures that ] : + k ] + δ † † ) algebra at level N p ( = ( obeying ] = [ Z,Z su Z,Z i † † i a cd [ 0 :[ ϕ | . The quantum version of the Gauss’ law constraint ( ,Z i a ) group indices are contracted; this is determined by the mat † i a ab ϕ ϕ N Z [ p =1 i X and =1 N a X † ab = 1+1 WZW conformal field theory. The smoking gun for the emerg Z ]. Meanwhile, the level determines the charge under U(1) d , ). In the quantum theory, the individual matrix and vector en ), gives 2.1 2.2 . However, the extra complications in the quantum theory mea k Both the classical and quantum matrix models exhibit the Kac This shift of the level can already be seen in the quantum vers The key point is that the U( become operators, obeying the canonical commutation relat i a difference between the two appearsclassical matrix only model to has be an a shift of the lev We choose a reference state by acting with commutator [ level unable to complete theidentities proof which of we present the below. existence of the algebra; relations ( physical states must carry. Taking the trace of Gauss’ law, a ϕ We’ll see that the current algebra arises, in part, due to the equation ( interpreted as the requirement that physical states are SU( freedom to obey related to the Kostka polynomials. Other appendices describethe our details convent of the current algebra computation. 2 The current algebra The purpose of this paper is to explain how the in normal ordered form as but not the way that U( Here the : : determines the order in which operators appear — w which are the lead characters in our story. Appendix We will see belowalgebra. that a similar normal ordering issue shifts dependent. In particular, Gauss’ law of the matrix model ( of a WZW modelconstruct such is, an of algebra from course, the a matrix current model algebra. degrees of fr In this sectio JHEP08(2016)007 ) 0. ld .) dy N p ( (2.6) (2.5) . We + O m k m, n < limit. We level → ∞  N kl δ ij d commutators of the δ e to a possible rescaling. 1 p current algebra in the ma- escale the positive-graded 0 e latter, the commutation − ) if we restrict to ). (We also show this struc- il ≥ e we will see the difference   δ ndix, in the Poisson brackets n 2.5 k 2.6 m + jk 0 ϕ he full Kac-Moody algebra. We δ m il . This same expression holds in by acting with fewer than m ˜  J i Z N 0 ) imply that these currents give a 0 † k kj | ) parts. The U(1) currents are simply m < δ n, ϕ p 0 denotes the grading. m + 2.2 ij ij − ˜ ≥ m δ m J positive graded n 1 p 2 Z + m m m/ − kj km δ ˜ − j J = tr ϕ ) since the overall scaling is a power of il  – 5 – δ | † m ) + m N  n 2.5 ˜ m ) Z i | J ji ) currents. Here too there is a normalisation issue, + † i p p p J ϕ m il + ] =  J = n k i kl ( kj ˜ m J δ ij = 0. These are trickier to compute because now the con-  , J − m m ij = ij ˜ ˜ n corrections. Moreover, the operators in this equation shou J ] = 0. This holds for any J [ + m n ij ) is heartening, our interest really lies in the full Kac-Moo ij m kj J J /N , 2.5 J m, n > il are flavour indices, while m δ ˜ ( J i . These too obey the graded Lie algebra ( ˜ J ∼ theory. In that setting we obtain an algebra at the unshifted ] with ] ) adjoint-valued currents are , . . . , p ] = [ n n p kl n ) should be replaced by classical Poisson brackets. − kl ˜ J J , = 1 ) comes into play. However, things simplify somewhat in the J , 2.2 , m ij m means up to 1 m 2.1 ij ˜ J [ J classical J ∼ i, j, k The rest of this section is devoted to the derivation of ( The central charge of the U(1) current is harder to pin down du Of course, the central term only arises when we consider mixe While the result ( It is simple to show that the commutators ( act on states that are constructed from the vacuum straint ( creation operators. For this reason, we focus on the SU( and similarly for form [ then define the negative graded currents as will see that only these rescaled currents will give rise to t currents but one that will turn out to be uniquely fixed. To this end, we r Here will show that the currents obey the Kac-Moody algebra Here Note that these still obey the algebra ( both the quantum theoryrelations and ( the classical theory where, in th ture arises perturbatively, inof a the sense made clear in the appe algebra and, inbetween particular, classical the and quantum central theories. extension term. Her representation of half of the Kac-Moody algebra, while [ It is straightforward to construct generators of the 2.1 The currents while the SU( trix model. The problem factorises into U(1) and SU( JHEP08(2016)007 † i ϕ i ϕ (2.7) (2.8) (2.9) (2.10) i  ld instead i ϕ ··· jc phys  ld | ϕ l phys kl ϕ | ϕ i δ ! l r jc i ϕ bc − ϕ ) r ] 1 r ) + − n − phys j † 1 − bd | phys l 1 m ϕ − | om in the final term l ssion. We deal with r − ϕ ,Z Z r m r − ϕ s m − m ld 1 ac r Z − i − 1 Z 1 1 − ϕ 1 − Z ] terms between currents − s [ 1 − − − † m jc − m n − n † kb m 1 † ϕ Z † phys t term combines nicely with Z ] m ϕ s | − Z s Z Z n ) s n − Z r Z,Z † † ia ) bd − † 1 ′ s − 1 † i N ϕ ), it can be written as Z 1 − Z − l † i − ,Z ϕ ( 1 n − ′ . If we don’t include the normal n + + † ϕ ϕ i n m − 2.8 † i ac ad † j 1 p n ϕ Z m ) Z Z † − † k ϕ [ s Z ] =0 Z ′ † acting on a physical state. Then, after + † r − X Z † i ϕ m m n † kb ] l phys Z † k † p ϕ | )( n 0 and look at ϕ Z r ′ ϕ l i ) ] Z n † ia + Z † i p Z,Z † ϕ n ϕ [ = ( ( † s ϕ s ϕ s k 1 Z † Z,Z † i + † [ ( † k − =0 jk s s + Z ,Z Z s k Z X n is key: it will become the central term in the † † δ ϕ r r j r – 6 – m 1 m, n > il ] Z Z Z l ϕ Z n δ + +( Z − phys r r ] appearing here with some combination of Z † i =0 † i ) | [ l ϕ † r 1 m X m p Z Z ϕ † i ϕ − n  ϕ − † i † i ( =0 † Z ] l 1 ϕ † i

s 1 X n + ϕ ϕ n n − Z ϕ =0 ϕ − 1 † ) to write † =0 1 1 Z,Z i † kb s X n k † s n k X n − =0 Z † − − =0 =0 1 ϕ ϕ 1 r X s s † k ,Z X X n n m 2.9 , ϕ − Z − =0 † ia =0 1 1 j ϕ m r X r m X ). Using the expansion ( m m ϕ ] + il − − ϕ =0 =0 Z † δ ] = r r Z [ X X m m − ) arises from the commutator [ m † i † i n = = 2.7 − † Z ϕ ϕ = = = i † ld 2.6 Z,Z jk jk ,Z [ i ϕ ϕ δ δ  m jc ). However, Gauss’ law is not an identity between operators; Z = = ϕ [ ] phys ] = [ | n l 2.3 n † bd − ϕ kl ] ˜ ), then the constraint is written as ,Z n J † , m ac 2.3 m Z ij ,Z [ ˜ J m [ † kb ) group indices are contracted in the obvious manner apart fr Z ϕ [ † i † ia N ϕ ϕ Our first goal is to simplify the two commutators in this expre ordering in ( using Gauss’ law ( All U( where we’ve written them explicitly. some manipulation, we can use ( To this end, we consider the operator We’re going to replace the factor [ travelling in opposite directions. We take it holds only when evaluated on physical states The novelty in deriving ( 2.2 Deriving the Kac-Moody algebra them in turn. For the first, we write This term above proportional to ( algebra. We’ll come backthe to second this commutator shortly. in ( Meanwhile, the firs JHEP08(2016)007 l ϕ n ′ − i m . Doing ϕ l r ′ . In each Z i † i − δ ; similarly, jk 1 indices. At ϕ δ il − n δ m i, l  and Z ′ i i 1 N ii ) ) ) − l l δ n p ϕ ϕ , † p . It can be neglected ) r r + j Z jk ld − − ); it differs only in its r δ 1 1 ϕ k . ϕ r , ( − − Z ′ − il † i ) m m somewhat involved, so we  1 2.10 δ rity: we separate the two j ϕ − Z Z spect to the pairs, as do half of the trace- ϕ past 1 ∼ s s m r n > m − − − ) =0 − l t to † kb 1 1 al matrices, the identity reads Z r X m 1 s ϕ − − ϕ − il r − n n δ 1 † † m − hich we neglect on the grounds that 1 − ); this is our central term. We again Z Z jk Z ′ n † i − δ -traceless part of this expression. † i s limit. This follows from the following † ϕ m − n ϕ ) are both proportional to kl Z 1 ) 2.10 )( Z N † k )( p ′ − ) are both proportional to s ′ i ϕ i 2.7 n p − † ϕ + ϕ 1 )( s 2.7 s ′ Z † − k i † † k ( n ϕ Z † Z ϕ s r r – 7 – † ∼ Z )( Z † i Z l l ′ Z † i † i r ϕ ) make it much more challenging, though we have ϕ ϕ ϕ ϕ s r Z identity. This time the identity takes a different form )( ( ( † ′ ′ − † i 2.2 i 1 Z jk limit. We’re left with jk N ϕ ϕ r arises from commuting δ − ( corrections. Further, we are neglecting a trace, propor- δ s r † Z il N kl − − † i Z δ Z δ 1 h ϕ r /N ( − 1 . In what follows, we will make the natural assumption that Z h n − ′ =0 † † i 1 s m X n − ϕ =0 Z 1 s ( . We have not been able to extend the proof to the quantum case X n r − , 1 =0 1 A Z r X n − m =0 and † i − =0 s X n r X 1 p ϕ m ≥ 1 n 1 − =0 − ∼ m r =0 X m r X m term above is very close to that appearing in ( 1 p Θ Θ = for jk ϕ − on both sides. A similar expression holds when l nδ ) ϕ il p again means up to 1 δ m , the traceless component is subleading; we have + Z ∼ n k N † ( The four- The proof of this identity in the classical theory is already It remains only to discuss the second term in ( ) indices and overall sign. Adding the two together gives Z p † i ϕ traceless terms, leaving only traceless-tracelessthey terms w are subleading in the large this we find that the products of traces cancel between the two case, the two terms combine together in the large identity: Identity 1: We can manipulate thedouble sums index into structure their trace to and exploit traceless parts this with simila respec simply because we are ultimately interested in the the second term above and the second term in ( relegate it to appendix To proceed, we need a secondin large the classical and quantum theories. Evaluated on classic large tional to checked it for small decompose it into the trace and traceless components with re where the additional commutators ( U( The first term above and the third term in ( where this identity generalises directly to the quantum case. where the term proportional to JHEP08(2016)007 . – α 17 # limit; il . The δ ular, it N jk δ results in a → ∞ i ). limit, up to ]. Because this ot been able to ϕ mn N 16 2.6 N mn kn δ δ  + p j arises due to the shift ϕ p mn ), can then be analysed + n δ the large metry in the continuum k − dels in the model can be related [ n ory. Nonetheless, as before m rix model. In the limit of ). 3.19  − , can typically be evaluated ctly for all values of ising from the gauge field Z oody algebra ( 1 a number of simple examples. odel in two dimensions which † k N 2.6 N ( )  ϕ p il n k tion ( p δ +  − 1 k N l ( symmetric representation [ identity now reads ) ϕ p  th n p N p . − + kN A m , k ∼ , ( n ′ Z i n † i  ϕ ≥ ϕ p r ≥ – 8 – jk − m 1 δ ∼ m ′ − approach, the hurdle is to find a correct way to implement " i m n for ϕ N for , and does not seem to have a direct interpretation in r  Z p 1 − 1 which sits in the N − current algebra at level ) , we have − n α † p 1 , we will show that this partition function is proportional kl m p − δ Z + Z p r 1 ˆ k A Z modulo − ( ′ → ∞ † i n and †  ϕ N , it shifts the saddle point in a complicated manner. We have n limit. However, the nature of this limit is subtle; in partic ij Z 1 N r ∼ δ N ). The corresponding large − =0 ) gauge field ] ). One can show that integrating out the fundamental matter N r Z X l m ′ N † i 2.4 ϕ 2.4 ϕ n † 1 Z − =0 † k r X m , ϕ seen in ( j ϕ k m → Z † i Putting all of this together, we arrive at our final result. In In fact, our formula for the partition function of the matrix There is a well-established machinery for solving matrix mo Meanwhile, in the quantum theory there is an extra term which p If one tries to take the standard large ϕ [ 1 ] to the partition function of a certain integrable lattice m + depends on the value of terms of a saddle point of the original matrix integral. We do not have a generalthe proof existence of of this the result new in factor the can quantum be the checked numerically in We present a proof of this identity in appendix Written in terms of currents, this is equivalent to the Kac-M terms proportional to to a character of the chiral gives rise to conformal field theory with affine Lie algebra sym representation scales with directly in the large 3 The partition function In this section, welarge compute particle the number, partition function of the mat k evaluate the partition function in this approach. the level constraint ( 19 Identity 2 (Classical Version): Wilson line for the SU( resulting formula for the partition function, given in equa the usual route is through the path integral which, at large by finding an appropriate saddle point for the Wilson lines ar Here we will do better and compute the partition function exa Identity 2 (Quantum Version): JHEP08(2016)007 ) ) b N 6= cter. (3.2) (3.1) (3.3) a ). This p gives the ). We call r N and the chem- U( β (for some ⊂ . Let’s deal with ) and SU( i 1 nian gives N 0 − non-invariant states b | N excitations, weighted ) is ω † i +1 a 1.1 ϕ , but instead in the non- ilbert space is simple to ω perature for all possible this in mind, we introduce . They have ∆ = 0. Taking i and i i i etry, U(1) J i odel ( ) global symmetry. Including † . 0 | i x p r Z i † βµ phys p and | =1 e ϕ i Y Z b † i g in the constraints imposed by the i a ! = i x ϕ ∆ i a /ω i J q i a µ ω x 1 µ 1 operators acting on ], and the results therein lead to a closed p qω − =1 i p X =1 = Tr 1 i 17 and − X † a i − for some 1 ϕ p − =1 βH i Y βω Z – 9 – − † − +1 i =1 ∆ or e N e =1 N x Y Z ω Y a gives the contribution to the partition function a,b operators lie in the adjoint representation of U( are integers labelling each state. † ab tr =

+1 a i i = Z = Z 0 . ω q ω J | = ϕ r Z ) = Tr i † = Z i Z ϕ H q, x phys ,...,N ( | ). They carry quantum numbers of Z H p = 1 a limit of the matrix model partition function as an affine chara N with operators transform in the fundamental of both U( for the U(1) Cartan elements of the SU( a i ϕ ω µ respectively. Evaluated on any physical state, the Hamilto i µ and ) gauge symmetry. Our strategy is to first enumerate all gauge If we ignore the restrictions of gauge invariance, then the H Our interest is in the partition function All the complexity in the problem lies not in the Hamiltonian Our partition function will depend on both the (inverse) tem ω N and ∆ = 1. Taking the trace over states of the form each species of operator in turn. The contribution to the partition function of the form and are singlets under SU( define: it consists of any number of where Tr is the trace over all states, trivial structure of the physicalU( Hilbert space originatin where the quantum numbers ∆ and the trace over states of the form by these fugacities and only later projectfurther onto fugacities the for each gauge Cartan invariant element subset. of the With gauge symm Meanwhile, the The Hamiltonian is trivial: it simply counts the number of formula for the large ical potentials means that they come with a factor these chemical potentials, the Hamiltonian for the matrix m limit. This limit has been studied in detail in [ JHEP08(2016)007 ) ) ) k 2 p ck N 3 ) or , ) (3.4) 2 λ 5 ( ℓ , appears 1 j y. We will ) and SU( ). This is an , . . . , λ N 2 λ ( of the partition. ). This can be , λ Y and the fugacities 1 N ϕ λ a Z U( w Z length = ( 3) can alternatively be ⊂ Z o expand the integrand , les of the partition. We will entary facts about these λ 3 r the partition (7 ) ical states must be SU( tric functions. For further  groups of U( , b c 5 re on the group manifold of N ts, , ω ω = 0 5 rge. ]. As symmetric functions are plex plane for each integration , ures of the latter. expanded in terms of symmetric − weight 21 1 )+1 Young diagram }| λ  j ( ℓ c 6= = is the number of times that b Y i > λ j λ ! ) λ a ( +1 1 : ℓ k a λ dω ω ≥ ) requires that physical states carry charge , is called the boxes. Each row is aligned so that the left-most i ≥ i i – 10 – I λ |{ λ 2.4 1 ... πi 1 ≥ 3 i 2 ) = λ λ P ( =1 N ≥ j Y a = 2 m

) but are singlets under SU( | λ ! λ N | 1 ≥ N row contains 1 in the denominator ensures that the only contributions we pi ) of the positive integer λ th λ i ) = ( i +1 j k m ω q, x ) where the exponent of each entry indicates its multiplicit ( 2 can be represented graphically by a is a non-increasing sequence of non-negative integers, 3 Z ; i.e. , λ λ λ 2 ) of non-zero elements in the sequence is called the 5 λ , ( 1 ℓ multiplicity for the set of all partitions. partition ) in a suitable basis of polynomials. The integration variab P ). The factor of A partition Our strategy for evaluating the partition function will be t The A 3.4 are invariant under permutations corresponding to the Weyl N i 3.1 A digressionIn on this symmetric subsection functions wedetails review and some proofs standard facts of about the statements symme reviewed below see [ respectively. This means that the partitionpolynomials. function can be Before proceedingfunctions. we pause to review some elem We can specify a partition either by listing its non-zero par of ( up in the contour integral are those with correct overall cha by specifying multiplicities. For example the partition (7 boxes sit under each other. For example, the Young diagram fo imposed by contour integration, giving us the expression in the partition x use this notation extensively below. where the contour of integrationvariable. is the Here unit the circle product inU( the factor com arises from the Haar measu array of boxes where the looks like this: write The sum of all the elements, labelled by partitions we will begin by reviewing basic feat under the U(1) centre of U( written as (7 The number We now impose the requirements of gauge invariance. The phys singlets. Further, the level constraint ( JHEP08(2016)007 s (3.5) . The and of the ) with , . . . , n µ | 2 r ) for the , variables X µ ) r, s variables. | n n ( n ic functions σ = 1 = ( i x invariant under n i and thus the sum S λ,µ x , . . . , x δ n ∈ S ough infinite dimen- (2) = σ σ ∀
in    -plane for 1 , so , x i n  − ). Finally, we also define are is vectors are labelled by µ n x i j of the monomial 2 ric functions of X denote a set of (1) an orthonormal basis: x x lative to the top left hand X 1 ) T σ } } , n 1 ). λ x ( n n 2 n − paces of finite dimension cor- , we define the Schur function µ ( . . . x µ σ plex µ n 3 X 1 g 2 , m ( 3  µ 2 ) g ) = ∈ P x n . . . x 1 . . . x } X }| , . . . , x µ µ 1 2 ) i>j i i ( Y 1 = (5 . x µ 2 (2) 2 . λ x λ } n X r µ σ x ) corresponds to the weight ≥ { T 1 ( ) labelled by their coordinate µ n 1 = 1) x m ) , . . . , x j g µ 1 − n X 2 2 µ = { ) therefore contains boxes ( λ x − (1) 1 3 . In particular, one possible choice of basis r, s µ ! λ σ { , µ σ i X , x n . . . x ( i ) is any polynomial of the X 2 ( i x 1 1 : m σ , . . . , λ 2 n 1 5 Y x x ≤ µ 2 = ( dx , µ n ≥ ( = , . . . , x ≥ i 1 X x ) 1 f – 11 – = 1 1 x /S } j C µ − 2 µ 1 n acting on the variables µ ( s I is obtained by interchanging the rows and columns = |{ S x ℓ X ] = , , . . . , x n , x ∈ X λ
λ r 2 1    πi 1 ) = σ S { [ 2 − 1 n n σ σ ( , x T i x σ . For each partition 1 λ n { ) = n x =1 S parts: i Y on an arbitrary function ( ∈ X X f n

σ ( n µ S ! , . . . , x 1 n m ) = ∈ ). Explicitly the non-zero parts of ) = by (2) λ . For example, (7 X σ 0 σ ( X variables 1 ) contains boxes ( ( ≥ i ≡ λ Y λ f of the partition µ , x n µ Z ( s T Y (1) λ , m Schur functions σ ) = λ x T with at most P → λ m : ( h f . One can easily define an inner product on the space of symmetr ) and, for each value of µ n ∈ P λ ( µ denotes the transpose , . . . , ℓ denotes the stabiliser of the monomial 1 µ n − , . . . , ℓ S = 1 X The set of all symmetric polynomials forms a vector space. Th The Another possible set of basis vectors for the space of symmet We now turn to symmetric functions. Let i = 1 with respect to which the monomial symmetric functions form is taken over distinct permutations permutation degree of the monomial symmetric function is provided by the A symmetric function Here action of a permutation vectors are monomial symmetric functions, given by of the Young diagram the action of the permutation group Concretely the set Here the contour of integration is the unit circle C in the com We will frequently use the shorthand notation and for the function r partitions sional, it is naturallyresponding written to as symmetric a direct polynomials sum of of fixed vector degree. subs The bas specifying the row andcorner column of the respectively diagram. of the The Young diagram diagram re JHEP08(2016)007 ) ), C X ( (3.8) (3.7) n, µ ( s Cauchy gl of degree variables. we have n X } λ,µ m is the precisely δ = λ,µ fold with the Haar , each monomial in
. ). Each such repre- are all positive inte- 1 K λ defined by } epresentation theory , . . . , y C − 1 R S Schur function y unctions in i as the product of Schur X λ,µ Schur functions provide n, { , symmetric power of the ( n, m K h λ mber { µ = gl s expressed by the th s in their close relation to the tation and the correspond- s e precisely, each monomial ) (3.9) j ne can construct a “matrix” Y ) o bases by writing min ) Y ( µ X ( ( λ ℓ > ) (3.6) s on is just the familiar orthogonal- µ and λ ) ) s n X λ } . Equivalently, the representation is ) corresponds to a family of ( X ( ) and that of the permutation group. n is essentially the character of the cor- n  ( ⊗ ℓ µ n i λ j } ( 3.6 s ≤ x x n m u denote the ... ) as the character of − j λ λ λ,µ . ⊗ , . . . , x ( X λ n λ 1 1 2 ℓ K s µ x R  ), is a polynomial in the variables = , . . . , x { ). The finite dimensional, irreducible represen- n j 1 µ j X n 6= – 12 – x X ( i y Y ( = ⊗ i { µ u is the multiplicity of the irreducible representation 1 x 1 µ X ! m = ) = i n − i λ,µ of length X x X 1 dx ). As discussed in more detail below, the Schur function K ( λ λ . The non-vanishing entries of ) = λ | ( C s =1 m µ . Thinking of Y j µ I ( Y | . This correspondence is a consequence of the famous Schur- T n πi λ 1 =1 = i Y 2 | R variables λ | n n appearing on the r.h.s. of ( =1 i Y µ

m ! 1 n ≡ Kostka numbers S ) are inherited from those of its complexification i n µ ( is zero unless u ) characters with respect to integration over the group mani , s n λ . For any two sets of variables s λ,µ h ) which will be important below. Let K ), evaluated on the n Like the monomial symmetric functions discussed above, the Although not immediately apparent from this definition, the The Schur functions form a complete basis for the symmetric f The Kostka numbers also have a second interpretation in the r ( in the decomposition of the tensor product X giving the explicit linear transformation between these tw u . The significance of the Schur functions for our problem lies ( | λ λ µ the Schur polynomial correspondsing to coefficient a is weight simply ofsymmetric function the the represen multiplicity of this weight. Mor gers, known as Here Weyl duality between the representation theory of K a basis for the vector space of symmetric functions. Indeed o responding representation labelled by the Young diagram s sentation is labelled by a partition representation theory of the Lie algebra tations of functions in the summand will vanish identically for The sum on the right-hand side can be taken over all partition like the monomial symmetric function In the group theoretic context describedity above, of this U( relati | measure. The completeness ofidentity the Schur functions as a basis i weights related by the action of the Weyl group. The Kostka nu They are orthonormal with respect to a modified inner product the common multiplicity of these weights in R fundamental representation. Then of JHEP08(2016)007 . | λ ) is | q ; (3.13) (3.12) (3.10) (3.11) on the X . ( P n λ as defined i . For any λ n P , q λ h have exactly of degree
t . . . x . 1 X 2 q − λ 2 = 0. Can we find partition function x X    1    q λ 1    g i j  x ial in i i j j x x i j ) x x x x x x q = q X − ( λ − lex parameter − − 1 f 1 1 X polynomials discussed above 1  ion, each term in in the partition    her with the adjoint partition   ns, which are orthogonal with i j j h respect to the inner product j i j x x x x ) q <λ i>j Y λ i Y we define ( − λ − j n 1 λ n 1 m n
λ,µ  j λ n  ϕ δ ∈ P j n in the special case q j 1 ) λ 6= 6= λ i . . . x Hall-Littlewood polynomials q ≥ S 1 − i j i N 2 . . . x 1 Q , − λ 2 Q 2 Q h x λ 2 = ) 1 (1 x λ λ 1 =1 m ! P 1 ( Y j ℓ i i – 13 – x λ 1 i ) as µ − x x ) we define =    n dx with integer coefficients. One might instead choose    ,P ϕ σ X m λ C ( 3.11 q ) in terms of symmetric functions. Because of the σ n ϕ I g P = S λ h n ∈ 3.4 λ X πi 1 σ /S 2 N n 6= 0? In fact the λ S X 1 q ) and ∈ N n =1 σ i Y X (

), is a homogeneous polynomial in the variables f ) = q ) = ! q ; 1 q n ; ; X X ( ≡ X ( λ ( λ P λ P i P P f, g h , it will be convenient to introduce yet another inner produc Z ) denotes the multiplicity of the positive integer Z λ ( j ; however then the basis functions would no longer be polynom m As already mentioned, we have the orthogonality property As stated above, our goal will be to evaluate the matrix model P i , It is useful to rewrite the definition ( space of symmetric functions depending on an arbitrary comp to normalise these functionsh to achieve orthonormality wit itself a polynomial in the parameter above. As before, two symmetric functions this property. Moreover, many ofare the generalised properties in of a the nice Schur way. For each partition where and An even more striking fact is that, with the given normalisat presence in this integrand offunction the Haar measure factor, toget where the sum is over distinct permutations of the monomial Note that our new inner product reduces to The normalisation factor is given by a new set ofrespect basis to functions, new generalising measure the with Schur functio by expanding the integrand of ( JHEP08(2016)007 λ ). µ and ( , for 0 T } . Here ≥ n B Z ), the cor- ∈ 3.7 , . . . , x 1 x { asis from Schur to eralisation of the for all partitions ]) and they will play = = 1 in the definition, count the number of site. Remarkably, the λ,µ 21 q ) (3.14) for some ∆ q X th K ; i λ,µ ∆ egarded as a graded gen- ) defined in ( the Heisenberg spin chain q Y K ( µ system provides an efficient ) and the third follows from the s given in appendix ( λ um for a spin chain with peri- P T (1) = ) (3.15) ) n explicit combinatoric formula 3.15 at the q q ; ; i λ,µ ), the Hall-Littlewood polynomial µ ) are polynomials in the parameter , the Kostka polynomial assigns an X X ]. ion ( K q n ( ( λ ) as the Hilbert space of a spin chain ( λ 22 µ µ 3.11 [ ( P ] with leading coefficient equal to unity. = 0. On setting P λ,µ ) ) λ T q q [ q = 1: K ( ( n λ q b − λ,µ ] is the monomial symmetric function defined ) for µ K λ [ µ (see e.g. chapter III.6 of [ X X – 14 – n ( ] (see eq. (2.6) in chapter III of this reference). The first µ m λ = 2 X 21 s ) receives a contribution j in the tensor product j q y y i ( i λ ) = x R qx λ,µ X . | ( − of degree ) where K − µ λ ) to each irreducible component of the tensor product | 1 ) is now generalised to q s 1 X ). From the definition ( ( = q λ 3.6 | λ, µ , the matrix elements ( =1 m ) ) spin in the representation Y j m λ | λ n ( ( j ∈ P n =1 u Kostka polynomials i Y m µ ϕ . 1) = , 1 λ ≥ µ,ν . We now have j X, δ } ( λ Q m P ) = 0 unless . q ) = ( (0) = µ q ( ) sites with a , . . . , y ) reduces to the Schur function λ,µ λ,µ λ 1 µ q b ) above. Again we can find a “matrix” describing the change of b ( y All non-zero coefficients are positive integers. K and They are polynomials in They reduce to the Kostka numbers for K ; ℓ { The completeness of the resulting basis is expressed in a gen 3.5 X The last two listed properties follow easily from the definit • • • • • ( 2 = λ . They are known as in ( we also find with these spins. Inodic fact ∆ boundary is conditions. the The essentially lattice Bethe moment ansatz solution of this energy ∆ precisely corresponds to one of the Hamiltonians of where occurrences of the representation q For each choice of invertibility of the changetwo of properties basis are proven highly in non-trivial [ and were first proven in responding Kostka polynomial a central role in ourfor evaluation the of the Kostka partition polynomials function.we due will A to list Kirillov some and of Reshetikhin their i main features: P Hall-Littlewood. Relation ( Cauchy identity. As before we consider two sets of variables with each such occurrence. Hence as we vary the partition It is useful to think of the representation space of These properties ensure thateralisation the of Kostka the polynomials Kostka can numbers. be r As the Kostka numbers integer-valued “energy” ∆( Y JHEP08(2016)007 . . ) ) 5 ]. | λ q q µ ( ; | but R 23 Hall- X ρ λ,µ Thus, ( ′ variables 4 K ). Indeed Q t , ( n λ λ,µ K of degree of length greater } λ n ). The partition ) where ∆ is the µ ( ) representation 3.4 we have omials, we see that of any length so we can expand it n T ( } ) µ , . . . , x u } for Schur functions [ m 1 p s of symmetric functions in X ). For our purposes, this x S ( q i { λ λ ; , s R ) (3.17) h X ) ) does not vanish identically = q gether we learn that ( , . . . , y q q ; , . . . , x rtitions ) (3.16) 1 pin chain defined on the tensor ′ µ ; ; 2 defined in ( ). Further, for each y nomials and leads directly to the X Y with positive integer coefficients. l-Littlewood polynomial in X Q Y { ( X ) ( ( , x q ( ρ ] Z 3.7 ρ 1 λ Y ′ µ = P ( x in the mathematical literature. s P 24 ) ) Q λ { ) q g the combinatorial algorithm of appendix B. , Y q s ; q ( ) ( 23 = X [ X ( λ,µ λ,ρ ( and ′ ρ is the character of a λ X K K appearing in the Kostka polynomial } Q s λ ) vanishes identically for partitions n s µ λ ρ λ,ρ λ X ) is non-zero only for irreducible representations X – 15 – . X X ) defined in ( q 3.16 ( µ m ( = = = . ) = , . . . , x . Milne polynomials λ,µ n T ≤ 1 q j B ; x ) y K ≤ i { ρ ) fugacities ) X x ( 1 ( p µ ℓ = ( ( ′ µ − ℓ u Q 1 X for each occurrence of the irrep . with respect to the inner product =1 m µ Y j ∆ q P > n n =1 ) ). The final sum on the r.h.s. can be taken over all partitions i Y has the following key property: it is adjoint to the ordinary µ ( ℓ ′ µ ). 3.16 µ Q ( T Modified Hall-Littlewood polynomials ) and ( 3 ) is a homogeneous polynomial in the variables q 3.15 ; , there is no such constraint for the partition n As discussed above, each Schur function To evaluate the partition function we will need one more clas Using the properties of the Schur functions and Kostka polyn X occurring in the tensor product Note that in some references the definition of the modified Hal These are also sometimes referred to as This follows because, although the r.h.s. of ( ( 5 3 4 ′ µ λ Importantly, this definition yields a non-zero answer for pa we evaluate several examples of this type in the following usin R is nevertheless restricted to the case receives a contribution than using ( function is symmetric in the For any two sets of variables unlike the other symmetric functions defined above, known as polynomial is defined by the formula for partitions with explicit formulae given in appendix combinatoric description of the corresponding Kostka poly Meanwhile, the Kostka polynomial product space has a natural interpretation as the partition function of a s appropriate spin chain Hamiltonian. Putting these facts to the summand will vanish unless 3.2 Back toWe the are partition now function ready to compute the partition function Q The polynomial Littlewood polynomials Moreover the coefficients are themselves polynomials in JHEP08(2016)007 . ) N
q kN ; . The (3.18) of the 1 ctions. k − = ( ), which Ω λ ) fugacities ) N N . k p ( k, p, N ( P 0 ≤ ) E ) q undamental-valued λ ( auchy identity in the ), as an inner product ℓ ) holds for all positive (Ω; ) . However, as explained λ q he definition of the Hall P 3.13 and 3.19 nds to the character of a ) (3.19)   ∈ P (Ω; b a b a model is λ X ate energy ω λ ω ω ω ( kN P unction of the U( q λ ) s using ( q − =
− ; ) | 1 q q ) 1  ; X ( ) ( 1  N b 1. In this special case, the formulae non-zero parts each equal to ′ λ N k − = 6 ), the resulting expansion determines ( k a a,b | Q ( p Ω ≤ N λ, Q ) Q ) = λ λ K | X N ( appearing in the expansion of the partition k λ ℓ ! ( | = λ a q P P a X i i ω ) – 16 – x dω = ) a N j for the adjoint-valued field will be left unexpanded k of the global symmetry and the particle number , and k a ω q C 1 ( Z ω I p 1 = 1. In this case there is only one partition − − Z ,P kN ) to write 1 p 1 λ =1 πi 1 (1 N a 2 P = ) global symmetry present in the matrix model spectrum. h p | 3.12 Q p =1 =1 i N Y N λ Y =1 j N | ) Y a U( q =1 N = Y

1 a , the rank ⊂ − k Our result for the partition function ( ! Z ) = 1 ) p N (1 ]. q ϕ ; 1 Z ) denotes the partition with × × X ( N 1, and compare it with our expectations based on the analysis ) ) ) k q q and we may thus expand it in terms of a suitable set of basis fun N ; ; k ′ ( } X X | ≪ N Q ( ( q | ′ λ ′ λ ) q Q Q ) to expand the factor of the integrand corresponding to the f ( 1 . . . , ω N λ λ 1 ϕ X X 3.17 ω { To proceed to the answer by the shortest path, we will use the C We begin with the abelian case = = = Z simply corresponds to the leading power of function for form ( ground state given in [ In the remainder of this section we will extract the ground st resulting integral over the variables Ω can then be written, integral values of the level, fields as Thus our final result for the partition function of the matrix The integrand of the partition function is also a symmetric f where the sum on the right-hand side runs over all partitions where, as above, ( finite-dimensional irreducible representation of U( Ω = in terms of Schur functions. As each Schur function correspo Ground state energy. as part of thepolynomials integration in measure. its second form For the ( next step, we use t the multiplets of the SU( In contrast, the corresponding factor above the summand vanishes unless we have which satisfies the conditions JHEP08(2016)007 . ) ) q N ( ible = 1 T ) 3.20 N q (3.20) k ( λ, K . The cor- p rm constraint. = 1 reads . This partition ), we deduce that p X ) representation ( p ) specialises for ]. λ ( = 1 where an explicit q s ( u ,...,N 15 n of the Kostka polyno- ) a polynomial k 2 N , mber where each power of k 6 ( ible λ, ) as mentioned above. ) ) = 1 l information contained in the j j λ K ent in the tensor product. ( q q l ]. The remaining factor in ( means we actually have an infi- 1 1  ) − − k ] > ℓ 1) ) responding to all possible products 1 ) q p x ) ( ( − N (1 (1 µ − ) h k ( ⊗ j ( H ℓ q [ q ) with =1 N =1 N n in the tensor product l N j Y ( j − q ... − Z λ 1 Q simplify, giving N 1) ⊗  ] − 2 k (1 B ) = k T ) N 1 is somewhat richer; it also depends on the q λ ( λ p [ =1 ( ( N j ) N n Y ) = – 17 – ≥ ⊗ 2 k q Y N ∈ Q k q k x p ( ,N , p ) . Thus the partition function for 1 ) = i kN q x = µ ) = kN k, ( ( q ( ) 1) ( N = 0 N K T ) can be nonzero when E H − (1 t =1 ( i λ, p ( . This in turn coincides with the multiplicity of the irreduc 1 ) K Z λ,µ ≥ . We will start with the easiest case N i K ) Cartan elements. To understand the form of this partition k B p ( P λ, symmetric power of the fundamental representation K th ) specified by the partition ] = k p µ ( [ = 1 matrix model was previously computed in [ u n p ) multiplied by the corresponding Schur function acts as a fugacity for the U(1) charge which is fixed by the D-te for the SU( 3.19 1 i x ) is the hook-length polynomial given by x independent single-trace operators tr( q ( = N H x copies of the The partition function for general To go further we will need to use the combinatoric descriptio Here we see explicitly that N 6 appears with a non-negative integer coefficient. As the Kostk where, as above, formula is available. Here we have is a plethystic exponential accounting forof excitations the cor where The ground state energy, where for the Kostka polynomial given in appendix the full spectrum of the matrix model transforms in the reduc representation of responding Kostka polynomial isq a gradation of the Kostka nu More precisely, the overall prefactor of mials given in appendix nite number of copiespartition of function this is representation. the The energy additiona of each irreducible compon of appears in ( agrees with the identification of the ground state given in [ to the Kostka number function for the function, we start by recalling that the Kostka polynomial fugacities JHEP08(2016)007 ) ) ≃ th for ) 1.2 M ,M , the p L M B p U( + = ( ]. Lp 1 T 0 λ , obtained by = λ ) corresponding N λ ( Y and transforms in an mplicated, the gener- antisymmetric power of kN LM ). Explicitly the non-zero th ) λ p + ( 1. Writing h the results of [ M is is the representation ( p Y h the properties of the ground ≥ 1)
i of the partition -fold symmetrisation of the kLM − k M . These restrictions correspond to
, . . . , x 1 p 0 is the length of the hook passing T i − }| + L M i p 1 is x λ ( ) p ( − ≤ > L p ) ≥ 1) 1) ) for all 1 2 kL j M | ≪ λ ( − ,L p − λ ( − q , + 1 i p | = ℓ M transpose ( L s M ≤  ,L 1 ( 1 : ) of the Young diagram ) ) ≥ M k L T i − i X and ≥ + 1) λ 2 k r, s r – 18 – + ,M +1) j = L L L N − ( |{  p = ( (( kL ( ) = T T s s = (  = λ x ) λ | = , the minimum occurs for the partition  n T i λ N 0 ) corresponding to a , which coincides with the | ( + n = denotes the λ 1 is straightforward. As we discuss in appendix 0 p λ and that r 0 N E k, p, N ) = T λ λ q ( N λ 0 M < p k > E = ) = , p, N | Z ≃ x λ ( and . To find the ground state energy of the model we must therefore (1 | 1 h 0 L λ = E | ]. 1 and T ) = . The Schur polynomial corresponds to the representation of λ . The resulting ground state has U(1) charge | T λ M λ M ( . over partitions with + x ) with U(1) charge + are λ p T Lp Lp , . . . , ℓ λ SU( = = × = 1 As we described above, Although the formulae for the Kostka polynomials are more co N N i corresponding to demanding that to the partition Thus the leading term in the partition sum for with vacuum energy non-negative integers minimise the quantity as we vary through box for that we mentioned in the introduction; it is in agreement wit minimum energy is obtained for the partition and takes the value interchanging the rows and columns of the Young diagram for Here the product is over the boxes parts of antisymmetric power of the fundamental representation. Th for the fundamental. Again, thisstate yields complete discussed agreement in wit [ irreducible representation of SU( U(1) alisation of this analysis to JHEP08(2016)007 , ) ) p ). ). p p p p ( ( ( u u su ). As 3.22 and 1 rows ) which M − of ( 3.19 p ntations of . Given an 3.30 p al symmetry with ≤ is divisible by ) . Setting λ (1) centre of N . This means that ( u affine Lie algebra at p ℓ → ∞ ) (3.21) antisymmetric repre- p ˆ L → ∞ X ng at most A ) corresponding to the th ( p λ mod ( with N M = 0, so s u s quantum numbers under s λ ) (3.22) N M X kN/p kLM garding the difference between ; − ion function of the WZW model. q + ( w p ) q k,M 1) ( ) R ) appearing in the sum on the r.h.s. ). When ˆ . In contrast, representations of − Z N p χ k X ( ( L M ) fugacities; this will be explained below. ( limit of the partition function ( ) λ λ, is the fugacity for the < p j p s kN/p + L rather than the original partition function q p K ) 2 k w 1 ˜ of SU( λ ˆ 0 − Z ( Lp λ E ℓ X – 19 – → ∞ q . . . x (1 = ) = 0 rows or to partitions k,M 2 E = denotes the character of the x N , we therefore take the limit R p =1 N − ∞ 1 Y j L q Z x -fold symmetrisation of the limit. The main result of this section is ( k,M k ) = having and the SU( k, p, N j = R ( ˆ q and ˜ λ Z χ 0 ), we obtain a unique irreducible representation of (1) charges of the states in the spectrum relative to those w 1 X p − E u ( u is the → ∞ →∞ = (1 lim N 0 N =1 M < p N k,M E Y j R = ). Recall that the finite-dimensional, irreducible represe ). The remainder of this section is devoted to the derivation p ˆ ( Z 1.2 6= 0, su M ⊕ ) symmetry depend sensitively on the value of p (1) . In the following it will be convenient to decompose the glob u λ ≃ ) is the character of the irreducible representation of ) fugacities labelled by associated to the representation p ) p k As discussed above, each Schur function It is also convenient to factorise the partition function as ) are in one-to-one correspondence with Young diagrams havi ( 3.21 p u ( or, equivalently with partitions correspond to diagrams with at most irreducible representation of su as of ( thus encodes the energies and of the ground state. As we will see, it is sentation, namely ( this is the vacuum character which coincidesMeanwhile, with for the partit is the ground state energy and Here the “roughly” refersthe to U( a slight notational subtlety re The key part of the result is that says (roughly) that we saw in thethe previous global section, U( the ground state energy and it partition which has a non-singular In this section we will investigate the 3.3 The continuum limit The reduced partition function level where in order to get a sensible limit, we must hold this value fixed a held fixed. for non-negative integers JHEP08(2016)007 , ) 1 ). p − ˜ p X Q kN ) for with ( A ˜ λ (3.24) (3.25) i s + ( h | ≤ . Given 3.21 ˜ λ ˜ λ such that | lently, the = ) = kM ˜ λ X and < p + ( in the variables λ ) p | s ˜ λ λ ( | ) (3.26) kLp reducible represen- ℓ ) more explicit (see ˜ X C mod ( = . Each such Λ has an ˜ λ appearing in ( | p, s ) is a function of variables ( tions and the characters + W with λ : ) kM N sl } q Λ . In the following we will ˜ } λ ( k , in which case we set 1) ) R ( Q = = p ∈ L /p | ons − N | 1 1 p k  of ˜ λ ( − denote the corresponding ( = | − ,   ) p Λ w | Λ p ) p ) A kM j λ R ˜ λ | h ( ) ( = 1, which implies Q Λ ˜ ,..., | − X R j ) ( p ˜ ) when λ +( j z | ) (3.23) ˜ λ (1) ˜ , . . . , x ˜ x λ ( character p s  1 Λ Λ /p K Q j − { =1 1 p 1 /p is divisible by 3.24 ... | Y j p ψ − λ 2

q L ( L 1) = (1 ( ( kL − L L ( µ −
L
, 1 2 1 + 1 )) 2 k )) L =  s M ( M ) of the Young diagram M ) ) for non-negative integers L λ − k
−
− ) = (0) ( ) = 1 2 P + 1) + P ν )) )) r, s Y in the configuration by a factor of r ( – 27 – ( + Y M ∈ L K K ) L L x − − − K + ) ) = = ( )( p p , p, N ( kL ( T s ( ν K K L +1)+ L x k, p, N λ (1 +1)+ ) = ( ( Lp ( 0 L 1 with q L 1 0 k − − ( = (( + )( E )( , p, N = − E p r 0 K H K ( (1 M p λ λ − N − 0 L ( ) = M ) = E M } ( } (( (( ν 1 = ) = ν 1 { k = x { ( − ( ( j c ) h ) ( c ( 0 ) where 0 . According to the recipe we must find a sequence of partitions λ λ = M = 0 ( ( ) − ) λ and p K +1 K ( λ ( = K ν ,L ν X . Thus the new ground state energy is ≥ λ . , . . . , ℓ j M k x 2 , = 1 ) is the hook-length polynomial given by | , q + 1) ) ( ) and K L ( H N = 0 ν | As explained in the text, the minumum for the case K = (( = (1 0 gives a ground state energy to the partition λ through box Here the product is over the boxes for It is instructive toµ reproduce this result from the general f where and one simply scales each partition The above configuration has a straightforward generalisati set which is the result stated in the main text. with non-negative occupation numbers which minimises the c The vacancy numbers arelinearly remain with non-negative and the charge o we set see that this is achieved by maximising the number of parts in One may then check that the corresponding occupation number JHEP08(2016)007 1 } ˆ Λ 1 − p can − with A 1 (C.2) (C.1) } − 1 p ˆ lie in the A − i , . . . p i which is a there is an h h ij ij ht lattice Λ Λ = 0 δ δ V R i , . . . p ; ) and its Affine i of . Each weight of ] = ] = C j j ˆ Λ = 1 , f i i 0 generate an p, V i , f ( , f Ψ ; i i | , e i sl e i e [ [ h i > , f subalgebra. i { = 1 , e 1 } i , − , − } 1 p h j j p on space 1 { e f A − n expansion A ji ji − ing ˆ A A − − . For each weight of i . Finite-dimensional, irreducible rep- Λ ) nδ Ψ ) , . . . , p i i | ( V ) , . . . , p ( i ] = i ] = ( + j j ψ are characterised by a highest weight ) = 1 Λ i i , f + Λ = 1 ( , f 1 = i i ψ , i ˆ Λ − h h i ) i , i [ 1 p i [ (0) ) ( ˆ i ˆ Ψ ψ − =1 ˆ Λ A ( | i p X Λ 1 – 28 –
Λ { − i =0 = 0 { of i , p , X h ) ≥ Z j i j ˆ Z ( Λ Ψ = e e Λ ˆ Λ ji R ji . We then have a basis vector ˆ we have Chevalley generators Ψ = ˆ R A A Z 1 − = Span ∈ p ] = = Span ] = ˆ j j A is the imaginary root. The fundamental weights of i W ψ + W δ , e , e L i i L h h [ [ Cartan matrix. Weights of each irreducible representation where are fundamental weights of the global 1 , , ) − n i p ( 1. . A are labelled by a highest weight Λ, lying in the positive weig 1 − ] = 0 − ] = 0 and Λ j p j i , we work in a Chevalley basis with generators ˆ R A ˆ 1 ψ , h , h i i is the affine Cartan matrix. The basis elements with − is the p h h [ [ , . . . , p ij ij A with Dynkin labels ˆ 0, where Λ A A = 1 W The integrable representations For the affine Lie algebra For i > i L with non-negative Dynkin labels, and have the representati be written as subalgebra. Weights of an integrable representation have a where for for integers with brackets for simultaneous eigenvector of the Cartan generators satisfy counterpart where of brackets Here we give our conventions for the simple Lie algebra C (Affine) Lie algebra conventions whose basis vectors are the fundamental weights Λ resentations element weight lattice We denote the corresponding representation space JHEP08(2016)007 ]. , ]. is a ] , , , ˆ Λ s ressible SPIRE V ommons IN SPIRE of ][ IN [ i , ˆ Ψ ry | , , , , cond-mat/9809384 tional quantum Hall (1989) 108 [ hep-th/0306266 [ redited. ]. B 326 ]. , (1999) 8084 Remarks on the canonical SPIRE SPIRE i IN i (2004) 046 IN ˆ Ψ ]. ˆ ][ Ψ | | ][ B 59 i n 02 Nucl. Phys. ]. ˆ ψ ]. , = ]. = i ]. i ˆ Ψ | SPIRE ˆ JHEP Ψ ) SPIRE | , IN 0 – 29 – SPIRE
IN ]. ]. i [ L Phys. Rev. ][ SPIRE IN ( [ h ). The corresponding basis vector , IN ˆ [ Λ ˆ Λ A matrix model for non-Abelian quantum Hall states R C.2 SPIRE SPIRE cond-mat/9811352 R arXiv:1508.00580 [ − [ New class of non-Abelian spin-singlet quantum Hall states cond-mat/9602113 IN IN [ [ [ Edge excitations of paired fractional quantum Hall states ), which permits any use, distribution and reproduction in ]. (1989) 351 Nonabelions in the fractional quantum Hall effect ]. Quantum Hall effect in supersymmetric Chern-Simons theorie (1990) 12838 Beyond paired quantum Hall states: parafermions and incomp Many body systems with non-Abelian statistics (1992) 1711 121 SPIRE hep-th/0103013 [ Quantum Hall states as matrix Chern-Simons theory IN (1999) 5096 SPIRE [ (1992) 615 (1991) 362 B 6 B 41 IN CC-BY 4.0 [ (1996) 13559 (2015) 235125 82 This article is distributed under the terms of the Creative C 1, and the derivation or grading operator, with The quantum Hall fluid and noncommutative Chern-Simons theo B 374 B 360 Theory of the edge states in fractional quantum Hall effects − B 53 B 92 Quantum field theory and the Jones polynomial Chiral Luttinger liquid and the edge excitations in the frac (2001) 011 A quantum Hall fluid of vortices ]. Phys. Rev. 04 , , . . . , p SPIRE IN = 1 Int. J. Mod. Phys. Phys. Rev. states in the first[ excited Landau level Phys. Rev. Lett. Nucl. Phys. Nucl. Phys. quantization of the Chern-Simons-Witten theory JHEP hep-th/0101029 Phys. Rev. Commun. Math. Phys. arXiv:1603.09688 states has an expansion of the form ( i ˆ Λ [1] N. Dorey, D. Tong and C. Turner, [8] B. Blok and X.G. Wen, [9] G.W. Moore and N. Read, [6] A.P. Polychronakos, [7] L. Susskind, [3] D. Tong and C. Turner, [4] E. Witten, [5] S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, [2] D. Tong, [11] E. Ardonnes and K. Schoutens, [13] X.-G. Wen, [14] M. Milovanovic and N. Read, [12] X.G. Wen, [10] N. Read and E. Rezayi, Open Access. Attribution License ( for simultaneous eigenvector of the Cartan generators, with References any medium, provided the original author(s) and source are c R JHEP08(2016)007 e , C. R. ]. , , in HE . SPIRE IN , (in French), ynomials ][ onformal anomaly , (1994) 38 (1995) 61 ]. 32 ]. 118 , Amer. Math. Soc., , SPIRE , Oxford Mathematical 254 ]. IN SPIRE ][ IN hep-th/0106072 [ ][ SPIRE IN ]. ][ Hall-Littlewood functions and Kostka-Foulkes ]. SPIRE (2001) 006 IN Contemp. Math. – 30 – 07 , ]. ][ hep-th/9504052 [ ]. Sur un conjecture de H.O. Foulkes The Bethe ansatz and the combinatorics of Young ´m LotharingienS´em. Combin. . Prog. Theor. Phys. Suppl. hep-th/9505083 Finite noncommutative Chern-Simons with a Wilson [ , , SPIRE JHEP q-alg/9512027 Kostka polynomials and energy functions in solvable lattic Crystalizing the spinon basis Crystalline spinon basis for RSOS models [ SPIRE , IN Critical theory of quantum spin chains [ IN math/9803006 [ (1996) 179 ]. (1988) 925 (1996) 395 (1978) 323. 41 178 arXiv:0812.4531 [ SPIRE (1997) 547 (1986) 746 IN A 11 3 (1987) 5291 ][ 288A Symmetric functions and Hall polynomials 56 Edge excitations of the Chern Simons matrix theory for the FQ New combinatorial formula for modified Hall-Littlewood pol Dilogarithm identities B 36 Universal term in the free energy at a critical point and the c (2009) 100 J. Sov. Math. Sel. Math. , 07 , -series from a contemporary perspective hep-th/9408113 Phys. Rev. tableaux Phys. Rev. Lett. polynomials in representation theory [ Monographs, Oxford University Press, Oxford U.K. (1995). Acad. Sci. Paris Commun. Math. Phys. Int. J. Mod. Phys. JHEP line and the quantum Hall effect models q Providence RI U.S.A. (2000) [ [23] A.N. Kirillov, [25] A.N. Kirillov and N.Yu. Reshetikhin, [26] I. Affleck, [27] I. Affleck and F.D.M. Haldane, [24] B. J. Leclerc D´esarm´enien, and J.-Y. Thibon, [21] I.G. Macdonald, [22] A. Lascoux and M.P. Schutzenberger, [18] A. Nakayashiki and Y. Yamada, [19] A. Nakayashiki and Y. Yamada, [20] A.N. Kirillov, [16] B. Morariu and A.P. Polychronakos, [17] A. Nakayashiki and Y. Yamada, [15] I.D. Rodriguez,