Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
aa r X i v : . [ m a t h . R T ] J u l CLASSIFYING RELAXED HIGHEST-WEIGHT MODULES FORADMISSIBLE-LEVEL BERSHADSKY–POLYAKOV ALGEBRAS
ZACHARY FEHILY, KAZUYA KAWASETSU AND DAVID RIDOUT
Abstract.
The Bershadsky–Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebrasassociated to sl and their simple quotients have a long history of applications in conformal field theory and string theory.Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules forall admissible but nonintegral levels, significantly generalising the known highest-weight classifications [1, 2]. In particular,we prove that the simple Bershadsky–Polyakov algebras with admissible nonintegral k are always rational in category O , whilstthey always admit nonsemisimple relaxed highest-weight modules unless k + ∈ š > .
1. Introduction
Background.
The Bershadsky–Polyakov algebras BP k , k ∈ ƒ , are among the simplest and best-known nonregularW-algebras [3, 4]. They may be characterised [5] as the minimal (or subregular) quantum hamiltonian reductions of thelevel- k universal affine vertex algebras V k ( sl ) . Here, we are interested in their representation theories and, in particular,those of their simple quotients BP k .When k + ∈ š > , BP k is known to be rational [1], meaning that the representation theory is semisimple and thatthere are finitely many simple BP k -modules, up to isomorphism. More recently, the representation theory of BP k wasexplored for certain other levels in [2]. There, the highest-weight modules were classified and some nonhighest-weightmodules were described. These works both relied on explicit formulae for singular vectors in BP k . Here, we shallextend these classifications to more general levels where the singular vector method is unavailable. Instead, we shallexploit the properties [6, 7] of minimal quantum hamiltonian reduction.In particular, we are interested in the relaxed highest-weight theory of the simple Bershadsky–Polyakov algebras BP k . Relaxed highest-weight modules are a type of generalised highest-weight module [8–10] that have been shownto be essential ingredients in the study of many nonrational vertex operator algebras, particularly the admissible-levelaffine ones associated with sl [8, 10–17], their affine cousins [16–25] and other close relatives [26, 27]. We thereforeexpect them to play a central role in Bershadsky–Polyakov representation theory and, indeed, in the representationtheory of most nonrational W-algebras.Here, we classify the simple relaxed highest-weight BP k -modules, in both the untwisted and twisted sectors,when k is admissible and nonintegral, leaving the much more difficult nonadmissible and integral cases for futureinvestigations. This classification includes the highest-weight classification as a special case. We also show that thereare nonsemisimple relaxed highest-weight BP k -modules when k is admissible, nonintegral and 2 k + < š > . In acompanion paper [28], these relaxed modules are constructed from the highest-weight modules of the Zamolodchikovalgebra [29], the regular W-algebra associated to sl , using the inverse quantum hamiltonian reduction procedure of[16, 30]. This results in beautiful character formulae for the relaxed BP k -modules, generalising those found in [14, 17]for L k ( sl ) and L k ( osp ( | )) .1.2. Results.
Our strategy in classifying relaxed highest-weight BP k -modules starts from the highest-weight classifi-cation. The idea for the latter is to use Arakawa’s celebrated results on minimal quantum hamiltonian reduction [7].However, we must first establish a subtle technical result concerning the surjectivity of the minimal reduction functor.This is the content of our first main result. Main Theorem 1 (Theorem 4.8) . Let k be an admissible nonintegral level. Then, every simple (untwisted) highest-weight BP k -module may be realised as the minimal quantum hamiltonian reduction of a simple highest-weight L k ( sl ) -module. In [1], explicit singular vector formulae are used to prove this theorem when 2 k + ∈ š > . Our general proof also usessingular vectors, but is necessarily very different because explicit formulae are no longer available.Given this result, it is straightforward to classify the simple (untwisted and twisted) highest-weight BP k -modulesand determine how they are related to one another. For this, write k + = uv , where u > v > Σ u , v of b sl weights λ = λ I − uv λ F satisfying λ I ∈ P u − > , λ F ∈ P v − > and λ F ,
0. Here, P ℓ > denotes thedominant integral weights of b sl whose level is ℓ . Main Theorem 2.
Let k be admissible and nonintegral. Then: (a) [Theorem 4.9] The isomorphism classes of the simple untwisted and twisted highest-weight BP k -modules, H λ and H tw λ , are each in bijection with Σ u , v . The connection between the b sl weights λ ∈ Σ u , v and the native BP k data isgiven explicitly in Equations (4.2) and (4.9) . (b) [Theorem 4.10] Every (untwisted or twisted) highest-weight BP k -module is simple, so BP k is rational in theBernšte˘ın–Gel’fand–Gel’fand category O k . (c) [Proposition 4.13] The module conjugate to H λ , λ ∈ Σ u , v , is H µ , where µ = [ λ , λ , λ ] ∈ Σ u , v . The moduleconjugate to H tw λ is highest-weight if and only if λ F = , in which case it is H tw ν , where ν = [ λ − uv , λ , λ + uv ] ∈ Σ u , v . (d) [Proposition 4.14] The spectral flow of the untwisted (twisted) highest-weight module labelled by λ ∈ Σ u , v ishighest-weight if and only if λ F = , in which case it is the untwisted (twisted) highest-weight module labelled by [ λ − uv , λ + uv , λ ] ∈ Σ u , v . This then generalises the highest-weight classifications of [1], when 2 k + ∈ š > , and [2], for k = − and − . Werefer to Section 2.2 for an introduction to the conjugation and spectral flow functors referred to above.To extend the highest-weight classification to simple twisted relaxed highest-weight modules, we adapt the method-ology developed in [22] for affine vertex algebras. This uses Mathieu’s notion of a coherent family [31], extending itfrom semisimple Lie algebras to the twisted Zhu algebra of BP k . Let Γ u , v consist of the b sl weights λ ∈ Σ u , v satisfying λ F ,
0. Writing k + = uv as above, it follows that Γ u , v is empty unless v >
3. Moreover, Γ u , v admits a free š -actiongenerated by λ
7→ [ λ − uv , λ , λ + uv ] (Lemma 4.19). Main Theorem 3 (Theorem 4.20) . Let k be admissible and nonintegral. Then: (a) The isomorphism classes of the simple twisted relaxed highest-weight BP k -modules R tw [ j ] , λ form families that are inbijection with Γ u , v / š . The connection between the b sl weights λ ∈ Γ u , v and the native BP k data is given explicitlyin Equations (4.9) and (4.16) . (b) The members of each of these families are indexed by all but three of the cosets [ j ] ∈ ƒ / š , the exceptions beingdetermined as the images of the š -orbit of λ under (4.9) . (c) The module conjugate to R tw [ j ] , λ is R tw [− j ] , µ , where µ = [ λ − uv , λ + uv , λ ] ∈ Γ u , v . Moreover, the spectral flow of each R tw [ j ] , λ is never a relaxed highest-weight module.Our final main result relates to the existence of nonsemisimple relaxed highest-weight BP k -modules when v > [ j ] in each family of simple relaxed modulesabove. However, there are two ways of filling each hole, each way related to the other by taking contragredient duals.This is very similar to the analogous nonsemisimple picture conjectured in [10, 14], and proven in [16, 17], for L k ( sl ) .In the case at hand, we establish this picture by combining a mix of the theory developed in [17, 22] with the rationalityof BP k in category O k (Theorem 4.10). This seems robust and we expect it to generalise to higher-rank cases. Main Theorem 4 (Theorem 4.24) . Let k be admissible and nonintegral. Then: (a) Every λ ∈ Γ u , v defines two indecomposable nonsemisimple relaxed highest-weight BP k -modules R tw , + [ j ] , λ and R tw , −[ j ] , λ ,where j is determined from λ by (4.9) . (b) R tw , + [ j ] , λ has a submodule isomorphic to the conjugate of H tw µ , where µ = [ λ , λ − uv , λ + uv ] ∈ Γ u , v , and its quotientby this submodule is isomorphic to H tw λ . The structure of R tw , −[ j ] , λ is similar, but with submodule and quotientexchanged. ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 3
Conjugation and spectral flow works as for the simple relaxed modules, except that the conjugate of a + -type module isof − -type (and vice versa). These nonsemisimple modules prove that BP k has a nonsemisimple module category. They,along with their spectral flows, are also the building blocks (the “atypical standards”) for the resolutions that underpinthe so-called standard module formalism [32, 33] for modular transformations and Verlinde formulae for nonrationalvertex operator algebras. We intend to return to this in a forthcoming paper [34].1.3. Outline.
We start by defining the universal Bershadsky–Polyakov vertex operator algebras BP k and their simplequotients BP k in Section 2.1. It is worthwhile remarking that we choose the conformal structure so that the chargedgenerators G ± have equal conformal weight . Equivalently, the Heisenberg field is a Virasoro primary. Accordingly,we study both untwisted and twisted BP k -modules. Section 2.2 then introduces the all-important conjugation andspectral flow automorphisms and explains how they lift to invertible functors of appropriate categories of BP k -modules.Happily, the untwisted and twisted sectors of the categories of interest are related by spectral flow.In Section 3, we embark on the first part of the journey: to understand how to identify BP k -modules, untwistedand twisted, relaxed and highest-weight. After defining these types of modules, we introduce Zhu algebras anddetermine that of BP k in Proposition 3.8. This leads to an easy classification of untwisted highest-weight BP k -modules(Theorem 3.11). The more-involved twisted classification (Theorem 3.23) is then detailed. For this, we review theidentification [1] of the twisted Zhu algebra with a central extension of a Smith algebra [35] (Proposition 3.15) andclassify the simple weight modules, with finite-dimensional weight spaces, of this extension in Theorem 3.22. For lateruse, we also introduce coherent families of modules, following [31], for the twisted Zhu algebra.The hard work then begins in Section 4 where we convert these classification results for the universal Bershadsky–Polyakov algebras BP k into the corresponding results for their simple quotients BP k . Section 4.1 is devoted to MainTheorem 1, first reviewing the highest-weight theory of the simple affine vertex operator algebra L k ( sl ) [36, 37] andsome basic, though deep, results about minimal quantum hamiltonian reduction [5–7]. The actual proof of this crucialresult is deferred to Appendix A so as not to disrupt the flow of the arguments too much.From this, we immediately deduce the classification of highest-weight BP k -modules, as in Main Theorem 2. Theremainder of Section 4.2 then addresses how the highest-weight modules are related by the conjugation and spectralflow functors. This will be important for the standard module analysis in [34]. Section 4.3 then lifts this classificationto simple relaxed highest-weight BP k -modules, establishing Main Theorem 3. The existence of nonsemisimple relaxedhighest-weight modules, hence Main Theorem 4, is the focus of Section 4.4.In Section 5, we conclude by illustrating our classification results with some examples. The rational cases with v = BP − / and the slightly more involved example BP / . The latter is interesting because it has a simple currentextension that may be regarded as a bosonic analogue of the N = L ( sl ) , and four weight fields.We also study three nonrational examples. Two, namely BP − / and BP − / , were already discussed in [2] and herewe take the opportunity to explicitly extend their highest-weight classifications to the full relaxed classifications. Wefinish by describing the example BP − / which we believe has not been analysed before. After describing its relaxedhighest-weight modules explicitly, we note an interesting fact: it seems to admit a simple current extension isomorphicto the minimal quantum hamiltonian reduction of L − / ( g ) . It follows then that this g W-algebra should have a š -orbifold isomorphic to BP − / , as well as a š -orbifold isomorphic to L / ( sl ) [14, 38]. Acknowledgements.
We thank Dražen Adamović, Tomoyuki Arakawa and Thomas Creutzig for interesting discussionsrelated to the research reported here.ZF’s research is supported by an Australian Government Research Training Program (RTP) Scholarship.KK’s research is partially supported by MEXT Japan “Leading Initiative for Excellent Young Researchers (LEADER)”,JSPS Kakenhi Grant numbers 19KK0065 and 19J01093 and Australian Research Council Discovery Project DP160101520.DR’s research is supported by the Australian Research Council Discovery Project DP160101520 and the AustralianResearch Council Centre of Excellence for Mathematical and Statistical Frontiers CE140100049.
Z FEHILY, K KAWASETSU AND D RIDOUT
2. Bershadsky–Polyakov algebras
Bershadsky–Polyakov vertex operator algebras.
We begin by defining one of the families of vertex operatoralgebras that we will study here.
Definition 2.1.
Given k ∈ ƒ , k , − , the level- k universal Bershadsky–Polyakov algebra BP k is the vertex operatoralgebra with vacuum that is strongly and freely generated by fields J ( z ) , G + ( z ) , G − ( z ) and L ( z ) satisfying the followingoperator product expansions: (2.1) L ( z ) L ( w ) ∼ − ( k + )( k + ) ( k + )( z − w ) + L ( w )( z − w ) + ∂ L ( w )( z − w ) , L ( z ) J ( w ) ∼ J ( w )( z − w ) + ∂ J ( w )( z − w ) , L ( z ) G ± ( w ) ∼ G ± ( w )( z − w ) + ∂ G ± ( w )( z − w ) , J ( z ) J ( w ) ∼ ( k + ) ( z − w ) , J ( z ) G ± ( w ) ∼ ± G ± ( w )( z − w ) , G ± ( z ) G ± ( w ) ∼ , G + ( z ) G − ( w ) ∼ ( k + )( k + ) ( z − w ) + ( k + ) J ( w )( z − w ) + J J : ( w ) + ( k + ) ∂ J ( w ) − ( k + ) L ( w ) z − w . This family of vertex operator algebras was first described in [3, 4] where it was constructed via a new type of quantumhamiltonian reduction from the corresponding family of universal affine vertex operator algebras V k ( sl ) associated to sl . In the general framework of quantum hamiltonian reductions [5], BP k is the minimal reduction corresponding totaking the nilpotent of sl to be a root vector.From (2.1), we see that the conformal weights of the generating fields J ( z ) , G + ( z ) , G − ( z ) and L ( z ) are 1, , and 2,respectively, whilst the central charge is(2.2) c = − ( k + )( k + ) k + . We shall expand the homogeneous fields of BP k in the form(2.3) A ( z ) = Õ n ∈ š − ∆ A + ε A A n z − n − ∆ A , where ∆ A is the conformal weight of A ( z ) and ε A = , if ∆ A ∈ š + and A ( z ) is acting on a twisted BP k -module (seeSection 3 below), and ε A = Proposition 2.2.
The commutation relations of the modes of the generating fields of BP k are (2.4) [ L m , L n ] = ( m − n ) L m + n − ( k + )( k + ) k + m − m δ m + n , , [ L m , J n ] = − nJ m + n , [ L m , G ± s ] = (cid:16) m − s (cid:17) G ± m + s , [ J m , J n ] = k + mδ m + n , , [ J m , G ± s ] = ± G ± m + s , [ G ± r , G ± s ] = , [ G + r , G − s ] = J J : r + s − ( k + ) L r + s + ( k + )( r − s ) J r + s + ( k + )( k + ) r − δ r + s , . Here, the indices m and n always take values in š while r and s take values in š + , if acting on an untwisted module,and in š , if acting on a twisted module. We call the (unital associative) algebra generated by the modes of the fieldsof BP k the untwisted mode algebra U , in the first case, and the twisted mode algebra U tw , in the latter case. Each is acompletion of the corresponding algebra generated by the modes of the generating fields. Definition 2.3. • A fractional level k ∈ ƒ for the Bershadsky–Polyakov algebras is one that is not critical, meaning that k , − , andfor which BP k is not simple. • The level- k simple Bershadsky–Polyakov vertex operator algebra BP k is the unique simple quotient of BP k . According to [39, Thms. 0.2.1 and 9.1.2], the fractional levels are precisely the k satisfying(2.5) k + = uv , where u ∈ š > , v ∈ š > and gcd { u , v } = . ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 5 If k is fractional, then we shall refer to BP k as a Bershadsky–Polyakov minimal model and favour the alternative notation BP ( u , v ) . We note that the central charge of the minimal model BP ( u , v ) takes the form(2.6) c = − ( u − v )( u − v ) uv = − ( u − v ) uv . Whilst the central charge is invariant under exchanging uv with vu , the corresponding simple vertex operator algebrasare not isomorphic. We shall see this explicitly when we analyse their representation theories in Section 4.2.2. Automorphisms.
There are two types of automorphisms of BP k that will prove useful for the classification resultsto follow: the conjugation automorphism γ and the spectral flow automorphisms σ ℓ , ℓ ∈ š . It is easy to verify thattheir actions, given below on the generating fields, indeed preserve the operator product expansions (2.1). Proposition 2.4.
There exist conjugation and spectral flow automorphisms γ and σ ℓ , ℓ ∈ š , of the vertex algebraunderlying BP k . They are uniquely determined by the following actions on the generating fields: (2.7) γ ( J ( z )) = − J ( z ) , γ ( G + ( z )) = + G − ( z ) , γ ( G − ( z )) = − G + ( z ) , γ ( L ( z )) = L ( z ) , σ ℓ ( J ( z )) = J ( z ) − k + ℓ z − , σ ℓ ( G + ( z )) = z − ℓ G + ( z ) , σ ℓ ( L ( z )) = L ( z ) − ℓ z − J ( z ) + k + ℓ z − , σ ℓ ( G − ( z )) = z + ℓ G − ( z ) The σ ℓ with ℓ , γ σ ℓ = σ − ℓ γ , though we do not have γ = . Proposition 2.5.
Conjugation and spectral flow act on the modes of the generating fields J ( z ) , G + ( z ) , G − ( z ) and L ( z ) of BP k as follows: (2.9) γ ( J n ) = − J n , γ ( G + r ) = + G − r , γ ( G − r ) = − G + r , γ ( L n ) = L n , σ ℓ ( J n ) = J n − k + ℓ δ n , , σ ℓ ( G + r ) = G + r − ℓ , σ ℓ ( G − r ) = G − r + ℓ , σ ℓ ( L n ) = L n − ℓ J n + k + ℓ δ n , . An extremely useful observation is that if we extend the definition of σ ℓ to allow ℓ ∈ š + , then we see that σ / exchanges the twisted and untwisted mode algebras U and U tw introduced above.Our main application for these automorphisms is to construct new BP k -modules from old ones. This amounts toapplying the automorphism (or its inverse) before acting with the representation morphism. As we prefer to keeprepresentations implicit, we implement this twisting notationally through the language of modules as follows. Givena BP k -automorphism ω and a BP k -module M , define ω ∗ ( M ) to be the image of M under an (arbitrarily chosen)isomorphism ω ∗ of vector spaces. The action of BP k on ω ∗ ( M ) is then defined by(2.10) A ( z ) · ω ∗ ( v ) = ω ∗ ( ω − ( A ( z )) v ) , A ( z ) ∈ BP k , v ∈ M . In other words, ω ( A ( z )) · ω ∗ ( v ) = ω ∗ ( A ( z ) v ) . In view of this, we shall hereafter drop the star that distinguishes theautomorphism ω from the corresponding vector space isomorphism ω ∗ .Each BP k -automorphism thus lifts to an invertible functor on a suitable category of BP k -modules. The examples wehave in mind are the category W k of weight modules, with finite-dimensional weight spaces (see Definition 3.1 below),and the analogous category W tw k of twisted modules. In particular, the conjugation and spectral flow automorphismslift to invertible endofunctors that provide an action of the infinite dihedral group on W k and W tw k . Extending the aboveformulae for σ ℓ to allow ℓ ∈ š + , we see that the lift of σ / moreover defines an equivalence between W k and W tw k .We remark that one of the important consistency requirements for building a conformal field theory from a modulecategory over a vertex operator algebra is that it is closed under twisting by automorphisms, especially conjugation. Z FEHILY, K KAWASETSU AND D RIDOUT
3. Identifying Bershadsky–Polyakov modules
Our aim is to classify the simple relaxed highest-weight modules, untwisted and twisted, for the Bershadsky–Polyakov minimal models BP ( u , v ) . In order to have well-defined characters, necessary to construct partition functionsin conformal field theory, we shall also require that the weight spaces of these simple relaxed highest-weight modulesare finite-dimensional. By [40], it therefore suffices to classify the simple weight modules, with finite-dimensionalweight spaces, of the untwisted and twisted Zhu algebras of BP ( u , v ) .A direct assault on this classification seems quite difficult. Our alternative strategy is threefold: First, we understandthe classification for certain associative algebras which have the untwisted and twisted Zhu algebras of BP ( u , v ) asquotients. (These algebras turn out to be the untwisted and twisted Zhu algebras of the universal Bershadsky–Polyakovvertex operator algebras BP k , but this is inessential to the argument.) This allows us to identify BP ( u , v ) -modules interms of data for these more easily understood associative algebras. Second, we use Arakawa’s results [7] on minimalquantum hamiltonian reductions to directly obtain the highest-weight classification for the BP ( u , v ) , at present onlyknown for v = BP k -modules. Of course, all BP ( u , v ) -modules are a priori BP k -modules.3.1. Relaxed highest-weight BP k -modules. In Section 2.1, we introduced the untwisted mode algebra U of theuniversal Bershadsky–Polyakov vertex operator algebra BP k and its twisted version U tw . Any BP k -module is obviouslya U -module and, similarly, any twisted BP k -module is a U tw -module. As these algebras are graded by conformal weight(eigenvalue of [ L , −] ), we have the generalised triangular decompositions(3.1) U = U < ⊗ U ⊗ U > and U tw = U tw < ⊗ U tw0 ⊗ U tw > , where U < , U and U > are the unital subalgebras generated by the modes A n , for all homogeneous A ( z ) ∈ BP k , with n < n = n >
0, respectively (and similarly for their twisted versions).
Definition 3.1. • A vector v in a twisted or untwisted BP k -module M is a weight vector of weight ( j , ∆ ) if it is a simultaneouseigenvector of J and L with eigenvalues j and ∆ called the charge and conformal weight of v , respectively. Thenonzero simultaneous eigenspaces of J and L are called the weight spaces of M . If M has a basis of weight vectorsand each weight space is finite-dimensional, then M is a weight module . • A vector in an untwisted BP k -module is a highest-weight vector if it is a simultaneous eigenvector of J and L that isannihilated by the action of U > . An untwisted BP k -module generated by a single highest-weight vector is called an untwisted highest-weight module . • A vector in a twisted BP k -module is a highest-weight vector if it is a simultaneous eigenvector of J and L that isannihilated by G + and the action of U tw > . A twisted BP k -module generated by a single highest-weight vector is calleda twisted highest-weight module . • A vector in a twisted or untwisted BP k -module is a relaxed highest-weight vector if it is a simultaneous eigenvectorof J and L that is annihilated by the action of U tw > or U > , respectively. A BP k -module generated by a single relaxedhighest-weight vector is called a relaxed highest-weight module . As every BP ( u , v ) -module is also a BP k -module (with k + = uv ), these definitions descend to BP ( u , v ) -modules in theobvious way.A simple consequence of these definitions is that an untwisted relaxed highest-weight vector of BP k is automatically ahighest-weight vector. We shall therefore be concerned with classifying untwisted highest-weight modules and twistedrelaxed highest-weight modules. The name “relaxed highest-weight module” was originally coined in [9] for the simpleaffine vertex operator algebra L k ( sl ) and now seems to be quite widespread. Such modules had, however, appeared inearlier works such as [8]. Here, we follow the definition proposed for quite general vertex operator algebras in [10]. ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 7
From the actions of the conjugation and spectral flow automorphisms, given explicitly in (2.9) and (2.10), we deducethe following useful facts.
Proposition 3.2. • If M is a twisted or untwisted BP k -module and v ∈ M is a weight vector of weight ( j , ∆ ) , then γ ( v ) and σ ℓ ( v ) areweight vectors in γ ( M ) and σ ℓ ( M ) of weights (− j , ∆ ) and ( j + k + ℓ, ∆ + j ℓ + k + ℓ ) , respectively. • Let M be an untwisted BP k -module. Then, v ∈ M is a highest-weight vector of weight ( j , ∆ ) if and only if σ / ( v ) isa highest-weight vector in the twisted module σ / ( M ) of weight ( j + k + , ∆ + j + k + ) . • M is a simple untwisted highest-weight BP k -module if and only if σ / ( M ) is a simple twisted highest-weight BP k -module. In particular, to classify all simple highest-weight BP k -modules, it is enough to only classify the untwisted ones.We remark that there are simple weight BP k -modules that are not highest-weight, nor even relaxed highest-weight.In particular, if M is a simple relaxed highest-weight BP k -module, then σ ℓ ( M ) is simple and weight, but is usually onlyrelaxed highest-weight for a few choices of ℓ . We believe, however, that the simple objects of the categories W k and W tw k of untwisted and twisted weight BP k -modules are all spectral flows of simple relaxed highest-weight BP k -modules.3.2. The untwisted Zhu algebra.
The main tools that we shall use to classify Bershadsky–Polyakov modules are thefunctors induced between these modules and those of the corresponding (untwisted) Zhu algebra. Although originallyintroduced by Zhu [40], the idea behind this unital associative algebra was already well-known to physicists (see [41]for example). Here, we use a (slightly restricted) abstract definition that is based on the physicists’ “zero-modes actingon ground states” approach to Zhu algebras. We refer to [10, App. B] for further details (and motivation).Suppose that V is a vertex operator algebra with conformally graded mode algebra U = U < ⊗ U ⊗ U > , as in (3.1).Let U ′ > denote the ideal of U > generated by the modes A n (so that U > = ƒ ⊕ U ′ > as vector spaces). Definition 3.3.
The untwisted Zhu algebra of V is the vector space (3.2) Zhu [ V ] = U U ∩ ( UU ′ > ) , equipped with the multiplication (defined for homogeneous A of conformal weight ∆ A and extended linearly) (3.3) (cid:2) A (cid:3) (cid:2) B (cid:3) = (cid:2) A B (cid:3) = ∞ Õ n = (cid:18) ∆ A n (cid:19) (cid:2) ( A − ∆ A + n B ) (cid:3) , where (cid:2) U (cid:3) is the image in Zhu [ V ] of U ∈ U . Zhu defined two functors between the categories of V - and Zhu [ V ] -modules. We shall refer to them as the Zhufunctor and the Zhu induction functor. The first is quite easy to define. Definition 3.4.
The
Zhu functor assigns to any V -module M , the Zhu [ V ] -module Zhu [ M ] = M U ′ > , the subspace of M whose elements are annihilated by U ′ > . The second is not so easily defined, but morally amounts to inducing a
Zhu [ V ] -module, treating it as a U -moduleequipped with a trivial U ′ > -action, and taking a quotient that imposes, among other things, the generalised commutationrelations (Borcherds relations) of V . The details may be found in [40, 42]. Proposition 3.5 ([40]) . There exists a functor, which we call the
Zhu induction functor , that assigns to any
Zhu [ V ] -module N a V -module Ind [ N ] such that Zhu [ Ind [ N ]] ≃ N . The Zhu functor is thus a left inverse of the Zhu induction functor, at the level of isomorphism classes of modules.However, it is not a right inverse in general. Nevertheless, it is if we restrict to a certain class of simple V -modules. Definition 3.6.
A (twisted or untwisted) V -module M is lower-bounded if it decomposes into (generalised) eigenspacesfor the Virasoro zero-mode L and the corresponding eigenvalues are bounded below. If M is lower-bounded, then the(generalised) eigenspace of minimal L -eigenvalue is called the top space of M and will be denoted by M top . Z FEHILY, K KAWASETSU AND D RIDOUT If M is a lower-bounded V -module, then M top is naturally a Zhu [ V ] -module. In fact, it may be identified with Zhu [ M ] if M is also simple, though this will not be true in general. Simple lower-bounded V -modules have the following property. Theorem 3.7 ([40]) . Zhu [−] and
Ind [−] induce a bijection between the sets of isomorphism classes of simple lower-bounded V -modules and simple Zhu [ V ] -modules. To classify the simple lower-bounded V -modules, it is therefore sufficient to classify the simple Zhu [ V ] -modules andapply Ind [−] . We remark that for V = BP k or BP ( u , v ) , the simple lower-bounded weight modules coincide preciselywith the simple relaxed highest-weight modules.The first order of business is therefore to get information about the untwisted Zhu algebra Zhu (cid:2) BP k (cid:3) . Proposition 3.8.
Zhu (cid:2) BP k (cid:3) is a quotient of ƒ [ J , L ] . Proof.
Since the fields G ± ( z ) have half-integer conformal weights, they do not have zero modes when acting on untwistedmodules. More generally, only the (homogeneous) fields of integer conformal weight have zero modes. Express thezero mode of such a field as a linear combination of monomials in the modes of the generating fields J ( z ) , G ± ( z ) and L ( z ) . Next, use the commutation relations to order the modes so that the mode index weakly increases from left to right— it is easy to see that this is always possible despite the nonlinear nature of the commutation relations (2.4). Nowremove any monomial which contains a positive mode. The image of the zero mode in Zhu (cid:2) BP k (cid:3) is thus a polynomialin (cid:2) J (cid:3) and (cid:2) L (cid:3) . Since (cid:2) L (cid:3) is central in Zhu (cid:2) BP k (cid:3) , the multiplication (3.3) of Zhu (cid:2) BP k (cid:3) matches that of ƒ [ J , L ] . Thereis therefore a surjective homomorphism ƒ [ J , L ] → Zhu (cid:2) BP k (cid:3) determined by J (cid:2) J (cid:3) and L (cid:2) L (cid:3) . (cid:4) It is in fact easy to show that
Zhu (cid:2) BP k (cid:3) ≃ ƒ [ J , L ] , though we will not need this result in what follows.3.3. Identifying simple untwisted highest-weight BP k -modules. Having identified
Zhu (cid:2) BP k (cid:3) as a quotient of thefree abelian algebra ƒ [ J , L ] , we may identify its finite-dimensional simple modules as ƒ [ J , L ] -modules. Definition 3.9. A ƒ [ J , L ] -module is said to be weight if J and L act semisimply and their simultaneous eigenspaces areall finite-dimensional. The simple weight modules of ƒ [ J , L ] are therefore one-dimensional. We shall denote them by ƒ v j , ∆ , where λ and ∆ are the eigenvalues of J and L , respectively, on v j , ∆ . As every simple Zhu (cid:2) BP k (cid:3) -module must also be simple as a ƒ [ J , L ] -module, we arrive at our first identification result. Proposition 3.10.
Every simple weight
Zhu (cid:2) BP k (cid:3) -module, and hence every simple weight Zhu [ BP ( u , v )] -module, isisomorphic to some ƒ v j , ∆ , where λ , ∆ ∈ ƒ . Proposition 3.5 and Theorem 3.7 then guarantee that if ƒ v j , ∆ is a Zhu (cid:2) BP k (cid:3) -module, then there exists a simple untwisted BP k -module H j , ∆ which is uniquely determined (up to isomorphism) by the fact that its top space is isomorphic to ƒ v j , ∆ (as a ƒ [ J , L ] -module). As this top space is one-dimensional, H j , ∆ is a highest-weight module. Theorem 3.11.
Every simple untwisted relaxed highest-weight BP k -module, and hence every simple untwisted relaxedhighest-weight BP ( u , v ) -module, is isomorphic to some H j , ∆ , where λ , ∆ ∈ ƒ . Note that there will be other simple weight BP k - and BP ( u , v ) -modules such as those obtained from the H j , ∆ by applyingspectral flow. Simple nonweight modules also exist in general [2], but they will not concern us here.3.4. The twisted Zhu algebra.
The theory that extends Zhu algebras and functors to twisted modules was developedindependently, and in different levels of generality, by Kac and Wang [43] and by Dong, Li and Mason [44]. From thepoint of view of “zero modes acting on ground states” however, the twisted story is almost identical to the untwistedone. This is discussed in detail in [45, App. A].Given a vertex operator algebra V with twisted mode algebra U tw = U tw < ⊗ U tw0 ⊗ U tw > , let U tw > ′ be the ideal of U tw > generated by the modes A n . Then, the twisted Zhu algebra and twisted Zhu functor of V may be characterised as follows. Definition 3.12.
ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 9 • The twisted Zhu algebra of V is the vector space (3.4) Zhu tw (cid:2) V (cid:3) = U tw0 U tw0 ∩ ( U tw U tw > ′ ) , equipped with the multiplication defined in (3.3) , but where (cid:2) U (cid:3) is now the image in Zhu tw (cid:2) V (cid:3) of U ∈ U tw0 . • The twisted Zhu functor assigns to any twisted V -module M the Zhu tw (cid:2) V (cid:3) -module Zhu tw (cid:2) M (cid:3) = M U tw > ′ of elements of M that are annihilated by U tw > ′ . The obvious analogues of Zhu’s theorems for the twisted setting then hold.
Theorem 3.13 ([44]) . • There exists a twisted Zhu induction functor that takes a
Zhu tw (cid:2) V (cid:3) -module N to a V -module Ind tw (cid:2) N (cid:3) satisfying Zhu tw (cid:2) Ind tw (cid:2) N (cid:3) (cid:3) ≃ N . • Zhu tw (cid:2) − (cid:3) and Ind tw (cid:2) − (cid:3) induce a bijection between the sets of isomorphism classes of simple lower-bounded twisted V -modules and simple Zhu tw (cid:2) V (cid:3) -modules. Again, the simple lower-bounded twisted weight V -modules coincide with the simple twisted relaxed highest-weightmodules when V = BP k or BP ( u , v ) .Our aim is to show that Zhu tw (cid:2) BP k (cid:3) is a quotient of some reasonably accessible associative algebra. In contrastto the untwisted case detailed in Section 3.2, the fields G ± ( z ) do have zero modes when acting on twisted modules.We therefore expect that Zhu tw (cid:2) BP k (cid:3) will be more complicated than Zhu (cid:2) BP k (cid:3) — in particular, we expect it to benonabelian — and so its representation theory will be more interesting. Definition 3.14.
Let Z k denote the (complex) unital associative algebra generated by J , G + , G − and L , subject to L being central and (3.5) [ J , G ± ] = ± G ± , [ G + , G − ] = f k ( J , L ) , where f k ( J , L ) = J − ( k + ) L − ( k + )( k + ) . Proposition 3.15.
Zhu tw (cid:2) BP k (cid:3) is a quotient of Z k . Proof.
Every homogeneous field of BP k has a zero mode when acting on a twisted module. As in the proof ofProposition 3.8, it follows that the zero modes of the generating fields have images that generate Zhu tw (cid:2) BP k (cid:3) . The factthat the generator (cid:2) L (cid:3) is central is standard [43, 44], but is also easy to verify directly in this case.We therefore start by using (3.3) to compute the products of the images of J and G ± in Zhu tw (cid:2) BP k (cid:3) : (cid:2) J (cid:3) (cid:2) G ± (cid:3) = ∞ Õ n = (cid:18) n (cid:19) (cid:2) ( J n − G ± ) (cid:3) = (cid:2) ( J G ± ) (cid:3) + (cid:2) ( J − G ± ) (cid:3) = ± (cid:2) G ± (cid:3) + (cid:2) : JG ± : (cid:3) , (3.6) (cid:2) G ± (cid:3) (cid:2) J (cid:3) = ∞ Õ n = (cid:18) / n (cid:19) (cid:2) ( G ± n − / J ) (cid:3) = (cid:2) ( G ±− / J ) (cid:3) + (cid:2) ( G ±− / J ) (cid:3) (3.7) = (cid:2) ( J − G ± ) (cid:3) ± (cid:2) ( ∂ G ± ) (cid:3) ± (cid:2) G ± (cid:3) = (cid:2) : JG ± : (cid:3) . Here, we have noted that G ±− / J = G ±− / J − = J − G ±− / ∓ G ±− / = : JG ± : ∓ ∂ G ± , that G ±− / J = ∓ G ± (similarly) andthat ( ∂ G ± ) = − G ± . With the surjection induced by A (cid:2) A (cid:3) , A = J , G ± , L , this proves the first relation in (3.5). Thesame method works for the second relation; we omit the somewhat more tedious details. (cid:4) It turns out that Z k is in fact isomorphic to Zhu tw (cid:2) BP k (cid:3) , though again we do not need this for what follows. Onecan establish this isomorphism by combining the fact that Zhu tw (cid:2) BP k (cid:3) is known [46] to be isomorphic to the finiteW-algebra associated to sl and the minimal nilpotent orbit, while an explicit presentation of this finite W-algebra isgiven in [47]. Either way, Z k is a central extension of a Smith algebra, the latter being studied in [35] as examples ofassociative algebras generalising the universal enveloping algebra of sl . The representation theory of Z k is thereforequite tractable, a fact that we shall exploit in the next section. Identifying simple twisted relaxed highest-weight BP k -modules. As in the untwisted case, we wish to identifysimple
Zhu tw (cid:2) BP k (cid:3) -modules as Z k -modules. For this, we need a classification of the simple Z k -modules. As Z k is“ sl -like”, similar classification methods may be used. We shall mostly follow the approach presented in [48] for sl .To begin, a triangular decomposition for Z k is given by(3.8) Z k = ƒ [ G − ] ⊗ ƒ [ J , L ] ⊗ ƒ [ G + ] . The existence of this decomposition is an easy extension of [35, Cor. 1.3], which guarantees a Poincaré–Birkhoff–Witt-style basis for Z k . The analogue of the Cartan subalgebra of sl is then spanned by J and L . Definition 3.16. • A vector in a Z k -module is a weight vector of weight ( j , ∆ ) if it is a simultaneous eigenvector of J and L witheigenvalues j and ∆ , respectively. The nonzero simultaneous eigenspaces of J and L are called the weight spaces .If the Z k -module has a basis of weight vectors and its weight spaces are all finite-dimensional, then it is a weightmodule . • A vector in a Z k -module is a highest-weight vector ( lowest-weight vector ) if it is a weight vector that is annihilatedby G + (by G − ). A highest-weight module ( lowest-weight module ) is a Z k -module that is generated by a singlehighest-weight vector (by a single lowest-weight vector). • A weight Z k -module is dense if its weights coincide with the set [ j ] × { ∆ } , for some coset [ j ] ∈ ƒ / š and some ∆ ∈ ƒ . We note that Z k possesses a “conjugation” automorphism γ defined by(3.9) γ ( J ) = − J , γ ( G + ) = + G − , γ ( G − ) = − G + , γ ( L ) = L . Conjugating a highest-weight Z k -module of highest weight ( j , ∆ ) then results in a lowest-weight module of lowest weight (− j , ∆ ) and vice versa. The structures of highest- and lowest-weight Z k -modules are therefore equivalent.To construct highest-weight Z k -modules, we realise them as quotients of Verma Z k -modules. Let Z > k denote the(unital) subalgebra of Z k generated by J , L and G + . Let ƒ j , ∆ , with j , ∆ ∈ ƒ , be the one-dimensional Z > k -module, spannedby v , on which we have Jv = jv , Lv = ∆ v and G + v =
0. The Verma Z k -module V j , ∆ is then the induced module Z k ⊗ Z > k ƒ j , ∆ , as usual. It is easy to check that V j , ∆ is a highest-weight module with highest-weight vector v = ⊗ v andone-dimensional weight spaces of weights ( j − n , ∆ ) , n ∈ š > . Let H j , ∆ denote the unique simple quotient of V j , ∆ .For convenience, we define(3.10) h n k ( J , L ) = n − Õ m = f k ( J − m , L ) = n (cid:18) n − n ( J + ) + ( J + J + ) − ( k + ) L − ( k + )( k + ) (cid:19) , where the f k were defined in (3.5). Proposition 3.17. • The Verma module V j , ∆ is simple, so H j , ∆ = V j , ∆ , unless h n k ( j , ∆ ) = for some n ∈ š > . • Verma Z k -modules may have at most three composition factors. Exactly one of these is infinite-dimensional. • If h n k ( j , ∆ ) = for some n ∈ š > and N is the minimal such n , then H j , ∆ ≃ V j , ∆ (cid:14) V j − N , ∆ and dim H j , ∆ = N . Proof.
The first statement follows easily by noting that every proper nonzero submodule of V j , ∆ is generated by asingular vector of the form ( G − ) n v , n ∈ š > . The condition to be a singular vector is0 = G + ( G − ) n v = n − Õ m = ( G − ) n − − m [ G + , G − ]( G − ) m v = n − Õ m = ( G − ) n − − m f k ( J , L )( G − ) m v (3.11) = n − Õ m = ( G − ) n − f k ( J − m , L ) v = ( G − ) n − n − Õ m = f k ( j − m , ∆ ) v = h n k ( j , ∆ )( G − ) n − v . Since h n k is a cubic polynomial in n , there can be at most three roots in š > , hence at most three highest-weight vectors.The remaining statements are now clear. (cid:4) ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 11
Unlike sl , there exist nonsemisimple finite-dimensional Z k -modules. Examples include the highest-weight modulesobtained by quotienting a Verma module with three composition factors by its socle.This proposition completes the classification of finite-dimensional Z k -modules and highest-weight Z k -modules. Toobtain the analogous classification of lowest-weight Z k -modules, we apply the conjugation automorphism γ . Theconjugate of a simple Verma module V j , ∆ is the lowest-weight Verma module of lowest weight (− j , ∆ ) . However, if V j , ∆ is not simple and N is the smallest positive integer such that h N k ( j , ∆ ) =
0, then the conjugate of H j , ∆ is isomorphicto H N − j − , ∆ .It remains to determine the simple weight Z k -modules that are neither highest- nor lowest-weight. Such modulesare necessarily dense. As for sl , the classification of simple dense Z k -modules is greatly simplified by identifying thecentraliser C k of the subalgebra ƒ [ J , L ] in Z k . Lemma 3.18.
The centraliser C k is the polynomial algebra ƒ [ J , L , G + G − ] . Proof.
Note first that G + G − obviously commutes with J , by (3.5). Consider a Poincaré–Birkhoff–Witt basis of Z k givenby elements of the form J a L b ( G + ) c ( G − ) d , for a , b , c , d ∈ š > . It is easy to check that such a basis element belongs to C k if and only if c = d . To show that J , L and G + G − generate C k , it therefore suffices to show that ( G + ) c ( G − ) c may bewritten as a polynomial in J , L and G + G − , for each c ∈ š > .Proceeding by induction, this is clear for c =
0. So take c > ( G + ) c − ( G − ) c − is a polynomial in J , L and G + G − . Then, the commutation rules (3.5) give ( G + ) c ( G − ) c = ( G + G − )( G + ) c − ( G − ) c − + G + [( G + ) c − , G − ]( G − ) c − = ( G + G − )( G + ) c − ( G − ) c − + c − Õ n = ( G + ) n f k ( J , L )( G + ) c − − n ( G − ) c − . (3.12)The first term on the right-hand side is a polynomial in J , L and G + G − , by the inductive hypothesis. For the remainingterms, note that as L is central and G + J = ( J − ) G + , we have ( G + ) n J = ( J − n )( G + ) n and hence(3.13) c − Õ n = ( G + ) n f k ( J , L )( G + ) c − − n ( G − ) c − = c − Õ n = f k ( J − n , L )( G + ) c − ( G − ) c − , which is likewise a polynomial in J , L and G + G − . (cid:4) Recall that the weight spaces of a simple weight Z k -module are simple C k -modules (see [48, Lem. 3.4.2] for example).The fact that C k is abelian now gives the following result. Proposition 3.19.
The weight spaces of a simple weight Z k -module are one-dimensional. To understand these weight spaces, one therefore needs to know the eigenvalues of J , L and G + G − on a given simpleweight Z k -module. The latter will vary with the weight ( j , ∆ ) in general, so it is convenient to note that we may replace G + G − by a central element of Z k , something like a Casimir operator, whose eigenvalue is therefore constant. Lemma 3.20.
The element (3.14) Ω = G + G − + G − G + + J + J − J (cid:18) ( k + ) L + ( k + )( k + ) (cid:19) is central in Z k and we have γ ( Ω ) = − Ω and C k = ƒ [ J , L , Ω ] . Proof.
We start by noting that(3.15) [ G + G − , G + ] = − G + f k ( J , L ) = − G + (cid:16) J − ( k + ) L − ( k + )( k + ) (cid:17) . Since [ J n , G + ] = G + (( J + ) n − J n ) , we can cancel the terms appearing on the right-hand side (starting with 3 J ) byadding counterterms to G + G − . In this way, we arrive at an element e Ω ∈ Z k that commutes with J , G + and L :(3.16) e Ω = G + G − + J − J + J − J (cid:18) ( k + ) L + ( k + )( k + ) (cid:19) . By using G + G − = G − G + + f k ( J , L ) , we obtain a second expression for e Ω . Adding the two expressions, we see that(3.17) Ω = e Ω + ( k + ) L + ( k + )( k + ) also commutes with J , G + and L . But, the explicit form (3.14) shows that it also commutes with G − because theconjugation automorphism (3.9) gives γ ( Ω ) = − Ω . (cid:4) By (3.14), the eigenvalue of Ω on a highest-weight vector ( + ) or lowest-weight vector ( − ) of weight ( j , ∆ ) is given by(3.18) ω ± j , ∆ = ( j ± ) (cid:18) j ( j ± ) − ( k + ) ∆ − ( k + )( k + ) (cid:19) . These eigenvalues satisfy the following relations:(3.19) ω −− j , ∆ = − ω + j , ∆ = ω + − j − , ∆ . We note that the first equality is consistent with conjugation.We now construct dense Z k -modules by induction. Let ƒ j , ∆ , ω be a one-dimensional C k -module, spanned by v , onwhich we have Jv = jv , Lv = ∆ v and Ω v = ωv , for some j , ∆ , ω ∈ ƒ . Define the induced module R j , ∆ , ω = Z k ⊗ C k ƒ j , ∆ , ω and note that a basis of R j , ∆ , ω is given by v = ⊗ v and the ( G ± ) n v with n ∈ š > . The weights therefore coincide with [ j ] × { ∆ } and so R j , ∆ , ω is a dense Z k -module generated by v . Proposition 3.21. • For each n ∈ š > , ( G − ) n + v is a highest-weight vector of R j , ∆ , ω if and only if ω = ω + j − n − , ∆ . • For each n ∈ š > , ( G + ) n + v is a lowest-weight vector of R j , ∆ , ω if and only if ω = ω − j + n + , ∆ . • The dense Z k -module R j , ∆ , ω is simple if and only if ω , ω + i , ∆ (equivalently ω , ω − i , ∆ ) for any i ∈ [ j ] . • R j , ∆ , ω has at most four composition factors. If it is not simple, then one composition factor is infinite-dimensionalhighest-weight and another is infinite-dimensional lowest-weight; any other composition factors are finite-dimensional. Proof.
The existence criteria for highest- and lowest-weight vectors is straightforward calculation using (3.19). Thesimplicity of R j , ∆ , ω is equivalent to the absence of highest- and lowest-weight vectors. However, ω , ω − j − n , ∆ for all n ∈ š > implies that ω , ω + j − n − , ∆ for all n ∈ š > , by (3.19). Combining with ω , ω + j + n , ∆ for all n ∈ š > , we getthe desired condition. The statements about composition factors now follow from the fact that ω − ω ± i , ∆ is a cubicpolynomial in i , so it can have at most three roots i ∈ [ j ] . (cid:4) It follows from this proposition that we have isomorphisms R j , ∆ , ω ≃ R j + , ∆ , ω when these modules are simple. We shalltherefore denote these simple dense Z k -modules by R [ j ] , ∆ , ω , where [ j ] ∈ ƒ / š . Theorem 3.22.
Every simple weight Z k -module is isomorphic to one of the modules in the following list of pairwise-inequivalent modules: • The finite-dimensional highest-weight modules H j , ∆ with j , ∆ ∈ ƒ such that h n k ( j , ∆ ) = for some n ∈ š > . • The infinite-dimensional highest-weight modules H j , ∆ = V j , ∆ with j , ∆ ∈ ƒ such that h n k ( j , ∆ ) , for all n ∈ š > . • The infinite-dimensional lowest-weight modules γ ( H j , ∆ ) = γ ( V j , ∆ ) with j , ∆ ∈ ƒ such that h n k ( j , ∆ ) , for all n ∈ š > . • The infinite-dimensional dense modules R [ j ] , ∆ , ω with [ j ] ∈ ƒ / š and ∆ , ω ∈ ƒ such that ω , ω + i , ∆ for any i ∈ [ j ] . Proof.
The classification was already completed after Proposition 3.17 for the first three cases, that is when the simpleweight module has either a highest- or lowest-weight (or both). If the simple weight module has no highest- or lowest-weight, choose an arbitrary weight space. This is a simple C k -module, hence it is one-dimensional (Proposition 3.19)and spanned by v say. As there are no highest- or lowest-weight vectors, G + and G − act freely on v and so the simpleweight module is dense and so isomorphic to one of the R [ j ] , ∆ , ω in the list. (cid:4) As in the untwisted case, the fact that
Zhu tw (cid:2) BP k (cid:3) is a quotient of Z k means that every simple Zhu tw (cid:2) BP k (cid:3) -moduleis also simple as a Z k -module. Theorem 3.13 then guarantees that every simple weight Zhu tw (cid:2) BP k (cid:3) -module M corresponds to a simple twisted relaxed highest-weight BP k -module M = Ind tw (cid:2) M (cid:3) which is uniquely determined (upto isomorphism) by the fact that its top space is isomorphic to M (as a Z k -module). ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 13
Theorem 3.23.
Every simple twisted relaxed highest-weight BP k -module, and hence every simple twisted relaxedhighest-weight BP ( u , v ) -module, is isomorphic to one of the modules in the following list of pairwise-inequivalentmodules: • The highest-weight modules H tw j , ∆ with j , ∆ ∈ ƒ such that h n k ( j , ∆ ) = for some n ∈ š > . • The highest-weight modules H tw j , ∆ = V tw j , ∆ with j , ∆ ∈ ƒ such that h n k ( j , ∆ ) , for all n ∈ š > . • The conjugate highest-weight modules γ ( H tw j , ∆ ) = γ ( V tw j , ∆ ) with j , ∆ ∈ ƒ such that h n k ( j , ∆ ) , for all n ∈ š > . • The relaxed highest-weight modules R tw [ j ] , ∆ , ω with [ j ] ∈ ƒ / š and ∆ , ω ∈ ƒ such that ω , ω + i , ∆ for all i ∈ [ j ] . Again, we remark that spectral flow will allow us to construct simple twisted weight BP k -modules that are not relaxedhighest-weight, in general.3.6. Coherent families.
A crucial observation of Mathieu [31] concerning simple dense g -modules, for g a simpleLie algebra, is that they may be naturally arranged into coherent families. Here, we extend this observation to dense Z k -modules in preparation for showing that it also extends to Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules. While Mathieu’s generalresults rely heavily on the properties of his twisted localisation functors, our discussion of this simple case will be quiteelementary. Definition 3.24. A coherent family of Z k -modules is a weight module C for which: • L and Ω act as multiples, ∆ and ω respectively, of the identity on C . • There exists d ∈ š > such that for all j ∈ ƒ , the dimension of the weight space C ( j , ∆ ) of weight ( j , ∆ ) is d . • For each U ∈ C k , the function taking j ∈ ƒ to tr C ( j , ∆ ) U is polynomial in j . Coherent families are highly decomposable. Indeed, a coherent family of Z k -modules necessarily has the form(3.20) C = Ê [ j ]∈ ƒ / š C [ j ] . If all of the C [ j ] are semisimple as Z k -modules, then C is said to be semisimple . If any of the C [ j ] are simple as Z k -modules, then C is said to be irreducible . It follows immediately from Proposition 3.19 that the common dimension d of the weight spaces of an irreducible coherent family of Z k -modules is 1.We would like to form a coherent family of Z k -modules by summing over some collection of dense modules R [ j ] , ∆ , ω , [ j ] ∈ ƒ / š , whilst holding ∆ and ω fixed. However, this is mildly ambiguous because there will always be at least one [ j ] (generically three) for which the corresponding element in the collection will not be simple and so we should thenspecify precisely which module we mean. For such j , we shall specify this in three distinct ways (though there areothers). • The first is to define R [ j ] , ∆ , ω to be R ss j , ∆ , ω , where the semisimplification M ss of a (finite-length) module M is thedirect sum of its composition factors. This is well-defined as R ss j , ∆ , ω ≃ R ss j + , ∆ , ω . • An alternative is to define R [ j ] , ∆ , ω to be R + [ j ] , ∆ , ω = R j + , ∆ , ω , where we choose j + ∈ [ j ] to have smaller real part thanthose of the solutions i ∈ [ j ] of ω = ω + i , ∆ . This ensures that R + [ j ] , ∆ , ω has no lowest-weight vectors. • We may instead define R [ j ] , ∆ , ω to be R −[ j ] , ∆ , ω = R j − , ∆ , ω , where we choose j − ∈ [ j ] to have larger real part than thoseof the solutions i ∈ [ j ] of ω = ω − i , ∆ . This ensures that R −[ j ] , ∆ , ω has no highest-weight vectors.For each of the three choices above, we take the direct sum of the R [ j ] , ∆ , ω over [ j ] ∈ ƒ / š . The result is easily verifiedto be an irreducible coherent family of Z k -modules. It will be denoted by C ss ∆ , ω , C + ∆ , ω or C − ∆ , ω , respectively. The firstis semisimple, whilst the second is nonsemisimple with G + acting injectively and the third is nonsemisimple with G − acting injectively. It is easy to check that the conjugates of these irreducible coherent families are(3.21) γ ( C ss ∆ , ω ) ≃ C ss ∆ , − ω , γ ( C + ∆ , ω ) ≃ C − ∆ , − ω and γ ( C − ∆ , ω ) ≃ C + ∆ , − ω . For classifying simple BP ( u , v ) -modules, the semisimple coherent families C ss ∆ , ω are most suitable. Note that C ss ∆ , ω is the unique irreducible semisimple coherent family of Z k -modules on which L acts as multiplication by ∆ and Ω actsas multiplication by ω , up to isomorphism. We shall return to C + ∆ , ω and C − ∆ , ω in Section 4.4 when considering theexistence of nonsemisimple BP ( u , v ) -modules. Proposition 3.25. • Every simple weight Z k -module embeds into a unique irreducible semisimple coherent family. • Every irreducible semisimple coherent family of Z k -modules contains an infinite-dimensional highest-weight sub-module. Proof.
By Theorem 3.22, a simple dense Z k -module M is isomorphic to some R [ j ] , ∆ , ω , where [ j ] ∈ ƒ / š and ∆ , ω ∈ ƒ satisfy ω , ω + i , ∆ for any i ∈ [ j ] . As R ss [ j ] , ∆ , ω = R [ j ] , ∆ , ω , we have an embedding M ֒ → C ss ∆ , ω . The target is obviouslyunique, up to isomorphism, since no other irreducible semisimple coherent family has the correct L - and Ω -eigenvalues.A simple highest-weight Z k -module M is isomorphic to H j , ∆ , for some j , ∆ ∈ ƒ . Take ω = ω + j , ∆ , so that R j , ∆ , ω isnot simple and there is a highest-weight vector of weight ( j , ∆ ) in R ss j , ∆ , ω , by Proposition 3.21. This vector generates acopy of H j , ∆ , so we again have an embedding M ֒ → C ss ∆ , ω with unique target.Finally, if M is a simple lowest-weight Z k -module, then we have an embedding γ ( M ) ֒ → C ss ∆ , ω for some unique ∆ , ω ∈ ƒ . By (3.21), we have M ֒ → C ss ∆ , − ω . This covers all possibilities, by Theorem 3.22, so the first statement isestablished.For the second, a given irreducible semisimple coherent family C ss ∆ , ω is uniquely specified by choosing ∆ , ω ∈ ƒ . As ω − ω + i , ∆ is a cubic polynomial in i , there is at least one solution in ƒ , i = j say. Then, R j , ∆ , ω is not simple and has aninfinite-dimensional highest-weight submodule, by Proposition 3.21, hence so does R ss j , ∆ , ω ⊂ C ss ∆ , ω . (cid:4)
4. Modules of the simple admissible-level Bershadsky–Polyakov algebras
Recall [49] that if I is an ideal of a vertex operator algebra V , then Zhu [ V / I ] ≃ Zhu [ V ]/ Zhu [ I ] . If J k denotes themaximal ideal of BP k , then classifying the relaxed highest-weight modules of BP k = BP k / J k is then just a matter ofclassifying those of BP k and then testing which have Zhu-images annihilated by Zhu (cid:2) J k (cid:3) . The twisted classificationthen follows, roughly speaking, from spectral flow. Unfortunately, it is hard to compute Zhu (cid:2) J k (cid:3) in general.Instead, we shall combine Arakawa’s celebrated classification [37] of the highest-weight modules of all simpleadmissible-level affine vertex operator algebras L k ( g ) , specialised to g = sl , with his results [7] on minimal quantumhamiltonian reduction. The result will be a classification of the highest-weight modules for the Bershadsky–Polyakovminimal models from which we will extract the full (twisted and untwisted) relaxed highest-weight classification.4.1. Admissible-level sl minimal models. Recall from (2.5) the fractional levels of BP k and their parametrisationin terms of u and v . These are also the fractional levels for the affine vertex operator algebras associated to sl — V k ( sl ) is not simple [39, Thm. 0.2.1] when k is a fractional level. For such k , the simple quotient will be denoted by L k ( sl ) = A ( u , v ) . Definition 4.1. An admissible level k for the affine vertex operator algebras associated to sl , and the Bershadsky–Polyakov algebras, is a fractional level for which u > . Every highest-weight module for the affine Kac–Moody algebra b sl is a V k ( sl ) -module [49]. Let L λ denote thesimple highest-weight b sl -module of highest weight λ = λ ω + λ ω + λ ω , where the λ i are the Dynkin labels andthe ω i are the fundamental weights. To be a level- k module, we must have λ + λ + λ = k . Let P ℓ > denote the set ofdominant integral level- ℓ weights of b sl , that is the set of weights λ satisfying λ i ∈ š > and λ + λ + λ = ℓ . This setis obviously empty unless ℓ ∈ š > . Let w i , i = , ,
2, denote the Weyl reflection corresponding to the simple root α i of b sl .The following definition specialises that of [36] to b sl (see also [50, App. 18.B]). Definition 4.2.
Let k be an admissible level. A level- k admissible weight λ of b sl is one of the form (4.1) λ = w · (cid:16) λ I − uv λ F , w (cid:17) , where w ∈ { , w } is a Weyl transformation of sl , · is the shifted Weyl group action, λ I ∈ P u − > , λ F , w ∈ P v − > and λ F , w > . A weight of the form (4.1) will be called a w = or w = w admissible weight according as to which w isused. ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 15
We remark that one may allow w to range over the full Weyl group, adding appropriate restrictions on the λ F , w , butthis gives no further admissible weights. In fact, every set of w = w ′ admissible weights is equal to either the w = or w = w sets and, moreover, these two sets are disjoint [51, Prop. 2.1].Arakawa’s highest-weight classification for affine vertex operator algebras now specialises as follows. Theorem 4.3 ([37]) . For k admissible, the simple level- k highest-weight module L λ is an A ( u , v ) -module if and onlyif λ is admissible. Denote by H (−) the minimal quantum hamiltonian reduction functor [5] taking V k ( sl ) -modules to BP k -modules,so that H ( V k ( sl )) = BP k . For definiteness, we take the nilpotent element of sl defining this functor to be the negativehighest-root vector f θ . We assemble some useful results about this functor, specialised to our setting. Theorem 4.4. • [6, Thm. 6.3] If K λ denotes the Verma module of V k ( sl ) with highest weight λ , then H ( K λ ) is isomorphic to theVerma module V j , ∆ of BP k with (4.2) j = λ − λ and ∆ = ( λ − λ ) − ( λ + λ ) (cid:0) ( k + ) − λ − λ (cid:1) ( k + ) . • [7, Thm. 6.7.4] H ( L λ ) = if and only if λ ∈ š > . For λ < š > , we have instead H ( L λ ) ≃ H j , ∆ , where j and ∆ are given by (4.2) . • [7, Cor. 6.7.3] The restriction of H (−) to the category b O k of level- k b sl -modules is exact. • H (−) induces a surjection from the set of isomorphism classes of simple highest-weight V k ( sl ) -modules to the unionof { } and the set of isomorphism classes of simple highest-weight BP k -modules. Moreover, there are at most twoinequivalent L λ mapping onto the same H j , ∆ . Proof.
We only prove the last assertion. It follows from the second assertion above and by inverting (4.2) to obtain twosolutions ( λ , λ ) for each ( j , ∆ ) . We have to ensure that at least one solution gives λ < š > . But, a simple calculationgives(4.3) λ = k − λ − λ = − ± p ( k + ) ∆ + ( k + ) − j , so the zeroth Dynkin labels of the two solutions sum to − (cid:4) Definition 4.5.
For k admissible, we shall call a level- k weight λ of b sl surviving if it is admissible and λ < š > .Theorem 4.4 then ensures that H ( L λ ) is nonzero (and is moreover a simple BP k -module). Lemma 4.6. • Every w = w admissible weight is surviving. • A w = admissible weight λ is surviving if and only if λ F , > . • w · gives a ( j , ∆ ) -preserving bijection between the w = surviving weights and the w = w admissible weights. • If λ and µ are distinct w = surviving weights, then H ( L λ ) and H ( L µ ) are not isomorphic. Proof.
The zeroth Dynkin label of a level- k admissible b sl -weight λ has one of the following two forms:(4.4) λ = λ I − uv λ F , ( w = ) or λ = λ I + λ I − uv (cid:16) λ F , w + λ F , w (cid:17) + w = w ) . Consider first a w = admissible weight λ . Since λ F , ∈ P v − > , we clearly have λ ∈ š if and only if λ F , =
0. On theother hand, a w = w admissible weight λ necessarily has 0 < λ F , w + λ F , w < v , since λ F , w ∈ P v − > and λ F , w >
1. Itfollows that the Dynkin label λ can never be an integer in this case. The first two statements are thus established.For the third, let µ be a level- k weight. Explicit calculation shows that the Dynkin labels of w · w · µ are(4.5) h µ − uv , µ , µ + uv i . Let λ = w · (cid:0) λ I − uv λ F , w (cid:1) be a w = w admissible weight. Then, w · λ has the form µ = µ I − uv µ F , with(4.6) µ I = h λ I , λ I , λ I i and µ F , = h λ F , w + , λ F , w , λ F , w − i . It is easy to see that µ I ∈ P u − > and µ F , ∈ P v − > , so µ is a w = admissible weight. Moreover, µ F , > µ is surviving. Since w · (−) is clearly self-inverse, we have the desired bijection between w = surviving weights and w = w admissible weights. To show that it is ( j , ∆ ) -preserving, we show that the functions j ( λ ) and ∆ ( λ ) defined by(4.2) are invariant under λ w · λ . This is clear from ( w · λ ) = k + − λ and ( w · λ ) = k + − λ .Finally, let λ and µ be surviving weights and suppose that H ( L λ ) ≃ H ( L µ ) , so that j ( λ ) = j ( µ ) and ∆ ( λ ) = ∆ ( µ ) .We have just seen that λ and w · λ always give the same j and ∆ . But, if λ is a w = surviving weight, then µ = w · λ is a w = w surviving weight. Since the intersection of the sets of w = and w = w admissible weightsis empty [51, Prop. 2.1], we have λ , µ . As there are at most two weights corresponding to a given choice of j and ∆ (Theorem 4.4), this shows that there are never two distinct w = surviving weights giving the same j and ∆ . (cid:4) In what follows, a surviving weight shall be understood to mean a w = surviving weight unless otherwise indicated.The set of ( w = ) surviving level- k weights will be denoted by Σ u , v . We shall also start dropping the label w from λ F , w , understanding that we mean w = unless otherwise indicated.Let I k denote the maximal ideal of V k ( sl ) , so that L k ( sl ) = V k ( sl )/ I k . If k is an admissible level, then Theorem 4.3says that I k · L λ = λ is an admissible weight. If, in addition, v >
2, then(4.7) H ( L k ( sl )) = H ( L k ω ) ≃ H , = BP k , by Theorem 4.4. Moreover, the exactness of H (−) means that the maximal ideal J k of BP k is then isomorphic to H ( I k ) .It follows that H ( L λ ) is a BP k -module if and only if H ( I k ) · H ( L λ ) = H (−) corresponds to tensoring with a ghost vertex operator superalgebra G , graded by the fermionic ghostnumber, and taking the degree-0 cohomology with respect to a given differential (see Appendix A.1 for the details).Denote the cohomology class of a (degree-0) cocycle a by [ a ] (we trust that this notation will not be confused with thenotation for Zhu algebra images in Section 3). Given (degree-0) cocycles a and v of the BRST complexes I k ⊗ G and L λ ⊗ G , respectively, the action of [ a ] ∈ H ( I k ) on [ v ] ∈ H ( L λ ) is given by [ a ] · [ v ] ≡ [ a ]( z )[ v ] = [ a ( z ) v ] ∈ H ( I k · L λ ) .For λ admissible, we therefore obtain(4.8) H ( I k ) · H ( L λ ) ⊆ H ( I k · L λ ) = . This proves the following assertion.
Proposition 4.7.
Let k be admissible with v > . If L λ is an L k ( sl ) -module, then H ( L λ ) is a BP k -module. This also motivates the following assumption, which we shall understand to be in force for everything that follows.
Assumption 1.
In what follows, we shall restrict to fractional levels k = − + uv with u > and v > . The restriction on u means that k is an admissible level for sl , whilst the restriction on v guarantees that the minimal quantum hamiltonianreduction of L k ( sl ) = A ( u , v ) is BP k = BP ( u , v ) ( for u > , we have H ( A ( u , )) = instead). Of course, to obtain a classification of simple highest-weight BP k -modules from Arakawa’s classification of simplehighest-weight L k ( sl ) -modules (Theorem 4.3), we need a converse of Proposition 4.7. This is much more subtle. Theorem 4.8.
Let k be as in Assumption 1. Then, every simple highest-weight BP k -module is isomorphic to the minimalquantum hamiltonian reduction of some simple highest-weight L k ( sl ) -module. Note that if λ ∈ š > , then H ( L λ ) = BP k -module, irrespective of whether or not it is an L k ( sl ) -module. It istherefore enough to show that if λ < š > and L λ is not a BP k -module, then H ( L λ ) is not a BP k -module. Equivalently,we must show that λ < š > and I k · L λ , H ( I k ) · H ( L λ ) ,
0. We defer the somewhat intricate proof ofthis assertion to Appendix A.4.2.
Simple highest-weight BP ( u , v ) -modules. From Theorems 4.3 and 4.8 and Lemma 4.6, we conclude that the H ( L λ ) , with λ ∈ Σ u , v , form a complete set of mutually nonisomorphic simple untwisted highest-weight modules forthe Bershadsky–Polyakov minimal model vertex operator algebra BP ( u , v ) (assuming that the level is as in Assumption 1).The charge ( J -eigenvalue) j and conformal weight ( L -eigenvalue) ∆ of the highest-weight vector of H ( L λ ) was given ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 17 in (4.2): H ( L λ ) ≃ H j , ∆ . This is then a classification of the simple untwisted highest-weight BP ( u , v ) -modules.Moreover, Proposition 3.2 extends this to a classification of their twisted cousins. Theorem 4.9.
Let k be as in Assumption 1. Then: • Every simple untwisted highest-weight BP ( u , v ) -module is isomorphic to one of the H j , ∆ , where j and ∆ are determinedfrom the Dynkin labels of a unique surviving weight λ ∈ Σ u , v by (4.2) . • Every simple twisted highest-weight BP ( u , v ) -module is isomorphic to one of the H tw j , ∆ , where j and ∆ are determinedfrom the Dynkin labels of a unique surviving weight λ ∈ Σ u , v by (4.9) j = λ − λ + k + and ∆ = ( λ − λ ) − ( λ + λ ) (cid:0) ( k + ) − λ − λ (cid:1) ( k + ) + λ − λ + k + . Moreover, the H j , ∆ and H tw j , ∆ determined by the surviving weights are all mutually nonisomorphic. In light of this classification, we let H λ = H j , ∆ and H tw λ = H tw j , ∆ , where j and ∆ are given in terms of λ ∈ Σ u , v by(4.2) and (4.9), respectively. Note that this implies that(4.10) H tw λ ≃ σ / ( H λ ) , by Proposition 3.2. With this new notation, the vacuum module H , is identified as H λ , where λ = [ k , , ] has λ I = [ u − , , ] and λ F = [ v − , , ] .We record the following strengthening of Theorem 4.9, following [52, Thm. 10.10], for later use. Theorem 4.10.
Let k be as in Assumption 1. Then, every highest-weight BP ( u , v ) -module, untwisted or twisted, issimple. Proof.
We prove this for untwisted modules as the twisted case follows immediately from (4.10) and the invertibilityof spectral flow. Since the simple quotient of any highest-weight BP ( u , v ) -module H is isomorphic to some H λ with λ ∈ Σ u , v , by Theorem 4.9, it is enough to show that H cannot have a composition factor isomorphic to H µ for some µ ∈ Σ u , v distinct from λ . Indeed, it is enough to show that the Verma module V λ = V j , ∆ of BP k does not have such acomposition factor.Recall that K λ denotes the Verma module of V k ( sl ) of highest weight λ and let (cid:2) K λ : L ν (cid:3) denote the multiplicitywith which L ν appears as a composition factor of K λ . By Theorem 4.4, quantum hamiltonian reduction takes K λ to V λ and only L µ and L w · µ are sent to H µ . As reduction is exact, we must have (cid:2) V λ : H µ (cid:3) = (cid:2) K λ : L µ (cid:3) + (cid:2) K λ : L w · µ (cid:3) (noting that µ and w · µ are distinct since µ ∈ Σ u , v ).It follows that if V λ has H µ , µ , λ , as a composition factor, then K λ has either L µ or L w · µ as a composition factor.But, λ , µ and w · µ are all admissible b sl -weights (corresponding to w = , and w , respectively, see Lemma 4.6),hence they are dominant. This is therefore impossible by the linkage principle for Verma b sl -modules. (cid:4) Because the Bernšte˘ın–Gel’fand–Gel’fand category O u , v of level- k BP ( u , v ) -modules admits contragredient duals, itfollows from Theorem 4.10 that every extension between H λ and H µ , with λ , µ , splits. It is likewise easy to seethat a nonsplit self-extension of H λ requires a nonsemisimple action of J or L (which is forbidden in O u , v ). O u , v is thus semisimple and, by Theorem 4.9, has finitely many isomorphism classes of simple objects. We may thereforesummarise this as follows: BP ( u , v ) is rational in category O u , v .In order to extend the highest-weight classification of Theorem 4.9 to twisted relaxed highest-weight BP ( u , v ) -modules, we need to know when the top space ( H tw j , ∆ ) top = Zhu tw (cid:2) H tw j , ∆ (cid:3) is infinite-dimensional. The condition for thisis beautifully succinct when expressed in terms of surviving weights. Proposition 4.11.
The top space of the simple twisted highest-weight BP ( u , v ) -module H tw λ is finite-dimensional if andonly if λ F = . When λ F = , the dimension of this top space is λ I + . Proof.
By Proposition 3.17, ( H tw j , ∆ ) top is finite-dimensional if and only if h n k ( j , ∆ ) = n ∈ š > and, if it isfinite-dimensional, then the dimension is the smallest such n . Substituting (4.9) into the definition (3.10) of h n k andsimplifying, we find that(4.11) h n k ( j , ∆ ) = n ( n − λ − ) (cid:16) n + λ + − uv (cid:17) . The only possible roots in š > are thus n = λ + n = uv − λ −
1. As λ = λ I − uv λ F , the former requires λ ∈ š so λ F = n = λ I + ∈ š > . On the other hand, the latter requires n = −( λ I + ) + uv ( λ F + ) which is only an integerif λ F = v −
1. However, this contradicts λ F ∈ P v − > and λ F > (cid:4) Corollary 4.12.
Given k as in Assumption 1, there are (up to isomorphism): • ( u − )( u − ) v ( v − ) simple untwisted highest-weight BP ( u , v ) -modules; • ( u − )( u − )( v − ) simple twisted highest-weight BP ( u , v ) -modules with finite-dimensional top spaces; • ( u − )( u − )( v − )( v − ) simple twisted highest-weight BP ( u , v ) -modules with infinite-dimensional top spaces; In particular, there are no simple twisted highest-weight BP ( u , v ) -modules with infinite-dimensional top spaces when v =
2. This is in accord with the fact that the BP ( u , ) with u > γ of BP ( u , v ) , given in (2.9), negates J and preserves L . At the level oftheir eigenvalues, this is effected in (4.2) by exchanging the Dynkin labels λ and λ of λ . The result of this exchangeis clearly still a surviving weight, by Lemma 4.6. Proposition 4.13.
For each λ ∈ Σ u , v , we have: • γ ( H [ λ , λ , λ ] ) ≃ H [ λ , λ , λ ] . • If λ F = , then γ ( H tw λ ) ≃ H tw µ , where µ = [ λ − uv , λ , λ + uv ] , hence µ I = [ λ I , λ I , λ I ] and µ F = [ λ F + , , λ F − ] .Otherwise, γ ( H tw [ λ , λ , λ ] ) is not highest-weight (though it is relaxed highest-weight). Proof.
The result of conjugating a simple untwisted highest-weight BP ( u , v ) -module is clear from the above remarks,because the top spaces are one-dimensional. For the twisted case, first note that the conjugate of H tw λ will be againhighest-weight if its top space is finite-dimensional (otherwise the top space of the conjugate module will be an infinite-dimensional lowest-weight Z k -module). By Proposition 4.11, this requires λ F =
0, hence λ = λ I . Assuming this,let j and ∆ denote the charge and conformal weight, respectively, of the highest-weight vector of H tw λ . Then, thehighest-weight vector of γ ( H tw λ ) has charge λ − j and conformal weight ∆ .We therefore need to find µ ∈ Σ u , v corresponding to these eigenvalues under (4.9). Solving for µ , we find twosolutions:(4.12) µ = λ − k − , µ = λ and µ = λ + k + , or µ = k + − λ , µ = − λ − µ = k + − λ . We know from the proof of Lemma 4.6 that only one of these is a w = surviving weight and the other is a w = w survivor obtained from the w = one by applying the shifted action of w . It is easy to check that the first solution isthe w = survivor by writing it in the form(4.13) µ = λ I − uv ( λ F + ) , µ = λ I and µ = λ I − uv ( λ F − ) . Indeed, λ F > µ I = [ λ I , λ I , λ I ] ∈ P u − > , µ F = [ λ F + , , λ F − ] ∈ P v − > and µ F >
1, hence that µ ∈ Σ u , v . (cid:4) It remains to determine when the spectral flow of a simple highest-weight BP ( u , v ) -module is another such module.By Proposition 3.2, it suffices to consider the untwisted case. Again the key is the finite-dimensionality of the top space: σ ( H λ ) will be highest-weight if and only if H tw λ = σ / ( H λ ) has a finite-dimensional top space, that is if and only if λ F = λ F = v denotes the highest-weight vector of H λ , then that of σ ( H λ ) iseasily checked to be ( G − / ) λ I σ ( v ) . We compute its charge and conformal weight, then determine the (unique w = )surviving weight that gives these eigenvalues under (4.2), as in the proof of Proposition 4.13. We thereby obtain thefollowing proposition. Proposition 4.14. If λ ∈ Σ u , v satisfies λ F = , then σ ( H λ ) ≃ H µ , where µ = [ λ − uv , λ + uv , λ ] ∈ Σ u , v , hence µ I = [ λ I , λ I , λ I ] and µ F = [ λ F + , λ F − , ] . If λ F , , then σ ( H λ ) is not highest-weight (nor relaxed highest-weight). Combining this with the dihedral relation (2.8) and Proposition 4.13, we obtain the following characterisation of thespectral flow orbit of a simple untwisted highest-weight BP ( u , v ) -module H λ . We recall from Proposition 3.2 that atwisted member σ ℓ + / ( H λ ) , ℓ ∈ š , of this orbit is highest-weight if and only if its untwisted predecessor σ ℓ ( H λ ) is. ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 19 · · · σ / σ / σ / λ F , λ F , σ / σ / σ / σ / σ / σ / λ F = λ F , , λ F , σ / σ / λ F = λ F , , λ F , σ / σ / σ / σ / σ / σ / λ F = [ , , v − ] σ / σ / λ F = [ v − , , ] σ / σ / λ F = [ , v − , ] σ / σ / σ / Figure 1.
A picture of the weights of the three types of spectral flow orbits through a simple highest-weight BP ( u , v ) -module with v >
3. The charge increases from left to right, whilst the conformalweight increases from top to bottom. The given constraints on the Dynkin labels of λ F must be satisfiedby the simple untwisted highest-weight BP ( u , v ) -module H λ appearing at that point in the orbit. Notethat the unpictured modules in each infinite orbit, indicated by · · · , are neither highest-weight norrelaxed highest-weight: their conformal weights are unbounded below. Theorem 4.15.
Take λ ∈ Σ u , v and define µ , ν , ¯ µ , ¯ ν ∈ Σ u , v by (4.14) µ I = [ λ I , λ I , λ I ] , µ F = [ λ F + , λ F − , ] , ¯ µ I = [ λ I , λ I , λ I ] , ¯ µ F = [ λ F + , , λ F − ] and ν I = [ λ I , λ I , λ I ] , ν F = [ , v − , ] , ¯ ν I = [ λ I , λ I , λ I ] , ¯ ν F = [ , , v − ] . • σ ( H λ ) is highest-weight if and only if λ F = . In this case, σ ( H λ ) ≃ H µ . • σ − ( H λ ) is highest-weight if and only if λ F = . In this case, σ − ( H λ ) ≃ H ¯ µ . • σ ( H λ ) is highest-weight if and only if λ F = [ , , v − ] . In this case, σ ( H λ ) ≃ H ν . • σ − ( H λ ) is highest-weight if and only if λ F = [ , v − , ] . In this case, σ − ( H λ ) ≃ H ¯ ν . • For | ℓ | ∈ š > , σ ℓ ( H λ ) is highest-weight if and only if v = . In this case, σ ± ( H λ ) ≃ H λ . Note that when v = , every λ ∈ Σ u , v has λ F = [ , , ] . The spectral flow orbits thus take the form(4.15) · · · σ / H λ σ / H tw λ σ / H µ σ / H tw µ σ / H ν σ / H tw ν σ / H λ σ / , where µ and ν are as in (4.14) (with µ F = ν F = [ , , ] ). We picture the v > spectral flow orbits in Figure 1.4.3. Simple relaxed highest-weight BP ( u , v ) -modules. As we noted in Theorem 3.11, every simple untwisted relaxedhighest-weight BP ( u , v ) -module is highest-weight. The classification of simple untwisted relaxed highest-weightmodules was therefore completed in Theorem 4.9. It remains to classify the simple twisted relaxed highest-weightmodules, specifically those whose top spaces are simple dense Z k -modules (those whose top spaces are simple lowest-weight Z k -modules are conjugates of the simple twisted highest-weight BP ( u , v ) -modules classified in Theorem 4.9).A simple twisted relaxed highest-weight BP k -module M is a BP k -module if and only if its top space M top = Zhu tw (cid:2) M (cid:3) is annihilated by Zhu tw (cid:2) J k (cid:3) , where J k denotes the maximal ideal of BP k . An obvious consequence of Theorem 4.9 isthat Zhu tw (cid:2) J k (cid:3) annihilates Zhu tw (cid:2) H tw λ (cid:3) ≃ H j , ∆ , with j and ∆ determined by λ as in (4.9), if and only if λ ∈ Σ u , v . Weextend this to the simple relaxed highest-weight modules R tw [ j ] , ∆ , ω of Theorem 3.23 using an argument similar to that of[22, Prop. 4.2]. Proposition 4.16.
The irreducible semisimple coherent family C ss ∆ , ω of Z k -modules is a Zhu tw (cid:2) BP ( u , v ) (cid:3) -module if andonly if one of its infinite-dimensional submodules is. Proof.
Obviously, C ss ∆ , ω being a Zhu tw (cid:2) BP ( u , v ) (cid:3) -module implies that every one of its submodules are too, in particularthe infinite-dimensional ones. To prove the converse, we lean heavily on the general methodology developed in [22] to classify relaxed highest-weight modules for affine vertex operator algebras, though the argument here is easier because the relevant coherentfamilies have one-dimensional weight spaces. The first step is to consider the subalgebra A k = Zhu tw (cid:2) J k (cid:3) ∩ C k , wherewe recall that C k = ƒ [ J , L , Ω ] (Lemma 3.20). The relevance is that a simple weight Zhu tw (cid:2) BP k (cid:3) -module M is a Zhu tw (cid:2) BP ( u , v ) (cid:3) -module if and only if A k annihilates some nonzero element of M . This fact is proved in exactly thesame way that [22, Lem. 4.1] is (see also [53]) and so we omit the details.We next note that the action of A k preserves each of the one-dimensional weight spaces of the irreducible semisimplecoherent family C ss ∆ , ω and that this action is polynomial: for each a ∈ A k ⊂ ƒ [ J , L , Ω ] , there is a polynomial p a in threevariables such that a acts on the weight space C ss ∆ , ω ( j , ∆ , ω ) as multiplication by p a ( j , ∆ , ω ) . Since ∆ and ω are fixed bythe choice of coherent family, we may regard p a as a single-variable polynomial.If we now assume that one of the infinite-dimensional submodules of C ss ∆ , ω is a Zhu tw (cid:2) BP ( u , v ) (cid:3) -module, then it isannihilated by Zhu tw (cid:2) J k (cid:3) and thus by A k . Thus, for every a ∈ A k , we have p a ( j , ∆ , ω ) = for infinitely many distinctvalues of j , whence p a (− , ∆ , ω ) must be the zero polynomial. But, then a annihilates all of C ss ∆ , ω , whence C ss ∆ , ω is a Zhu tw (cid:2) BP ( u , v ) (cid:3) -module. (cid:4) Note that the top space of every (simple) R tw [ j ] , ∆ , ω embeds into some irreducible semisimple coherent family and thatevery such family has an infinite-dimensional highest-weight submodule H j ′ , ∆ , by Proposition 3.25. From Theorem 4.9,we have classified all the simple highest-weight BP ( u , v ) -modules in terms of surviving weights. Proposition 4.16 thusdetermines the irreducible semisimple coherent families that are Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules and in this way we find allthe R tw [ j ] , ∆ , ω that are simple BP ( u , v ) -modules. Algorithmically, this classification proceeds as follows.Let Γ u , v denote the set of ( w = ) admissible b sl -weights λ of level k with λ F , , so that λ ∈ Σ u , v (Lemma 4.6), and λ F , , so that H tw λ has an infinite-dimensional top space (Proposition 4.11). Then, Γ u , v parametrises the isomorphismclasses of the simple highest-weight BP ( u , v ) -modules with infinite-dimensional top spaces. • For each λ ∈ Γ u , v , compute j and ∆ using (4.9), then substitute into (3.18) to compute ω :(4.16) ω = ω + j , ∆ = − ( λ − λ + k + )( λ + λ − k )( λ + λ − k − ) . This gives the eigenvalues of J , L and Ω on the highest-weight vector of ( H tw λ ) top . • Then, the R tw [ j ′ ] , ∆ , ω are, for all [ j ′ ] ∈ ƒ / š satisfying ω + i , ∆ , ω for every i ∈ [ j ′ ] , simple relaxed highest-weight BP ( u , v ) -modules (by Theorem 3.23 and Proposition 4.16) and all such modules are obtained, up to isomorphism, inthis way.As with the highest-weight BP ( u , v ) -modules classified in Section 4.2, it is convenient to let R tw [ j ] , λ = R tw [ j ] , ∆ , ω , where ∆ and ω are given in terms of λ by (4.9) and (4.16), respectively.We may now summarise this classification as follows. Theorem 4.17.
Let k be as in Assumption 1 and let j be such that R tw [ j ] , λ is simple. Then, R tw [ j ] , λ is a (twisted) BP ( u , v ) -module if and only if λ ∈ Γ u , v . In fact, we shall see that a complete classification does not require considering every possible weight λ ∈ Γ u , v . Firsthowever, we recall from Corollary 4.12 that there are no highest-weight BP ( u , v ) -modules with infinite-dimensional topspaces, hence Γ u , v = œ , when v = . Corollary 4.18.
Let k be as in Assumption 1 with v = . Then, every simple (twisted) relaxed highest-weight BP ( u , v ) -module is highest-weight. Again, this is consistent with the fact [1] that BP ( u , ) is rational for every u ∈ š > + . It is therefore convenient toslightly refine Assumption 1 as follows. Assumption 2.
In what follows, we shall restrict to fractional levels k = − + uv with u , v > . The levels of Assumption 2 are also known as nondegenerate admissible levels in the literature. We shall understandthat Assumption 2 is in force for the rest of this section.
ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 21
Given an irreducible semisimple coherent family C ss ∆ , ω of Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules, we ask how many inequivalentinfinite-dimensional highest-weight submodules it possesses. By Proposition 3.21, the direct summands R [ j ] , ∆ , ω arenot simple for at least one, and at most three, [ j ] ∈ ƒ / š and each nonsimple summand has precisely one infinite-dimensional highest-weight submodule. The answer to our question is therefore either one, two or three. In fact, for k as in Assumption 2, the answer is always three. Lemma 4.19. If k is as in Assumption 2, then each irreducible semisimple coherent family C ss ∆ , ω of Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules has precisely three infinite-dimensional highest-weight submodules. The map Γ u , v → ƒ given by λ
7→ ( ∆ , ω ) is thus -to- . Moreover, the highest weights λ = λ I − uv λ F of these three submodules are related by the following š -action: (4.17) · · · 7−→ [ λ , λ , λ ] 7−→ [ λ − uv , λ , λ + uv ] 7−→ [ λ , λ − uv , λ + uv ] 7−→ · · · , · · · 7−→ [ λ I , λ I , λ I ] 7−→ [ λ I , λ I , λ I ] 7−→ [ λ I , λ I , λ I ] 7−→ · · · , · · · 7−→ [ λ F , λ F , λ F ] 7−→ [ λ F + , λ F , λ F − ] 7−→ [ λ F , λ F + , λ F − ] 7−→ · · · . Proof.
It is easy to see from (4.17) that if λ ∈ Γ u , v , then so do its images under the š -action. The three highest-weight modules corresponding to the š -orbit are thus BP ( u , v ) -modules with infinite-dimensional top spaces if anyis. Moreover, substituting λ λ = k − λ − λ and λ λ + k + into (4.9) and (4.16) shows that ∆ and ω areinvariant under this š -action. The three highest-weight modules therefore arise as submodules of the same irreduciblesemisimple coherent family. These modules are mutually inequivalent because their highest weights can only coincideif λ I = λ I = λ I = u − and λ F = λ F = λ F + = v . But, this requires both u and v to be divisible by . (cid:4) From Corollary 4.12, we now have a precise count of the number of irreducible semisimple coherent families of
Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules. Each direct summand of such a family is the top space of a simple twisted relaxed highest-weight BP ( u , v ) -module, by Theorem 3.13. With Equation (2.9), Lemma 3.20, and Proposition 4.16, we have thefollowing theorem. Theorem 4.20.
Let k be as in Assumption 2. Then: • There are (cid:12)(cid:12) Γ u , v (cid:12)(cid:12) = ( u − )( u − )( v − )( v − ) irreducible semisimple coherent families of Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules C ss ∆ , ω , up to isomorphism. • The families of twisted relaxed highest-weight BP ( u , v ) -modules R tw [ j ] , λ = R tw [ j ] , ∆ , ω are in -to- correspondence with Γ u , v / š , where š acts freely as in (4.17) . • For each λ ∈ Γ u , v , the twisted relaxed highest-weight module R tw [ j ] , λ is a simple BP ( u , v ) -module for all cosets [ j ] ∈ ƒ / š except three, namely the three distinct cosets that contain a root i of the polynomial ω + i , ∆ − ω . • The conjugate of the simple twisted relaxed highest-weight BP ( u , v ) -module R tw [ j ] , ∆ , ω is γ ( R tw [ j ] , ∆ , ω ) ≃ R tw [− j ] , ∆ , − ω . Note that if ( ∆ , ω ) corresponds to a coherent family of Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules, then the conjugation functorrequires that so must ( ∆ , − ω ) . In fact, it is easy to check that ∆ is invariant and ω is antiinvariant under the š -action [ λ , λ , λ ] ↔ [ λ − uv , λ , λ + uv ] , that is(4.18) [ λ I , λ I , λ I ] ←→ [ λ I , λ I , λ I ] , [ λ F , λ F , λ F ] ←→ [ λ F + , λ I , λ I − ] , which obviously preserves belonging to Γ u , v . With (4.17), this defines an action of W = S on Γ u , v . The orbits clearlyhave length unless ω = , in which case Lemma 4.19 forces them to have length . It is easy to check that this isconsistent with the explicit factorisation of ω given in (4.16).We remark that the spectral flow images σ ℓ ( R tw [ j ] , λ ) , ℓ , , of these simple twisted relaxed highest-weight BP ( u , v ) -modules are likewise simple BP ( u , v ) -modules, but they are not relaxed highest-weight because their conformal weightsare not bounded below.4.4. Nonsimple relaxed highest-weight BP ( u , v ) -modules. In Section 3.6, we introduced three classes of irreduciblecoherent families of Z k -modules. The first, the semisimple class, was the key ingredient in the classification argumentsof the previous section. Here, we will analyse the other two classes in order to demonstrate the existence of certain nonsemisimple twisted relaxed highest-weight BP ( u , v ) -modules, assuming that k is as in Assumption 2. We will alsodescribe the structure of these nonsemisimple modules in terms of short exact sequences.Consider therefore the irreducible nonsemisimple coherent family C ± ∆ , ω of Z k -modules on which G ± acts injectively.Recall that its simple direct summands are the R [ j ] , ∆ , ω , for all but (up to) three [ j ] ∈ ƒ / š , and that its nonsimple directsummands are denoted by R ±[ j ] , ∆ , ω . We begin by determining the structure of these nonsimple Z k -modules in the caserelevant to studying BP ( u , v ) -modules. Proposition 4.21.
Let λ ∈ Γ u , v and let j , ∆ and ω be defined by (4.9) and (4.16) . Then, the nonsimple Z k -module R ±[ j ] , ∆ , ω has exactly two composition factors, H j , ∆ and γ ( H − j − , ∆ ) , both of which are Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules. Moreover, wehave the following nonsplit short exact sequences: (4.19) −→ γ ( H − j − , ∆ ) −→ R + [ j ] , ∆ , ω −→ H j , ∆ −→ , −→ H j , ∆ −→ R −[ j ] , ∆ , ω −→ γ ( H − j − , ∆ ) −→ . Proof.
We only consider R + [ j ] , ∆ , ω as the argument for R −[ j ] , ∆ , ω is identical. First, note that H j , ∆ is an infinite-dimensional Zhu tw (cid:2) BP ( u , v ) (cid:3) -module, by Theorem 4.9. The irreducible semisimple coherent family C ss ∆ , ω is thereforea Zhu tw (cid:2) BP ( u , v ) (cid:3) -module too, by Proposition 4.16, hence so is the lowest-weight module γ ( H − j − , ∆ ) ⊂ R ss [ j ] , ∆ , ω . As R ss [ j ] , ∆ , ω is the semisimplification of R + [ j ] , ∆ , ω , they have the same composition factors. To demonstrate that there are nomore factors beyond the two already found, it suffices to show that H − j − , ∆ is infinite-dimensional.Since the conjugate of H − j − , ∆ is a Zhu tw (cid:2) BP ( u , v ) (cid:3) -module, H − j − , ∆ must correspond to some µ ∈ Σ u , v , byTheorem 4.9. Proceeding as in the proof of Lemma 4.6, we find that the unique solution is µ = [ λ , λ − uv , λ + uv ] ,hence µ I = [ λ I , λ I , λ I ] and µ F = [ λ F , λ F + , λ F − ] . Because µ F = λ F + , , it follows that µ ∈ Γ u , v and so H − j − , ∆ is infinite-dimensional, as desired. This establishes the first exact sequence in (4.19). It is clearly nonsplit because G + acts injectively on R + [ j ] , ∆ , ω . (cid:4) At this point, it is not clear if the R ±[ j ] , ∆ , ω corresponding to λ ∈ Γ u , v are Zhu tw (cid:2) BP ( u , v ) (cid:3) -modules, even though theircomposition factors are. We settle this using a simplified version of the argument of [22, Thm. 5.3]. Proposition 4.22.
Let λ ∈ Γ u , v and let j , ∆ and ω be defined by (4.9) and (4.16) . Then, the nonsimple Z k -module R ±[ j ] , ∆ , ω is a Zhu tw (cid:2) BP ( u , v ) (cid:3) -module. Proof.
Again, we shall only detail the argument for R + [ j ] , ∆ , ω . Recall that J k denotes the maximal ideal of BP k and so Zhu tw (cid:2) J k (cid:3) · H j , ∆ = , by virtue of H j , ∆ being a Zhu tw (cid:2) BP ( u , v ) (cid:3) -module. From the first exact sequence in (4.19), weconclude that Zhu tw (cid:2) J k (cid:3) · R + [ j ] , ∆ , ω ⊆ γ ( H − j − , ∆ ) .As Z k is noetherian (this is an easy generalisation of [35, Cor. 1.3]), so is its quotient Zhu tw (cid:2) BP k (cid:3) (Proposition 3.15).The ideal Zhu tw (cid:2) J k (cid:3) ⊂ Zhu tw (cid:2) BP k (cid:3) is therefore generated by a finite number of elements a , . . . , a n which we may,without loss of generality, choose to be eigenvectors of J . Let j i denote the J -eigenvalue of a i , i = , . . . , n .Choose j ′ ∈ [ j ] such that j ′ j − max { j , . . . , j n } . Then, a i takes the J -eigenspace of R + [ j ] , ∆ , ω of eigenvalue j ′ into the J -eigenspace of γ ( H − j − , ∆ ) of eigenvalue j ′ + a i j . But, the eigenvalues of J acting on γ ( H − j − , ∆ ) are bounded belowby j + , hence a i annihilates the J -eigenspace of R + [ j ] , ∆ , ω of eigenvalue j ′ , for each i . It follows that Zhu tw (cid:2) J k (cid:3) annihilatesthis eigenspace. But, this eigenspace generates R + [ j ] , ∆ , ω , hence Zhu tw (cid:2) J k (cid:3) (being an ideal) annihilates R + [ j ] , ∆ , ω . (cid:4) By Zhu-induction (Theorem 3.13), one may construct from each
Zhu tw (cid:2) BP ( u , v ) (cid:3) -module R ±[ j ] , ∆ , ω a twisted BP ( u , v ) -module whose twisted Zhu image (its top space) is R ±[ j ] , ∆ , ω . Consider the submodule of this induced module obtainedby summing all the submodules whose intersection with the top space R ±[ j ] , ∆ , ω is zero. Quotienting by this submoduleresults in a twisted BP ( u , v ) -module, which we shall denote by R tw , ±[ j ] , λ = R tw , ±[ j ] , ∆ , ω , that has R ±[ j ] , ∆ , ω as its top space andhas the property that its nonzero submodules intersect this top space nontrivially. In a sense, R tw , ±[ j ] , ∆ , ω is the smallest BP ( u , v ) -module whose top space is R ±[ j ] , ∆ , ω .The R tw , ±[ j ] , λ are clearly nonsemisimple, because their top spaces are. This proves the following result. Theorem 4.23.
When k is as in Assumption 2, the simple vertex operator algebra BP ( u , v ) admits nonsemisimplemodules. In physical language, the corresponding minimal model conformal field theory is logarithmic . ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 23
As we have mentioned before, the Bershadsky–Polyakov minimal models corresponding to BP ( u , ) , with u ∈ š > + ,were shown to be rational in [1].Our final task is then to determine the structure of these nonsemisimple BP ( u , v ) -modules. For this, it is convenientto introduce new modules W tw , ±[ j ] , λ = W tw , ±[ j ] , ∆ , ω that are obtained by treating R ±[ j ] , ∆ , ω as a module over the twisted modealgebra U tw of (3.1), letting U tw > act as , and then inducing to a U tw -module. It follows that W tw , ±[ j ] , λ is a “relaxed Verma” BP k -module whose top space is R ±[ j ] , ∆ , ω . In a sense, it is the largest BP k -module with this top space.As such, we may consider the sum N tw , ±[ j ] , λ of all the submodules of W tw , ±[ j ] , λ whose intersection with the top space R ±[ j ] , ∆ , ω is zero. Because this top space is nonsemisimple, N tw , ±[ j ] , λ is a proper submodule of the maximal submodule M tw , ±[ j ] , λ of W tw , ±[ j ] , λ . Its utility lies in the fact that it provides an alternative construction of the BP ( u , v ) -module R tw , ±[ j ] , λ :(4.20) R tw , ±[ j ] , λ ≃ W tw , ±[ j ] , λ . N tw , ±[ j ] , λ . This exploits the fact that R tw , ±[ j ] , λ is, in a sense, the smallest BP k -module with top space R ±[ j ] , ∆ , ω .We now proceed in an analogous fashion to [17, Sec. 4]. Theorem 4.24.
Let k be as in Assumption 2 and let λ ∈ Γ u , v define j , ∆ and ω via (4.9) and (4.16) . We then have thefollowing nonsplit short exact sequences of BP ( u , v ) -modules: (4.21) −→ γ ( H tw − j − , ∆ ) −→ R tw , + [ j ] , ∆ , ω −→ H tw j , ∆ −→ , −→ H tw j , ∆ −→ R tw , −[ j ] , ∆ , ω −→ γ ( H tw − j − , ∆ ) −→ . Proof.
Once again, we only give the argument for R tw , + [ j ] , ∆ , ω . First, note that the twisted Verma module V tw j , ∆ is clearlyisomorphic to the quotient W tw , + [ j ] , ∆ , ω (cid:14) γ ( V tw − j − , ∆ ) , by (4.19) and the exactness of induction. Hence, H tw j , ∆ is also a quotientand (4.20) gives(4.22) R tw , + [ j ] , ∆ , ω M tw , + [ j ] , ∆ , ω (cid:14) N tw , + [ j ] , ∆ , ω ≃ W tw , + [ j ] , ∆ , ω M tw , + [ j ] , ∆ , ω ≃ H tw j , ∆ , since relaxed highest-weight modules have unique irreducible quotients. Thus, H tw j , ∆ is a quotient of R tw , + [ j ] , ∆ , ω .Next, note that the (unique) maximal submodule of γ ( V tw − j − , ∆ ) is γ ( V tw − j − , ∆ ) ∩ N tw , + [ j ] , ∆ , ω , because the only submoduleof γ ( V tw − j − , ∆ ) intersecting its top space nontrivially is γ ( V tw − j − , ∆ ) itself. We therefore have(4.23) γ ( H tw − j − , ∆ ) = γ ( V tw − j − , ∆ ) γ ( V tw − j − , ∆ ) ∩ N tw , + [ j ] , ∆ , ω ≃ γ ( V tw − j − , ∆ ) + N tw , + [ j ] , ∆ , ω N tw , + [ j ] , ∆ , ω , which is clearly a submodule of W tw , + [ j ] , ∆ , ω (cid:14) N tw , + [ j ] , ∆ , ω ≃ R tw , + [ j ] , ∆ , ω . Thus, γ ( H tw − j − , ∆ ) embeds into R tw , + [ j ] , ∆ , ω .To demonstrate exactness of the first sequence of (4.21), we note that(4.24) R tw , + [ j ] , ∆ , ω γ ( H tw − j − , ∆ ) ≃ W tw , + [ j ] , ∆ , ω γ ( V tw − j − , ∆ ) + N tw , + [ j ] , ∆ , ω ≃ V tw j , ∆ (cid:0) γ ( V tw − j − , ∆ ) + N tw , + [ j ] , ∆ , ω (cid:1) (cid:14) γ ( H tw − j − , ∆ ) using (4.20) and (4.23). This shows that R tw , + [ j ] , ∆ , ω (cid:14) γ ( H tw − j − , ∆ ) is a twisted highest-weight BP ( u , v ) -module. ByTheorem 4.10, it is simple and therefore isomorphic to H tw j , ∆ , by (4.22). This completes the proof. (cid:4)
5. Examples
We conclude by illustrating the above classification results with some specific examples of Bershadsky–Polyakovminimal models. The examples with v = extend the results of [1] whilst the ( u , v ) = ( , ) and ( , ) examples extendthose of [2]. Example: BP ( , ) . For k = − , the central charge of the minimal model is c = . Since λ I ∈ P > = {[ , , ]} and λ F ∈ P > is constrained by λ F > so that λ F = [ , , ] , we only have λ = [ , , ] − [ , , ] = [ k , , ] . There istherefore a unique simple untwisted highest-weight module H − ω / = H , and a unique simple twisted highest-weightmodule H tw − ω / = H tw , (up to isomorphism). This is clearly the trivial minimal model. λ I λ I λ I (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1)(cid:0) − , (cid:1)(cid:0) − , (cid:1) (cid:0) , (cid:1) γ (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , − (cid:1) (cid:0) − , (cid:1) (cid:0) , (cid:1) γ Figure 2.
The charges and conformal weights ( j , ∆ ) of the untwisted (left) and twisted (right)simple highest-weight BP ( , ) -modules, arranged by the Dynkin labels of the integral parts λ I of thecorresponding surviving weights λ . The subscript on the twisted labels gives the dimension of thetop space. Conjugation γ is indicated by reflection about the dashed line and spectral flow σ by ◦ anticlockwise rotation about each triangle’s centre. j ∆ J G + G + G − G − T − L ∂ J : J J : T + EF ∂ E : JE : ∂ F : J F : ...... Figure 3.
The states with conformal weight ∆ of the “ N = -like” vertex operator algebra A = H − , ⊕ H , ⊕ H , that extends BP ( , ) . Here, T + = G −− / G + and T − = G + − / G − . Example: BP ( , ) . For k = − , the central charge is instead c = and we have λ I ∈ P > and λ F = [ , , ] . Thereare thus (cid:12)(cid:12) P > (cid:12)(cid:12) = simple untwisted highest-weight modules and so simple twisted highest-weight modules, all withfinite-dimensional top spaces. We illustrate these modules in Figure 2, arranging them according to λ I and listingthe charges and conformal weights of their highest-weight vectors. We also indicate the effect of the conjugation andspectral flow automorphisms in this arrangement. Example: BP ( , ) . We discuss one further minimal model with v = , that with k = and c = − . This time, thereare (cid:12)(cid:12) P > (cid:12)(cid:12) = simple untwisted highest-weight modules and, of course, each has a single twisted cousin. As alwayswhen v = , the top spaces are all finite-dimensional and the fractional part λ F of the corresponding b sl -weights is [ , , ] .An interesting feature of this minimal model is that the (integer) spectral flows of the vacuum module H , correspondto λ I = [ , , ] and [ , , ] , hence ( j , ∆ ) = ( , ) and (− , ) . Recalling that spectral flows of the vacuum module arealways simple currents [54], it follows that BP ( , ) admits an order- simple current extension A . Moreover, if E and F denote the highest-weight vectors of the simple current modules H , and H − , , respectively, then it is easy to checkthat E , F and J define a (nonconformal) embedding of the sl minimal model A ( , ) = L ( sl ) into A .Defining G + = G −− / E and G − = G + − / F , we see that A has four linearly independent fields of conformal weight and that they decompose into two sl -doublets ( G − , G + ) and ( G − , G + ) . A may thus be regarded as some sort ofbosonic analogue of the N = superconformal vertex operator superalgebra, see Figure 3. However, a major differenceis that the elements E , J , F , G ± G ± and L do not strongly generate A . For example, the singular part of the operatorproduct expansion of G + ( z ) and G − ( w ) is a simple pole whose coefficient is the ( j , ∆ ) = ( , ) field corresponding to T − = ( G + − / ) F . ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 25 λ F λ F λ F (cid:0) , (cid:1) (cid:0) − , − (cid:1)(cid:0) − , − (cid:1)(cid:0) , − (cid:1)(cid:0) , − (cid:1) (cid:0) , − (cid:1) γ (cid:0) − , − (cid:1) (cid:0) − , − (cid:1) ∞ (cid:0) − , − (cid:1) ∞ (cid:0) , − (cid:1) (cid:0) , − (cid:1) (cid:0) − , − (cid:1) ∞ γ Figure 4.
The charges and conformal weights ( j , ∆ ) of the untwisted (left) and twisted (right) simplehighest-weight BP ( , ) -modules, arranged by the Dynkin labels of the fractional parts λ F of thecorresponding surviving weights λ . The subscript on the twisted labels gives the dimension of the topspace. Conjugation γ is indicated by reflection about the dashed line, restricted to the modules withfinite-dimensional top spaces (the conjugate of a highest-weight module with an infinite-dimensionaltop space is not highest-weight).It is nevertheless easy to explore the representation theory of A . The set of (isomorphism classes of) simpleuntwisted highest-weight BP ( , ) -modules decomposes into spectral flow orbits: of length and one fixed point.It is easy to check from the charges and conformal weights that only four of these orbits define untwisted A -modules.There are therefore precisely simple untwisted A -modules:(5.1) A = H − , ⊕ H , ⊕ H , , H − , / ⊕ H , − / ⊕ H , / , H − , − / ⊕ H , / ⊕ H , − / and H , − / . One can also classify the simple twisted A -modules, but now there are several more twisted sectors to consider. Example: BP ( , ) . Consider next the Bershadsky–Polyakov minimal model with k = − and c = − . This modelarises as the p = member of a series B p of interesting vertex operator algebras constructed in [55]. As λ I ∈ P > and λ F ∈ P > satisfies λ F > , there are (cid:12)(cid:12) P > (cid:12)(cid:12) = simple untwisted highest-weight modules and simple twistedhighest-weight modules, of which have finite-dimensional top spaces. We illustrate these in Figure 4 as we did for BP ( , ) , but arranging the data according to λ F instead of λ I . One can check that this recovers the highest-weightclassification of [2].In this illustration, the spectral flow functor σ is again represented by a ◦ anticlockwise rotation, but does notpreserve being highest-weight (because v , ). Indeed, the three spectral flow orbits through the simple highest-weight BP ( , ) -modules are(5.2) · · · σ / H , − / σ / H tw − / , − / σ / , · · · σ / H / , − / σ / H tw , − / σ / H − / , − / σ / H tw − / , − / σ / , · · · σ / H / , − / σ / H tw / , − / σ / H , σ / H tw − / , − / σ / H − / , − / σ / H tw − / , − / σ / , where the · · · indicate simple BP ( , ) -modules that are not highest-weight.The three simple twisted highest-weight modules with λ F > have infinite-dimensional top spaces. They also sharethe same conformal weight ∆ = − and ω -parameter ω = ω + j , ∆ = , the latter computed as in (4.16). It thereforefollows that BP ( , ) admits one family of simple twisted relaxed highest-weight modules R tw [ j ] , − / , , j , − , − , − ( mod 1 ) , as per Theorem 4.20. As a consistency check, substituting k = − and ∆ = − into (3.18) indeed gives(5.3) ω + j , − / − ω = ( j + )( j + j + ) − = ( j + )( j + )( j + ) , as expected.This family was first constructed in [2, Thm. 7.2], though four exceptional values of j ( mod 1 ) were given thereinstead of three. Here, we have also proven that there are no other families. We also note that Theorem 4.24 provesthe existence of six nonsemisimple twisted relaxed highest-weight BP ( , ) -modules, each characterised by a nonsplit (cid:0) , (cid:1) (cid:0) , (cid:1)(cid:0) − , (cid:1) (cid:0) − , (cid:1)(cid:0) − , − (cid:1)(cid:0) − , (cid:1)(cid:0) , (cid:1) (cid:0) , (cid:1)(cid:0) , − (cid:1) (cid:0) − , − (cid:1) (cid:0) , (cid:1) (cid:0) − , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , − (cid:1) (cid:0) − , − (cid:1) ∞ (cid:0) − , − (cid:1) ∞ (cid:0) − , − (cid:1) ∞ Figure 5.
The charges and conformal weights ( j , ∆ ) of the untwisted (left) and twisted (right) simplehighest-weight BP ( , ) -modules, arranged by the Dynkin labels of the integral (small-scale) andfractional (large-scale) parts λ F of the corresponding surviving weights λ . The subscript on thetwisted labels gives the dimension of the top space.short exact sequence: −→ γ ( H tw − / , − / ) −→ R tw , + [− / ] , − / , −→ H tw − / , − / −→ , −→ γ ( H tw − / , − / ) −→ R tw , + [− / ] , − / , −→ H tw − / , − / −→ , −→ γ ( H tw − / , − / ) −→ R tw , + [− / ] , − / , −→ H tw − / , − / −→ , (5.4a) −→ H tw − / , − / −→ R tw , −[− / ] , − / , −→ γ ( H tw − / , − / ) −→ , −→ H tw − / , − / −→ R tw , −[− / ] , − / , −→ γ ( H tw − / , − / ) −→ , −→ H tw − / , − / −→ R tw , −[− / ] , − / , −→ γ ( H tw − / , − / ) −→ . (5.4b)There are other nonsemisimple BP ( , ) -modules. In particular, there exist staggered (logarithmic) modules onwhich J acts semisimply but L has Jordan blocks of rank . This follows from the well-known fact [56, 57] thatstaggered modules exist for the triplet vertex operator algebra W ( , ) of central charge − . The connection is that thecoset of BP ( , ) = B by the Heisenberg subalgebra generated by J is the singlet algebra I ( , ) [55] and that the latterhas W ( , ) as an (infinite-order) simple current extension [58]. We shall not study these staggered BP ( , ) -moduleshere, but intend to investigate them more generally in a sequel. Example: BP ( , ) . The minimal model with k = − and c = − was also studied in [2]. This time, we have λ I ∈ P > and λ F ∈ P > , hence there are (cid:12)(cid:12) P > (cid:12)(cid:12)(cid:12)(cid:12) P > (cid:12)(cid:12) = simple untwisted highest-weight modules. Moreover, of the simple twistedhighest-weight modules have finite-dimensional top spaces whilst the top spaces of the other are infinite-dimensional.We arrange the highest-weight data in an sl -covariant fashion in Figure 5, making the scale for λ I significantly smallerthan that for λ F to improve clarity. It follows that there is again only one family of generically simple relaxed highest-weight BP ( , ) -modules. This family must therefore be closed under conjugation and so ω = . This can of course bechecked explicitly using (4.16).Along with the simple twisted relaxed highest-weight BP ( , ) -modules R tw [ j ] , − / , , j , − , − , − ( mod 1 ) , wealso deduce the existence of six nonsemisimple twisted relaxed highest-weight BP ( , ) -modules, characterised by thefollowing nonsplit short exact sequences: −→ γ ( H tw − / , − / ) −→ R tw , + [− / ] , − / , −→ H tw − / , − / −→ , −→ γ ( H tw − / , − / ) −→ R tw , + [− / ] , − / , −→ H tw − / , − / −→ , −→ γ ( H tw − / , − / ) −→ R tw , + [− / ] , − / , −→ H tw − / , − / −→ , (5.5a) ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 27 (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1)(cid:0) − , (cid:1)(cid:0) − , (cid:1) (cid:0) , (cid:1) (cid:0) − , (cid:1) (cid:0) − , (cid:1) (cid:0) , (cid:1)(cid:0) − , (cid:1)(cid:0) − , (cid:1)(cid:0) − , (cid:1)(cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1)(cid:0) , (cid:1)(cid:0) − , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) − , (cid:1) (cid:0) − , (cid:1) (cid:0) , (cid:1) (cid:0) − , (cid:1) ∞ (cid:0) − , − (cid:1) ∞ (cid:0) , (cid:1) ∞ (cid:0) − , − (cid:1) ∞ (cid:0) − , (cid:1) ∞ (cid:0) − , − (cid:1) ∞ (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) , (cid:1) (cid:0) − , (cid:1) (cid:0) , (cid:1) Figure 6.
The charges and conformal weights ( j , ∆ ) of the untwisted (left) and twisted (right) simplehighest-weight BP ( , ) -modules, arranged by the Dynkin labels of the integral (small-scale) andfractional (large-scale) parts λ F of the corresponding surviving weights λ . The subscript on thetwisted labels gives the dimension of the top space. −→ H tw − / , − / −→ R tw , −[− / ] , − / , −→ γ ( H tw − / , − / ) −→ , −→ H tw − / , − / −→ R tw , −[− / ] , − / , −→ γ ( H tw − / , − / ) −→ , −→ H tw − / , − / −→ R tw , −[− / ] , − / , −→ γ ( H tw − / , − / ) −→ . (5.5b)As with the case ( u , v ) = ( , ) discussed above, there are other nonsemisimple BP ( , ) -modules, in particular thereare staggered (logarithmic) modules (as was already noted in [2]). We review the argument briefly for completeness.First, note [2, Sec. 5.2] that the Bershadsky–Polyakov minimal model vertex operator algebra BP ( , ) embeds in thesymplectic bosons vertex operator algebra B (also known as the bosonic ghost system, βγ ghosts and the Weyl vertexalgebra) with c = − . We recall that B is strongly generated by β and γ , both of conformal weight , subject to theoperator product expansions(5.6) β ( z ) β ( w ) ∼ ∼ γ ( z ) γ ( w ) and β ( z ) γ ( w ) ∼ − z − w . An embedding BP ( , ) ֒ → B is then given by(5.7) J
13 : βγ : , G + √ βββ : , G − √ βββ : , L ( : ∂ βγ : − : ∂ γ β : ) . This suggests, and it is easy to check [2, Prop. 5.9], that BP ( , ) is (isomorphic to) the š -orbifold of B correspondingto the automorphism e π i J . As B is known [27, 59, 60] to admit a family of staggered modules, each member related tothe others by spectral flow, so does BP ( , ) . In fact, BP ( , ) admits three such families. Example: BP ( , ) . We conclude with the Bershadsky–Polyakov minimal model with k = − and c = . With λ I ∈ P > and λ F ∈ P > , there are (cid:12)(cid:12) P > (cid:12)(cid:12) | P > | = simple untwisted highest-weight modules and the twisted highest-weight modulesdivide into with finite-dimensional top spaces and with infinite-dimensional top spaces. We illustrate the highest-weight data in Figure 6. There are thus two families of generically simple twisted relaxed highest-weight modules, onewith ∆ = and one with ∆ = − . As these conformal weights differ, each family must be closed under conjugationand so we have ω = for both (again). We therefore have simple twisted relaxed highest-weight BP ( , ) -modules R tw [ j ] , / , , j , − , − , ( mod 1 ) , and R tw [ j ] , − / , , j , − , − , − ( mod 1 ) , along with the nonsemisimple versionsguaranteed by Theorem 4.24.An interesting feature of this minimal model is the existence of modules H ± / , corresponding to λ I = [ , , ] , [ , , ] and λ F = [ , , ] . These are not spectral flows of the vacuum module, but we nevertheless conjecture that they aresimple currents generating an order- simple current extension C of BP ( , ) . As with BP ( , ) (and assuming this j ∆ J G + G − G + G − L ∂ J : J J : EF : EE :: F F : ∂ E : JE : ∂ F : J F : ...... Figure 7.
The states with conformal weight ∆ of the extended algebra C = H − / , ⊕ H , ⊕ H / , of BP ( , ) . Here, G + = G + − / F and G − = G −− / E , whilst : EE : and : F F : are proportional to ( G + − / ) F and ( G −− / ) E , respectively.conjecture), the highest-weight vectors E and F of H / , and H − / , , respectively, generate a copy of an sl minimalmodel, this time A ( , ) = L / ( sl ) . But unlike the situation for BP ( , ) , the embedding A ( , ) ֒ → C is conformal.We recall from [14, Sec. 10], see also [61], that A ( , ) has a simple current whose top space is the four-dimensionalsimple sl -module with conformal weight . We therefore conjecture that this order- simple current extension of A ( , ) is isomorphic to C , illustrating the low-conformal weight states of C in Figure 7 for convenience (and notingthat the A ( , ) Cartan element H is identified with J ). This extended vertex operator algebra was conjectured to be theminimal quantum hamiltonian reduction of L − / ( g ) in [14, Sec. 10]. This was settled affirmatively in [38, Thm. 6.8].The conjectured embeddings BP ( , ) ֒ → C ← ֓ A ( , ) may be tested through representation theory. Indeed, A ( , ) has two simple highest-weight modules with finite-dimensional top spaces, in addition to the vacuum andsimple current module. Their direct sum may be identified with the simple C -module H − / , / ⊕ H , / ⊕ H / , / .Likewise, there are four simple highest-weight A ( , ) -modules with infinite-dimensional top spaces and they combineto give two simple C -modules H − / , / ⊕ H − / , / ⊕ H / , / and H − / , − / ⊕ H − / , − / ⊕ H − / , − / . The storyis predictably similar for the relaxed highest-weight modules.We finish by noting that BP ( , ) also admits staggered (logarithmic) modules because A ( , ) does [16, 56], seealso [21]. In fact, we expect that staggered BP ( u , v ) -modules exist for all v > and hope to return to this in the future. Appendix A. Proof of Theorem 4.8
In this appendix, we adopt the notation of Section 4.1 and assume throughout that λ < š > so that H ( L λ ) , (andthat the level k is as in Assumption 1). With these assumptions, the aim is to prove the following assertion:(A.1) I k · L λ , ⇒ H ( I k ) · H ( L λ ) , . H (−) is a cohomological functor which involves tensoring with a ghost vertex operator superalgebra whose vacuumelement will be denoted by | i . With this, we shall prove (A.1) by exhibiting elements χ ∈ I k and v ∈ L λ for which χ ⊗ | i and v ⊗ | i are (degree- ) closed elements of the appropriate BRST complexes and the (clearly closed) element χ n v ⊗ | i is not exact, for some n ∈ š . Using brackets to denote cohomology classes, [ χ n v ⊗ | i] then gives a nonzeroelement of H ( I k ) · H ( L λ ) :(A.2) [ χ ⊗ | i] · [ v ⊗ | i] ≡ [ χ ⊗ | i]( z )[ v ⊗ | i] = [ χ ( z ) v ⊗ | i] , . As noted at the end of Section 4.1, this amounts to a proof of Theorem 4.8. To prove (A.1) however, we need to delve alittle deeper into the details of minimal quantum hamiltonian reduction for V k ( sl ) .A.1. Minimal quantum hamiltonian reduction.
Recall from [5] that the minimal quantum hamiltonian reductionfunctor H (−) computes the cohomology of the tensor product of a given V k ( sl ) -module with certain ghost vertexoperator superalgebras. Specifically, we need a fermionic ghost system F α for each positive root α ∈ ∆ + of sl and onebosonic ghost system B corresponding to the two simple roots α and α . Denoting the fermionic ghosts by b α and c α , ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 29 e θ e α e α h α h α f α f α f θ b α c α b α c α b θ c θ β γ − − − e j − − − − − e ∆
12 12
32 32
12 12 12 12
12 12
Table 1.
The ghost numbers , charges e j and conformal weights e ∆ of the generating fields of thevertex operator superalgebra V k ( sl ) ⊗ G . α ∈ ∆ + , and the bosonic ghosts by β and γ , we take the defining operator product expansions to be(A.3) b α ( z ) c α ( w ) ∼ z − w and β ( z ) γ ( w ) ∼ z − w , understanding that the remaining operator product expansions between ghost generating fields are regular. The tensorproduct of these ghost vertex operator superalgebras will be denoted by G = F α ⊗ F α ⊗ F θ ⊗ B , for convenience.We fix a basis of sl for the computations to follow. Let E ij denote the × matrix with in the ( i , j ) -th positionand zeroes elsewhere. Then, we set(A.4) e θ = E , e α = E , h α = E − E , f α = E , e α = E , h α = E − E , f α = E , f θ = E . Here, θ = α + α is the highest root of sl and we shall also set h θ = h α + h α = E − E .To define H ( M ) for a V k ( sl ) -module M , one first grades M ⊗ G by the fermionic ghost number, that is by the totalnumber of c -modes minus the total number of b -modes. Equivalently, the ghost number is the eigenvalue of the zeromode of the field Í α ∈ ∆ + : b α ( z ) c α ( z ) : . Next, one introduces [4, 5] the following fermionic field of ghost number :(A.5) d ( z ) = (cid:0) e θ ( z ) + (cid:1) c θ ( z ) + (cid:0) e α ( z ) + β ( z ) (cid:1) c α ( z ) + (cid:0) e α ( z ) + γ ( z ) (cid:1) c α ( z ) + : b θ ( z ) c α ( z ) c α ( z ) : . A straightforward computation verifies that d ( z ) d ( w ) ∼ . We then form a differential complex by requiring that d ( z ) ishomogeneous of conformal weight and equipping M ⊗ G with the differential d = d (which obviously squares to ).With (A.5), this requirement on d ( z ) requires that the conformal weight of c θ is also , whilst that of e θ is . Thelatter may be achieved by adding ∂ h θ to the standard Sugawara energy-momentum tensor T Sug. of V k ( sl ) . Whenthis is done, homogeneity and (A.5) now fix the conformal weight e ∆ of all the generating fields as in Table 1. Theenergy-momentum tensor of V k ( sl ) ⊗ G is thus(A.6) e L = T Sug. + ∂ h θ + Õ α ∈ ∆ + T F α + T B , where T F αi =
12 : ∂ b α i c α i + ∂ c α i b α i : , T F θ = : ∂ b θ c θ : and T B =
12 : ∂ γ β − ∂ βγ : . The central charge matches that of BP k , see (2.2):(A.7) kk + − k + + − − = − ( k + )( k + ) k + . As the notation suggests, e L is closed and its image in cohomology (that is, in H ( V k ( sl ) ⊗ G , d ) = H ( V k ( sl )) ≃ BP k )is L . Note that the “symmetric” deformation of adding ∂ h θ to T Sug. ensures this result. There are other deformationsconsistent with d being a differential — they correspond to adding a multiple of ∂ J to L . Speaking of which, the element(A.8) e J = ( h α − h α ) + : b α c α : − : b α c α : − : βγ : is likewise closed and its image in cohomology is J [5]. We give the charge ( e J -eigenvalue) of the generating fields of V k ( sl ) ⊗ G in Table 1 for completeness. We also note that(A.9) e G + = f α + : h α β : − : b α c θ : − : b α c α β : + b α c α β : + : b θ c θ β : + : ββγ : + ( k + ) ∂ β and e G − = f α − : h α γ : + : b α c θ : − b α c α γ : + : b α c α γ : − : b θ c θ γ : + : γγ β : − ( k + ) ∂ γ are both closed. Their images in cohomology are G + and G − , respectively [6]. We remark that deforming the energy-momentum tensor of V k ( sl ) means that we now have two distinct modeconventions for affine fields. Our convention will be that mode indices with respect to the deformed conformal weightwill be denoted with parentheses. Thus, for an affine generator a with deformed conformal weight e ∆ as in Table 1, weshall write(A.10) a ( z ) = Õ n ∈ š a n z − n − = Õ n ∈ š − e ∆ a ( n ) z − n − e ∆ . We shall not bother to so distinguish mode indices for ghost fields: their expansions will always be taken with respectto the conformal weights in Table 1.A.2.
The proof.
We start with a well-known fundamental result for the highest-weight vector v of L λ , recalling thatwe are assuming throughout that λ < š > and that k satisfies Assumption 1. Let | i denote the vacuum vector of G .By [7, Lem. 4.6.1 and Prop. 4.7.1], we have the following lemma. Lemma A.1.
For all n ∈ š > , ( e θ − ) n v ⊗ | i is closed and inexact. In particular, [ v ⊗ | i] , . We next consider the maximal ideal I k of V k ( sl ) . Lemma A.2. I k is generated by a single singular vector χ whose sl -weight and conformal weight are ( u − ) θ and ( u − ) v , respectively. Moreover, χ ⊗ | i is closed. Proof.
This follows easily from [36, Cor. 1], which says that the maximal submodule of a Verma module whose highestweight is admissible is generated by singular vectors of known weight. In our case, the highest weight is k ω (which isadmissible because k is) and the only generating singular vector that is nonzero in the quotient V k ( sl ) of this Vermamodule has weight w · ( k ω ) , where w is the Weyl reflection corresponding to the root − θ + v δ . Here, δ denotes thestandard imaginary root of b sl . This is χ and its sl - and conformal weights are now easily computed. The fact that χ ⊗ | i is closed again follows from χ being a highest-weight vector. (cid:4) In fact, χ ⊗ | i is also inexact, though we will not need to a priori establish this fact for what follows.We remark that a nice conceptual proof of [36, Cor. 1] starts from the celebrated fact that the submodule structureof a Verma module only depends on the corresponding integral Weyl group [62]. This structure is therefore the samefor all admissible highest-weight b sl -modules, irrespective of their level. In particular, this structure matches that ofa Verma whose simple quotient is integrable, integrability being equivalent to admissibility for simple highest-weightmodules with v = . However, the fact that the maximal submodule is generated by singular vectors is well-known inthe integrable case, see [63] or [64].Suppose now that χ ( z ) v = . Because χ generates I k , it follows that I k · v = . Since v generates L λ , as a V k ( sl ) -module, and I k is a two-sided ideal of V k ( sl ) , we get I k · L λ = . Thus, the hypothesis of (A.1), that L λ is notan L k ( sl ) -module, requires that χ n v , for some n ∈ š . As χ has sl -weight ( u − ) θ , our knowledge of the weightsof L λ lets us refine this requirement to χ −( u − )− i v , for some i ∈ š > . There is therefore a minimal N ∈ š > suchthat χ −( u − )− N v , .As L λ is simple, there therefore exists a Poincaré–Birkhoff–Witt monomial U ∈ U ( b sl ) such that U χ −( u − )− N v = v (rescaling χ if necessary). We choose an ordering for U so that(A.11) f αn < h αn < < e αn < < f αn > < h αn > < e αn > (obviously we may omit the h α and K ). This means, for example, that the f αn with n are ordered to the left whilethe e αn with n > are ordered to the right. For n > , we have e α χ = and e αn v = , hence(A.12) e αn χ −( u − )− N v = ( e α χ ) −( u − )− N + n v = . We may therefore assume that U contains no e αn -modes with n > . Similarly,(A.13) h αn χ −( u − )− N v = ( u − ) θ ( h α ) χ −( u − )−( N − n ) v = ELAXED HIGHEST-WEIGHT MODULES FOR THE BERSHADSKY–POLYAKOV ALGEBRAS 31 for n > , by the minimality of N . Thus, we may assume that U contains no h αn -modes with n > either. Finally, v is not in the image of any f αn , with n , h αn , with n < , or e αn , with n < . All these modes may therefore also beexcluded from U .Given a partition ξ = [ ξ > ξ > · · · ] , let ℓ ( ξ ) denote its length and | ξ | denote its weight. We write f αξ = f αξ f αξ · · · .Then, there exist partitions ξ , π and ρ such that U = f θξ f α π f α ρ and so(A.14) f θξ f α π f α ρ χ −( u − )− N v = v . Moreover, considering sl - and conformal weights gives(A.15) ℓ ( π ) = ℓ ( ρ ) , ℓ ( ξ ) + ℓ ( π ) = u − and | ξ | + | π | + | ρ | = u − + N . Lemma A.3.
Let T ( z ) , T ∈ sl , be an affine field and let U be a monomial in the negative root vectors f α of b sl . Then,the modes of the field ( U χ )( w ) satisfy (A.16) [ T m , ( U χ ) n ] = ( T U χ ) m + n , for all m , n ∈ š . Proof.
Observe that U χ is annihilated by the T m with m > . Consequently, the assertion follows easily from theoperator product expansion (cid:4) (A.17) T ( z )( U χ )( w ) ∼ ( T U χ )( w ) z − w . We apply Lemma A.3 to the left-hand side of (A.14), noting that the f -modes all annihilate v . The result is(A.18) f θξ f α π f α ρ χ −( u − )− N v = (cid:16) ( f θ ) ℓ ( ξ ) ( f α ) ℓ ( π ) ( f α ) ℓ ( ρ ) χ (cid:17) v , using (A.15). This looks complicated, but it allows us to determine the partitions ξ , π and ρ . Lemma A.4.
If any of the parts of ξ , π or ρ are greater than , then f θξ f α π f α ρ χ −( u − )− N v = . Proof.
Suppose that ξ has a part ξ i > (the argument is identical if π or ρ has a part greater than ). Then, we canform a new partition ξ ′ from ξ by subtracting from ξ i and reordering parts if necessary. Note that ℓ ( ξ ′ ) = ℓ ( ξ ) and | ξ ′ | = | ξ | − . Then, Lemma A.3 and N being minimal give = f θξ ′ f α π f α ρ χ −( u − )−( N − ) v = (cid:16) ( f θ ) ℓ ( ξ ′ ) ( f α ) ℓ ( π ) ( f α ) ℓ ( ρ ) χ (cid:17) −( u − ) + | ξ ′ | + | π | + | ρ |− N + v (A.19) = (cid:16) ( f θ ) ℓ ( ξ ) ( f α ) ℓ ( π ) ( f α ) ℓ ( ρ ) χ (cid:17) v . But, this is the right-hand side of (A.18). (cid:4)
Combining (A.14), which is manifestly nonzero, with Lemma A.4 now forces all parts of ξ , π and ρ to be . As partitionlengths and weights are now equal, the relations of (A.15) are easily solved to give | ξ | = u − − N and | π | = | ρ | = N .In particular, (A.14) now becomes(A.20) ( f θ ) u − − N ( f α ) N ( f α ) N χ −( u − )− N v = v . By rescaling χ again, if necessary, we arrive at following key result. Proposition A.5. If N is the minimal integer such that χ −( u − )− N v , , then (A.21) ( f α ) N ( f α ) N χ −( u − )− N v = ( e θ − ) u − − N v . The idea now is to use the fact that the right-hand side of (A.21) is inexact when tensored with | i (Lemma A.1) toprove that the same is true for χ −( u − )− N v . For this, we need to replace the action of f α and f α with elements thatpreserve exactness, for example any closed elements. Lemma A.6.
For all i , j ∈ š > , we have (A.22) (cid:0) e G + ( / ) (cid:1) i (cid:0) e G −( / ) (cid:1) j (cid:0) χ −( u − )− N v ⊗ | i (cid:1) = ( f α ) i ( f α ) j χ −( u − )− N v ⊗ | i . Proof.
We start with (A.9), which gives(A.23) e G −( / ) = f α ( / ) − Õ m ∈ š h α ( m ) γ − m + / + · · · = f α − Õ m ∈ š h α m γ − m + / + · · · , where the · · · stands for pure ghost terms. As these ghost terms annihilate | i , we have(A.24) e G −( / ) (cid:16) ( f α ) j χ −( u − )− N v ⊗ | i (cid:17) = ( f α ) j + χ −( u − )− N v ⊗ | i − ∞ Õ m = h α m ( f α ) j χ −( u − )− N v ⊗ γ − m + / | i , for any j ∈ š > . Now, m > implies that h α m v = , hence that(A.25) h α m ( f α ) j χ −( u − )− N v = [ h α m , ( f α ) j ] χ −( u − )− N v + ( f α ) j [ h α m , χ −( u − )− N ] v . The first commutator on the right-hand side is a sum of terms, each obtained from ( f α ) j by replacing one of the f α by − f α m + . However, each of these terms is by Lemma A.4. On the other hand, the second commutator is proportionalto χ −( u − )−( N − m ) , so it annihilates v by minimality of N . We therefore obtain(A.26) e G −( / ) (cid:16) ( f α ) j χ −( u − )− N v ⊗ | i (cid:17) = f α ( f α ) j χ −( u − )− N v ⊗ | i , from which we conclude inductively that (cid:0) e G −( / ) (cid:1) j (cid:0) χ −( u − )− N v ⊗ | i (cid:1) = ( f α ) j χ −( u − )− N v ⊗ | i , for all i ∈ š > .To deduce (A.22), we now repeat the argument by acting with e G + ( / ) on ( f α ) i ( f α ) j χ −( u − )− N v ⊗ | i . There are noessential differences between this case and that described above, so we omit the details. (cid:4) Corollary A.7. χ −( u − )− N v ⊗ | i is closed and inexact. Proof.
We have already seen that χ −( u − )− N v ⊗ | i is closed. Suppose therefore that it is exact. As [ d , e G ±( / ) ] = , since e G ± is closed, it now follows from Proposition A.5 and Lemma A.6 that(A.27) (cid:0) e G + ( / ) (cid:1) N (cid:0) e G −( / ) (cid:1) N (cid:0) χ −( u − )− N v ⊗ | i (cid:1) = ( f α ) N ( f α ) N χ −( u − )− N v ⊗ | i = ( e θ − ) u − − N v is also exact. But, this contradicts Lemma A.1. (cid:4) This corollary completes the proof of Theorem 4.8.
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School of Mathematics and Statistics, University of Melbourne, Parkville, Australia, 3010.
E-mail address : [email protected] (Kazuya Kawasetsu) Priority Organization for Innovation and Excellence, Kumamoto University, Kumamoto 860-8555, Japan.
E-mail address : [email protected] (David Ridout) School of Mathematics and Statistics, University of Melbourne, Parkville, Australia, 3010.
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