A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
aa r X i v : . [ m a t h . QA ] J u l A REALISATION OF THE BERSHADSKY–POLYAKOV ALGEBRASAND THEIR RELAXED MODULES
DRAˇZEN ADAMOVI ´C, KAZUYA KAWASETSU AND DAVID RIDOUT
Abstract.
We present a realisation of the universal/simple Bershadsky–Polyakov ver-tex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikovvertex algebras and an isotropic lattice vertex algebra. This generalises the realisationof the universal/simple affine vertex algebras associated to sl and osp (1 |
2) given in[2]. Relaxed highest-weight modules are likewise constructed, conditions for their ir-reducibility are established, and their characters are explicitly computed, generalisingthe character formulae of [38]. Introduction
Let g be a finite-dimensional complex basic classical simple Lie superalgebra and let V k ( g ) denote the corresponding universal affine vertex superalgebra of level k . Associ-ated to every nilpotent element f ∈ g , or rather to every orbit of nilpotent elements,there is a vertex superalgebra W k ( g ; f ) called a (universal) W-algebra. It is defined[27, 34] as the cohomology of the tensor product of V k ( g ) and a certain ghost vertex op-erator superalgebra. An important problem is to understand the representation theoryof W k ( g ; f ) and that of its simple quotient W k ( g ; f ).There are certain cases in which the representation theory of W k ( g ; f ) is relatively well-understood. In particular, W k ( g ; 0) = V k ( g ) and so W k ( g ; 0) is the simple affine vertexsuperalgebra L k ( g ). For admissible levels, the highest-weight modules of the latter wereclassified in [14] for g a simple Lie algebra. On the other hand, when f is principal and k is admissible and nondegenerate, the representation theory of W k ( g ; f ) was completelydetermined in [13], again for g nonsuper. Other results in this direction may be foundin [11, 17].Our interest here is in the so-called relaxed highest-weight modules [28, 46] that playan important role in the representation theory of certain classes of nonrational non- C -cofinite vertex superalgebras including the universal W-algebras and many of theirsimple quotients. The first classification result of this type addressed the simple relaxedhighest-weight modules of L k ( sl ) for k admissible [8]. Recently, similar classificationsfor other affine vertex superalgebras have started to appear [16, 21, 23, 30, 31, 39, 44, 50].Moreover, [37] explains how one can rigorously derive, for general affine vertex algebras,the relaxed classification from the highest-weight one. A natural question now is how toobtain relaxed classifications for nonrational non- C -cofinite W-algebras.One answer to this question is to use explicit singular vector formulae [6]. However,this is limited to a very small subset of W-algebras and levels. Another is to realisethe simple W-algebras using a coset construction, if one is available. Such constructionsare generally very difficult to prove, see [15] for example, and are thus far limited to asmall class of nilpotents (notably the principal ones). Nevertheless, coset constructionsprovide powerful tools to analyse the representation theory of certain W-algebras. Aparticularly tractable, but still important, special case concerns cosets by a Heisenbergsubalgebra for which there are general tools available [20]. This includes, for example,the nonunitary minimal models of the N = 2 superconformal algebras [22]. A somewhat more general approach is to use quantum hamiltonian reduction functorsto construct W-algebra modules from affine vertex superalgebra modules, when the latterare well-understood. More precisely, one can restrict the functor to category O , try toprove that the reduction functor is surjective onto category O for the W-algebra, andthen use the methods of [37] to extend this to a classification of relaxed modules.This approach is currently being tested in [26] for the simplest nonrational non- C -cofinite W-algebras, the Bershadsky–Polyakov algebras BP k = W k ( sl ; f min ) [19,43] with k nondegenerate admissible and f min minimal (take the lowest root vector of sl fordefiniteness). This uses a detailed understanding [9, 34, 35] of the minimal reductionfunctor. However, generalising this approach to other W-algebras will require a farbetter understanding of the corresponding reduction functors than is currently available.Here, we study the representation theory of the Bershadsky–Polyakov algebras using apromising alternative approach that has the benefit of constructing the relaxed highest-weight modules directly, by “inverting” the quantum hamiltonian reduction functor (see[47]). This method was pioneered in [2] for g = sl and osp (1 | L k ( sl ) was, for k / ∈ Z ≥ , there realised as a subalgebra ofthe tensor product of a simple Virasoro vertex algebra L Vir c and a lattice vertex algebra Πof indefinite type. Similarly, L k ( osp (1 | N = 1 superconformal vertex algebra L N =1 c , a free fermion and another lattice vertexalgebra (closely related to Π).Moreover, the known irreducible L k ( sl )-modules were constructed in [2] as submod-ules of the tensor product of an irreducible L Vir c -module M and an irreducible Π-moduleΠ r ( λ ). An especially nice observation is that the irreducible relaxed highest-weight mod-ules were realised directly as M ⊗ Π − ( λ ), where M is some irreducible highest-weight L Vir c -module. This neatly explains why the characters of these relaxed modules, pro-posed in [24] and proven in [38], have the well-known irreducible Virasoro characters asfactors. Analogous realisations for irreducible L k ( osp (1 | L Vir c and L N =1 c are the principal quantum hamiltonian reductions W k ( sl ; f pr )and W k ( osp (1 | f pr ), respectively. It is in this sense that tensoring with an appropriatevertex operator superalgebra inverts the reduction functor. In this paper, we extend theresults of [2] to the Bershadsky–Polyakov algebras BP k and BP k . The role of the quantumhamiltonian reductions will be played by the Zamolodchikov algebra Z k = W k ( sl ; f pr )and its simple quotient Z k [51]. Our results may therefore be regarded as not invertingthe principal (or minimal) reduction functor, but rather as inverting an (as yet unde-fined) affine version of the quantum hamiltonian reduction by stages functors introduced(for type A) in [42].Recall the lattice vertex algebra Π and its irreducible modules Π − ( λ ) (see Section 3.2for precise definitions). We shall prove the following results. • For all k , the universal Bershadsky–Polyakov algebra BP k is a vertex subalgebra ofZ k ⊗ Π (Theorems 3.6 and 7.3). • For all k such that 2 k + 3 / ∈ Z ≥ ∪ {− } , the simple Bershadsky–Polyakov algebra BP k is a vertex subalgebra of Z k ⊗ Π (Theorem 6.2). • If M is an irreducible highest-weight Z k -module, then M ⊗ Π − ( λ ) is an indecomposablerelaxed highest-weight BP k -module that is irreducible for almost all λ (Theorem 5.12). • If 2 k + 3 / ∈ Z ≥ ∪ {− } , then the previous assertion holds with Z k and BP k replacedby Z k and BP k (Theorem 6.3). • Every nonordinary irreducible (conjugate) highest-weight BP k -module may be con-structed as an explicitly given submodule of some M ⊗ Π − ( λ ) (Proposition 5.14). In this paper, we will not delve deeper into the representations of the Bershadsky–Polyakov algebras, leaving logarithmic (staggered) modules and Whittaker modules fora sequel. We will also not prove that our relaxed construction produces all the relaxedhighest-weight BP k -modules, up to isomorphism, noting only that this follows by com-paring with the classification results of [26]. Nevertheless, it would be very satisfyingto prove this completeness using the framework developed here and we intend to alsoaddress this in the sequel.Our success in generalising the results of [2] not only lends weight to the conjecturalexistence of affine reduction by stages functors, it also suggests a general program forelucidating the representation theory of a given (simple) W-algebra W k ( g ; f ). We shalldescribe this program, initially assuming that k is nondegenerate admissible for sim-plicity. The principal W-algebra W k ( g ; f pr ) is then rational [13] and its representationtheory is, in principle, known. Nilpotent orbits admit a natural partial ordering viainclusions of their closures with the principal orbit being the largest and the zero orbitthe smallest. The program then amounts to iteratively inverting the affine reduction bystages functors to construct the representations of W k ( g ; f ) from those of W k ( g ; f pr ).For g = sl , the poset of nilpotent orbits is totally ordered, with the minimal orbitlying between the zero and principal ones. Following the program described above meanschoosing a nondegenerate admissible level k and using the known representation theoryof the rational vertex operator algebra Z k = W k ( sl ; f pr ) to construct the representationsof BP k = W k ( sl ; f min ). This is the content of this paper. Continuing this program,we should next attempt to use these results to construct the representation theory of L k ( sl ) = W k ( sl ; 0). We intend to return to this in the future [4], comparing with theresults of the complementary approach of [14, 37].For degenerate admissible levels, we expect that the privileged role of the principalW-algebra as the starting point of the program will be replaced by the simple exceptionalW-algebras of [12, 25, 36], many of which have been recently proven to be rational [17].For example, if g = sl and k ∈ − + Z ≥ , then the principal reduction of L k ( sl ) is zeroand the exceptional W-algebra is BP k . The latter is rational [11] and so our programbegins here. If instead k ∈ Z ≥ , then both the principal and minimal reductions of L k ( sl ) are zero and the exceptional W-algebra is L k ( sl ) itself (which is also rational).We conclude this introduction with a brief outline of the contents of the paper. InSections 2 and 3, we first introduce our conventions for the Bershadsky–Polyakov alge-bras, as well as the Zamolodchikov algebras and the lattice vertex algebra Π. We thenverify that we have a homomorphism from the universal Bershadsky–Polyakov algebrato the tensor product of the universal Zamolodchikov algebra and Π in Section 3.3. Thefact that this is an embedding is proven in Section 4.We next construct relaxed highest-weight modules for the Bershadsky–Polyakov al-gebras, arguing in Section 5 that these modules are “almost-irreducible” which means,among other things, that for almost all values of the parameters that naturally specifythese modules, they are irreducible. We also give a precise criterion for irreducibility.This allows us to realise the simple Bershadsky–Polyakov algebra as a submodule of sucha relaxed module in Section 6 when the level satisfies 2 k + 3 / ∈ Z ≥ , thereby proving thesimple analogue of the embedding of Sections 3.3 and 4 for these levels. Finally, Section 7establishes critical-level analogues of these results. Notation.
Given a homogeneous field A ( z ) of conformal weight ∆ A , we define operators A n and A ( n ) , n ∈ Z , by the expansions(1.1) A ( z ) = X n ∈ Z − ∆ A A n z − n − ∆ A = X n ∈ Z A ( n ) z − n − . Acknowledgements.
We thank Thomas Creutzig and Zac Fehily for discussions relat-ing to the results presented here.D.A. is partially supported by the QuantiXLie Centre of Excellence, a project cofi-nanced by the Croatian Government and European Union through the European Re-gional Development Fund - the Competitiveness and Cohesion Operational Programme(KK.01.1.1.01.0004).KK’s research is partially supported by MEXT Japan “Leading Initiative for ExcellentYoung Researchers (LEADER)”, JSPS Kakenhi Grant numbers 19KK0065 and 19J01093and Australian Research Council Discovery Project DP160101520.DR’s research is supported by the Australian Research Council Discovery ProjectDP160101520 and the Australian Research Council Centre of Excellence for Mathemat-ical and Statistical Frontiers CE140100049.2.
Bershadsky–Polyakov algebras
In this section, we introduce the Bershadsky–Polyakov algebras as vertex algebras.They were originally defined independently by Polyakov [43] and Bershadsky [19] as non-principal quantum hamiltonian reductions of the universal affine vertex algebras V k ( sl )associated to sl . In the framework of Kac–Roan–Wakimoto [34], they are simultaneouslythe minimal and subregular reductions.We start by defining the universal Bershadsky–Polyakov vertex operator algebras BP k in terms of generators and relations. The Bershadsky–Polyakov vertex algebra at thecritical level k = − Definition 2.1.
For k = −
3, the universal Bershadsky–Polyakov vertex operator algebraBP k is the universal vertex algebra generated by fields L , J , G + and G − subject to thefollowing operator product expansions:(2.1) J ( z ) J ( w ) ∼ k + 33( z − w ) , J ( z ) G ± ( w ) ∼ ± G ± ( w ) z − w ,L ( z ) G + ( w ) ∼ G + ( w )( z − w ) + ∂G + ( w ) z − w , L ( z ) G − ( w ) ∼ G − ( w )( z − w ) + ∂G − ( w ) z − w ,L ( z ) J ( w ) ∼ − k + 33( z − w ) + J ( w )( z − w ) + ∂J ( w ) z − w , G ± ( z ) G ± ( w ) ∼ ,L ( z ) L ( w ) ∼ c BP k z − w ) + 2 L ( w )( z − w ) + ∂L ( w ) z − w ,G + ( z ) G − ( w ) ∼ ( k + 1)(2 k + 3)( z − w ) + 3( k + 1) J ( w )( z − w ) + 3: J ( w ) J ( w ): + (2 k + 3) ∂J ( w ) − ( k + 3) L ( w ) z − w . The central charge is(2.2) c BP k = − k + 1)(2 k + 3) k + 3 . As always, BP k has a unique simple quotient and we shall denote it by BP k . Remark 2.2.
Both BP k and BP k are Z ≥ -graded by the eigenvalue of the zero mode L because the conformal weights of G ± ( z ) are 1 and 2, respectively. This asymmetry isalso reflected in the fact that J ( z ) fails to be quasiprimary. This failure may be rectifiedby instead choosing the conformal vector to be(2.3) e L = L − ∂J. Both G + and G − will then have conformal weight . The only downside is that BP k andBP k are now Z ≥ -graded by the eigenvalue of e L . Unless otherwise indicated, we shallkeep L as the conformal vector.The commutation relations of the modes are easily computed from the operator prod-uct expansions (2.1). We record them for convenience.(2.4) [ J m , J n ] = 2 k + 33 mδ m + n, , [ J m , G ± n ] = ± G ± m + n , [ L m , G + n ] = − nG + m + n , [ L m , G − n ] = ( m − n ) G − m + n , [ L m , J n ] = − nJ m + n − k + 33 m − m δ m + n, , [ G ± r , G ± s ] = 0 , [ L m , L n ] = ( m − n ) L m + n − (2 k + 3)( k + 1) k + 3 m − m δ m + n, , [ G + m , G − n ] = 3: J J : m + n − ( k + 3) L m + n + (cid:0) km − (2 k + 3)( n + 1) (cid:1) J m + n + ( k + 1)(2 k + 3) m − m δ m + n, . The associative algebra of modes specified by these relations admits a useful familyof automorphisms called spectral flow automorphisms . These may be lifted to maps onBP k , following [41], by introducing(2.5) Λ( ℓJ, z ) = z − ℓJ ∞ Y n =1 exp (cid:18) ( − n n ℓJ n z − n (cid:19) and defining the result of acting with the spectral flow map σ ℓ , ℓ ∈ Z , on a field A ( z ) ofBP k to be(2.6) σ ℓ ( A ( z )) = Y (cid:0) Λ( ℓJ, z ) A, z (cid:1) , where Y is the vertex map of BP k . In particular, we have(2.7) σ ℓ ( G ± ( z )) = z ∓ ℓ G ± ( z ) , σ ℓ ( J ( z )) = J ( z ) − k + 33 ℓz − ,σ ℓ ( L ( z )) = L ( z ) − ℓz − J ( z ) + 2 k + 33 ℓ ( ℓ + 1)2 z − . One can check explicitly that spectral flow preserves the defining operator product expan-sions (2.1) (and the vacuum) of BP k . These spectral flows may, moreover, be extended to ℓ ∈ Z if we allow half-integer modes for G ± ( z ) as when acting on twisted BP k -modules.Let M be a BP k -module. By twisting the action of BP k on M by the spectral flowmap σ − ℓ , we may give M a new structure as a BP k -module. We shall denote this newBP k -module by σ ℓ ( M ). Denoting its elements by σ ℓ ( v ), where v ∈ M , the twisted actionis explicitly realised as(2.8) A · σ ℓ ( v ) = σ ℓ (cid:0) σ − ℓ ( A ) v (cid:1) . In this way, spectral flow lifts to invertible endofunctors on the category of BP k -modules.Consequently, σ ℓ ( M ) is irreducible if and only if M is.3. Realisation of BP k Here, we realise the universal Bershadsky–Polyakov algebra BP k as a vertex subalgebraof the tensor product of the principal quantum hamiltonian reduction of V k ( sl ), whichwe shall refer to as the Zamolodchikov algebra, and an isotropic lattice vertex algebraΠ. We first introduce these vertex algebras and some of their modules. The Zamolodchikov algebras Z k and Z k . As with the universal Bershadsky–Polyakov algebras, the universal Zamolodchikov algebras [51] may also be defined interms of generators and relations.
Definition 3.1.
For k = −
3, the universal Zamolodchikov vertex operator algebra Z k is the universal vertex algebra generated by fields T and W subject to the followingoperator product expansions:(3.1) T ( z ) T ( w ) ∼ c Z k z − w ) + 2 T ( w )( z − w ) + ∂T ( w ) z − w ,T ( z ) W ( w ) ∼ W ( w )( z − w ) + ∂W ( w ) z − w ,W ( z ) W ( w ) ∼ ( k + 3) (cid:20) w )( z − w ) + ∂ Λ( w ) z − w (cid:21) + A (cid:20) c Z k z − w ) + 2 T ( w )( z − w ) + ∂T ( w )( z − w ) + ∂ T ( w )( z − w ) + ∂ T ( w ) z − w (cid:21) . Here, the central charge is(3.2) c Z k = − k + 5)(4 k + 9) k + 3and we have defined(3.3) A = − ( k + 3) (3 k + 4)(5 k + 12)6 and Λ = : T T : − ∂ T, for convenience. The unique simple quotient of Z k will be denoted by Z k . Remark 3.2.
The definition of Z k given above differs from the standard one in that wehave renormalised the field W ( z ) by a factor of √ A . This removes a singularity in thestandard definition when c Z k = − , hence k = − or − . At this central charge, W and Λ belong to the maximal ideal of Z − / = Z − / . The simple quotient is in fact theVirasoro minimal model M (2 ,
5) (also known as the Yang–Lee model).3.2.
The vertex algebra Π and its modules. Consider the abelian Lie algebra h =span C { a, b } , equipped with the bilinear form defined by(3.4) h a, a i = − h b, b i = 1 and h a, b i = 0 . For convenience, we let(3.5) c = a − b and d = a + b, noting that these elements of h are isotropic: h c, c i = h d, d i = 0. The group algebra C [ Z c ] = span C { e nc : n ∈ Z } then becomes an h -module with action(3.6) he nc = h h, c i e nc , h ∈ h . Let H denote the Heisenberg vertex algebra associated to h . Definition 3.3.
Let Π denote the lattice vertex algebra H ⊗ C [ Z c ], where the action of h ∈ h on C [ Z c ] is identified with the action of the zero mode h of h ( z ) ∈ H . We equipΠ with the conformal structure given by(3.7) t ( z ) = 12 : c ( z ) d ( z ): + 2 k + 33 ∂c ( z ) − ∂d ( z ) , so that a and b both have conformal weight 1, whilst the weight of e nc , n ∈ Z , is n . Thecentral charge is(3.8) c Π k = 2 + 8(2 k + 3) . Vertex algebras like Π were studied in [18], under the name “half-lattice vertex al-gebras”, as were their representation theories. We refer to [2, Sec. 4] for a convenientsummary. Here, we record the following operator product expansion for future conve-nience:(3.9) e c ( z ) e − c ( w ) = + c ( w )( z − w ) + 12 (cid:0) : c ( w ) c ( w ): + ∂c ( w ) (cid:1) ( z − w ) + · · · . Before introducing the Π-modules relevant to what follows, we discuss spectral flowfor Π. In analogy with (2.5), we set [41](3.10) Λ( ℓj, z ) = z − ℓj ∞ Y n =1 exp (cid:18) ( − n n ℓj n z − n (cid:19) , j = b + k + 33 c, and define the action of the spectral flow map γ ℓ , ℓ ∈ Z , on a field A ( z ) of Π by(3.11) γ ℓ ( A ( z )) = Y (Λ( ℓj, z ) A, z ) , where Y is now the vertex map of Π. The reason for taking the particular element j ∈ h will become clear in Section 3.3.With this, it is easy to verify that spectral flow acts on the generators of Π as follows:(3.12) γ ℓ ( a ( z )) = a ( z ) − k +33 ℓz − ,γ ℓ ( b ( z )) = b ( z ) − k ℓz − , γ ℓ ( e nc ( z )) = z − ℓn e nc ( z ) ( n ∈ Z ) . The γ ℓ thus preserve operator product expansions (and the vacuum) of Π, but do notpreserve its conformal structure:(3.13) γ ℓ ( t ( z )) = t ( z ) − ℓz − j ( z ) + 2 k + 33 ℓ ( ℓ + 1)2 z − . As in Section 2, twisting by spectral flow defines invertible functors on the category ofΠ-modules. This therefore associates to every ℓ ∈ Z and every (irreducible) Π-module M a new (irreducible) Π-module γ ℓ ( M ). Again, this generalises to ℓ ∈ Z and twistedmodules.The Π-modules of interest here may be obtained by considering the C [ Z c ]-modulegenerated by e h ∈ C [ h ] and inducing. For later convenience, we shall generally write h as a linear combination of the basis vectors j , defined in (3.10), and c . It will also beconvenient to introduce(3.14) i = d − j = a − k + 33 c. We then have h j, j i = k +33 = − h i, i i , h j, c i = 1 = h i, c i and h i, j i = 0.For each r ∈ Z and λ ∈ C , define(3.15) Π r ( λ ) = Π · e rj + λc . Recall that an indecomposable module is positive-energy, with respect to a given con-formal structure, if its conformal weights are bounded below. Its top space is then theeigenspace corresponding to the minimal conformal weight (should it exist).
Proposition 3.4. • For r ∈ Z and λ ∈ C , Π r ( λ ) is an irreducible (untwisted) Z -graded Π -module. • For r ∈ Z + and λ ∈ C , Π r ( λ ) is an irreducible Z -graded ( e π √− i -twisted) Π -module. • In both cases, we have Π r ( λ ) ∼ = Π r ( λ + n ) for all n ∈ Z . Otherwise, the Π r ( λ ) aremutually inequivalent. • c acts on Π r ( λ ) as r times the identity and the e nc − n ( r +1) , n ∈ Z , act injectively. • Π r ( λ ) is positive-energy if and only if r = − . Π − ( λ ) is thus a relaxed highest-weight Π -module and its top space Π − ( λ ) top has conformal weight k +33 . • Twisting the action of Π by the spectral flow maps γ ℓ gives (see [3, Prop. 4.1] for asimilar calculation) (3.16) γ ℓ (Π r ( λ )) ∼ = Π r + ℓ ( λ ) . The parameter r in Π r ( λ ) is therefore a spectral flow index while λ represents the eigen-value of i . The isomorphism class of Π r ( λ ) thus only depends on r and the image of λ in C / Z . Obviously, the vacuum module is Π (0).It is also straightforward to determine the characters of the relaxed highest-weightmodules Π − ( λ ). Proposition 3.5.
Let δ ( z ) = P n ∈ Z z n as usual. Then, the character of Π − ( λ ) is (3.17) ch (cid:2) Π − ( λ ) (cid:3) ( y, z ; q ) = tr Π − ( λ ) y c z i q t − c Π / = y − z λ η ( q ) δ ( z ) . Realisation.
The (easily verified) central charge relation c BP k = c Z k + c Π k suggeststhat the three vertex operator algebras BP k , Z k and Π might be related. The followingresult determines this relation precisely. Theorem 3.6.
For k = − , there is an injective vertex operator algebra homomorphism φ k : BP k → Z k ⊗ Π , uniquely determined by (3.18) G + ⊗ e c , J ⊗ j, L T ⊗ + ⊗ t,G − (cid:0) W + ( k + 2)( k + 3) ∂T (cid:1) ⊗ e − c + ( k + 3) T ⊗ i − e − c − ⊗ (cid:0) i − + 3( k + 2) i − i − + 2( k + 2) i − (cid:1) e − c . Here, i and j were defined in (3.14) and (3.10) , respectively.Sketch of proof that φ k is a vertex operator algebra homomorphism. Because BP k is uni-versal, it suffices to show that the operator product expansions (2.1) of the generators J , L and G ± match those of their φ k -images. This can be checked explicitly from the defin-ing operator product expansions (3.1) of Z k and those, for example (3.9), of Π. We usedthe computer algebra package OPEdefs [48] for this purpose, but the computations arealso easily performed by hand.For example, the coefficient of the third-order pole of G + ( z ) G − ( w ) in (2.1) is de-termined by G +2 G − = ( k + 1)(2 k + 3) whilst the corresponding calculation for their φ k -images proceeds as follows. First, note that(3.19) φ k ( G + ) φ k ( G − ) = − ⊗ (cid:2) e c , i − + 3( k + 2) i − i − + 2( k + 2) i − (cid:3) e − c , because e c annihilates both e − c and i − e − c . Since [ e cm , i n ] = − e cm + n and e c − e − c = , thisindeed evaluates to φ k ( G + ) φ k ( G − ) = ⊗ (cid:0) e c i − + 3( k + 2) e c i − + 2( k + 2) e c − (cid:1) e − c (3.20) = ⊗ (cid:0) − e c i − − k + 2) e c − + 2( k + 2) e c − (cid:1) e − c = ⊗ (cid:0) − k + 2) + 2( k + 2) (cid:1) e c − e − c = ( k + 1)(2 k + 3) . Similar computations determine that all the singular coefficients match, hence that φ k is a homomorphism. (cid:3) Remark 3.7.
Comparing the spectral flow maps σ ℓ of BP k and γ ℓ of Π, we see that theexplicit realisation (3.18) requires σ ℓ = id ⊗ γ ℓ . In the definition (3.10) of the spectral flowmaps of Π, we could have replaced j by any h ∈ h and still preserved the operator productexpansions. However, the above realisation singles out h = j as being particularly usefulfor our purposes. We will prove that φ k is injective in Section 4. Granting this, it follows from Theo-rem 3.6 that for k = −
3, any Z k ⊗ Π-module is a BP k -module, by restriction. Combiningthis with Proposition 3.4, we obtain a construction of many positive-energy BP k -modules.Proposition 3.5 then gives their characters. Corollary 3.8.
Suppose that k = − and that M is a Z k -module with q -character ch (cid:2) M (cid:3) ( q ) = tr M q T − c Z k / . Then, the BP k -module R M ( λ ) = M ⊗ Π − ( λ ) has character ch (cid:2) R M ( λ ) (cid:3) ( z ; q ) = tr M ⊗ Π − ( λ ) z J q L − c BP k / (3.21) = ch (cid:2) M (cid:3) ( q ) ch (cid:2) Π − ( λ ) (cid:3) (cid:0) z (2 k +3) / , z ; q (cid:1) = z λ − (2 k +3) / ch (cid:2) M (cid:3) ( q ) η ( q ) δ ( z ) . When M is irreducible, we shall show in Section 5 that the R M ( λ ) are indecomposablerelaxed highest-weight modules that are irreducible for almost all λ .4. Injectivity of φ k In this section, we show that the homomorphism φ k : BP k → Z k ⊗ Π is injective, for k = −
3, and thereby prove Theorem 3.6. Recall that a partition is a finite sequence ofpositive integers µ = ( µ , µ , . . . , µ ℓ ) of length ℓ = ℓ ( µ ) ∈ Z ≥ satisfying(4.1) µ ≥ µ ≥ · · · ≥ µ ℓ . The weight of the partition µ is defined to be | µ | = µ + µ + · · · + µ ℓ . Let P denote theset of all partitions.Given a partition µ ∈ P of length ℓ and an element A of a vertex algebra, we introduce(whenever it makes sense) the convenient notation(4.2) A + µ = A µ ℓ · · · A µ A µ , A − µ = A − µ A − µ · · · A − µ ℓ ,A (+ µ ) = A ( µ ℓ ) · · · A ( µ ) A ( µ ) , A ( − µ ) = A ( − µ ) A ( − µ ) · · · A ( − µ ℓ ) , recalling the conventions for mode indices in (1.1). We shall also write µ + n (and − µ − n )in the above to indicate that every part of µ ∈ P should be increased by n ∈ Z ≥ . Thefollowing lemma is now clear from universality (see [35, Thm. 4.1(b)]). Lemma 4.1. • The universal Bershadsky–Polyakov algebra BP k has a Poincar´e–Birkhoff–Witt-typebasis B BP = { J ( − µ ) G +( − ν ) L ( − ρ ) G − ( − σ ) : µ, ν, ρ, σ ∈ P} . • The universal Zamolodchikov algebra Z k likewise has a Poincar´e–Birkhoff–Witt-typebasis B Z = { T ( − µ ) W ( − ν ) : µ, ν ∈ P} . Because the lattice vertex operator algebra Π restricts, as an H -module, to an infinitedirect sum of Fock modules (one for each e nc ∈ Π), we get our third basis.
Lemma 4.2.
The lattice vertex operator algebra Π has a Poincar´e–Birkhoff–Witt-typebasis B ′ Π = { j ( − µ ) c ( − ν ) e nc : µ, ν ∈ P and n ∈ Z } . This basis will, however, need some finessing. Let S m ( c ), m ∈ Z ≥ , denote the Schurfunction in the (commuting) variables c ( − n ) , n ∈ Z ≥ , corresponding to the partition( m ). Equivalently, these functions may be defined by the following special case of theCauchy identity:(4.3) ∞ Y n =1 exp (cid:16) c ( − n ) n z n (cid:17) = ∞ X m =0 S m ( c ) z m . In particular, we have(4.4) S ( c ) = 1 , S ( c ) = c ( − and S ( c ) = 12 (cid:0) c ( − + c − (cid:1) . For general m ∈ Z ≥ , the S m ( c ) have the form(4.5) S m ( c ) = 1 m c ( − m ) + [terms quadratic and higher in the c ( − n ) with n < m ] . Proposition 4.3.
The set B Π = { j ( − µ ) e c ( − ν − e nc : µ, ν ∈ P and n ∈ Z } is also a basisof Π .Proof. It follows easily from (4.3) and c being isotropic that(4.6) e c ( − m − e nc = ( S m ( c ) e ( n +1) c if m ≥ ℓ = ℓ ( ν ), we therefore have e c ( − ν − e nc = e c ( − ν − · · · e c ( − ν ℓ − e nc = S ν ( c ) · · · S ν ℓ ( c ) e ( n + ℓ ) c (4.7) = c ( − ν ) · · · c ( − ν ℓ ) e ( n + ℓ ) c ν · · · ν ℓ + · · · = c ( − ν ) e ( n + ℓ ) c ν · · · ν ℓ + · · · , where + · · · indicates terms whose c -degrees are greater than ℓ . Composing with j ( − µ ) , itfollows that the elements of B Π are linearly independent. Moreover, they span Π becausean obvious inductive triangularity argument shows that (4.7) may be inverted and thussolved for the basis elements of B ′ Π . (cid:3) Remark 4.4.
Note that an element of the form j ( − µ ) e c ( − ν ′ ) e n ′ c ∈ Π, µ, ν ′ ∈ P and n ′ ∈ Z ,may be identified with one of the basis vectors of B Π by isolating any parts of ν ′ equalto 1. If there are m such parts, let ν ′′ be the partition obtained from ν ′ by removingthem. Then, ν ′′ has no part equal to 1 so it may be written as ν + 1 for some (unique)partition ν . Setting n = m + n ′ , we get the desired form:(4.8) j ( − µ ) e c ( − ν ′ ) e n ′ c = j ( − µ ) e c ( − ν ′′ ) (cid:0) e c ( − (cid:1) m e n ′ c = j ( − µ ) e c ( − ν − e nc . We now prove that φ k is injective. Proof of Theorem 3.6.
Recall from the explicit realisation (3.18) that(4.9) φ k ( J ( − n ) ) = ⊗ j ( − n ) ,φ k ( G +( − n ) ) = ⊗ e c ( − n ) ,φ k ( L ( − n ) ) = T ( − n ) ⊗ + [terms not involving T ]and φ k ( G − ( − n ) ) = ∞ X m =0 W ( − n + m ) ⊗ e − c ( − m − + [terms not involving W ].We will show that the images of the BP k basis vectors in B BP are linearly independent.These images have the form φ k ( J ( − µ ) G +( − ν ) L ( − ρ ) G − ( − σ ) ) = T ( − ρ ) W ( − σ ) ⊗ j ( − µ ) e c ( − ν ) (cid:0) e − c ( − (cid:1) ℓ ( σ ) + · · · (4.10) = T ( − ρ ) W ( − σ ) ⊗ j ( − µ ) e c ( − ν ) e − ℓ ( σ ) c + · · · , where + · · · indicates a linear combination of similar terms that have either fewer T -modes, fewer W -modes, or have the same number of T - and W -modes but also havesome e − c ( − m − -modes with m ≥
1. In the latter case, the W -partition σ is replaced byanother of the same length but strictly lower weight. The c → − c analogues of (4.6) and (4.7) show that the action of any e − c ( − m − maybe expressed in terms of the action of c -modes, hence in terms of actions of e c -modes.We can thereby rewrite any term with an e − c ( − m − as a linear combination of basis termsfrom B Z ⊗ B Π . The point is that this rewriting will not change the j -, T - and W -modes,in particular it will not change the fact that the corresponding W -partition has weightstrictly lower than | σ | .The term exhibited on the right-hand side of (4.10) is therefore the unique term, whenexpressed in the basis B Z ⊗ B Π , with ℓ ( µ ) j -modes, ℓ ( ρ ) T -modes and ℓ ( σ ) W -modescorresponding to a partition of weight | σ | . The images on the left-hand side are thereforelinearly independent, as desired, hence φ k is injective. (cid:3) Almost-irreducibility
Our next aim is to study the irreducibility of the relaxed highest-weight BP k -modules R M ( λ ), introduced in Corollary 3.8. More specifically, we wish to show that R M ( λ ) is“almost-irreducible” (to be defined shortly) when M is an irreducible highest-weight Z k -module. Here, we first prepare the groundwork for this by proving that the Π-modulesΠ − ( λ ) are almost-irreducible as modules over a certain vertex operator subalgebra U . Definition 5.1.
Let V be a vertex operator algebra and M an Z ≥ -graded V -modulewith top component M top . • We say that M is top-generated if M is generated by M top . • We say that M has only top-submodules if every nonzero submodule of M has a nonzerointersection with M top . • We say that M is almost-irreducible if it is top-generated and has only top-submodules.One motivation for introducing almost-irreducibility is to isolate a class of moduleswhose irreducibility is determined by its top space. Recall that the top space M top ofa module over a vertex operator algebra V is naturally a module over the Zhu algebra A ( V ) [52]. The action of A ( V ) is of course nothing but the action of the zero modes A , A ∈ V , on the top space. Proposition 5.2. If M is an almost-irreducible V -module and M top is an irreducible A ( V ) -module, then M is irreducible.Proof. Let N be a nonzero submodule of M . Since M has only top-submodules, N ∩ M top is a nonzero A ( V )-module. Since M top is irreducible, we have M top ⊆ N . Finally, M being top-generated forces M ⊆ N , hence N = M . (cid:3) Another motivation is to model modules obtained by Zhu-induction [40, 52]. Moreprecisely, this induction functor constructs a vertex operator algebra module from a Zhualgebra module in such a way that the top space of the former coincides with the latter.If we now quotient the former by the sum of all submodules whose intersection with thetop space is zero, then the result has the same top space but is now almost-irreducible.In this sense, almost-irreducibility captures the notion of the “smallest” vertex operatoralgebra module with a given top space.5.1.
Almost-irreducibility of Π − ( λ ) . Let U be the vertex subalgebra of Π generatedby j = b + k +33 c and e c . Our aim is to show that the Π − ( λ ) are almost-irreducible as U -modules. Recall that(5.1) Π − ( λ ) top = span C { e − j +( λ + n ) c : n ∈ Z } . We first prove top-generation. This follows easily by determining an appropriate basis us-ing the method of Proposition 4.3. An obvious basis (the analogue of that of Lemma 4.2) is(5.2) { j − µ c − ν e − j +( λ + n ) c : µ, ν ∈ P and n ∈ Z } , where we recall the notation of (4.2). Lemma 5.3.
For every λ ∈ C , the set (5.3) { j − µ e c − ν e − j +( λ + n ) c : µ, ν ∈ P and n ∈ Z } is a basis of Π − ( λ ) .Proof. First, generalise (4.6) to(5.4) e c − m e − j +( λ + n ) c = ( S m ( c ) e − j +( λ + n +1) c if m ≥ S m ( c ) are the Schur functions defined in (4.3). The assertion now follows usingthe same argument as in the proof of Proposition 4.3. (cid:3) Proposition 5.4.
For every λ ∈ C , Π − ( λ ) is top-generated as a U -module. To show that Π − ( λ ) has only top-submodules, we need a preparatory lemma. Itfollows easily from the commutation relations [ j m , c n ] = mδ m + n, , [ e cm , j n ] = − e cm + n and[ e cm , c n ] = [ e cm , e cn ] = 0 ( m, n ∈ Z ), as well as the formula(5.5) e cn e − j + λc = δ n, e − j +( λ +1) c ( n ∈ Z ≥ ). Lemma 5.5. • For every µ, ν ∈ P and λ ∈ C , we have (5.6) e c + µ j − µ c − ν e − j + λc = ( − ℓ ( µ ) c − ν e − j +( λ + ℓ ( µ )) c = 0 . Moreover, if µ ′ = µ satisfies either ℓ ( µ ′ ) > ℓ ( µ ) or ℓ ( µ ′ ) = ℓ ( µ ) and | µ ′ | ≥ | µ | , then e c + µ ′ j − µ c − ν e − j + λc = 0 . • For every ν ∈ P and λ ∈ C , we have (5.7) j + ν c − ν e − j + λc = ℓ ( ν ) Y i =1 ν i · e − j + λc = 0 . Moreover, if ν ′ = ν satisfies | ν ′ | ≥ | ν | , then j + ν ′ c − ν e − j + λc = 0 . Proposition 5.6.
For every λ ∈ C , the U -module Π − ( λ ) has only top-submodules.Proof. Let w be a nonzero element of Π − ( λ ). It suffices to show that there exists a linearcombination of products of modes of U mapping w to a nonzero element of Π − ( λ ) top .Since i acts semisimply on Π − ( λ ), we may assume that i w = ( λ + n ) w for some n ∈ Z .Then, w has the form(5.8) w = X µ,ν ∈P C µ,ν j − µ c − ν e − j +( λ + n ) c , for some C µ,ν ∈ C such that C µ,ν = 0 for all but finitely many ( µ, ν ) ∈ P × P .Since w = 0, the set S of partitions µ for which there exists some ν ∈ P such that C µ,ν = 0 is nonempty. We may therefore choose µ ′ ∈ S of maximal weight among theelements of S of maximal length. By Lemma 5.5, we have a nonzero vector(5.9) w ′ = e c + µ ′ w = ( − ℓ ( µ ′ ) X ν ∈P C µ ′ ,ν c − ν e − j +( λ + n + ℓ ( µ ′ )) c = 0 . The set S ′ consisting of those ν ∈ P for which C µ ′ ,ν = 0 is clearly nonempty, hence ithas an element ν ′ of maximal weight. By Lemma 5.5 again, we have(5.10) j + ν ′ w ′ = ( − ℓ ( µ ′ ) C µ ′ ,ν ′ ℓ ( ν ′ ) Y i =1 ν ′ i · e − j +( λ + n + ℓ ( µ ′ )) c = 0 . As j + ν ′ e c + µ ′ w is a nonzero element of Π − ( λ ) top , the proof is complete. (cid:3) Corollary 5.7. Π − ( λ ) is almost-irreducible as a U -module. Remark 5.8.
On the other hand, Π − ( λ ) is never irreducible as a U -module. Thisfollows from observing that the generators j and e c of U have i -eigenvalues 0 and 1,respectively, so U has no elements that reduce the i -eigenvalue. Each e − j +( λ + n ) c , n ∈ Z ,in Π − ( λ ) top thus generates a distinct U -submodule.5.2. Almost-irreducibility of R M ( λ ) . Recall from Section 3.3 that each Z k -module M yields a BP k -module R M ( λ ) = M ⊗ Π − ( λ ). If M is irreducible and highest-weight,then the top space of R M ( λ ) is clearly(5.11) R M ( λ ) top = M top ⊗ Π − ( λ ) top . It shall prove convenient to identify BP k with its φ k -image in Z k ⊗ Π, as per Theorem 3.6.Recalling the explicit form of this realisation, we introduce f W ∈ BP k so that(5.12) f W = G + − G − = W ⊗ + α∂T ⊗ + βT ⊗ ( i − c ) − ⊗ ω, where α and β are ( k -dependent) constants and ω is an element of Π. Their preciseforms will not be needed in what follows. Theorem 5.9. If M is a weight Z k -module that has only top-submodules, then the relaxedhighest-weight BP k -module R M ( λ ) also has only top-submodules.Proof. Assume that N is a nonzero BP k -submodule of R M ( λ ) and choose a weight vector w ∈ N . Since the vertex operator subalgebra U ⊂ Π is generated by j and e c , ⊗ U is generated by ⊗ j = J and ⊗ e c = G + . Hence, ⊗ U ⊂ BP k . Since Π − ( λ ) hasonly top-submodules as a U -module (Proposition 5.6), it follows that w may be sent toa nonzero element of M ⊗ Π − ( λ ) top by acting with BP k . Hence, there exists nonzero w ∈ N of the form(5.13) w = u ⊗ v top , where u ∈ M and v top ∈ Π − ( λ ) top .Our aim now is to construct a nonzero element in R M ( λ ) top from w , by acting withBP k . We do this by recursively defining weight vectors w n = u n ⊗ v top , n = 1 , , . . . , in N until we achieve our aim. Here is the definition: • If there exists m > T m u n = 0, choose the maximal such m (for definitenessonly) and set(5.14) w n +1 = L m w n = T m u n ⊗ v top + u n ⊗ t m v top = T m u n ⊗ v top . This is then a nonzero element of N and we have u n +1 = T m u n . • If T m u n = 0 for all m >
0, but there exists m > W m u n = 0, then choosethe maximal such m and set w n +1 = f W m w n (5.15) = W m u n ⊗ v top − α ( m + 1) T m u n ⊗ v top + β ∞ X r =0 T m + r u n ⊗ ( i − r − c − r ) v top − u n ⊗ ω m v top = W m u n ⊗ v top . This is again a nonzero element of N , this time with u n +1 = W m u n . • If T m u n = W m u n = 0 for all m >
0, then u n generates a highest-weight submodule of M . As M has only top-submodules, the intersection of this submodule with M top isnonzero, hence we must have u n ∈ M top . Thus, w n ∈ R M ( λ ) top is a nonzero elementof N and we are done.This recursion has to terminate because the conformal weight of u n +1 is strictly less thanthat of u n and M is positive-energy. There therefore exists n such that w n ∈ R M ( λ ) top is nonzero and so R M ( λ ) has only top-submodules. (cid:3) Theorem 5.10. If M is a top-generated weight Z k -module, then the relaxed highest-weight BP k -module R M ( λ ) is also top-generated.Proof. Consider the submodule of R M ( λ ) generated by R M ( λ ) top = M top ⊗ Π − ( λ ) top ,denoting it by N . Clearly, N contains the elements of the form u top ⊗ v top , where u top ∈ M top and v top ∈ Π − ( λ ) top . We shall first show that this remains true when u top isreplaced by any u ∈ M . As M is top-generated, it is spanned by the elements obtainedfrom the u top by acting iteratively with the T − m and W − m , m ∈ Z > . It therefore sufficesto assume that u ⊗ v top ∈ N , for some u ∈ M , and then show that upon replacing u by T − m u or W − m u , the result is still in N .Suppose then that u ⊗ v top ∈ N for all v top ∈ Π − ( λ ) top . Acting with L − m , m ∈ Z > ,then gives T − m u ⊗ v top + u ⊗ t − m v top ∈ N . Since the U -module Π − ( λ ) is top-generated(Proposition 5.4), u ⊗ t − m v top may be obtained from u ⊗ v top ∈ N by acting with U -modes.But ⊗ U ⊂ BP k , so we have u ⊗ t − m v top ∈ N and hence T − m u ⊗ v top ∈ N .Similarly, acting with f W − m , m ∈ Z > , results in W − m u ⊗ v top + α ( m − T − m u ⊗ v top (5.16) + β ∞ X r =0 T − m + r u ⊗ ( i − r − c − r ) v top − u ⊗ ω − m v top ∈ N. As before, Π − ( λ ) being top-generated implies that u ⊗ ω − m v top ∈ N whilst the previousargument gives T − m u ⊗ v top ∈ N . An obvious hybrid of these arguments then shows that T − m + r u ⊗ ( i − r − c − r ) v top ∈ N and so we conclude that W − m u ⊗ v top ∈ N as well.It follows that M ⊗ Π − ( λ ) top ⊂ N . One more appeal to Π − ( λ ) being top-generatedthen forces M ⊗ Π − ( λ ) ⊆ N , completing the proof. (cid:3) Corollary 5.11. If M is an almost-irreducible weight Z k -module, then the relaxed highest-weight BP k -module R M ( λ ) is also almost-irreducible. Irreducibility of R M ( λ ) . We now show that when M is an irreducible highest-weight Z k -module, almost all of the relaxed highest-weight BP k -modules R M ( λ ) that wehave constructed are irreducible. This gives an a posteriori justification for the term“almost-irreducible”.We recall that a highest-weight vector for BP k is a J - and L -eigenvector that isannihilated by G +0 and all the A n , A ∈ BP k , with n >
0. The definition of a conju-gate highest-weight vector is then obtained by replacing the condition of annihilation by G +0 with annihilation by G − . A conjugate highest-weight module is then one which isgenerated by a single conjugate highest-weight vector. Theorem 5.12.
Let M be an irreducible highest-weight Z k -module whose highest-weightvector u has T -eigenvalue ∆ and W -eigenvalue w . Then: • The relaxed highest-weight BP k -module R M ( λ ) is always indecomposable. • R M ( λ ) is irreducible if and only if the polynomial (5.17) p ∆ ,wk ( x ) = w − ( k + 2)( k + 3)∆ + (cid:2) ( k + 3)∆ − k + 2) (cid:3) x + 3( k + 2) x − x has no roots in the coset λ + Z . • R M ( λ ) has no highest-weight vectors and its conjugate highest-weight vectors are pre-cisely the u ⊗ e − j + µc with µ ∈ λ + Z satisfying p ∆ ,wk ( µ ) = 0 .Proof. By Proposition 5.2 and Corollary 5.11, R M ( λ ) is irreducible if R M ( λ ) top is anirreducible A (BP k )-module. As the latter has one-dimensional weight spaces, it is irre-ducible if and only if both G +0 and G − act bijectively. From the realisation (3.18), thebasis (5.1) of R M ( λ ) top and Equation (5.5), we see that G +0 always acts bijectively:(5.18) G +0 ( u ⊗ e − j +( λ + n ) c ) = u ⊗ e c e − j +( λ + n ) c = u ⊗ e − j +( λ + n +1) c = 0 .R M ( λ ) top is therefore uniserial, meaning that its submodules S i form a linear chain underinclusion: 0 ⊂ S ⊂ S ⊂ · · · ⊂ R M ( λ ) top .Suppose that R M ( λ ) was decomposable, hence that R M ( λ ) = N ⊕ N ′ for nonzerosubmodules N and N ′ . Then, N ∩ R M ( λ ) top and N ′ ∩ R M ( λ ) top are both nonzero sub-modules of R M ( λ ) top , because R M ( λ ) has only top-submodules (Theorem 5.9). But, theirintersection is clearly zero, in contradiction to R M ( λ ) top being uniserial. We thereforeconclude that R M ( λ ) is indecomposable.Now, any highest-weight vector of R M ( λ ) must be in R M ( λ ) top as otherwise the sub-module it generates would contradict Theorem 5.9. But, (5.18) shows that R M ( λ ) top hasno highest-weight vectors. The situation is similar for conjugate highest-weight vectorsexcept that a calculation somewhat more involved than (5.18) gives(5.19) G − ( u ⊗ e − j +( λ + n ) c ) = p ∆ ,wk ( λ + n ) u ⊗ e − j +( λ + n − c . If p ∆ ,wk ( λ + n ) = 0 for any n ∈ Z , then G − acts bijectively and so R M ( λ ) is irreducible.On the other hand, p ∆ ,wk ( λ + n ) = 0 for some n ∈ Z implies that u ⊗ e − j +( λ + n ) c is aconjugate highest-weight vector in R M ( λ ) generating a nonzero proper submodule. (cid:3) Irreducible submodules of R M ( λ ) . The result of the previous section explicitlyrealises irreducible relaxed highest-weight BP k -modules. We now show that we can simi-larly realise irreducible highest-weight BP k -modules by analysing submodules of R M ( λ ),when the latter is reducible. Proposition 5.13.
Suppose that M is an irreducible highest-weight Z k -module withhighest-weight vector u and that R M ( λ ) is reducible. Choose µ ∈ λ + Z with maxi-mal real part such that u ⊗ e − j + µc is a conjugate highest-weight vector. Then, u ⊗ e − j + µc generates an irreducible conjugate highest-weight submodule of R M ( λ ) .Proof. By Theorem 5.12, R M ( λ ) being reducible implies that there exist conjugatehighest-weight vectors of the form u ⊗ e − j + µc , so we may indeed choose µ maximal.Then, u ⊗ e − j + µc is, up to nonzero multiples, the unique conjugate highest-weight vectorin the conjugate highest-weight submodule C = BP k · ( u ⊗ e − j + µc ) of R M ( λ ). Moreover, A (BP k ) · ( u ⊗ e − j + µc ) is an irreducible A (BP k )-submodule of R M ( λ ) top .Choose a nonzero element v ∈ C . Then, D = BP k · v ⊆ C is a nonzero submodule of R M ( λ ) and thus D ∩ R M ( λ ) top = 0, by Theorem 5.9. Since(5.20) D ∩ R M ( λ ) top ⊆ C ∩ R M ( λ ) top = A (BP k ) · ( u ⊗ e − j + µc )the irreducibility of the latter implies that u ⊗ e − j + µc ∈ D , whence D = C . This showsthat every nonzero element of C is cyclic, so this submodule is irreducible. (cid:3) Note that if µ is chosen maximal, as in Proposition 5.13, then the irreducible conjugatehighest-weight submodule generated by u ⊗ e − j + µc has an infinite-dimensional top space. Proposition 5.14.
Every irreducible conjugate highest-weight BP k -module whose topspace is infinite-dimensional may be explicitly realised as a submodule of R M ( λ ) , forsome irreducible highest-weight Z k -module M and some λ ∈ C .Proof. Let C be an arbitrary irreducible conjugate highest-weight BP k -module withinfinite-dimensional top space and let v denote its conjugate highest-weight vector. Then,a basis of C top is given by the ( G +0 ) n v with n ≥ u is the highest-weight vector of some irreducible highest-weight Z k -module M and∆ and w are its T - and W -eigenvalues, then Proposition 3.4 and Theorem 3.6 give(5.21) J ( u ⊗ e − j + µc ) = (cid:18) µ − k + 33 (cid:19) u ⊗ e − j + µc and L ( u ⊗ e − j + µc ) = (cid:18) ∆ + 2 k + 33 (cid:19) u ⊗ e − j + µc . We may therefore arrange for u ⊗ e − j + µc to have the same J - and L -eigenvalues as v by choosing µ and ∆ in accordance with (5.21). Moreover, if we specialise λ to µ and w to the unique root of the linear (in w ) polynomial p ∆ ,wk ( µ ), see (5.17), then u ⊗ e − j + µc isa conjugate highest-weight vector in R M ( λ ), by Theorem 5.12.This shows that u ⊗ e − j + µc generates a conjugate highest-weight submodule V of R M ( λ ) whose irreducible quotient is isomorphic to C . It moreover generates a A (BP k )-module W = V ∩ R M ( λ ) top with basis u ⊗ e − j +( µ + n ) c , n ≥
0, whose irreducible quotientis isomorphic to the A (BP k )-module C top . Comparing bases, we see that W ∼ = C top isirreducible and so µ must be the maximal solution in λ + Z of p ∆ ,wk ( µ ) = 0 (otherwise, V would have another conjugate highest-weight vector in V ∩ R M ( λ ) top contradictingirreducibility). But now Proposition 5.13 shows that V is irreducible, hence V ∼ = C asrequired. (cid:3) Of course, explicit realisations of the irreducible conjugate highest-weight modules leadto explicit realisations of the irreducible highest-weight modules as well, via the conjuga-tion functor of BP k . Similar to spectral flow functors, this arises from the automorphismof the mode algebra corresponding to(5.22) G + ( z ) G − ( z ) , J ( z )
7→ − J ( z ) − k + 33 z − ,G − ( z )
7→ − G + ( z ) , L ( z ) L ( z ) − ∂J ( z ) − J ( z ) z − . This then realises all the irreducible highest-weight BP k -modules with infinite-dimensionaltop spaces, but as submodules of the conjugates of the R M ( λ ). Remark 5.15.
Suppose now that p ∆ ,wk has at least two roots in λ + Z and let µ and µ ′ < µ be the maximal and next-to-maximal root, respectively. Then, Proposition 5.13shows that u ⊗ e − j + µc generates an irreducible conjugate highest-weight module N of R M ( λ ) while Theorem 5.12 shows that u ⊗ e − j + µ ′ c generates a conjugate highest-weightmodule N ′ that contains N . Despite the fact that N ′ top /N top is a finite-dimensionalirreducible A (BP k )-module, it does not necessarily follow that N ′ /N is an irreducibleBP k -module.The issue here is that N ′ , and hence R M ( λ ), may contain a subsingular vector which,in the language developed here, would generate a submodule that is not top-generated.The upshot is that one can use such quotients to identify (conjugate) highest-weightmodules with finite-dimensional top spaces but that this does not amount to a concreterealisation. Instead, one can employ spectral flow. Corollary 5.16.
Given an irreducible highest-weight BP k -module N , one of the followingpossibilities occurs: • N may be realised as a submodule of σ ℓ ( R M ( λ )) ∼ = M ⊗ Π ℓ − ( λ ) , for some ℓ ∈ Z ≥ ,some irreducible highest-weight Z k -module and some λ ∈ C . • The σ − ℓ ( N ) have finite-dimensional top spaces for all ℓ ∈ Z ≥ .Proof. This follows from the assertion, easily checked using (2.7) and (2.8), that thespectral flow map σ − takes a highest-weight vector to a conjugate highest-weight vector.For example, G + n v = 0 for n ≥ G + n +1 σ − ( v ) = σ − (cid:0) σ ( G + n +1 ) v (cid:1) = σ − (cid:0) G + n v (cid:1) = 0 .σ − ( N ) is thus an irreducible conjugate highest-weight module. If its top space is finite-dimensional, then it is also a highest-weight module and so we may apply σ − to itshighest-weight vector.Iterating, we find that either the σ − ℓ ( N ), with ℓ ∈ Z ≥ , all have finite-dimensional topspaces or we arrive at an irreducible conjugate highest-weight module σ − ℓ ( N ) with aninfinite-dimensional top space. In the latter case, σ − ℓ ( N ) embeds into some R M ( λ ), byProposition 5.14. Since spectral flow functors are invertible, we conclude that N embedsinto σ ℓ ( R M ( λ )), as desired. (cid:3) It follows that we can realise any given irreducible highest-weight BP k -module, aslong as its negative spectral flow orbit does not consist exclusively of modules withfinite-dimensional top spaces. A generic orbit will not have this property and thereforeirreducible highest-weight modules are generically realisable. However, there are somehighest-weight modules that cannot be realised in this way. In particular, the irreduciblehighest-weight BP k -modules with 2 k + 3 ∈ Z ≥ are examples because their top spacesare always finite-dimensional (see Remark 6.4 below).We conclude with an example that illustrates these realisations. As in [49], let usparametrise the highest weights of the irreducible highest-weight Z k -modules by(5.24) ∆ = ( r − ts ) + ( r − ts )( r ′ − ts ′ ) + ( r ′ − ts ′ ) t − ( t − t and w = (cid:0) r − ts − ( r ′ − ts ′ ) (cid:1)(cid:0) r − ts ) + ( r ′ − ts ′ ) (cid:1)(cid:0) r − ts + 2( r ′ − ts ′ ) (cid:1) , where r, r ′ , s, s ′ ∈ C and t = k + 3. By direct calculation, we see that the polynomial in(5.17) factorises. Lemma 5.17.
We have (5.25a) p ∆ ,wk ( x ) = − ( x − x )( x − x )( x − x ) , where (5.25b) x = t − − r − ts − ( r ′ − ts ′ )3 ,x = t − r − ts ) + r ′ − ts ′ ,x = t − − r − ts + 2( r ′ − ts ′ )3 so x − x = r − ts,x − x = r ′ − ts ′ ,x − x = r + r ′ − t ( s + s ′ ) . Example 5.18.
One sees that if k / ∈ Z , s = s ′ = 1 and r, r ′ ∈ Z > , then the roots x i , i = 1 , ,
3, lie in different cosets of C / Z . Each in therefore maximal in the sense ofProposition 5.13, so this proposition and Theorem 5.12 show that, for each i = 1 , , C i = BP k · ( u ⊗ e − j + x i c ) is an irreducible conjugate highest-weight BP k -module with an infinite-dimensional topspace. Applying spectral flow, it follows that σ ( C i ) is an irreducible highest-weightmodule, for each i . Detailed calculation shows that σ ( C ) and σ ( C ) always have finite-dimensional top spaces, indeed of dimensions r ′ and r respectively. The situation for σ ( C ) is more subtle: if 3 t is an integer larger than r + r ′ , then the top space is finite-dimensional (with dimension 3 t − r − r ′ ); otherwise it is infinite-dimensional.In our forthcoming publications [7], we shall present a detailed study of the structureof R M ( λ ).6. Realisation of the vertex algebra BP k and its relaxed modules Recall from Theorem 3.6 that we have established an embedding φ k of the univer-sal Bershadsky–Polyakov algebra BP k , k = −
3, in the tensor product of the universalZamolodchikov algebra Z k and the lattice vertex algebra Π. It is natural to ask if thisrealisation descends to the simple quotients, that is if BP k embeds in Z k ⊗ Π. We shallshow that the answer is frequently, but not always, yes.The answer is obviously yes if BP k is already simple. By [33, Thms. 0.2.1 and 9.1.2],BP k is not simple if and only if the (noncritical) level k satisfies(6.1) k + 3 = p ′ p , for some coprime p ∈ Z ≥ and p ′ ∈ Z ≥ . For these levels, we consider the projection π k : Z k → Z k and the composition(6.2) ψ k : BP k φ k ֒ −→ Z k ⊗ Π π k ⊗ id −−− ։ Z k ⊗ Πof vertex operator algebra morphisms. Since ψ k maps the vacuum of BP k to the vacuumof Z k ⊗ Π, it is not zero. We shall investigate when im ψ k is simple.The following lemma is a version of Lemma 8.1 from [6]. Lemma 6.1.
The vector ( G + − ) n , n > , is singular in BP k if and only if either n = k +2 and k ∈ {− , , , , . . . } or n = 2( k + 2) and k ∈ {− , − , − , , , . . . } .Proof. This is a straightforward computation using the commutation relations (2.4).Since the J m , L m , G − m +1 and G + m − , with m >
0, clearly annihilate ( G + − ) n , we onlyneed to calculate G − ( G + − ) n . This is easy (and has already appeared in the proof of[6, Lem. 8.1]): (cid:3) (6.3) G − ( G + − ) n = − n ( n − k − n − k − G + − ) n − . Theorem 6.2. BP k embeds into Z k ⊗ Π if and only if k + 3 / ∈ Z ≥ .Proof. Suppose that im ψ k has a nonzero proper ideal I . Then, I is a submodule ofZ k ⊗ Π = Z k ⊗ Π (0). Here, we may regard I and Z k ⊗ Π as BP k -modules, by (6.2),noting that this makes ψ k into a BP k -module homomorphism.Applying the spectral flow map σ − = id ⊗ γ − of BP k ( γ − is a spectral flow map ofΠ, see (3.16)), it follows that σ − ( I ) is a nonzero submodule of(6.4) (1 ⊗ γ − )(Z k ⊗ Π (0)) ∼ = Z k ⊗ Π − (0) = R Z k (0) . Since Z k is an irreducible Z k -module, it has only top-submodules. The BP k -module R Z k (0) therefore also has only top-submodules, by Theorem 5.9. Thus,(6.5) R Z k (0) top ∩ σ − ( I ) = 0 . In other words, there exists n ∈ Z such that ⊗ e − j + nc ∈ σ − ( I ). Applying σ = id ⊗ γ ,we conclude that ⊗ e nc ∈ I . As I ⊂ im ψ k and im ψ k is a homomorphic image of BP k , I has nonnegative conformalweights. Thus, n ∈ Z ≥ . However, n = 0 implies that I = im ψ k , a contradiction.Therefore, we have(6.6) ψ k (cid:0) ( G + − ) n (cid:1) = ⊗ e nc ∈ I, for some n ∈ Z > , using the explicit realisation of Theorem 3.6.Now choose n ∈ Z > minimal such that (6.6) holds. Then, ψ k (cid:0) ( G + − ) n (cid:1) is annihi-lated by G − , because the result must be proportional to ψ k (cid:0) ( G + − ) n − (cid:1) which is 0 byminimality. But, it is also annihilated by the J m , L m , G − m +1 and G + m − , with m >
0. Weconclude that ψ k (cid:0) ( G + − ) n (cid:1) is a singular vector in I (regarded as a BP k -module). As ψ k (cid:0) ( G + − ) n − (cid:1) = ⊗ e ( n − c is nonzero, so is ( G + − ) n ∈ BP k .If 2 k + 3 / ∈ Z ≥ , then this is impossible by Lemma 6.1 and hence im ψ k is simple. Onthe other hand, if k does have this form, then there exists n ∈ Z > such that ⊗ e nc isa singular vector in im ψ k generating a proper nonzero ideal. Thus, im ψ k is not simplein this case. (cid:3) A consequence of Theorems 5.12 and 6.2 is that we get families of relaxed highest-weight modules for the simple Bershadsky–Polyakov vertex operator algebra BP k , atleast when p = 1 , Theorem 6.3.
Assume that k + 3 / ∈ Z ≥ and that M is an irreducible highest-weight Z k -module M whose highest-weight vector u has T -eigenvalue ∆ and W -eigenvalue w .Then: • R M ( λ ) is an indecomposable BP k –module. • R M ( λ ) is irreducible if and only if the polynomial p ∆ ,wk , defined in (5.17) , has no rootsin the coset λ + Z . • R M ( λ ) has no highest-weight vectors and its conjugate highest-weight vectors are pre-cisely the u ⊗ e − j + µc for which µ ∈ λ + Z satisfies p ∆ ,wk ( µ ) = 0 . Remark 6.4. • When k = − , Z − / = C and so Theorem 6.2 gives one family of relaxed highest-weight BP − / -modules. This family was first constructed in [6, Thm. 7.2]. • When k = − , we also have Z − / = C , hence one family of relaxed highest-weightBP − / -modules. This family may be constructed by noting that BP − / is a Z -orbifold of the rank-1 bosonic ghost system [6, Prop. 5.9] and that this ghost systemadmits a family of relaxed highest-weight modules [45]. • When p = 2, so k ∈ {− , − , , . . . } , BP k is rational [11]. It therefore has no suchfamilies of relaxed highest-weight modules. • When k = −
1, BP − is isomorphic to the rank-1 Heisenberg vertex operator algebra[5]. It therefore also has no such families of relaxed highest-weight modules. • For k ∈ Z ≥ , the relations ( G ±− ) k +2 = 0 in BP k ([6] and Lemma 6.1 above) imply thatthe Zhu algebra A (BP k ) has only finite-dimensional irreducible modules, more preciselymodules of dimension at most k + 2. As there are no infinite-dimensional irreducible A (BP k )-modules, BP k likewise has no such families of relaxed highest-weight modules. Remark 6.5.
In the case of the affine vertex algebras V k ( sl ) and L k ( sl ), the irre-ducibility of the relaxed highest-weight modules was discussed in [2] and [38], usingdifferent techniques. As a consequence of our results (with some minor modifications),we can now give a new proof of the irreducibility of these modules. In particular, [2] showed that all the relaxed highest-weight L k ( sl )-modules could berealised in the form E λr,s = M r,s ⊗ Π − ( λ ), where(6.7) k + 2 = p ′ p , for some coprime p, p ′ ∈ Z ≥ , and, for r = 1 , . . . , p − s = 1 , . . . , p ′ − M r,s is the irreducible highest-weightVirasoro module of central charge and conformal weight(6.8) c p,p ′ = 1 − p − p ′ ) pp ′ and h r,s = ( p ′ r − ps ) − ( p − p ′ ) pp ′ , respectively. Completely analogous arguments to those resulting in Theorems 5.12and 6.3 then prove that E λr,s is irreducible if and only if λ / ∈ λ ± r,s + Z , where λ ± r,s isas in [2, Sec. 7]. 7. Critical-level results
A critical level definition of the Bershadsky–Polyakov algebra was investigated in [10,32].
Definition 7.1.
At the critical level k = −
3, the Bershadsky–Polyakov vertex algebraBP − is the universal vertex algebra generated by fields S , J , G + and G − subject to thefollowing operator product expansions:(7.1) J ( z ) J ( w ) ∼ − z − w ) , J ( z ) G ± ( w ) ∼ ± G ± ( w ) z − w , G ± ( z ) G ± ( w ) ∼ ,S ( z ) G ± ( w ) ∼ , S ( z ) J ( w ) ∼ , S ( z ) S ( w ) ∼ ,G + ( z ) G − ( w ) ∼ z − w ) − J ( w )( z − w ) + 3 : J ( w ) J ( w ) : − ∂J ( w ) − S ( w ) z − w . Remark 7.2.
We can formally obtain the definition of BP − given above by substituting S = ( k + 3) L into Definition 2.1 and then setting k = − V − ( sl ) by Z ( V − ( sl )). It is a commutative vertex algebragenerated by two fields [27](7.2) S ( z ) = X n ∈ Z S n z − n − and S ( z ) = X n ∈ Z S n z − n − . Define the operator d ∈ End( Z ( V − ( sl ))) by(7.3) [ d, S mn ] = − nS mn , n ∈ Z , m = 2 , . By setting d = 0, this gives Z ( V − ( sl )) the structure of a Z ≥ -graded vertex algebra.Direct calculation now gives the following critical-level version of Theorem 3.6. Theinjectivity follows in exactly the same way as in Section 4. Theorem 7.3.
At the critical level k = − , there is an injective vertex algebra homo-morphism φ − : BP − → Z ( V − ( sl )) ⊗ Π , uniquely determined by (7.4) G + ⊗ e c , J ⊗ j, S S ⊗ ,G − (cid:18) S − ∂S (cid:19) ⊗ e − c + S ⊗ i ( − e − c − ⊗ (cid:0) i − − i ( − i ( − + 2 i ( − (cid:1) e − c . Remark 7.4.
It is proved in [27] that Z ( V − ( sl )) is isomorphic to the critical-levelprincipal W-algebra W − ( sl ; f pr ) = Z − . Since Z ( V − ( sl )) is a commutative vertex algebra generated by S ( z ) and S ( z ), itsirreducible modules are 1-dimensional and parametrised by χ , χ ∈ C (( z )) such that(7.5) χ m ( z ) = X n ∈ Z χ m ( n ) z − n − m , m = 2 , , and S mn acts as multiplication by χ m ( n ) on the irreducible module. We shall thereforedenote the irreducible modules by L χ ,χ .Consider the “relaxed” BP − -module(7.6) R χ ,χ ( λ ) = L χ ,χ ⊗ Π − ( λ )(actually, this module is of Wakimoto type [29]). The question of when this moduleis irreducible may be treated using methods from [1] and we hope to study this inforthcoming publications.Here, we consider the case in which R χ ,χ ( λ ) is Z ≥ -gradable. Since Theorem 7.3gives BP − the grading defined by d + t , where t is the conformal vector of Π (seeSection 3.2), R χ ,χ ( λ ) will be Z ≥ -gradable if and only if L χ ,χ is gradable by d . This,in turn, requires that the S mn , m = 2 , n ∈ Z , act trivially unless n = 0. Wetherefore conclude that R χ ,χ ( λ ) is Z ≥ -gradable if and only if(7.7) χ ( z ) = ∆ z and χ ( z ) = wz , for some ∆ , w ∈ C . Moreover, R χ ,χ ( λ ) top = L χ ,χ ⊗ Π − ( λ ) top will then be an A (BP − )-module. Theorem 7.5.
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J. Amer. Math. Soc. ,9:237–302, 1996.(Draˇzen Adamovi´c)
Department of Mathematics, Faculty of Science, University ofZagreb, Bijeniˇcka 30
E-mail address : [email protected] (Kazuya Kawasetsu) Priority Organization for Innovation and Excellence, KumamotoUniversity, 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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