Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models

The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels $k \in \mathbb{Q}$. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level $\widehat{\mathfrak{sl}}(2)$ models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu's algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.


INTRODUCTION
We begin by briefly summarising some historical details regarding fractional level sl (2) models and the application of Jack symmetric functions to conformal field theory. The work presented here forms a part of of an ambitious project aimed at elucidating the properties of general fractional level models as fundamental examples of logarithmic conformal field theories. A short digression on the notion of admissibility follows as we use the term in a non-standard manner as compared to much of the literature. This also provides us with an opportunity to fix some notation. Finally, we outline the main results of the research reported here.
For the Virasoro algebra, the admissible central charges c are then precisely those for which there exist coprime integers p, p ′ ∈ Z ≥2 satisfying c = 13 − 6(t + t −1 ), where t = p p ′ . These central charges correspond, of course, to the Virasoro minimal models which are commonly denoted by M p, p ′ . We will also denote the simple Virasoro vertex operator algebra of this (admissible) central charge by M p, p ′ , for convenience.
For sl (2), the structure theory of its Verma modules [41] leads to the following characterisation: Proposition 1.1. The admissible levels k of the universal vertex operator algebras of sl (2) are precisely those for which there exist coprime integers u ∈ Z ≥2 and v ∈ Z ≥1 satisfying k = −2 + t, where t = u v .
Needless to say, the admissible levels of sl (2) are precisely those of the fractional level sl (2) models (including those of non-negative integer level, for convenience). To emphasise the analogy with the Virasoro minimal models M p, p ′ , we shall denote by A 1 u, v both the fractional level sl (2) model with k = −2 + u v and the corresponding simple vertex operator algebra. We regard the A 1 u, v as the minimal models of sl (2). Note that A 1 k + 2, 1 requires k ∈ Z ≥0 and is therefore just the level k Wess-Zumino-Witten model on SU 2 .
It should be clear now that the focus of our interest is not so much on the universal vertex operator algebras themselves, but rather on their admissible level simple quotients. The point is that these simple quotients will have constrained representation theories, due to their additional defining relations, about which we expect to be able to prove classification theorems. The representation theory of the universal vertex operator algebras is, on the other hand, unconstrained by additional relations so that (almost) every g-module is allowed. 2 To complete the analogy between Virasoro minimal models and fractional level sl (2) models, we consider the modules of the simple quotient vertex operator algebras.
Definition. Suppose that i ∈ I is admissible for a given family {V i } of universal vertex operator algebras. Then, any module of the simple quotient of V i is said to be an admissible module of V i .
Note that the admissible modules of the universal Virasoro vertex operator algebra are precisely the central charge c = 13 − 6( p p ′ + p ′ p ) modules of the Virasoro minimal model M p, p ′ . These are the highest weight Virasoro modules corresponding to the entries in the Kac table. The admissible modules of the universal sl (2) vertex operator algebras are the subject of this article. As noted above, they were first classified in [36,37]. We remark that they can also be associated with the entries of a table with similar properties to the Virasoro Kac table, see [15].
Finally, note that the definition of admissible module given above is not the original definition of Kac and Wakimoto [2], who originally defined admissible highest weight modules, for arbitrary affine Kac-Moody algebras, in terms of criteria that guaranteed a character formula, generalising that of Weyl-Kac, and good modular properties.
The vertex operator algebra definition given above is certainly more general and, in our opinion, more fundamental. However, the two definitions of admissibility coincide for highest weight sl (2)-modules. A generalised version of this coincidence for higher rank affine Kac-Moody algebras has recently appeared in [42].
1.3. Outline. We close with a brief outline of the contents of this paper. Section 2 is a pedagogical introduction to the notion of relaxed highest weight theory, crucial for studying the A 1 u, v models. The idea actually reduces to a special case of parabolic (also called parahoric or generalised) highest weight theory, but the connection to vertex operator algebras via Zhu's algebra is so important that we feel it warrants separate consideration. This is then followed by a detailed discussion of the relaxed highest weight modules of the three Lie algebras used in the remainder of the paper: The Heisenberg algebra, the (bosonic) β γ ghost algebra and, of course, sl (2). Section 3 opens with a brief review of the Wakimoto free field realisation of sl (2) and the corresponding Wakimoto modules. We pay particular attention to the screening fields and operators as a means to motivate the usage of symmetric function theory. We then discuss the construction of certain singular vectors in highest weight and relaxed highest weight sl (2)-modules using Jack symmetric function technology to deduce explicit formulae 2 One should only exclude modules, such as the adjoint module, that lead to operator product expansions with essential singularities.
for these (Theorem 3.1). We believe that these formulae are new: Even in the standard highest weight case, the only similar result we are aware of is an old paper of Kato and Yamada [43] where an integral formula is derived and evaluated in special cases using Schur polynomials. For singular vectors of relaxed highest weight modules, the only other formulae we know are of the Malikov-Feigin-Fuchs (complex power) form [39].
One consequence of this singular vector study is that at admissible levels, Wakimoto's construction yields a free field realisation of the universal vertex operator algebra of sl (2) rather than of its simple quotient A 1 u, v . This is important as it means that the singular vector of the universal vacuum module is accessible in the Wakimoto realisation, hence it may be exploited to such ends as determining the spectrum of A 1 u, v -modules. We first obtain an upper bound on the highest weight spectrum (Corollary 4.2) using a surprisingly effortless calculation that combines the form of the vacuum singular vector with the specialisation formula for Jack symmetric functions.
This bound turns out to be saturated, but to prove this we must address the more involved relaxed highest weight spectrum. In this case, a few more symmetric function manipulations allow us to identify an explicit presentation for Zhu's algebra (Theorem 4.4) in terms of generators and relations.
It is now easy to classify the simple relaxed highest weight modules of A 1 u, v , reproducing in an elegant fashion the results of [36,37]. We emphasise that our proofs are, to the best of our knowledge, independent of those which have appeared in the literature. We also deduce that highest weight A 1 u, v -modules are semisimple, giving a new proof of an old result that (essentially) appeared in [2]. In contrast, A 1 u, v is shown to admit nonsemisimple relaxed highest weight modules with two composition factors and these are characterised using short exact sequences. We moreover conjecture that these modules, together with their simple analogues, exhaust the indecomposable relaxed highest weight A 1 u, v -modules, remarking that proving this would require, among other things, a more detailed knowledge of the submodule structure of relaxed highest weight sl (2)-modules. Finally, we prove that the Virasoro zero mode L 0 acts semisimply on all relaxed highest weight A 1 u, v -modules, independent of our conjecture on the indecomposable spectrum, and discuss briefly how this is consistent with the expectation that, for v > 1, the A 1 u, v model is a logarithmic conformal field theory.
Appendix A gives a brief, but thorough, introduction to the aspects of symmetric function theory, in particular, those relating to Jack symmetric functions, that are used in the text. This material is all standard and may be found in [28]. We have also included a brief introduction to Zhu's algebra in Appendix B, concentrating on motivating it as an abstract version of the algebra of zero modes acting on the space of "ground states" of a relaxed highest weight module. This appendix is aimed at physicists, in particular, it uses physics conventions for Fourier expansions, but we hope that it will also prove useful to mathematicians.

GENERALISING HIGHEST WEIGHT THEORY
In this section, we consider a generalisation of highest weight theory that we will qualify as relaxed. Originally introduced for sl (2) in order to study a correspondence relating sl (2)-modules to those over the N = 2 superconformal algebra [39], relaxed highest weight modules have since appeared as necessary constituents of the SL 2; R Wess-Zumino-Witten model [40], in admissible level fusion rules [7,11], in relations to logarithmic minimal models [10,44,45], in demonstrating the modular invariance of admissible level theories [14,15], and in the full description of bosonic β γ ghosts [11,13]. Necessity aside, we feel that from some points of view, particularly that of Zhu's algebra [46], discussed in Appendix B, it is more natural to consider these relaxed modules instead of the standard highest weight modules that one typically encounters in rational conformal field theory.
2.1. Relaxed highest weight theory. We recall that the formalism of highest weight theory for a Lie algebra g is built from a triangular decomposition This is a vector space direct sum of subalgebras of g in which the Cartan subalgebra h is abelian and acts semisimply, through the adjoint action, on both g − and g + . In particular, h, g ± ⊆ g ± . Moreover, the subalgebras g − and g + are assumed to be antiequivalent in that there exists a (linear) order two antiautomorphism, the adjoint †, satisfying g † ± = g ∓ . The elements of h are supposed to be fixed by †, meaning that each x ∈ h is self-adjoint [47]. This is, however, a little too restrictive; we instead only demand that the adjoint preserves the Cartan subalgebra.
Since h is abelian, this implies that each x ∈ h is normal: x, x † = 0. Note that being a linear antiautomorphism means that (ax) † = ax † and x, y † = y † , x † , for all a ∈ C and x, y ∈ g.
Given such a triangular decomposition, one defines a highest weight vector in a given g-module to be a simultaneous eigenvector of the elements of h which is annihilated by all the elements of g + . Any module generated by a single highest weight vector is called a highest weight module. Conspicuous examples include the Verma mod- where Ug denotes the universal enveloping algebra of g, Ub that of the Borel subalgebra b = h ⊕ g + , λ ∈ h * is the highest weight, and C λ is the one-dimensional b-module upon which g + acts as 0 and each x ∈ h acts as λ (x) ∈ C. The highest weight vector generating V λ is 1 ⊗ Ub 1 λ , where 1 denotes the unit of Ug (and 1 λ that of C λ ∼ = C).
The Lie algebras g that one typically encounters in conformal field theory have triangular decompositions. However, they also have more structure in that they are graded by the semisimple action of the Virasoro zero mode L 0 so that, if g n denotes the ad (L 0 )-eigenspace of eigenvalue −n, then g m , g n ⊆ g m+n . Moreover, the Cartan subalgebra h is generally chosen to include L 0 , which we will always assume is self-adjoint, hence it follows that g † m = g −m . Given this structure, the following definition is natural: Definition. Let g be a Lie algebra with triangular decomposition (2.1) (where the Cartan subalgebra may include elements that are not self-adjoint). If there exists L 0 ∈ h such that g = n g n , where g n is the eigenspace of ad (L 0 ) of eigenvalue −n, and L 0 is the zero mode of a subalgebra of g isomorphic to the Virasoro algebra, then we will say that g is conformally graded.
The most obvious example is the Virasoro algebra itself which is clearly conformally graded with g n = span {L n }, for n ∈ Z \ {0}, g 0 = span {L 0 ,C} and g n = {0} otherwise. This shows that the usual, but somewhat confusing, convention that L 0 , L n = −L n is responsible for g n having ad (L 0 )-eigenvalue −n in the above definition.
Definition. Given a conformally graded Lie algebra g, its relaxed triangular decomposition is where g < = n<0 g n and g > = n>0 g n . A relaxed highest weight vector is then a simultaneous eigenvector of h ⊆ g 0 that is annihilated by g > . A relaxed highest weight module is a module that is generated by a single relaxed highest weight vector. The relaxed Borel subalgebra is g ≥ = g 0 ⊕ g > and a relaxed Verma module is a g-module where M is a simple weight module of g 0 upon which g > acts as 0.
These definitions have obvious analogues for Lie superalgebras and other more general structures, but we will not need this level of generality in what follows.
In this article, we will only consider triangular decompositions of a conformally graded Lie algebra g that satisfy g < ⊆ g − and g > ⊆ g + . Thus, positive modes are always in g + and negative modes are always in g − . When g 0 = h, the relaxed triangular decomposition then coincides with the (unique) triangular decomposition of this type. We will shortly see examples of relaxed highest weight modules which are not highest weight modules in the usual sense. First, however, we mention that relaxed highest weight modules may be identified as generalised highest weight modules with respect to the parabolic subalgebra g ≥ = g 0 ⊕ g > . We recall that a parabolic subalgebra is any subalgebra that contains a Borel subalgebra (see [47,48] for a quick overview of parabolic subalgebras).
It will be occasionally convenient to take this a step further and introduce a category of modules, generalising the well known category O, that contains the relaxed highest weight modules of g. This is essentially (a variant of) the parabolic category O discussed, for example, in [48]. The explicit details of this category are not essential for understanding the results to follow, but we shall devote a few paragraphs to explaining their physical motivation.
Definition. Given a conformally graded Lie algebra g = g < ⊕ g 0 ⊕ g > , define the relaxed category R to consist of the g-modules M satisfying the following axioms: • M is finitely generated.
• M is a weight module (the action of the Cartan subalgebra h is semisimple).
• The action of g > is locally nilpotent: For each v ∈ M, the space Ug > · v is finite-dimensional.
The morphisms are the g-module homomorphisms between these modules, as usual.
All highest weight and relaxed highest weight modules belong to category R. Moreover, if g has finite-dimensional root spaces, then it follows that each module in M will have finite-dimensional weight spaces. Another important consequence of these axioms is that every (non-zero) module in category R possesses a relaxed highest weight vector, hence that every simple category R module is a relaxed highest weight module. One can, and should, ask whether the mathematical axioms that we impose on category R will end up excluding modules relevant for applications. This is an important question and the answer is that they do exclude relevant modules, but in a well-controlled manner. Our motivation for introducing this category is that we want to classify the modules of the admissible level k vertex operator algebra A 1 u, v by identifying these modules as sl (2)modules. For the physical application of investigating the corresponding conformal field theories, we must insist that the category of A 1 u, v -modules be closed under the conjugation operation of sl (2) (see Section 2.2) and fusion. Moreover, we want the characters to behave well under modular transformations so that one can identify modular invariant partition functions and compute (Grothendieck) fusion rules from a Verlinde-type formula.
Category O is not sufficient for these purposes, in particular, the conjugate of an A 1 u, v -module from category O need not lie in category O. Relaxing to category R alleviates this problem and has recently been shown [15] to lead to characters with excellent modular behaviour, provided that one extends the category again to take into account twists by the so-called spectral flow automorphisms. The upshot is that these spectrally-flowed modules are not in category R, but the twisting is very well understood, justifying the above statement that the exclusion of these physically relevant modules is under control.
Axiomatically, accounting for the spectrally-flowed modules would require weakening the local nilpotency axiom above. However, this axiom has the advantage that category R provides a very natural setting for the important, and very useful, technology of Zhu's algebra, discussed in Appendix B. We restrict to weight modules because the fusion coproduct formulae [49] for sl (2) show that the fusion product of two weight modules will again be weight. Similarly, the conjugate of a weight module is weight and omitting non-weight modules does not restrict the characters in any way. Moreover, being weight does not preclude the Virasoro zero mode L 0 from acting non-semisimply as is required in logarithmic conformal field theories. To summarise, the relaxed category R is a rich source of modules for affine Kac-Moody (super)algebras that appears to be even more relevant to conformal field theory than the much more familiar category O.

Examples.
In this paper, we will use the Wakimoto free field realisation to study the relaxed highest weight modules of sl (2) and determine which of these are modules for the admissible level vertex operator algebras and v ∈ Z ≥1 are coprime and the level k is determined by u v = t = k + 2. We will therefore need to investigate the (relaxed) highest weight theory of the Heisenberg algebra H, the c = 2 bosonic β γ ghost system G and sl (2) itself. This investigation constitutes the rest of the section.
The Heisenberg algebra H. We will use the same notations and conventions for the Heisenberg algebra as in [33]. The free boson vertex operator algebra is generated by a single bosonic field a(z), defined by the operator product expansion With the standard Fourier expansion, a(z) = ∑ n∈Z a n z −n−1 , the operator product expansion implies the following commutation relations: a m , a n = mδ m+n,0 1, m, n ∈ Z. (2.4) The Heisenberg algebra H is then the infinite-dimensional Lie algebra spanned by the a n and 1, the latter being identified with the unit of the universal enveloping algebra of H, as usual. We will assume that 1 acts as the identity operator on any H-module. This is only a minor restriction since a simple rescaling of the generators lets this operator act as multiplication by any non-zero number.
As is well known, the free boson vertex operator algebra admits a one-parameter family of conformal structures. We will write the energy-momentum tensor and central charge in the form where α parametrises the conformal structure. We note that α → ∞ reproduces the standard free boson central charge c bos. = 1. In Wakimoto's construction, this would correspond to k → ∞, so it is permissible to ignore this case. It is worth recalling that a(z) is not a Virasoro primary for α finite; instead we have The Fourier expansion T bos. (z) = ∑ n∈Z L bos. n z −n−2 defines the Virasoro modes and it is easy to check that the Lie algebra g spanned by the a n , L bos. n and 1 is conformally graded. It is likewise easy to check that (for finite α) the only adjoint on the Heisenberg algebra consistent with the standard Virasoro adjoint (L bos.
Since g 0 = span {a 0 , L 0 , 1} is abelian, it coincides with the Cartan subalgebra h, hence relaxed highest weight theory reduces to ordinary highest weight theory for the Heisenberg algebra. We remark that this is one example where we cannot insist that the Cartan subalgebra consist of self-adjoint elements.
The highest weight theory of the Heisenberg Lie algebra is well known. The Verma modules are known as Fock spaces and are parametrised by the a 0 -eigenvalue p of the highest weight vector. They are simple for all p ∈ C. We will denote the Fock spaces by F p and their corresponding highest weight vectors by u p . In accordance with (2.7), the module conjugate to the Fock space F p is F −p−2/α .
The ghost algebra G. For the β γ ghost system, we follow the notations and conventions of [13]. The ghost vertex operator algebra is generated by two bosonic fields, β (z) and γ(z), whose operator product expansions are From these, one constructs a Heisenberg field J(z) and an energy-momentum tensor T gh. (z) by These give β (z) and γ(z) Heisenberg weights +1 and −1, and conformal weights 1 and 0, respectively. We remark that J(z) is not normalised as in (2.3), nor is it primary with respect to T gh. (z): As with the free boson, there is actually a one-parameter family of conformal structures; T gh. (z) has been chosen so that the ghost fields β (z) and γ(z) have the required conformal weights. This choice also fixes the central charge of the ghost system to be c gh. = 2.
The Fourier expansions β (z) = ∑ n∈Z β n z −n−1 and γ(z) = ∑ n∈Z γ n z −n now yield the commutation relations β m , β n = 0, γ m , β n = δ m+n,0 1, γ m , γ n = 0; m, n ∈ Z. (2.11) The infinite-dimensional Lie algebra G, spanned by the β n , γ n and 1, is called the ghost Lie algebra. Again, 1 is identified with the unit of UG and we assume that it acts as the identity on all G-modules. As with the Heisenberg algebra, we will extend this algebra by the modes J n , L gh.
n ∈ UG, defined by J(z) = ∑ n∈Z J n z −n−1 and T gh. The Cartan subalgebra h = span J 0 , L gh. 0 , 1 is a proper subalgebra of g 0 = span β 0 , γ 0 , J 0 , L gh. 0 , 1 , so the relaxed and ordinary highest weight theories of the β γ ghost system do not coincide. In this case, the ghost adjoint implies that all the elements of h are self-adjoint.
We start with the ordinary highest weight theory corresponding to the triangular decomposition in which β 0 ∈ g + annihilates the highest weight vector v and γ 0 ∈ g − need not. It turns out that this yields a unique Verma module because these conditions imply that J 0 v = L gh.
0 v = 0 (and 1 acts as the identity, as always). This module is simple; in fact, we may take it to be the ghost vacuum module G because these conditions also imply that L gh. −1 v = 0. Of course, one can also take the triangular decomposition in which γ 0 annihilates the highest weight vector and β 0 need not. 3 The resulting Verma module is the simple module c G conjugate to G, obtained by twisting the action of G by c, the (order 4) conjugation automorphism that sends β n to γ n and γ n to −β n .
The relaxed highest weight theory of the ghost system is significantly more interesting as we no longer require that β 0 (or γ 0 ) annihilates the highest weight vector. In fact, we may induce from a fairly arbitrary simple weight module of g 0 . Given that J 0 and L gh. 0 , as abstract elements of g 0 , are going to be identified with elements of UG and that the g 0 -module weight vectors are going to be identified with the relaxed highest weight vectors of the induced module, we may restrict to g 0 -modules on which the action of J 0 is identified with that of γ 0 β 0 and the action of L gh. 0 is always 0. Proposition 2.1 [13,Prop. 1]. A simple weight module over g 0 upon which J 0 = γ 0 β 0 and L gh. 0 = 0 is isomorphic to one of the following: • The module G generated by a vector v which is annihilated by β 0 and thus also J 0 . This module has a basis of weight vectors v j , j ∈ Z ≤0 , where J 0 v j = jv j .
• The module c G , conjugate to G, generated by a vector v which is annihilated by γ 0 and thus J 0 v = v. This module has a basis of weight vectors v j , j ∈ Z ≥1 , where J 0 v j = jv j .
• The modules G q , where q ∈ C \ Z, each of which is generated by a vector v satisfying J 0 v = qv; it follows that no non-zero vector is annihilated by β 0 or γ 0 . The eigenvalues of J 0 = γ 0 β 0 all lie in q + Z and these modules have a basis of weight vectors v j , j ∈ q + Z, satisfying J 0 v j = jv j . The modules G q and G q+1 are isomorphic.
Additionally, one can consider the indecomposable g 0 -modules G + 0 and G − 0 that likewise have a basis of weight vectors v j , j ∈ Z, satisfying J 0 v j = jv j , but they are not simple. They are determined (up to isomorphism) by the following non-split short exact sequences: Inducing the g 0 -modules G, c G and G q (q / ∈ Z) therefore results in relaxed Verma modules for G. The first two give the simple ghost vacuum module G and its conjugate c G , respectively; the last gives new modules G q which are also simple and satisfy G q ∼ = G q+1 . We may similarly induce the non-simple g 0 -modules G + 0 and G − 0 to obtain non-simple G-modules G + 0 and G − 0 which are likewise determined (up to isomorphism) by the following non-split short exact sequences: The affine Kac-Moody algebra sl (2). We consider the universal vertex operator algebra of non-critical level k = −2 generated by three fields e(z), h(z) and f (z) satisfying the operator product expansions (2.16) and no other (independent) relations. We denote this vertex operator algebra by V k . The maximal proper ideal of V k is non-trivial if and only if k is admissible. Moreover, this ideal is generated by a single primary field (singular vector) [41]; we do not set this primary field to zero. Note that we have chosen the sl 2 basis {e, h, f } to be consistent with previous work, [12] in particular, where it was necessary to tailor the basis to the sl 2; R adjoint rather than the (more traditional) su 2 adjoint. This is reflected in the signs appearing in the formulae for and these make span {e n , h n , f n , 1} into a Lie algebra which we denote by sl (2). Once again, we assume that the unit 1 ∈ U sl (2) acts as the identity on each sl (2)-module.
The standard conformal structure of V k is uniquely determined by requiring that e(z), h(z) and f (z) are Virasoro primaries of conformal weight 1 (this structure exists for all k = −2). The Sugawara construction then gives the explicit form of the energy-momentum tensor as where t = k + 2. With T z = ∑ n∈Z L n z −n−2 , one finds that the modes L n generate a copy of the Virasoro algebra with central charge c = 3 − 6/t. The Lie algebra g spanned by the e n , h n , f n , L n and 1 is then conformally graded with Cartan subalgebra h = span {h 0 , L 0 , 1}. We have chosen the sl 2 basis so that the sl 2; R adjoint becomes Again, the Cartan subalgebra consists of self-adjoint elements.
The highest weight theory of sl (2) is well known. The standard triangular decomposition splits the zero modes so that e 0 ∈ g + annihilates highest weight vectors but f 0 ∈ g − need not. Then, the Verma modules V λ are parametrised by the sl 2 -weight (h 0 -eigenvalue) λ ∈ C of the highest weight vector because it follows from (2.18) that its conformal weight is then given by These Verma modules need not be simple.
by acting with f 0 on the (generating) highest weight vector of V 0 , is the vacuum module; it carries the structure of the universal vertex operator algebra V k defined by (2.16). As with the ghosts, one can also consider the triangular decomposition in which f 0 annihilates highest weight vectors but e 0 need not. 4 The Verma modules with respect to this decomposition are then the conjugates w V λ of the Verma modules V λ , where the (order 2) conjugation automorphism w sends e n to − f n and h n to −h n .
is non-abelian, it strictly contains h, so the relaxed highest weight theory of sl (2) is strictly more general. We note that inducing from a g 0 -module reduces to choosing an sl 2 -module because we require that 4tL 0 acts as h 2 0 − 2e 0 f 0 − 2 f 0 e 0 (this is how 4tL 0 acts on relaxed highest weight vectors). The analogue of Proposition 2.1 is then the classification of weight modules for sl 2 (see [50], for example): Proposition 2.2. The simple weight modules of sl 2 are exhausted by the following: • The (λ + 1)-dimensional modules L λ , with λ ∈ Z ≥0 . The module L λ has a basis of weight vectors w m , where m = λ , λ − 2, . . . , −λ and h 0 w m = mw m . It is both highest and lowest weight.
Inducing each of the simple sl 2 -modules now yields relaxed Verma modules for sl (2). More precisely: This induced module need not be simple; its simple quotient will be denoted by L λ . These simple modules are self-conjugate: • Inducing R λ ;∆ , with λ , ∆ ∈ C satisfying 4t∆ = µ (µ + 2) for any µ ∈ λ + 2Z, results in a new relaxed Verma module that we shall denote by R λ ;∆ ; its simple quotient will be denoted by E λ ;∆ . As above, there are isomor- We emphasise that whereas the simple highest weight modules (and their conjugates) are characterised by a single parameter, the highest weight, the E λ ;∆ require two parameters in general. The three classes of simple sl (2)modules L λ , D λ and w D −λ , and E λ ;∆ are distinguished by their relaxed highest weight vectors: L λ has finitely many (λ + 1 in fact); D λ and w D −λ have infinitely many, but their sl 2 -weights are bounded above and below, respectively; E λ ;∆ has infinitely many with no bound on the sl 2 -weights.

THE WAKIMOTO FREE FIELD REALISATION
A free field realisation of sl (2) for any level was constructed by Wakimoto in [20]. It shows that fields e z , h z and f z , satisfying the operator product expansions (2.16), may be constructed in terms of a free boson and a pair of bosonic ghosts. We review this and the screening operator formalism of free field theories, concluding by deriving explicit formulae, in terms of Jack symmetric polynomials, for certain (relaxed) singular vectors in (relaxed) Wakimoto modules over sl (2). A corollary of this analysis is that for admissible levels, Wakimoto's construction describes the universal vertex operator algebra V k of sl (2), rather than its simple quotient A 1 u, v .

The Wakimoto construction.
There is a one-parameter family of realisations of sl (2) in terms of tensor products of the free boson and ghost fields. Given the operator product expansions (2.3) and (2.8), it is straightforward to verify that defining (we omit the tensor products for notational simplicity) reproduces the sl (2) operator product expansions (2.16) with the level k = t − 2 being related to the parameter α by α 2 = 2t. Moreover, the sl (2) energy-momentum tensor (2.18) and central charge then decompose as identifying α with the deformation parameter in the free boson conformal structure (2.5).
Definition. We define the Wakimoto vertex operator algebra to be the tensor product of the Heisenberg and ghost vertex operator algebras, equipped with the conformal structure given in (3.2).
Recall that when k is admissible, the universal sl (2) vertex operator algebra is not simple, but has a unique maximal ideal that is generated by a single primary field (singular vector). One of the results of this section (Corollary 3.2) is that this field is non-zero in the Wakimoto vertex operator algebra.
The Wakimoto free field realisation endows the tensor product of a Heisenberg Fock space and a ghost module with the structure of an sl (2)-module, by restriction. We refer to such modules as Wakimoto modules, distinguishing at least four types: • The highest weight Wakimoto modules W p = F p ⊗ G, recalling that G denotes the ghost vacuum module.
• The conjugate highest weight Wakimoto modules c W p = F p ⊗ c G , obtained by conjugating the ghost module.
• The relaxed highest weight Wakimoto modules The structure of the highest weight Wakimoto modules W p was determined in [4]. To the best of our knowledge, the relaxed modules W p;q have not previously been considered in the literature.
Consider now the tensor product of a Heisenberg highest weight vector u p and the ghost vacuum v, denoting it by p = u p ⊗ v for convenience. The free field realisations (3.1) of the sl (2) fields imply that p is an sl (2) highest weight vector of sl 2 -and conformal weight where we recall that α 2 = 2t. Similarly, the tensor product p; q = u p ⊗ v q of u p with a relaxed highest weight vector v q for the ghosts is an sl (2) relaxed highest weight vector of sl 2 -and conformal weight This shows that all the simple sl (2)-modules, highest weight and relaxed, may be realised as subquotients of Wakimoto modules.

Screening fields and operators.
We begin by recalling the construction of vertex operators for the free boson. For this, one extends the Heisenberg algebra by introducing a generatorâ satisfying The vertex operators V p (z), parametrised by p ∈ C, are then defined by A standard computation shows that the vertex operators are free boson primaries of Heisenberg weight p and conformal weight 1 2 p p + 2 α , by virtue of the operator product expansions For later use, we record that the composition of k vertex operators is given by We now define the notion of a screening field.

Definition.
A screening field Q(w) for a vertex operator algebra V is a field, generally not belonging to V itself, which has the property that the singular terms of the operator product expansions of each of the generating fields of V with Q(w) are total derivatives in w.
Our definition of a screening operator, in a moral sense at least, is then a vertex operator algebra module homomorphism that can be constructed from screening fields. This means that a screening operator commutes with the fields, and hence the mode algebra, of the vertex operator algebra. The standard way of constructing screening operators is as the residues (zero modes) of screening fields; these are guaranteed to commute with all the vertex operator algebra fields, provided that the residues are well defined. Their chief application stems from the fact that they map (relaxed) highest weight vectors to (relaxed) highest weight vectors and thereby explicitly construct (relaxed) singular vectors. We remark that if the screening field is a Virasoro primary (excluding the identity field), then its operator product expansion with the energy-momentum tensor forces its conformal weight to be 1.
The operator product expansions of the free field sl (2) fields (3.1) with a free boson vertex operator are easily computed to be For a vertex operator to be a non-trivial screening field, its Heisenberg weight would have to satisfy 1 2 p p + 2 α = 1. However, the above operator product expansions show that these vertex operators are not screening fields for sl (2).
It follows that the zero mode is a screening operator, whenever the contour around 0 actually closes. This will be the case when Q [1] acts on a state for which the relevant Fourier expansion of Q(z) has only integer powers of z. Equivalently, Q [1] has a well defined action on a given state if and only if the operator product expansion of Q(z) with the corresponding field is a Laurent series. For example, Q [1] only acts on p = u p ⊗ v, the tensor product of a Heisenberg vacuum u p with the ghost vacuum v, This shows that the screening field Q(z) only defines module homomorphisms, hence constructs singular vectors, for certain vertex operator algebra modules.
To construct more module homomorphisms, it is natural to consider products of screening fields. To check that such products also yield screening operators, it is convenient to use the language of differential forms. Suppose then that j(z) is a vertex operator algebra field and that Q i (w i ) is a screening field, so that for some (n-dependent) P i (w i ). Then, is exact, hence the commutator vanishes upon integrating over a closed cycle. The generalisation to more than two screening fields is immediate.
We therefore ask the question of when there exists a closed cycle over which some given product of the screening fields Q(z) = V −2/α (z)β (z) can be integrated to define sl (2)-homomorphisms. We introduce the shorthand and consider the action of this product of screening fields on the Wakimoto module W p whose Heisenberg highest weight is p . Using (3.8), this action can be written in the form where p m (z) denotes a power sum and the overline p m (z) = p m z −1 1 , . . . , z −1 r denotes variable inversion (see Appendix A for our conventions for power sums and other symmetric polynomials). The action of Q [r] (z) on the other Wakimoto modules with Heisenberg highest weight p, for example the relaxed highest weight module W p;q , is identical -the ghost weight q is not changed.
Up to an unimportant phase factor, which we suppress, the first two factors on the right-hand side of (3.16) are ∏ 1≤i< j≤r where we recall that α 2 = 2t. The second factor on the right-hand side of this expression therefore isolates all the (potential) non-integer powers of the z i in (3.16), hence a closed cycle over which Q [r] (z) may be integrated will exist precisely when the common exponent of the z i in this factor is an integer, s ∈ Z say. This requires the Heisenberg weight p of the Wakimoto module W to have the form We remark that the multivalued function appearing on the right-hand side of (3.17), is just the integration kernel of the inner product (A.18) with respect to which the Jack symmetric polynomials are orthogonal. Setting p = p r,s , (3.16) now takes the form Again, the action on W p r,s ;q is identical except that the right-hand side now acts on W p −r,s ;q .
3.3. Singular vectors. The existence of cycles over which the product Q [r] (z) of screening fields may be integrated follows from the same arguments used in the analogous question for the free field realisation of the universal Virasoro vertex operator algebra, because the multivalued function G r (z;t) is the same in both cases. This question was answered for the Virasoro case (see Theorem A.4) by Tsuchiya and Kanie [26] to whom we refer for further details. We will use the cycles [∆ r ] that they construct in what follows, but normalised so that G r (z;t) dz 1 · · · dz r z 1 · · · z r = 1. (3.21) We mention that there are various explicit constructions of cycles, over which screening operators can be integrated, in the conformal field theory literature, but that the symmetric function literature uses different constructions again. However, Cohen and Varchenko [51] showed that, up to normalisation, there is only one non-trivial homology class of cycles when the integrand is G r (z;t) times a symmetric function, bar some restrictions on t, and thus the various constructions in the literature are all essentially equivalent. 5 The normalised cycles [∆ r ] let us construct sl (2)-homomorphisms (screening operators) from the products Q [r] (z) of screening fields. Our aim is to use these homomorphisms to explicitly construct singular vectors in (relaxed) highest weight modules over sl (2) using symmetric function technology (we refer to Appendix A for a primer on what is needed). Of course, we need to verify that the screening operators are not just zero maps.
To this purpose, we introduce an algebra isomorphism ρ δ , for each δ ∈ C * , from the algebra Λ of symmetric polynomials in infinitely many variables to the universal enveloping algebra UH − of the negative subalgebra of the Heisenberg algebra. This isomorphism is given, on the power sum generators, by ρ δ (p m (y)) = δ a −m . (3.22) For convenience, we will always take ρ δ to act only on symmetric polynomials of the variables y i . For the ghosts, we analogously define injective linear maps σ r and σ r from the algebra Λ r of symmetric polynomials in r variables to the ghost universal enveloping algebra UG. These maps are defined on the basis {g t ν : ℓ(ν) ≤ r} dual to the symmetric monomials m ν by We will only take these maps to act on symmetric polynomials in the x i . Here, the partition ν may have length ℓ(ν) strictly less than r; in this case, we pad the partition with zeroes so that the images σ r (g t ν ) and σ r (g t ν ) are padded by β −1 or β 0 , respectively, so that the right hand sides of (3.23) each consist of r factors.
With these maps, we can prove that the Q [r] (z) yield non-trivial screening operators and derive explicit formulae for certain (relaxed) singular vectors of the (relaxed) Verma modules over sl (2).
where Q t ν denotes the symmetric polynomials dual to the Jack polynomials (see Appendix A). The Q [r] -images of p r,s ( p r,s ; q ) in W p −r,s (W p −r,s ;q ) are therefore (relaxed) singular vectors, for s ≤ −1 (s ≤ 0).
Proof. We first show the non-triviality of Q [r] . Let p −r,s and p −r,s ; q denote the functionals dual to p −r,s and p −r,s ; q , respectively. Then, it follows from the ghost commutation relations (2.11) and adjoint (2.13 (3.27a) 5 It is because of these minor restrictions on t that we state our choice of class of cycle explicitly. These restrictions are only relevant for the precise statement of Theorem 3.1. A more detailed understanding of these cycles is not required for reading the remainder of this article.
Replacing the relaxed highest weight vector v q by the ghost vacuum v gives the same result, but for s ≥ 0 (because The non-triviality of Q [r] on W p r,s , for s ≥ 0, now follows from (3.20) and the normalisation of [∆ r ]: Here, we note that the a −m and a m , with m ≥ 1, appearing in (3.20) annihilate p −r,s and p r,s , respectively. A similar calculation gives the relaxed version for s ≥ 1: We note that the conformal weight of p −r,s ( p −r,s ; q ) is greater than that of p r,s ( p r,s ; q ) by rs. It follows is an sl (2)-module homomorphism and hence it preserves conformal weights.
We settle the non-triviality of Q [r] for the remaining values of s by explicitly computing the image of the (relaxed) highest weight vectors p r,s and p r,s ; q . In the former case, we obtain (p m (x) + p m (y)) p m (z) m p −r,s dz 1 · · · dz r z 1 · · · z r (if s = 0, this vanishes, in agreement with (3.26a), as the powers of the z i in the integrand are all positive) Here, we have used the finite-variable version of (A.16) to rewrite the sum over the symmetric monomials m ν and their duals g t ν as a product of exponentials of power sums. Then, we note that p m (x) + p m (y) is the power sum p m (x, y) in both x i and y j variables and use (A.16) again to expand the product of exponentials in terms of Jack polynomials and their duals. Finally, we use Proposition A.5 to identify ∏ i z −1−s i as a Jack polynomial in the z −1 i , then apply the orthogonality of Jack polynomials, and lastly note that the Jack polynomial norm follows from the normalisation of [∆ r ]. For s ≤ −1, the result is non-vanishing since ρ −α and σ r are injective. This proves the non-triviality of Q [r] on W p r,s for all s ∈ Z, as well as (3.26a).
A similar computation in the relaxed sector results in which is likewise non-vanishing for s ≤ 0 as ρ −α and σ r are injective. This proves (3.26b) and the non-triviality of Q [r] on W p r,s ;q , for all s ∈ Z.
Suppose now that the sl (2) level k is admissible: t = k + 2 = u v , where u ∈ Z ≥2 and v ∈ Z ≥1 are coprime. The sl (2) vacuum may be identified with the Wakimoto highest weight vector 0 = p 1,0 = p −u+1,−v ∈ W p −u+1,−v (we note the symmetry p r,s = p r+u,s+v ). Theorem 3.1 then guarantees that the sl (2)-module homomorphism Q [u −1] acts non-trivially on p u−1,−v to give a non-trivial singular vector in the vacuum module. The corresponding field then generates the non-trivial proper ideal of the universal vertex operator algebra, proving the following result: This is, of course, obvious if k is not admissible. What it means in the admissible level case is that calculations requiring the singular vector of the vacuum module of sl (2) may be equivalently carried out in the free field realisation using Jack symmetric polynomials.

CLASSIFYING ADMISSIBLE MODULES
In this section, we specialise to admissible levels k = t − 2, where t = u v and u ∈ Z ≥2 and v ∈ Z ≥1 are coprime. Then, the universal vertex operator algebra V k of sl (2) is not simple and the unique maximal proper ideal is generated by the field that corresponds to the singular vector of the vacuum module (see Section 2.2). As we saw in Theorem 3.1 and Corollary 3.2, the vacuum module may be constructed as a submodule of the Wakimoto vacuum module W 0 and its singular vector is then explicitly given by Quotienting by this maximal proper ideal, that is, setting the singular vector Q [u−1] p u−1,−v to zero, amounts to replacing the universal vertex operator algebra V k by its simple counterpart A 1 u, v . Our aim in this section is to use the explicit expression for the vacuum singular vector to classify the possible (relaxed) highest weights and thereby determine the spectrum of A 1 u, v -modules. By Corollary 3.2, these calculations may be performed in Wakimoto's free field realisation. More specifically, we will use symmetric polynomial technology to compute a generator of the annihilating ideal in Zhu's algebra. For readers unfamiliar with Zhu's algebra, we refer to Appendix B for motivation, basic definitions and a very short primer.
An old result of Frenkel and Zhu [52] states that Zhu's algebra Zhu V k for V k is nothing but the universal enveloping algebra U sl 2 of (non-affine) sl 2 (see Proposition B.3). By Proposition B.2, Zhu's algebra Zhu A 1 u, v for the quotient A 1 u, v is then the quotient of U sl 2 by the annihilating ideal generated by the representative of the singular vector Q [u−1] p u−1,−v . Since Zhu's algebra is filtered by conformal weight, whereas the conformal weight of e(z), h(z) and f (z) is 1 and the conformal weight of the singular vector is (u − 1)v, it follows that the image of the singular vector in Zhu V k is a polynomial in the sl 2 generators of total degree at most (u − 1)v. Furthermore, as Zhu's algebra is just the algebra of zero modes acting on relaxed highest weight vectors, the polynomial corresponding to the singular vector can be determined by evaluating the zero mode of the singular vector on general relaxed highest weight vectors, as in [53], since this is equivalent to evaluating the polynomial at infinitely many points.
However, the sl 2 -weight of the vacuum singular vector , which means that the corresponding field and its zero mode shifts the sl 2 -weight of any relaxed highest weight vector upon which it acts by this amount. It is far more convenient to work with a field that does not shift sl 2weights and so we instead consider the field corresponding to the sl 2 -weight 0 vector (4.1) Since u ≥ 2, we may renormalise this vector by dividing by the non-zero λ u−1,−v -dependent factors on the righthand side. The field corresponding to this renormalised vector is then We note that e u−1 0 acting on the vector f u−1 0 Q [u−1] p u−1,−v gives a non-zero multiple of the singular vector It follows that the annihilating ideal that we obtain from χ(w) will be the same as that which we would have obtained if we had instead worked with the singular vector directly.

Admissible highest weight modules.
In this section, we will evaluate the action of the zero mode of χ(w) on a general highest weight vector before moving on to the evaluation of χ(w) on relaxed highest weight vectors. This highest weight computation will not yet yield sufficient information to determine the image of χ(w) in Zhu's algebra Zhu V k as a polynomial in the sl 2 generators e, h and f , but it will constrain the weights of admissible highest weight vectors to a finite set. Every highest weight vector of V k may be realised as a highest weight vector of some Wakimoto module W p . It follows that the Heisenberg weights p of the admissible highest weight vectors of sl (2), that is, the highest weight vectors of the A 1 u, v -modules, are zeroes of The ghost contribution is easily evaluated by computing the operator product expansion of the ghost fields. Since v is the ghost vacuum, only the fully contracted part of the operator product expansion contributes: The contribution from the free boson part of the Wakimoto realisation is likewise easily determined. Up to non-zero constant factors, which obviously do not affect the zeroes of p χ(w) p , this contribution is where we have used (3.8), noting that all the terms involving the a n with n = 0 either annihilate the Heisenberg vacuum u p or its dual, and recognised the kernel of the symmetric polynomial inner product from (3.19).
Putting these contributions together, we find that the admissible Heisenberg weights of an sl (2) highest weight vector (in the Wakimoto free field realisation) must satisfy where we have recognised the product of the z i as a Jack polynomial using Proposition A.5 (recall that the overline indicates a symmetric polynomial in the inverse variables z −1 i ). We have also, again, neglected an overall non-zero constant factor. This expression has the form of an inner product for symmetric polynomials; to evaluate it, we only need to decompose the product over i into Jack polynomials. For this, we use specialisation in the form given in (A.28) with x i = −z i /w for i ≤ u − 1, x i = 0 for i > u − 1, and X = α p: Here, we have also used the homogeneity of the Jack polynomials (|τ| is the sum of the parts of the partition τ).
Using the orthogonality of Jack polynomials, (4.6) now becomes where the result of specialising the dual Jack polynomials Q t τ is given in Proposition A.5 (or (A.28)) and the arm and leg (co)lengths of (A.2) are easy to determine for the rectangular partition [v u−1 ]. As the denominators of the factors appearing in this expression are all strictly positive, we arrive at the constraint This proves the following result: Note that when v = 1, so that k = t − 2 = u − 2 ∈ Z ≥0 , the allowed sl 2 -weights λ r,s = λ r,0 = r − 1 belong to the set {0, 1, . . . , k}. These are, of course, the highest weights of the integrable sl (2)-modules and are well known to be the A 1 k + 2, 1 -modules that arise in the Wess-Zumino-Witten models defined on the Lie group SU 2 . At this point, we cannot say whether all these highest weight vectors do actually appear in an A 1 u, v -module.
For this, we need to work out the constraints on an arbitrary relaxed highest weight vector because it is these constraints which allow us to write down the generator of the annihilating ideal of Zhu's algebra. It is, however, well known that the Verma module V λ r,s , with r = 1, 2, . . . , u − 1 and s = 0, 1, . . . , v − 1, has infinitely many linearly independent singular vectors; however, none have sl 2 -weights belonging to the allowed set except the generator of sl 2 -weight λ r,s . It follows that there are only finitely many highest weight A 1 u, v -modules: Again, we have not yet proven that all these V k -modules are actually A 1 u, v -modules. However, it is germane to point out, at this point, that these highest weight modules are precisely the admissible modules first discussed by Kac and Wakimoto [2].

Admissible relaxed highest weight modules.
We now turn to the more intricate, but ultimately more rewarding, analysis of the relaxed highest weight modules of A 1 u, v . The goal is to determine the image of the field χ(w) in Zhu's algebra Zhu V k ∼ = U sl 2 as a polynomial I u,v (e, h, f ) in the sl 2 generators. Here, we identify the sl 2 generators e, h and f with the images of e(z), h(z) and f (z), respectively, in Zhu V k . Since Zhu's algebra is nothing but the algebra of zero modes acting on relaxed highest weight vectors (Appendix B), that describes the action of the zero mode χ 0 on relaxed highest weight vectors. We remark that as the sl 2 -weight of χ(w) is 0, the polynomial I u,v may be expressed as a polynomial in h 0 and the Virasoro zero mode L 0 . So as in the previous section, we evaluate matrix elements containing the field χ(w), but this time the "bra" and the "ket" will be relaxed highest weight vectors from a general relaxed Wakimoto module: The contribution from the free boson is exactly the same as in the non-relaxed case and was given in (4.5). The ghost contribution, however, requires more work. Wick's theorem lets us write this contribution in terms of contractions and normally-ordered products: Here, each factor of −z −1 i is the contraction of β (z i + w) and γ(w) and the factorials count how many such contractions are needed. As relaxed highest weight vectors are not necessarily annihilated by β 0 , the normally-ordered factor is quite non-trivial: Up to an overall non-zero constant factor, the total ghost contribution is therefore where e m denotes the m-th elementary symmetric polynomial and we have used the identity to get from the second to the third line. Combining this with the free boson contribution (4.5), the matrix element (4.10) is thus proportional to where we recall that elementary symmetric polynomials are examples of Jack polynomials (Proposition A.5). Using (A.25) and specialisation as in (4.7), but with X = α p + t, this reduces to In the last step, we have used (A.25) again and the definition (A.18) of the symmetric polynomial inner product. Proposition A.5 gives the norm squared of P t [1 m ] = e m and the specialisation of Q t µ as .

(4.17b)
Here, we have assumed that v > 1; if v = 1, then the denominator of (4.17b) is just ∏ m i=1 (t + m − i). Noting that the double product in the denominator of (4.17b) is a constant, independent of m, n, p and q, the matrix element (4.10) is thus proportional to where in addition to suppressing the denominator we have also suppressed an overall non-zero constant factor that arises when simplifying the binomial expressions.
The double product in this expression can be interpreted by noting that identified, then the matrix element (4.10) is given by In order to use this to find a generator of the annihilating ideal in Zhu's algebra, the sum factor in (4.20) needs to be expressible in terms of sl 2 data. To demonstrate this, we define a function f of u, p and q by For small values of u, it gives polynomials in λ p;q and ∆ p : Of course, α = √ 2t = 2u/v also depends upon u, but may be regarded as an independent variable for the following analysis because of its v-dependence.
Let g u,v (λ , ∆) denote the polynomial for which f p;q (u) = g u,v (λ p;q , ∆ p ) and let It is a simple corollary of (4.22) and (4.23) that g u,v has degree u − 1 as a polynomial in λ . If we regard ∆ as having degree 2, then the total degree of g u,v is also u − 1 and that of I u,v is therefore (u − 1)v.

Theorem 4.4. Zhu's algebra of A 1 u, v is given by the quotient
Proof. As noted above, the ideal of U sl 2 by which one quotients to get the Zhu algebra Zhu A 1 u, v is generated by the image of the null field χ(z). This image is a polynomial in e, f and h of total degree at most (u − 1)v. We have evaluated the action of the zero mode of χ(z) on a continuum of relaxed highest weight vectors and thus the image of χ(z) in Zhu's algebra is, up to non-zero constant factors, equal to the polynomial I u,v (h, T ) of total degree (u − 1)v.
Before we can use this presentation of Zhu's algebra of A 1 u, v , we need to know a little more about the zeroes of the polynomial g u,v .
We are now in a position to classify the simple weight modules over Zhu A 1 u, v . Recall that Zhu A 1 u, vmodules are automatically sl 2 -modules; by a weight module over Zhu A 1 u, v , we mean that it is a weight module over sl 2 . The classification of simple sl 2 weight modules was summarised in Proposition 2.2. We also recall that the quadratic Casimir operator acts as a scalar multiple of the identity on any simple sl 2 weight module.
Theorem 4.6. The following sl 2 -modules provide a complete list of the inequivalent isomorphism classes of simple weight modules of Zhu A 1 u, v : • The finite-dimensional highest and lowest weight modules L λ r,0 , where 1 ≤ r ≤ u − 1.
Proof. We first consider a simple finite-dimensional Zhu A 1 u, v -module M, which must therefore also be a finite-dimensional sl 2 -module. As the quadratic Casimir takes the value 1 2 (r 2 − 1) = 2t∆ r,0 on the r-dimensional simple sl 2 module, it follows that g u,v (h, T ) must act trivially on M because the remaining λ -independent factors of I u,v (λ , ∆) do not have ∆ r,0 as a root for any positive integer r. By Proposition 4.5, g u,v (m, ∆ r,0 ) = 0 if 1 ≤ r ≤ u − 1 and m = r − 1, r − 3, . . . , −r + 1. Conversely if r ≥ u, then g u,v (m, ∆ r,0 ) = 0 for some m = r − 1, r − 3, . . . , −1 + r, because the λ -degree of g u,v (λ , ∆ r,0 ) is u − 1, so there cannot be more than u − 1 zeroes. Thus, M must be isomorphic to one of the L λ r,0 for some 1 ≤ r ≤ u − 1.
Next, we consider a simple infinite-dimensional Zhu A 1 u, v -module M, which must therefore also be an infinite-dimensional sl 2 weight module. Because there must be an infinite number of weight vectors in M with distinct sl 2 -weights, g u,v (h, T ) cannot vanish identically on M. In order for I u,v (h, T ) to then vanish, T must act as multiplication by ∆ r,s for some (r, s) ∈ K(u, v). Referring to Proposition 2.2, it follows that the last three cases of Theorem 4.6 exhaust all the possible isomorphism classes for M.
The correspondence (Theorem B.1) between simple modules of a vertex operator algebra and its Zhu algebra then proves the following classification result: Theorem 4.7. The following sl (2)-modules provide a complete list of the inequivalent isomorphism classes of simple relaxed highest weight modules of A 1 u, v : • The highest weight modules L λ r,0 , where 1 ≤ r ≤ u − 1.
These are therefore the simple modules of the vertex operator algebra A 1 u, v which belong to the category R that was introduced in Section 2.1.
Zhu's correspondence also extends to non-simple Zhu A 1 u, v -modules. In particular, sl 2 admits reducible, but indecomposable, modules similar to the R λ ;∆ of Proposition 2.2 whenever 4t∆ = µ(µ + 2) for some µ ∈ λ + 2Z. (4.32) For ∆ = ∆ r,s , where (r, s) ∈ K(u, v), the only solutions are µ = r − 1 − ts = λ r,s and µ = −r − 1 + ts = λ u−r,v−s . As 0 < s < v, we find that λ r,s − λ u−r,v−s / ∈ 2Z, concluding that an indecomposable with ∆ = ∆ r,s may have at most one weight µ satisfying (4.32). Thus, there are [50] precisely two reducible, but indecomposable, sl 2 -modules R + λ r,s ;∆ r,s and R − λ r,s ;∆ r,s for each 1 ≤ r ≤ u − 1 and 1 ≤ s ≤ v − 1. They are determined (up to isomorphism) by the following non-split short exact sequences: Applying Zhu's construction now leads to the following result:  . We remark that we have not excluded the possibility that there exist relaxed highest weight modules over A 1 u, v that extend the E λ ,∆ r,s , or the E ± λ r,s ,∆ r,s , non-trivially; this would seem to require more information about the submodule structure of the relaxed Verma modules than is currently available, see [39,55]. However, the highest weight result given in Corollary 4.2 and the analogous results for the Virasoro minimal models suggest the following conjecture: Conjecture. The A 1 u, v -modules of Theorem 4.7 and Theorem 4.8 exhaust the indecomposable A 1 u, vmodules of category R.
We close by demonstrating that the non-semisimplicity of relaxed highest weight modules over A 1 u, v does not imply that the Virasoro mode L 0 acts non-semisimply, a fact that is of interest to logarithmic conformal field theory studies. This result requires the finite-dimensionality of the weight spaces of the category R modules, discussed in Section 2.1. 1) The image of the energy momentum tensor T in Zhu A 1 u, v ∼ = U sl 2 / I u,v (h, T ) acts semisimply on every weight module of Zhu A 1 u, v with finite-dimensional weight spaces.
2) The Virasoro zero mode L 0 acts semisimply on every relaxed highest weight module of A 1 u, v . Proof.
1) Let M be an indecomposable weight module of Zhu A 1 u, v on which (the image of) T acts non-semisimply. As T is proportional to the image of the quadratic Casimir Q, it follows that T has a single (generalised) eigenvalue on M. Then, by Weyl's theorem for sl 2 , M must be infinite-dimensional with an infinite number of distinct sl 2 weights. Let W be the submodule of M spanned by the eigenvectors of T . As W is non-zero, W must also be infinite-dimensional as it possesses an infinite number of distinct sl 2 -weights. Thus, W is an eigenspace of T with eigenvalue ∆ r,s , for some (r, s) ∈ K(u, v). Next, assume that there exists a generalised eigenvector v of T , so that (T − ∆ r,s )v = 0, with sl 2 -weight λ v . The existence of v would imply that I u,v (λ v , ∆) has a zero of order at least 2 at ∆ = ∆ r,s . Since there are only finitely many sl 2 -weights λ for which ∆ = ∆ r,s is a zero of I u,v (λ , ∆) of order at least 2, the quotient M/W must be finite-dimensional. But, the eigenvalue of T on v + W is then ∆ r,0 , for some 1 ≤ r ≤ u − 1, which is a contradiction. It follows that no such generalised eigenvectors exist, hence that T acts semisimply on M.
2) On any relaxed highest weight module over A 1 u, v , the action of L 0 on the relaxed highest weight vectors coincides with that of T on the corresponding Zhu A 1 u, v -module. As the latter action is semisimple, so is that of L 0 on the relaxed highest weight vectors. As these generate the whole module, L 0 acts semisimply.
We stress that this result does not imply that the conformal field theories corresponding to the vertex operator algebras A 1 u, v are non-logarithmic. Indeed, it has been known for some time that there are models with k = − 4 3 [7] and k = − 1 2 [9,11] that are logarithmic. The loophole is that we may also twist by the so-called spectral flow automorphisms which, when v = 1, lead to infinitely many new simple (and indecomposable) A 1 u, v -modules that do not belong to category R. For k = − 4 3 and k = − 1 2 , there exist indecomposable A 1 u, v -modules that are formed from relaxed highest weight modules from different spectral flow sectors and the action of L 0 on these is non-semisimple. In fact, these modules are staggered in the sense of [16,56]; a detailed discussion may be found in [14]. We expect that there exist staggered A 1 u, v -modules whenever v = 1, hence that the associated conformal field theories are logarithmic, and hope to report on this in the future. Macdonald's book [28]. In this appendix, we summarise the results from Chapters 1 and 6 of this book that we use freely throughout.
A.1. Partitions of integers. A number of bases of the ring of symmetric polynomials are indexed by partitions. We therefore fix some notation on partitions before going on to discuss symmetric polynomials. A partition λ = [λ 1 , . . . , λ m ] is a weakly descending sequence of positive integers called parts. The length ℓ(λ ) = m is the length of the sequence and the weight |λ | = ∑ i λ i is the sum over all elements of the sequence. Sometimes, it is convenient to define the λ i for i > ℓ(λ ) to be 0. A partition λ is often also referred to as a partition of the integer |λ |; [3, 3, 2, 1, 1, 1] is thus a partition of 11. It is customary to regard the empty partition [ ] as a partition of 0.
A convenient shorthand for partitions is to indicate repeated parts using a superscript. Thus, [3, 3, 2, 1, 1, 1] and [3 2 , 2, 1 3 ] denote the same partition. The multiplicity in λ of a given part i, that is, the superscript in the convenient shorthand notation, will be denoted by m λ (i). For every partition λ , one may then introduce the following number: These numbers play a small role in what follows, see (A.15) below for example. One also associates, to each partition λ , a diagram called a Young diagram. This consists of ℓ(λ ) rows of left-aligned boxes for which the length of the i-th row is λ i . We draw the first row at the top and the ℓ(λ )-th at the bottom. With this convention, the conjugate partition λ ′ is defined to be the partition whose Young diagram is the reflection of that of λ along the diagonal line from top left to bottom right. This reflection exchanges the lengths of the columns and the rows.
Each box b of the Young diagram of λ , generally written as b ∈ λ , may be parametrised by a pair b = (i, j), where i is the row number counted from top to bottom and j is the column number counted from left to right. Given a partition λ and a box b ∈ λ , the arm length a(b), the leg length l(b), the arm colength a ′ (b) and the leg colength l ′ (b) are the distances from b to the right, bottom, left and top edges of the Young diagram, respectively. In formulae, Finally, we remark that partitions admit a number of partial orderings, among which the dominance ordering, which we denote by ≥, will prove useful. Two partitions λ and µ satisfy λ ≥ µ if |λ | = |µ| and A.2. Symmetric polynomials. Let Λ n be the ring of n-variable polynomials with complex coefficients that are invariant under arbitrary permutations of the variables. Λ n is called the ring of symmetric polynomials in n variables. This ring is graded, where Λ k n is the space of homogeneous symmetric polynomials of degree k. Examples of symmetric polynomials include: 1) Power sums: The symmetric polynomial is called the k-th power sum. For each partition λ = (λ 1 , . . . , λ m ), we define 2) Elementary symmetric polynomials: The symmetric polynomials where the sum runs over all distinct permutations σ of α, are called the monomial symmetric polynomials or symmetric monomials for short. Given α, there is always precisely one distinct permutation σ whose entries are in descending order, hence by omitting any trailing zeroes, we may assume that α is a partition of length at most n.

1)
The power sums p k are algebraically independent for k ≤ n and generate Λ n : 2) The elementary symmetric polynomials are algebraically independent and generate Λ n : Λ n = C[e 1 , . . . , e n ]. (A.11) 3) The symmetric monomials m λ with ℓ(λ ) ≤ n form a basis of Λ n .
When working with symmetric polynomials, it is a remarkable fact that the number of variables is often irrelevant, assuming only that this number is sufficiently large. For this reason, it is convenient to work with symmetric polynomials in infinitely many variables. The ring of symmetric polynomials in infinitely many variables, x i say, is given by an inverse limit: Often the elements of Λ are distinguished by referring to them as symmetric functions. The projection π n : Λ → Λ n , (A. 13) defined by setting x i = 0 for all i > n, recovers the case of finitely many variables.
1) The power sums are algebraically independent for all k ≥ 1 and generate Λ: (A.14) In addition, the p λ form a basis of Λ as λ runs over all partitions of all non-negative integers.
2) The symmetric monomials m λ form a basis of Λ as λ runs over all partitions of all non-negative integers. The projection π n : Λ → Λ n satisfies π n (m λ ) = 0 if and only if ℓ(λ ) > n.
The basis of Λ which is most interesting for the purposes of this article is that consisting of the Jack symmetric polynomials. These polynomials are orthogonal with respect to the following inner product on Λ: In fact, there are infinitely many families of Jack polynomials, each forming a basis of Λ, parametrised by the complex number t ∈ C \ Q ≤0 appearing in this inner product. Analogous identities hold for finite numbers of variables x i and/or y j by projection. 2) For every partition λ and every t ∈ C \ Q ≤0 , there exists a unique basis of symmetric polynomials P t λ such that P t λ , P t µ t = 0, whenever λ = µ, and such that they satisfy upper-triangular decompositions into symmetric monomials, where u λ ,µ (t) ∈ C and u λ ,λ (t) = 1. Here, the sum runs over all µ that are dominated by λ . It follows that the P t λ are homogeneous symmetric polynomials of degree |λ |.
Definition. The P t λ determined by the previous proposition are called the Jack symmetric polynomials or, when working in the infinite-variable ring Λ, the Jack symmetric functions.
We remark that if t ∈ Q ≤0 , then some of the coefficients u λ ,µ (t) in (A.17) will diverge for certain λ , hence some of the Jack polynomials will not be defined.
The Jack polynomials P t λ satisfy a number of remarkable properties. Before we can list those that we shall require, we need to introduce one more inner product, this time that of two symmetric polynomials f , g ∈ Λ n of n variables: Here, the overline indicates that the arguments of the function have been inverted: f (x 1 , . . . , x n ) = f (x −1 1 , . . . , x −1 n ). We remark that the integral in (A.18) may be thought of as an n-variable generalisation of taking the residue at 0 of a meromorphic function. In particular, the integral vanishes if f and g are homogeneous of different degrees.
In [28], an inner product is defined for a deformation of the Jack polynomials (now) called the Macdonald polynomials. The definition is almost identical to (A.18), utilising a cycle [∆ n ] that is just a (normalised) product of n unit circles. However, there are some subtle ties when passing to the Jack polynomial limit and, in particular, this cycle is no longer suitable. Instead, we will use the (normalised) cycles constructed by Tsuchiya and Kanie which also have the advantage of being supported in the domains required for radially-ordered expansions of screening operators.
The normalised cycle [∆ n ] is then given by [∆ n ] = ∆ n /S n (t), so that 1, 1 n t = 1. We finish by summarising the properties of the Jack symmetric polynomials that we will use in this article. 2) The norm squared of P t λ with respect to the infinite variable inner product ·, · t is The corresponding dual basis of the Jack polynomials will be denoted by 3) For any X ∈ C, one defines a ring homomorphism Ξ X : Λ → C, called the specialisation map, by Ξ X (p k ) = X, for all k ≥ 1. The Jack polynomials and their duals specialise to 4) The projection onto n variables satisfies π n (P t λ ) = 0 if and only if ℓ(λ ) > n.
Suppose that λ satisfies ℓ(λ ) ≤ n and let λ + [m n ] denote the partition [λ 1 + m, . . . , λ n + m]. Then, in Λ n , Suppose that λ satisfies ℓ(λ ) ≤ n. Then, the norm squared of the P t λ with respect to the finite variable inner product ·, · n t is The specialisation map of Proposition A.5, item 3), seems somewhat mysterious at first glance and deserves some additional explanation. Our chief use for it is to expand products of the form ∏ i≥1 (1 − x i ) −X/t in terms of Jack polynomials. To do this, we consider (A.16) with y 1 = 1 and y i = 0 for all i > 1: It follows that we may now write where we take the specialisation map Ξ X to act only on the symmetric polynomials in the y i .

APPENDIX B. ZHU'S ALGEBRA
From any vertex operator algebra V, one can construct a unital associative algebra Zhu V called Zhu's algebra. The representation theory of this associative algebra is closely related that of V and is a crucial tool for classifying V-modules, especially simple V-modules. In this appendix, we motivate Zhu's algebra by using generalised commutation rules to relate it to the algebra of zero modes of the fields of V. From this point of view, the technology of Zhu may be regarded as an abstract formalisation of the "annihilating ideals" discussed in the physics literature [53]. A part of this motivational discussion may be found in [57]; another means of motivating Zhu's algebra via deforming the standard conventions for normal ordering is the subject of [58].
A weight module M of V is said to be N-graded if its decomposition into generalised L 0 -eigenspaces is bounded below, that is, if there exists an h ∈ C so that The space M = M 0 is often called the space of ground states; these states are clearly annihilated by every positive mode of every field. We remark that in the language of Section 2.1, the space M is spanned by relaxed highest weight vectors (though M may contain other such vectors that are not in M).
Our preferred means to motivate the construction of Zhu's algebra is the observation that a simple weight module M of V is, rather generally, completely determined by its space of ground states M. This space admits an action of the zero modes of the fields and Zhu's algebra is an abstract realisation of this action. To be more precise, consider [57] the generalised commutation relation obtained from the contour integral where A(z) and B(z) are two fields of V of conformal weights h A and h B , respectively. (We denote their modes by A j and B j , respectively, and the corresponding states by A and B.) Without the denominator in the integrand, the usual procedure of breaking the inner contour in two or employing the operator product expansion would lead to the commutation rules of the modes A 1 and B 0 . With the denominator, the generalised commutation relation is where (A j B) 0 denotes the zero mode of the field corresponding to the state A j B. Letting this generalised commutation relation act on a ground state v ∈ M (or a relaxed highest weight vector), we arrive at This motivates the definition [46] of Zhu's product * on the vertex operator algebra V (we identify its elements with the states of the vacuum module for convenience): The vacuum Ω is easily verified to be a (two-sided) unit with respect to this operation. To illustrate, we tabulate the products of the generators e, h and f of the level k universal vertex operator algebra V k of sl (2) Because the algebra of zero modes is associative, it follows that the failure of * to be associative on V must be explained by the discrepancies being mapped to zero: The resulting generalised commutation relation, applied to v ∈ M, takes the form where we have combined this relation with that of (B.4) in order to obtain the vanishing condition. This motivates the definition [46] of Zhu's other product • on V: It is clear from (B.10) and (B.11) that all elements of the form A • B belong to the kernel of every π M . Indeed, Zhu showed that the space O(V) spanned by the elements of this form is a two-sided ideal of V. There appears to be some confusion in the literature as to whether Zhu proved explicitly that O(V) is in fact the intersection of the kernels of the π M (for example, a remark amounting to this is stated without proof in [52]), but this result may be found in [59,App. A.2]. In any case, the quotient Zhu V = V/O(V) is what is now referred to as Zhu's algebra. It is a unital associative algebra with respect to Zhu's * product.
Consideration of the elements of O(V), for example, T + ∂ T = T • Ω ∈ O(V), shows that Zhu's algebra is not graded by conformal weight. It is, however, filtered by conformal weight in the sense that it has an increasing sequence of subspaces Zhu 0 V ⊆ Zhu 1 V ⊆ Zhu 2 V ⊆ · · · , where Zhu m V is the image in Zhu V of m n=0 V n (and V is here regarded as an N-graded module over itself).
It should now be clear that the space M of ground states of a V-module M is an Zhu V -module. The converse to this statement is of great importance to the representation theory of vertex operator algebras. Zhu actually gives a construction of a universal N-graded V-module M from a simple Zhu V -module M so that any N-graded V-module M ′ with M as its space of ground states is a quotient of M. If the vertex operator algebra V is rational, then its modules are semisimple and thus M is simple. However, for more general vertex operator algebras the universal M that Zhu constructs may be reducible. Moreover, one may also start with a non-simple Zhu V -module; the result is then always reducible.
We close this appendix with two results of Frenkel and Zhu that will be used in this paper.