The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector

We classify irreducible σ-twisted modules for the N = 1 super triplet vertex operator superalgebra SW(m) introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of σ-twisted modules are also determined. These results, combined with ou...

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Datum:	2008
Hauptverfasser:	Adamovic, D., Milas, A.
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Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1480112019-02-17T01:25:32Z The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector Adamovic, D. Milas, A. We classify irreducible σ-twisted modules for the N = 1 super triplet vertex operator superalgebra SW(m) introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of σ-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the SL(2,Z)-closure of the space spanned by irreducible characters, irreducible supercharacters and σ-twisted irreducible characters is (9m + 3)-dimensional. We present strong evidence that this is also the (full) space of generalized characters for SW(m). We are also able to relate irreducible SW(m) characters to characters for the triplet vertex algebra W(2m + 1), studied in [Adamovic D., Milas A., Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857]. 2008 Article The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector / D. Adamovic, A. Milas // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 30 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B69; 17B67; 17B68; 81R10 http://dspace.nbuv.gov.ua/handle/123456789/148011 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	We classify irreducible σ-twisted modules for the N = 1 super triplet vertex operator superalgebra SW(m) introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of σ-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the SL(2,Z)-closure of the space spanned by irreducible characters, irreducible supercharacters and σ-twisted irreducible characters is (9m + 3)-dimensional. We present strong evidence that this is also the (full) space of generalized characters for SW(m). We are also able to relate irreducible SW(m) characters to characters for the triplet vertex algebra W(2m + 1), studied in [Adamovic D., Milas A., Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857].
format	Article
author	Adamovic, D. Milas, A.
spellingShingle	Adamovic, D. Milas, A. The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Adamovic, D. Milas, A.
author_sort	Adamovic, D.
title	The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector
title_short	The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector
title_full	The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector
title_fullStr	The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector
title_full_unstemmed	The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector
title_sort	n = 1 triplet vertex operator superalgebras: twisted sector
publisher	Інститут математики НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/148011
citation_txt	The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector / D. Adamovic, A. Milas // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 30 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-07-11T03:47:47Z
last_indexed	2025-07-11T03:47:47Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 087, 24 pages The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector? Dražen ADAMOVIĆ † and Antun MILAS ‡ † Department of Mathematics, University of Zagreb, Croatia E-mail: adamovic@math.hr URL: http://web.math.hr/∼adamovic ‡ Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY 12222, USA E-mail: amilas@math.albany.edu URL: http://www.albany.edu/∼am815139 Received August 31, 2008, in final form December 05, 2008; Published online December 13, 2008 Original article is available at http://www.emis.de/journals/SIGMA/2008/087/ Abstract. We classify irreducible σ-twisted modules for the N = 1 super triplet vertex operator superalgebra SW(m) introduced recently [Adamović D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of σ-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the SL(2,Z)-closure of the space spanned by irreducible characters, irreducible supercharacters and σ-twisted irreducible characters is (9m + 3)-dimensional. We present strong evidence that this is also the (full) space of generalized characters for SW(m). We are also able to relate irreducible SW(m) characters to characters for the triplet vertex algebra W(2m+1), studied in [Adamović D., Milas A., Adv. Math. 217 (2008), 2664–2699, arXiv:0707.1857]. Key words: vertex operator superalgebras; Ramond twisted representations 2000 Mathematics Subject Classification: 17B69; 17B67; 17B68; 81R10 1 Introduction Many constructions and results in vertex algebra theory are easily extendable to the setup of vertex superalgebras by simply adding adjective “super”. Still, there are results that deviate from this “super-principle” and new ideas are needed compared to the non-super case. For example, modular invariance for vertex operator superalgebras requires inclusion of supercharacters of (untwisted) modules, and more importantly, the characters of σ-twisted modules [13], where σ is the canonical parity automorphism. Since the construction and classification of σ-twisted modules is more or less independent of the untwisted construction, many aspects of the theory need to be reworked for the twisted modules (e.g., twisted Zhu’s algebra [29]). In fact, even for the free fermion vertex operator superalgebra, construction of σ-twisted modules is far from being trivial (see [13] for details). Present work is a natural continuation of our very recent paper [5], where we introduced a new family of C2-cofinite N = 1 vertex operator superalgebra that we call the supertriplet family SW(m), m ≥ 1. In this installment we focus on σ-twisted SW(m)-modules and their irreducible characters. The σ-twisted SW(m)-modules are usually called modules in the Ramond ?This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html mailto:adamovic@math.hr http://web.math.hr/~adamovic mailto:amilas@math.albany.edu http://www.albany.edu/~am815139 http://www.emis.de/journals/SIGMA/2008/087/ http://arxiv.org/abs/0712.0379 http://arxiv.org/abs/0707.1857 http://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html 2 D. Adamović and A. Milas sector, while untwisted (ordinary) modules are referred to as modules in the Neveu–Schwarz sector, in parallel with two distinct N = 1 superconformal algebras. Of course, Ramond sector has been studied for various (mostly rational) vertex operators superalgebras (e.g., [24, 10, 27], etc.). We should also point out that Ramond twisted represen- tation of certain vertex W-algebras were recently investigated in [8] and [23]. Here are our main results: Theorem 1.1. The twisted Zhu’s algebra Aσ(SW(m)) is finite-dimensional with precisely 2m+1, non-isomorphic, Z2-graded irreducible modules. Consequently, SW(m) has 2m+1, non-isomor- phic, Z2-graded irreducible σ-twisted modules. By using embedding structure of N = 1 Feigin–Fuchs modules [20] we can also easily compute the characters of σ-twisted SW(m)-modules. Equipped with these formulas, results from [5] about untwisted modules and supercharacters, and transformation formulas for classical Jacobi theta functions we are able to prove: Theorem 1.2. The SL(2,Z)-closure of the space of characters, (untwisted) supercharacters and σ-twisted SW(m) characters is (9m+ 3)-dimensional. Conjecturally, this is also the space of certain generalized characters studied in [28] (strictly speaking pseudotraces are yet-to-be defined in the setup of σ-twisted modules). A closely related conjecture is Conjecture 1.1. Let Z(A) denote the center of associative algebra A and T (A) = A/[A,A] the trace group of A. Then dim(Z(A(SW(m)))) + dim(Z(Aσ(SW(m)))) = 9m+ 3, and dim(T (A(SW(m)))) + dim(T (Aσ(SW(m)))) = 9m+ 3, where A(SW(m)) is the untwisted Zhu’s associative algebra of SW(m) (cf. [5]). The conjecture would follow if we knew more about structure of logarithmic SW(m)-modules. Here is a short outline of the paper. In Section 2 we recall the standard results about the σ-twisted Zhu’s algebra, the free fermion vertex superalgebra F , and the construction of the σ-twisted F -module(s). In Section 4 we focus on σ-twisted modules for the super singlet vertex algebra introduced also in [5]. Sections 5 and 6 deal with the construction and classification of σ-twisted SW(m)-modules. Finally, in Section 7 we derive characters formulas for irreducible σ-twisted SW(m)-modules, discuss modular invariance, and relate our characters with the cha- racters for the triplet vertex algebra W(2m+ 1) (see for instance [4] and [15]). Needless to say, this paper is largely continuation of [4] and especially [5]. Thus the reader is strongly encouraged to consult [5], where we studied SW(m) in great details. 2 Preliminaries Let V = V0 ⊕ V1 be a vertex operator superalgebra, where as usual V0 is the even and V1 is the odd subspace (cf. [13, 17, 21]). Every vertex operator superalgebra has the canonical parity automorphism σ, where σV0 = 1 and σV1 = −1. This leads to the notion of σ-twisted V -modules, well recorded in the literature (see for example [14] and [29]). As in the untwisted case, a large part of representation theory of σ-twisted V -modules can be analyzed via the σ-twisted Zhu’s algebra whose construction we recall here. The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 3 Within the same setup, consider the subspace O(V ) ⊂ V , spanned by elements of the form Resx (1 + x)deg(u) x2 Y (u, x)v, where u ∈ V is homogeneous. It can be easily shown that Resx (1 + x)deg(u) xn Y (u, x)v ∈ O(V ) for n ≥ 2. Then, the vector space Aσ(V ) = V/O(V ) is equipped with an associative algebra structure via u ∗ v = Resx (1 + x)deg(u) x Y (u, x)v (see [29]). An important difference between the untwisted associative algebra A(V ) and Aσ(V ) is that the latter is Z2-graded, so Aσ(V ) = A0 σ(V )⊕A1 σ(V ). It is also not clear that Aσ(V ) 6= 0, while A(V ) is always nontrivial. We shall often use [a] ∈ Aσ(V ) for the image of a ∈ V under the natural map V −→ Aσ(V ). The result we need is [29]: Theorem 2.1. There is a one-to-one correspondence between irreducible Z≥0-gradable σ-twisted V -modules and irreducible Aσ(V )-modules. In the theorem there is no reference to graded Aσ(V )-modules. For practical purposes we shall need a slightly different version of the above theorem, because some modules are more natural if considered as Z2-graded modules (shorthand, graded modules). Theorem 2.2. There is a one-to-one correspondence between graded irreducible Z≥0-gradable σ-twisted V -module and graded irreducible Aσ(V )-module. Proof. The proof mimics the non-graded case, so we shall omit details. As in the non-graded case, by applying Zhu’s theory, from an irreducible graded Aσ(V )-module U we construct a σ- twisted graded V -module L(U) (but of course in the process of getting L(U) will be moding out by the maximal graded submodule). On the other hand, if M is an irreducible graded V -module, then the top component Ω(M) is clearly a graded Aσ(V )-module. Suppose that Ω(M) is not graded irreducible, therefore there is a graded submodule Ω(M ′). Then we let M ′ = U(V [σ])Ω′(M), where U(V [σ]) is the enveloping algebra of the Lie algebra V [σ]. But M ′ is a proper graded submodule of M , so we get a contradiction. � Our goals are to describe the structure of Aσ(SM(1)), where SM(1) is the super singlet vertex algebra [5], and to discuss Aσ(SW(m)) (in fact, we have a very precise conjecture about the structure of Aσ(SW(m))). Let us recall (see [5] for details) that both SM(1) and SW(m) are N = 1 superconformal vertex operator superalgebras, with the superconformal vector τ . In other words, if we let Y (τ, x) = ∑ n∈Z+1/2 G(n)x−n−3/2, then G(n) and L(m) close the N = 1 Neveu–Schwarz superalgebra. It is not hard to see that the following hold: (L(−m− 2) + 2L(−m− 1) + L(−m))v ∈ O(V ), m ≥ 2, (L(0) + L(−1))v ∈ O(V ), L(−m)v ≡ (−1)m((m− 1)(L(−2) + L(−1)) + L(0))v mod O(V ), m ≥ 2,∑ n≥0 (3 2 n ) G(−3/2− i+ n)v ∈ O(V ), i ≥ 1, (2.1) where in all formulas v is a vector in N = 1 vertex operator superalgebra V . 4 D. Adamović and A. Milas To illustrate how to use twisted Zhu’s algebra, let us classify Z-graded σ-twisted modules for the (neutral) free fermion vertex operator superalgebra F , used in [5]. The next result is known so we will not provide its proof here. Proposition 2.1. We have Aσ(F ) ∼= C[x]/ ( x2− 1 2 ) where x = [φ(−1/2)1], an odd generator in the associative algebra. This result in particular implies there are precisely two irreducible σ-twisted F -modules: M±. These two modules can be constructed explicitly. As vector spaces M± = ⊕n≥0Λn(φ(−1), φ(−2), . . . ), where Λ∗ is the exterior algebra, which is also a Z≥0-graded module for the Clifford algebra K spanned by φ(n), n ∈ Z with anti-bracket relations {φ(m), φ(n)} = δm+n,0. The only difference between M+ and M− is in the action of φ(0) on the one-dimensional top subspace M±(0). More precisely we have φ(0)\|M±(0) = ± 1√ 2 . However, notice that M± are not Z2-graded thus it is more natural to examine graded Aσ(F )-modules. There is a unique such module (up to parity switch), spanned by 1R and φ(0)1R. Thus M = ⊕n≥0Λn(φ(0), φ(−1), φ(−2), . . . ) = M+ ⊕M−. Then 1R is a cyclic vector in M , i.e., M = K.1R. Moreover, M± = K.1±, where 1± = 1R ± √ 2φ(0)1R. Next we describe the twisted vertex operators Y : F ⊗M −→M [[x1/2, x−1/2]]. Details are spelled out in [13], here we only give the explicit formula. Define first Y ( φ ( −n− 1 2 ) 1, x ) = 1 n! ( d dx )n (∑ m∈Z φ(m)x−m−1/2 ) , acting on M and use (fermionic!) normal ordering • • • • to define Y ( φ ( −n1 − 1 2 ) · · ·φ ( −nk − 1 2 ) 1, x ) = • •Y ( φ ( −n1 − 1 2 ) , x ) · · ·Y ( φ ( −nk − 1 2 ) , x ) • • , and extend Y by linearity on all of F (see [13] and [14] for details, especially about normal ordering). Then, we let Y (v, x) := Y (e∆xv, x), where ∆x = 1 2 ∑ m,n≥0 Cm,nφ ( m+ 1 2 ) φ ( n+ 1 2 ) x−m−n−1, Cm,n = 1 2 m− n m+ n+ 1 ( −1/2 m )( −1/2 n ) . Let us fix the Virasoro generator ωs = 1 2φ ( −3 2 ) φ ( −1 2 ) 1. Then e∆xωs = ωs + 1 16x −21. The following lemma will be important in the rest of the paper. The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 5 Lemma 2.1. Let G(x1, x2) = ∞∑ m=0 ∞∑ n=0 Cm,nx m 1 x n 2 , where Cm,n are as above. Then G(x1, x2) = 1 2 ( (1 + x1)1/2(1 + x2)−1/2 − 1 x2 − x1 + (1 + x1)−1/2(1 + x2)1/2 − 1 x2 − x1 ) ∈ Q[[x1, x2]]. Proof. The lemma follows directly from the identity (x2 − x1)G(x1, x2) = 1 2(1 + x1)1/2(1 + x2)−1/2 + 1 2(1 + x1)−1/2(1 + x2)1/2 − 1. which can be easily checked by expanding both sides as power series in x1 and x2. � 3 Highest weight representations of the Ramond algebra The N = 1 Ramond algebra R is the infinite-dimensional Lie superalgebra R = ⊕ n∈Z CL(n) ⊕⊕ m∈Z CG(m) ⊕ CC with commutation relations (m,n ∈ Z): [L(m), L(n)] = (m− n)L(m+ n) + δm+n,0 m3 −m 12 C, [G(m), L(n)] = ( m− n 2 ) G(m+ n), {G(m), G(n)} = 2L(m+ n) + 1 3 (m2 − 1/4)δm+n,0C, [L(m), C] = 0, [G(m), C] = 0. The representation theory of the N = 1 Ramond algebra has been intensively studied first in [22] and other papers (cf. [25, 12], etc.). Assume that (c, h) ∈ C2 such that 24h 6= c. Let LR(c, h)± denote the irreducible highest weight R-module generated by the highest weight vector v±c,h such that G(n)v±c,h = ± √ h− c/24 δn,0v ± c,h, L(n)v±c,h = hδn,0v ± c,h (n ≥ 0). These modules can be considered as irreducible σ-twisted modules for the Neveu–Schwarz ver- tex operator superalgebra (cf. [24]). Since LR(c, h)± are not Z2-graded (notice that v±c,h are eigenvectors for G(0)), it is more useful to consider graded modules. It is not hard to show that the direct sum LR(c, h) = LR(c, h)+ ⊕ LR(c, h)− is in fact a Z2-graded R-module. Indeed, for Z2-graded subspaces take LR(c, h)0 = U(R) ( v+ c,h + 1√ h− c/24 v−c,h ) and LR(c, h)1 = U(R) ( v+ c,h − 1√ h− c/24 v−c,h ) Since LR(c, h) does not contain non-trivial Z2-graded submodules, we shall say that this module is Z2-graded irreducible. 6 D. Adamović and A. Milas Remark 3.1. Details about construction of LR(c, h) and its relation to irreducible non-graded modules can be found in [12] and [19]. In Section 5 we shall present free fields realization of modules LR(c, h) and LR(c, h)±. For i, n,m ∈ Z≥0, let c2m+1,1 = 3 2 − 12m2 2m+ 1 , h1,3 = 2m+ 1 2 , h2i+2,2n+1 := (2i+ 2− (2n+ 1)(2m+ 1))2 − 4m2 8(2m+ 1) + 1 16 . Let L(c2m+1,1, h) be the irreducible highest weight module for the Neveu–Schwarz algebra with central charge c and highest weight h. Then L(c2m+1,1, 0) is a simple vertex operator superalgebra, LR(c, h)± are irreducible σ-twisted L(c2m+1,1, 0)-modules, and LR(c, h) is Z2- graded irreducible σ-twisted L(c2m+1,1, 0)-module. 3.1 Intertwining operators among twisted modules If V is a vertex operator superalgebra and W1, W2 and W3 are three V -modules then we consider the space of intertwining operators ( W3 W1 W2 ) . It is perhaps less standard to study intertwining operators between twisted V -modules, so we recall the definition here (see [29, p. 120]). Definition 3.1. Let W1, W2 and W3 be σi-twisted V -modules, respectively, where σi is a finite order automorphism of order νi, with common period T . An intertwining operator of type( W3 W1 W2 ) is a linear map Y(·, x) : W1 −→ End(W2,W3){x}, such that Y(L(−1)w, x) = d dx Y(w, x), w ∈W1 and Jacobi identity holds 1 Tx0 T−1∑ p=0 δ (( x1 − x2 x0 )−1/T ωp T ) Y ν3(σp 1v, x1)Y(w1, x2) − (−1)ij 1 Tx0 T−1∑ p=0 δ (( −x2 + x1 x0 )−1/T ωp T ) Y(w1, x2)Y ν2(σp 1v, x1) = 1 Tx2 T−1∑ p=0 δ (( x1 − x0 x2 )1/T ωp T ) Y(Y ν1(σp 2v, x0)w1, x2), where v ∈ Vi, w1 ∈ (W1)j , i, j ∈ Z2 and ωT is a primitive T -th root of unity. We shall also need the following result on the fusion rules. The proof is completely analogous to that of Proposition 4.1 of [5] (see also [18]). Proposition 3.1. For every i = 0, . . . ,m− 1 and n ≥ 1 we have: the space I ( LR(c2m+1,1, h)± L(c2m+1,1, h1,3) LR(c2m+1,1, h2i+2,2n+1)± ) The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 7 is nontrivial only if h ∈ {h2i+2,2n−1, h2i+2,2n+1, h2i+2,2n+3}, and I ( LR(c2m+1,1, h)± L(c2m+1,1, h1,3) LR(c2m+1,1, h2i+2,1)± ) is nontrivial only if h = h2i+2,3. Similarly, for every i = 0, . . . ,m− 1 and n ≥ 2 we have: the space I ( LR(c2m+1,1, h)± L(c2m+1,1, h1,3) LR(c2m+1,1, h2i+2,−2n+1)± ) is nontrivial only if h ∈ {h2i+2,−2n−1, h2i+2,−2n+1, h2i+2,−2n+3}, and I ( LR(c2m+1,1, h)± L(c2m+1,1, h1,3) LR(c2m+1,1, h2i+2,−1)± ) is nontrivial only if h ∈ {h2i+2,−3, h2i+2,−1}. We have analogous result for fusion rules of type( LR(c2m+1,1, h) L(c2m+1,1, h1,3) LR(c2m+1,1, h2i+2,2n+1) ) . Proof. The proof goes along the lines in [5] (cf. [26]), with some minor modifications due to twisting. In fact, in order to avoid the twisted version of Frenkel–Zhu’s formula we can proceed in a more straightforward fashion. Because all modules in question are irreducible and because all intertwining operators we are interested in are of the form ( W3 W1 W2 ) , where W1 is untwisted (so ν1 = 0), the left-hand side in the Jacobi identity looks like the ordinary Jacobi identity for intertwining operators. Thus we can use commutator formula and the null vector conditions( −L(−1)G ( −1 2 ) + (2m+ 1)G ( −3 2 )) v1,3 = 0, G ( −1 2 )( −L(−1)G ( −1 2 ) + (2m+ 1)G ( −3 2 )) v1,3 = 0, which hold in L(c2m+1,1, h 1,3) (here v1,3 is the highest weight vector), to study the matrix coefficient f(x) = 〈w′,Y(v1,3, x)w〉, where w and w′ are highest weight vectors in appropriate modules. This leads to differential equations for f(x), which can be solved. The general solution is a linear combination of power functions xs, where s is a rational number. The rest follows by interpreting s in terms of conformal weights for the three modules involved in Y. � 4 σ-twisted modules for the super singlet algebra SM(1) In this section σ2 will denote the parity automorphism of F . Recall also the Heisenberg vertex operator algebra M(1). Then, as in [5] we equip the space M(1) ⊗ F with a vertex super- algebra structure such that the total central charge is 3 2(1 − 8m2 2m+1). We define the following superconformal and conformal vectors: τ = 1√ 2m+ 1 ( α(−1)1⊗ φ ( −1 2 ) 1 + 2m1⊗ φ ( −3 2 ) 1 ) , ω = 1 2(2m+ 1) (α(−1)2 + 2mα(−2))1⊗ 1 + 1⊗ ω(s). 8 D. Adamović and A. Milas Recall also the singlet superalgebra SM(1) obtained as the kernel of the screening operator Q̃ = ResxY ( e −α 2m+1 ⊗ φ ( −1 2 ) , x ) acting from M(1) ⊗ F to M(1) ⊗ e− α 2m+1 ⊗ F , where 〈α, α〉 = 2m + 1. The vertex operator superalgebra SM(1) is generated by τ = G ( −3 2 ) 1 and H = Qe−α, where Q = ResxY ( eα ⊗ φ ( −1 2 ) , x ) . We will also use ω and Ĥ, where the latter is proportional to G ( −1 2 ) H (for details see [5]). Consider the automoprhism σ = 1⊗σ2 of M(1)⊗F , acting nontrivially on the second tensor factor. This automorphism plainly preserves SM(1), thus we can study σ-twisted SM(1)- modules. It is clear that M(1, λ)⊗M is a graded σ-twisted SM(1)-module, and M(1, λ)⊗M± are σ-twisted SM(1)-modules, where M(1, λ) is as in [4] and λ ∈ C. We would like to classify irreducible SM(1)-modules by virtue of Zhu’s algebra. As usual, we denote by [a] ∈ Aσ(SM(1)) the image of a ∈ SM(1). Lemma 4.1. Let v±λ = vλ⊗1± be the highest weight vector in M(1, λ)⊗M±, where 〈α, λ〉 = t. Then the twisted generators act as G(0) · v±λ = ± t−m√ 2(2m+ 1) v±λ , L(0) · v±λ = ( t(t− 2m) 2(2m+ 1) + 1 16 ) v±λ , H(0) · v±λ = ± 1√ 2 ( t− 1/2 2m ) v±λ , Ĥ(0) · v±λ = m− t 2m+ 1 ( t− 1/2 2m ) v±λ . Proof. Recall from [5] the formulas H = S2m(α)⊗ φ ( −1 2 ) + S2m−1 ⊗ φ ( −3 2 ) + · · ·+ 1⊗ φ ( −2m− 1 2 ) , Ĥ = φ(1/2)S2m+1(α)φ ( −1 2 ) + w = S2m+1(α) + S2m−1(α)⊗ φ ( −3 2 ) φ ( −1 2 ) + · · ·+ 1⊗ φ ( −2m− 1 2 ) φ ( −1 2 ) . As in [5] we have Sr(α)(0) · v±λ = ( t r ) v±λ . Then we get H(0) · v±λ = ± 1√ 2 2m∑ n=0 ( t n )( −1/2 2m− n ) v±λ = ± 1√ 2 ( t− 1/2 2m ) v±λ . The last formula is proven similarly by using e∆x operator. � Here are the main result of this section. The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 9 Theorem 4.1. The Zhu’s algebra Aσ(SM(1)) is an associative algebra generated by [τ ], [ω], [H] and [Ĥ], where the following relations hold (here [ω] is central): [τ ]2 = [ω]− c2m+1,1 24 , [τ ] ∗ [H] = [H] ∗ [τ ] = − √ 2m+ 1 2 [Ĥ], [H] ∗ [H] = 1 2 (√ 2(2m+ 1)[τ ] +m− 1/2 2m )2 , [H] ∗ [H] = Cm m−1∏ i=0 ( [ω]− h2i+2,1 )2 , Cm = 22m−1(2m+ 1)2m (2m)!2 , (4.1) [Ĥ] ∗ [Ĥ] = 4 2m+ 1 Cm ( [ω]− c2m+1,1 24 )m−1∏ i=0 ( [ω]− h2i+2,1 )2 . In particular, Zhu’s algebra is commutative. Proof. First notice (cf. [5]) that G ( −1 2 ) F = − √ 2m+ 1φ ( −1 2 ) F = − √ 2m+ 1F̂ and consequently G ( −1 2 ) H = − √ 2m+ 1Ĥ. We also have the relation( (2m+ 1)G ( −3 2 ) − L(−1)G ( −1 2 )) H = 0, because H = v1,3 is a highest weight vector. By using (2.1) we obtain [τ ] ∗ [H] = [ G ( −3 2 ) H ] + 3 2 [ G ( −1 2 ) H ] = [ 1 2m+ 1 L(−1)G ( −1 2 ) H ] + 3 2 [ G ( −1 2 ) H ] = 1 2 [ G ( −1 2 ) H ] = − √ 2m+ 1 2 [Ĥ]. On the other hand, by using skew-symmetry we also have [H] ∗ [τ ] = − [ Resxx −1exL(−1)Y (τ,−x)(1 + x)2m+ 1 2H ] = − [ Resxx −1e−xL(0)Y (τ,−x)(1 + x)2m+ 1 2H ] = − ([ G ( −3 2 ) H ] − [( 2m+ 1 2 − (2m+ 1) ) G ( −1 2 ) H] ) = − [ G ( −3 2 ) H ] − 1 2 [ G ( −1 2 ) H ] = 1 2 [ G ( −1 2 ) H ] . Combined, we obtain [τ ] ∗ [H] = [H] ∗ [τ ]. � Notice that relation (4.1) can be written as [H] ∗ [H] = Cm m−1∏ i=0 ( [τ ]2 − (2i+ 1− 2m)2 8(2m+ 1) )2 , Cm 6= 0. Let C[a, b] denote the Z2-graded complex commutative associative algebra generated by odd vectors a, b. 10 D. Adamović and A. Milas Theorem 4.2. The associative algebra Aσ(SM(1)) is isomorphic to the Z2-graded commutative associative algebra C[a, b]/〈H(a, b)〉, where 〈H(a, b)〉 is (two-sided) ideal in C[a, b], generated by H(a, b) = b2 − Cm m−1∏ i=0 ( a2 − (2i+ 1− 2m)2 8(2m+ 1) )2 . Proof. The proof is similar to that of Theorem 6.1 of [2]. First we notice that we have a sur- jective homomorphism Φ : C[a, b] → Aσ(SM(1)), a 7→ [τ ], b 7→ [H]. It is easy to see that KerΦ is a Z2-graded ideal. We shall now prove that KerΦ = 〈H(a, b)〉. Evidently the generating element H(a, b) is even, so Theorem 4.1 gives 〈H(a, b)〉 ⊂ Ker Φ. Assume now that K(a, b) ∈ Ker Φ. By using division algorithm we get K(a, b) = A(a, b)H(a, b) +R(a, b), where A(a, b), R(a, b) ∈ C[a, b] and R(a, b) has degree at most 1 in b. Assume that R(a, b) 6= 0. Then R(a, b) = A(a)b+B(a) for certain polynomials A,B ∈ C[x]. We also notice that R(a, b) ∈ Ker Φ. Since KerΦ is Z2-graded ideal, we can assume that R(a, b) is homogeneous. If R(a, b) is an even element we have that A has odd degree and B has even degree, and therefore deg(B)− deg(A) is an odd natural number. (4.2) The case when R(a, b) is odd element again leads to formula (4.2). As in [2] we now shall evaluate R(a, b) on Aσ(SM(1))-modules and get A ( t−m√ 2(2m+ 1) )( t− 1/2 2m ) + √ 2B ( t−m√ 2(2m+ 1) ) = 0 ∀ t ∈ C. This implies that deg(B) − deg(A) = 2m. Contradiction. Therefore R(a, b) = 0 and K(a, b) ∈ 〈H(a, b)〉. The proof follows. � Remark 4.1. By using the same arguments as in [2] and [5], we can conclude that every irreducible Z≥0-gradable σ-twisted SM(1)-module is isomorphic to an irreducible subquotient of M(1, λ)⊗M±. By using the structure of twisted Zhu’s algebra Aσ(SM(1)) and the methods developed in [3], we can also construct logarithmic σ-twisted SM(1)-modules. 5 The N = 1 Ramond module structure of twisted VL ⊗ F -modules In this section we shall assume that the reader is familiar with basic results on twisted repre- sentations of lattice vertex superalgebras. Details can be found in [9, 10, 16] and [29]. We shall use the same notation as in [5]. Let L = Zα be a rank one lattice with nondegenerate form given by 〈α, α〉 = 2m+1, wherem ∈ Z>0. Let VL be the corresponding vertex superalgebra. For i ∈ Z, we set γi = i 2m+ 1 α, γR i = α 2(2m+ 1) + i 2m+ 1 α. Then σ1 = exp[ πi 2m+1α(0)] is a canonical automorphism of order two of VL. The set {VγR i +L, i = 0, . . . , 2m} provides all the irreducible σ1-twisted VL-modules. The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 11 Remark 5.1. It is important to notice that σ1-twisted VL-module (VγR i +L, YγR i +L) can be con- structed from untwisted module (Vγi+L, Yγi+L) as follows (cf. [29]): VγR i +L := Vγi+L as vector space; YγR i +L(·, z) := Yγi+L ( ∆ ( α 2(2m+1) , z ) ·, z ) , where ∆(h, z) := zh(0) exp ( ∞∑ n=1 h(n) −n (−z)−n ) . Let F be the fermionic vertex operator superalgebra with central charge 1/2 and σ2 its parity map. Let M be the σ2 twisted F -module (cf. [14]). Then σ = σ1⊗σ2 is the parity automorphism of order two of the vertex superalgebra VL⊗F and VγR i +L ⊗M, i = 0, . . . , 2m are σ-twisted VL ⊗ F -modules. Let (W,YW ) be any σ-twisted VL⊗F -module. From the Jacobi identity for σ-twisted modules it follows that the coefficients of YW (τ, z) = ∑ n∈Z G(n)z−n−3 2 and YW (ω, z) = ∑ n∈Z L(n)z−n−2 define a representation of the N = 1 Ramond algebra. Recall that h2i+2,2n+1 = ((2i+ 2)− (2n+ 1)(2m+ 1))2 − 4m2 8(2m+ 1) + 1 16 . Proposition 5.1. Assume that n ∈ Z. Then eγ R i −nα ⊗ 1R is a singular vector for the N = 1 Ramond algebra R and U(R) ( eγ R i −nα ⊗ 1R ) ∼= LR ( c2m+1,1, h 2i+2,2n+1 ) = LR ( c2m+1,1, h 2i+2,2n+1 )+ ⊕ LR(c2m+1,1, h 2i+2,2n+1)−. Moreover, U(R) ( eγ R i −nα ⊗ 1± ) ∼= LR ( c2m+1,1, h 2i+2,2n+1 )± . As in [5] (see also [20] and [25]) we have the following result. Lemma 5.1. The (screening) operator Q = ResxYW ( eα ⊗ φ ( −1 2 ) , x ) commutes with the action of R. Remark 5.2. By using generalized (lattice) vertex algebras and their twisted representations one can also define the second screening operator Q̃ acting between certain σ-twisted VL ⊗ F - modules such that [Q̃,R] = 0, [Q, Q̃] = 0 (for details and some applications see [7]). 12 D. Adamović and A. Milas We shall first present results on the structure of σ-twisted VL⊗F -modules, viewed as modules for the N = 1 Ramond algebra. Each VL+γR i ⊗M is a direct sum of super Feigin–Fuchs modules via VL+γR i ⊗M = ⊕ n∈Z ( M(1)⊗ eγ R i +nα ) ⊗M. Since operators Qj , j ∈ Z>0, commute with the action of the Ramond algebra, they are actually (Lie superalgebra) intertwiners between super Feigin–Fuchs modules inside VL+γR i ⊗M . To simplify the notation, we shall identify eβ with eβ ⊗ 1R for every β ∈ L+ γR i . Assume that 0 ≤ i ≤ m− 1. If Qjeγ R i −nα is nontrivial, it is a singular vector of weight wt ( Qjeγ R i −nα ) = wt ( eγ R i −nα ) = h2i+2,2n+1. Since wt ( eγ R i +(j−n)α ) > wt ( eγ R i −nα ) if j > 2n, we conclude that Qjeγ R i −nα = 0 for j > 2n. One can similarly see that for m ≤ i ≤ 2m: Qjeγ R i −nα = 0 for j > 2n+ 1. The following lemma is useful for constructing singular vectors in VL+γR i ⊗M : Lemma 5.2. (1) Q2neγ R i −nα 6= 0 for 0 ≤ i ≤ m. (2) Q2n+1eγ R i −nα 6= 0 for m+ 1 ≤ i ≤ 2m. Proof. The proof uses the results on fusion rules from Proposition 3.1 and is completely ana- logous to that of Lemma 6.1 in [5]. � As in the Virasoro algebra case the N = 1 Feigin–Fuchs modules are classified according to their embedding structure. For the purposes of our paper we shall focus only on modules of certain types (Type 4 and 5 in [20]). These modules are either semisimple (Type 5) or they become semisimple after quotienting with the maximal semisimple submodule (Type 4). The following result follows directly from Lemma 5.2 and the structure theory of super Feigin– Fuchs modules, after some minor adjustments of parameters (cf. Type 4 embedding structure in [20]). Theorem 5.1. Assume that i ∈ {0, . . . ,m− 1}. (i) As an R-module, VL+γR i ⊗M is generated by the family of singular and cosingular vectors S̃ingi ⋃ C̃Singi, where S̃ingi = { u (j,n) i \| j, n ∈ Z≥0, 0 ≤ j ≤ 2n } ; C̃Singi = { w (j,n) i \| n ∈ Z>0, 0 ≤ j ≤ 2n− 1 } . These vectors satisfy the following relations: u (j,n) i = Qjeγ R i −nα, Qjw (j,n) i = eγ R i +nα. The submodule generated by singular vectors S̃ingi, denoted by RΛ(i+1), is isomorphic to ∞⊕ n=0 (2n+ 1)LR ( c2m+1,1, h 2i+2,2n+1 ) . The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 13 (ii) Let RΛ(i+ 1)± denote the submodule of VL+γR i ⊗M± generated by singular vectors Qj ( eγ R i −nα ⊗ 1± ) , n ∈ Z≥0, 0 ≤ j ≤ 2n. Then RΛ(i+ 1)± = ∞⊕ n=0 (2n+ 1)LR ( c2m+1,1, h 2i+2,2n+1 )± , and RΛ(i+ 1) = RΛ(i+ 1)+ ⊕RΛ(i+ 1)−. (iii) For the quotient module we have RΠ(m− i) := (VL+γR i ⊗M)/RΛ(i+ 1) ∼= ∞⊕ n=1 (2n)LR ( c2m+1,1, h 2i+2,−2n+1 ) . Moreover, we have RΠ(m− i) = RΠ(m− i)+ ⊕RΠ(m− i)−, where RΠ(m− i)± = (VL+γR i ⊗M)/RΛ(i+ 1)± = ∞⊕ n=1 (2n)LR ( c2m+1,1, h 2i+2,−2n+1 )± . Theorem 5.2. Assume that i ∈ {0, . . . ,m− 1}. (i) As an R-module, VL+γR m+i ⊗M is generated by the family of singular and cosingular vectors S̃ing ′ i ⋃ C̃Sing ′ i, where S̃ing ′ i = { u ′(j,n) i \| n ∈ Z>0, 0 ≤ j ≤ 2n− 1 } ; C̃Sing ′ i = { w ′(j,n) i \| j, n ∈ Z≥0, 0 ≤ j ≤ 2n } . These vectors satisfy the following relations: u ′(j,n) i = Qjeγ R m+i−nα, Qjw ′(j,n) i = eγ R m+i+nα. The submodule generated by singular vectors S̃ingi is isomorphic to RΠ(i+ 1) ∼= ∞⊕ n=1 (2n)LR ( c2m+1,1, h 2m−2i,−2n+1 ) . (ii) For the quotient module we have RΛ(m− i) ∼= (VL+γR i ⊗M)/RΠ(i+ 1) ∼= ∞⊕ n=0 (2n+ 1)LR ( c2m+1,1, h 2m−2i,2n+1 ) . 14 D. Adamović and A. Milas Theorem 5.3. (i) As an R-module, VL+γR 2m ⊗M is completely reducible and generated by the family of singular vectors S̃ing2m = { u (j,n) 2m := Qjeγ R 2m−nα \| n ∈ Z>0, j ∈ Z≥0, 0 ≤ j ≤ 2n− 1 } ; and it is isomorphic to RΠ(m+ 1) := VL+γR 2m ⊗M ∼= ∞⊕ n=1 (2n)LR ( c2m+1,1, h 4m+2,−2n+1 ) . (ii) Let RΠ(m+ 1)± be the submodule of VL+γR 2m ⊗M± generated by the singular vectors Qj ( eγ R 2m−nα ⊗ 1± ) , n ∈ Z>0, j ∈ Z≥0, 0 ≤ j ≤ 2n− 1. Then RΠ(m+ 1)± = ∞⊕ n=1 (2n)LR ( c2m+1,1, h 4m+2,−2n+1 )± , and RΠ(m+ 1) = RΠ(m+ 1)+ ⊕RΠ(m+ 1)−. Remark 5.3. In this section we actually constructed explicitly all the non-trivial intertwining operators from Proposition 3.1. 6 The σ-twisted SW(m)-modules Since SW(m) ⊂ VL ⊗ F is σ-invariant, then every σ-twisted VL ⊗ F -module is also a σ-twisted module for the vertex operator superalgebra SW(m). In this section we shall consider σ-twisted VL ⊗ F -modules from Section 5 as σ-twisted SW(m)-modules. In what follows we shall classify all the irreducible σ-twisted SW(m)-modules by using Zhu’s algebra Aσ(SW(m)). Following [4] and [5], we first notice the following important fact: Q2e−2α ∈ O(SW(m)). (6.1) Proposition 6.1. Let v±λ be the highest weight vector in M(1, λ)⊗M±. We have o ( Q2e−2α ) v±λ = Fm(t) v±λ , where t = 〈λ, α〉, and Fm(t) = Am ( t+m+ 1/2 3m+ 1 )( t− 1/2 3m+ 1 ) , where Am = (−1)m ( 2m m )( 4m+1 m ) . Proof. First we notice that Q2e−2α = w1 + w2, where w1 = ∞∑ i=0 eα−i−1e α i e −2α, w2 = ∞∑ i,j=0 eαi e α j e −2α ⊗ φ ( −i− 1 2 ) φ ( −j − 1 2 ) 1. The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 15 The proof of Proposition 8.3 from [5] gives that o(w1)v±λ = Resx1Resx2(x2 − x1)2m(1 + x1)t(1 + x2)t(x1x2)−4m−2vλ = Am ( t+m 3m+ 1 )( t 3m+ 1 ) v±λ , so it remains to examine o(w2). Recall Lemma 2.1, so that (x2 − x1)G(x1, x2) = 1 2 ( (1 + x1)1/2(1 + x2)−1/2 + (1 + x1)−1/2(1 + x2)1/2 − 2 ) . Now, we compute o(w2) · v±λ = ∑ i,j≥0 o ( eαi e α j e −2α ⊗ φ ( −i− 1 2 ) φ ( −j − 1 2 ) 1 ) · v±λ = ∑ i,j≥0 Resx1Resx2x i 1x j 2o ( Y (eα, x1)Y (eα, x2)e−2α ) o ( ψ ( −i− 1 2 ) ψ ( −j − 1 2 )) · v±λ = Resx1Resx2 ( ∑ i,j≥0 xi 1x j 2ci,jResx1Resx2(x2 − x1)2m+1(1 + x1)t × (1 + x2)t(x1x2)−4m−2v±λ ) = Resx1Resx2(G(x1, x2)(x2 − x1)2m+1(1 + x1)t(1 + x2)t)(x1x2)−4m−2v±λ = −Resx1Resx2(x2 − x1)2m(1 + x1)t(1 + x2)t(x1x2)−4m−2v±λ (6.2) + 1 2Resx1Resx2 ( (x2 − x1)2m(x1x2)−4m−2(1 + x1)t+1/2(1 + x2)t−1/2 ) v±λ (6.3) + 1 2Resx1Resx2 ( (x2 − x1)2m(x1x2)−4m−2(1 + x1)t−1/2(1 + x2)t+1/2 ) v±λ . (6.4) Now, observe that the expression in (6.2) is precisely −o(w1) ·v±λ , while (6.3) and (6.4) are equal. Consequently, o(w1 + w2) · v±λ = Resx1Resx2 ( (x2 − x1)2m(1 + x1)t−1/2(1 + x2)t+1/2(x1x2)−4m−2 ) v±λ . Now, we expand the generalized rational function in the last formula and obtain o ( Q2e−2α ) v±λ = 2m∑ k=0 (−1)k ( 2m k )( t+ 1/2 4m+ 1− k )( t− 1/2 2m+ 1 + k ) v±λ . The sum in the last formula can be evaluated as in [4] and [5]. We have 2m∑ k=0 (−1)k ( 2m k )( t+ 1/2 4m+ 1− k )( t− 1/2 2m+ 1 + k ) = Am ( t+m+ 1/2 3m+ 1 )( t− 1/2 3m+ 1 ) , where Am is above. The proof follows. � A direct consequence of Proposition 6.1 and relation (6.1) is the following important result: Theorem 6.1. In Zhu’s algebra Aσ(SW(m)) we have the following relation fR m([ω]) = 0, where fR m(x) = 3m∏ i=0 ( x− h2i+2,1 ) = ( m−1∏ i=0 ( x− h2i+2,1 )2)( 3m∏ i=2m ( x− h2i+2,1 )) . 16 D. Adamović and A. Milas In parallel with [5] we conjecture that fR m(x) is in fact the minimal polynomial of [ω] in Aσ(SW(m)). We have the following irreducibility result. The proof is similar to that of Theorem 3.7 in [4]. Theorem 6.2. (1) For every 0 ≤ i ≤ m−1, RΛ(i+1)± are Z≥0-gradable irreducible σ-twisted SW(m)-modules and the top components RΛ(i+1)±(0) are 1-dimensional irreducible Aσ(SW(m))-modules. (1′) For every 0 ≤ i ≤ m− 1, RΛ(i+ 1) is a graded irreducible σ-twisted SW(m)-module, and its top component RΛ(i+ 1)(0) is a graded irreducible Aσ(SW(m))-module. (2) For every 0 ≤ j ≤ m ,RΠ(j + 1)± are irreducible Z≥0-gradable σ-twisted SW(m)-modules and the top components SΠ(j+1)±(0) are irreducible 2-dimensional Aσ(SW(m))-modules. (2′) For every 0 ≤ j ≤ m ,RΠ(j + 1) is a graded irreducible σ-twisted SW(m)-module and its top component SΠ(j + 1)(0) is a graded irreducible Aσ(SW(m))-module. Corollary 6.1. The minimal polynomial of [ω] is divisible by m−1∏ i=0 (x− h2i+2,1) ( 3m∏ i=2m (x− h2i+2,1) ) . Moreover, both [ω] and [τ ] are units in Aσ(SW(m)). Consequently, SW(m) has no supersym- metric sector (i.e., there is no σ-twisted SW(m)-modules of highest weight c2m+1,1 24 ). By using similar arguments as in [4] and [5] we have the following result on the structure of Zhu’s algebra Aσ(SW(m)) (as in the proof of Theorem 4.1 we see that [τ ] is central). Proposition 6.2. The associative algebra Aσ(SW(m)) is generated by [E], [H], [F ], [τ ], [Ê], [Ĥ], [F̂ ] and [ω]. The following relations hold: [τ ] and [ω] are in the center of Aσ(SW(m)), [τ ]2 = [ω]− c2m+1,1 24 , [τ ] ∗ [X] = − √ 2m+ 1 2 [X̂], for X ∈ {E,F,H}, [Ĥ] ∗ [F̂ ]− [F̂ ] ∗ [Ĥ] = −2q([ω])[F̂ ], [Ĥ] ∗ [Ê]− [Ê] ∗ [Ĥ] = 2q([ω])[Ê], [Ê] ∗ [F̂ ]− [F̂ ] ∗ [Ê] = −2q([ω])[Ĥ], where q is a certain polynomial. Equipped with all these results we are now ready to classify irreducible σ-twisted SW(m)- modules. Remark 6.1. By using same arguments as in Proposition 5.6 of [1] we have that for C2-cofinite SVOAs, every week (twisted) module is admissible (see also [11]). Thus, the classification of irreducible σ-twisted SW(m)-modules reduces to classification of irreducible Z≥0-gradable modules. Theorem 6.3. (i) The set {RΠ(i)±(0), : 1 ≤ i ≤ m+ 1} ∪ {RΛ(i)±(0) : 1 ≤ i ≤ m} provides, up to isomorphism, all irreducible modules for Zhu’s algebra Aσ(SW(m)). The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 17 (ii) The set {RΠ(i)(0), : 1 ≤ i ≤ m+ 1} ∪ {RΛ(i)(0) : 1 ≤ i ≤ m} provides, up to isomorphism, all Z2-graded irreducible modules for Zhu’s algebra Aσ(SW(m)). Proof. The proof is similar to those of Theorem 3.11 in [4] and Theorem 10.3 in [5]. Assume that U is an irreducible Aσ(SW(m))-module. Relation fR m([ω]) = 0 in Aσ(SW(m)) implies that L(0)\|U = h2i+2,1 Id, for i ∈ {0, . . . ,m− 1} ∪ {2m, . . . , 3m}. Moreover, since [τ ] is in the center of Aσ(SW(m)), we conclude that G(0) also acts on U as a scalar. Then relation [τ ]2 = [ω]− c2m+1,1/24 implies that G(0)\|U = 1/2 + i−m√ 2(2m+ 1) Id or G(0)\|U = − 1/2 + i−m√ 2(2m+ 1) Id. Assume first that i = 2m + j for 0 ≤ j ≤ m. By combining Propositions 6.2 and Theorem 6.2 we have that q(h2i+2,1) 6= 0. Define e = 1√ 2q(h2i+2,1) [Ê], f = − 1√ 2q(h2i+2,1) [F̂ ], h = 1 q(h2i+2,1) [Ĥ]. Therefore U carries the structure of an irreducible, sl2-module with the property that e2 = f2 = 0 and h 6= 0 on U . This easily implies that U is a 2-dimensional irreducible sl2-module. Moreover, as an Aσ(SW(m))-module U is isomorphic to either RΠ(m+ 1− j)+(0) or RΠ(m+ 1− j)−(0). In the case 0 ≤ i ≤ m− 1, as in [4] we prove that U ∼= RΛ(i+ 1)+(0) or U ∼= RΛ(i+ 1)−(0). Let us now prove the second assertion. Let N = N0 ⊕ N1 be the graded irreducible Aσ(SW(m))-module. As above, we have that L(0)\|N = h2i+2,1 Id, for i ∈ {0, . . . ,m− 1} ∪ {2m, . . . , 3m}. Then N = N+ ⊕N−, where N± = spanC { v ± √ 2(2m+ 1) 1/2 + i−m G(0)v \|v ∈ N0 } and G(0)\|N± = ± 1/2 + i−m√ 2(2m+ 1) Id. By using assertion (i), we easily get that N± = RΛ(i + 1)±(0) (if 0 ≤ i ≤ m − 1) and N± = RΠ(3m+ 1− i)±(0) (if 2m ≤ i ≤ 3m). � Theorem 6.4. (i) The set {RΠ(i)± : 1 ≤ i ≤ m+ 1} ∪ {RΛ(i)± : 1 ≤ i ≤ m} provides, up to isomorphism, all irreducible σ-twisted SW(m)-modules. (ii) The set {RΠ(i) : 1 ≤ i ≤ m+ 1} ∪ {RΛ(i) : 1 ≤ i ≤ m} provides, up to isomorphism, all Z2-graded irreducible σ-twisted SW(m)-modules. 18 D. Adamović and A. Milas So the vertex operator algebra SW(m) contains only finitely many irreducible modules. But one can easily see that modules VL+γR i ⊗M and VL+γR m+i ⊗M (0 ≤ i ≤ m − 1) constructed in Theorems 5.1 and 5.2 are not completely reducible. Thus we have: Corollary 6.2. The vertex operator superalgebra SW(m) is not σ-rational, i.e., the category of σ-twisted SW(m)-modules is not semisimple. Remark 6.2. In our forthcoming paper [7] we shall prove that SW(m) also contains logarithmic σ-twisted representations. 7 Modular properties of characters of σ-twisted SW(m)-modules We first introduce some basic modular forms needed for description of irreducible twisted SW(m) characters. The Dedekind η-function is usually defined as the infinite product η(τ) = q1/24 ∞∏ n=1 (1− qn), an automorphic form of weight 1 2 . As usual in all these formulas q = e2πiτ , τ ∈ H. We also introduce f(τ) = q−1/48 ∞∏ n=0 (1 + qn+1/2), f1(τ) = q−1/48 ∞∏ n=1 (1− qn−1/2), f2(τ) = q1/24 ∞∏ n=1 (1 + qn). Let us recall Jacobi Θ-function Θj,k(τ, z) = ∑ n∈Z qk(n+ j 2k )2e2πikz(n+ j 2k ), where j, k ∈ 1 2Z. If j ∈ Z + 1 2 or k ∈ N + 1 2 it will be useful to use the formula Θj,k(τ, z) = Θ2j,4k(τ, z) + Θ2j−4k,4k(τ, z). Observe that Θj+2k,k(τ, z) = Θj,k(τ, z). We also let ∂Θj,k(τ, z) := 1 πi d dz Θj,k(τ, z) = ∑ n∈Z (2kn+ j)q(2kn+j)2/4k. Related Θ-functions needed for supercharacters are Gj,k(τ, z) = ∑ n∈Z (−1)nqk(n+ j 2k )2e2πikz(n+ j 2k ), The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 19 where j ∈ Z and k ∈ 1 2N. It is easy to see that Gj,k(τ, z) = Θ2j,4k(τ, z)−Θ2j−4k,4k(τ, z). Eventually, we shall let z = 0 so before we introduce Θj,k(τ) := Θj,k(τ, 0), Gj,k(τ) := Gj,k(τ, 0). Similarly, we define ∂Θj,k(τ) and ∂Gj,k(τ). The following formulas will be useful: Θj,k ( −1 τ , z τ ) = √ −iτ√ 2k e 4πikz2 τ 4k−1∑ j′=0 e−πij′j/kΘ2j′,4k(τ, z)  , ∂Θ′ j,k ( −1 τ , z τ ) = √ −iτ√ 2k e 4πikz2 τ 8kz 4k−1∑ j′=0 e−πij′j/kΘ2j′,4k(τ, z)  + τ √ −iτ√ 2k e 4πikz2 τ 4k−1∑ j′=0 e−πij′j/k∂Θ2j′,4k(τ, z)  . If we let now z = 0, we obtain Θj,k ( −1 τ ) = √ −iτ√ 2k 4k−1∑ j′=0 e −πijj′ k Θ2j′,4k(τ, 0), (7.1) and ∂Θ′ j,k ( −1 τ ) = τ √ −iτ√ 2k 4k−1∑ j′=0 e −πijj′ k ∂Θ2j′,4k(τ, 0). (7.2) For j ∈ Z and k ∈ N + 1 2 we now have: Θj,k ( −1 τ ) = √ −iτ 2k 2k−1∑ j′=0 e−iπjj′/kΘj′,k(τ), Θj,k(τ + 1) = eiπj2/2kGj,k(τ), (∂Θ)j,k(τ + 1) = eiπj2/2k(∂G)j,k(τ), (∂Θ)j,k ( −1 τ ) = τ √ −iτ/2k 2k−1∑ j′=1 e−iπjj′/k(∂Θ)j′,k(τ), where we used (7.1) and (7.2), together with Θj,k(τ) = Θj+2k,k(τ) and ∂Θj,k(τ) = ∂Θj+2k,k(τ). For j ∈ Z + 1 2 and k ∈ N + 1 2 the transformation formulas are slightly different: Θj,k ( −1 τ ) = √ −iτ 2k 2k−1∑ j′=0 e−iπjj′/kGj′,k(τ), Θj,k(τ + 1) = eiπj2/2kΘj,k(τ), (∂Θ)j,k(τ + 1) = eiπj2/2k(∂Θ)j,k(τ), (∂Θ)j,k ( −1 τ ) = τ √ −iτ/2k 2k−1∑ j′=1 e−iπjj′/k(∂G)j′,k(τ). 20 D. Adamović and A. Milas For a (twisted) vertex operator algebra module W we define its graded-dimension or simply character χW (τ) = tr\|W qL(0)−c/24. If V = Lns(c2m+1,0, 0) and W = LR(c2m+1,0, h 2i+2,2n+1), then (see [20], for instance) χLR(c2m+1,1,h2i+2,2n+1)(τ) = 2q m2 2(2m+1) − 1 16 f2(τ) η(τ) ( qh2i+2,2n+1 − qh2i+2,−2n−1 ) . (7.3) By combining Theorems 5.1, 5.2 and 5.3, and formula (7.3) we obtain Proposition 7.1. For i = 0, . . . ,m− 1 χRΛ(i+1)(τ) = 2 f2(τ) η(τ) ( 2i+ 2 2m+ 1 Θ m−i−1 2 , 2m+1 2 (τ) + 2 2m+ 1 (∂Θ) m−i−1 2 , 2m+1 2 (τ) ) , (7.4) χRΠ(m−i)(τ) = 2 f2(τ) η(τ) ( 2m− 2i− 1 2m+ 1 Θ m−i−1 2 , 2m+1 2 (τ)− 2 2m+ 1 (∂Θ) m−i−1 2 , 2m+1 2 (τ) ) . (7.5) Also, χRΠ(m+1)(τ) = 2 f2(τ) η(τ) Θ m+ 1 2 , 2m+1 2 (τ). As in [5], the characters of irreducible σ-twisted SW(m)-modules can be described by using characters of irreducible modules for the triplet vertex algebra W(p), where p = 2m + 1. Let Λ(1), . . . ,Λ(p), Π(1), . . . ,Π(p) be the irreducible W(p)-module. We have the following result: Proposition 7.2. (i) For 0 ≤ i ≤ m− 1, we have: χRΛ(i+1)(τ) = 2 χΛ(2i+2)(τ/2) f(τ) . (ii) For 0 ≤ i ≤ m, we have: χRΠ(m+1−i)(τ) = 2 χΠ(2m−2i+1)(τ/2) f(τ) . Now, we recall also formulas for irreducible SW(m) characters and supercharacters obtained in [5]. For i = 0, . . . ,m− 1 χSΛ(i+1)(τ) = f(τ) η(τ) ( 2i+ 1 2m+ 1 Θm−i, 2m+1 2 (τ) + 2 2m+ 1 (∂Θ)m−i, 2m+1 2 (τ) ) , (7.6) χSΠ(m−i)(τ) = f(τ) η(τ) ( 2m− 2i 2m+ 1 Θm−i, 2m+1 2 (τ)− 2 2m+ 1 (∂Θ)m−i, 2m+1 2 (τ) ) . (7.7) Also, χSΛ(m+1)(τ) = f(τ) η(τ) Θ0, 2m+1 2 (τ). (7.8) The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 21 For supercharacters we have: for i = 0, . . . ,m− 1 χF SΛ(i+1)(τ) = f1(τ) η(τ) ( 2i+ 1 2m+ 1 Gm−i, 2m+1 2 (τ) + 2 2m+ 1 (∂G)m−i, 2m+1 2 (τ) ) , (7.9) χF SΠ(m−i)(τ) = f1(τ) η(τ) ( 2m− 2i 2m+ 1 Gm−i, 2m+1 2 (τ)− 2 2m+ 1 (∂G)m−i, 2m+1 2 (τ) ) . (7.10) Also, χF SΛ(m+1)(τ) = f1(τ) η(τ) G0, 2m+1 2 (τ). (7.11) These characters and supercharacters can be expressed by using characters of W (2m + 1)- modules. Proposition 7.3. (i) For 0 ≤ i ≤ m, we have χSΛ(i+1)(τ) = χΛ(2i+1)( τ 2 ) f2(τ) , χF SΛ(i+1)(τ) = χΛ(2i+1)( τ+1 2 ) f2(τ) . (ii) For 0 ≤ i ≤ m− 1, we also have χSΠ(m−i)(τ) = χΠ(2m−2i)( τ 2 ) f2(τ) , χF SΠ(m−i)(τ) = χΠ(2m−2i)( τ+1 2 ) f2(τ) . Here Λ(i) and Π(2m+ 2− i), i = 1, . . . , 2m+ 1, are irreducible W(2m+ 1)-modules [4]. By combining transformation formulas for Θj,k(τ), ∂Θj,k(τ), formulas (7.4)–(7.8), (7.9)–(7.11) and Proposition 7.1 we obtain second main result of our paper. Theorem 7.1. The SL(2,Z) closure, called H, of the vector space determined by SW(m)- characters, SW(m)-supercharacters and σ-twisted SW(m)-characters has the following basis: f1(τ) η(τ) G0, 2m+1 2 (τ), f(τ) η(τ) Θ0, 2m+1 2 (τ), f2(τ) η(τ) Θ m+ 1 2 , 2m+1 2 (τ), f1(τ) η(τ) Gm−i, 2m+1 2 (τ), f(τ) η(τ) Θm−i, 2m+1 2 (τ), f2(τ) η(τ) Θ m−i−1 2 , 2m+1 2 (τ), f1(τ) η(τ) ∂Gm−i, 2m+1 2 (τ), f(τ) η(τ) ∂Θm−i, 2m+1 2 (τ), f2(τ) η(τ) ∂Θ m−i−1 2 , 2m+1 2 (τ), τ f1(τ) η(τ) ∂Gm−i, 2m+1 2 (τ), τ f(τ) η(τ) ∂Θm−i, 2m+1 2 (τ), τ f2(τ) η(τ) ∂Θ m−i−1 2 , 2m+1 2 (τ), where i = 0, . . . ,m− 1. In particular, the space is 9m+ 3 dimensional. 7.1 Modular differential equations for σ-twisted SW(m) characters Let us recall classical SL(2,Z) Eisenstein series (k ≥ 1): G2k(τ) = −B2k (2k)! + 2 (2k − 1)! ∞∑ n=1 n2k−1qn 1− qn , 22 D. Adamović and A. Milas and certain linear combination of level 2 Eisenstein series (k ≥ 1): G2k,1(τ) = B2k (2k)! + 2 (2k − 1)! ∑ n≥1 n2k−1qn 1 + qn , G2k,0(τ) = B2k(1/2) (2k)! + 2 (2k − 1)! ∞∑ n=1 (n− 1/2)2k−1qn−1/2 1 + qn−1/2 , where B2k(x) are the Bernoulli polynomials, and B2k are the Bernoulli numbers. A modular differential equation is an (ordinary) differential equation of the form:( q d dq )k y(q) + k−1∑ j=0 Hj(q) ( d d dq )i y(q) = 0, where Hj(q) are polynomials in Eisenstein series G2i, i ≥ 1, such that the vector space of solutions is modular invariant. It is known (cf. [30]) that C2-cofiniteness condition leads to certain modular differential equation satisfied by irreducible characters trMq L(0)−c/24. In some instances the degree of this differential equation is bigger than the number of irreducible characters. So it is not clear what k should be in general. In the case of Virasoro minimal models, the degree of the modular differential equation is precisely the number of (linearly independent) irreducible characters. If V is a vertex operator superalgebra one can also get modular differential equation sat- isfied by ordinary characters but with respect to the subgroup Γθ ⊂ SL(2,Z), where Hi are polynomials in G2i and G2i,0 (see [27] for the precise statement in the case of N = 1 minimal models). In [6] we proved that the C2-cofiniteness for the super triplet vertex algebra SW(m) gives rise to a differential equation of order 3m+ 1 satisfied by 2m+ 1 irreducible characters (additional m solutions can be interpreted as certain pseudotraces). By applying arguments similar to those in [6] it is not hard to prove Theorem 7.2. The irreducible σ-twisted SW(m) characters satisfy the differential equation of the form( d d dq )3m+1 y(q) + 3m∑ j=0 H̃j,0(q) ( q d dq )i y(q) = 0, where H̃j,0(q) are certain polynomials in G2i and G2i,1. As in the case of W(p)-modules and ordinary SW(m)-modules, we expect that additional m linearly independent solutions in Theorem 7.2 have interpretation in terms of σ-twisted pseudo- traces (cf. Conjecture 8.2 below). 8 Conclusion Here we gather a few more-or-less expected conjectures (especially in view of our earlier work [4] and [5]). The first one is concerned about the structure of Aσ(V ). As in the case of A(W(p)) and A(SW(m)), from our analysis in Chapter 6, we can show that in fact dim(Aσ(SW(m))) ≥ 10m+ 8. The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector 23 But this is well below the conjectural dimension, because 10m + 8-dimensional part cannot control possible logarithmic modules. Thus, as in the case of ordinary SW(m)-modules, we expect to have 2m non-isomorphic (non-graded) logarithmic modules with two-dimensional top component. This then leads to the following conjecture: Conjecture 8.1. For every m ∈ N, dim(Aσ(SW(m))) = 12m+ 8. If we assume the existence of m logarithmic modules so that Conjecture 8.1 holds true, then the following fact is expected. Conjecture 8.2. The vector space of (suitably defined) generalized σ-twisted characters is 3m+ 1-dimensional. Thus, generalized SW(m) characters, supercharacters and σ-twisted characters together should give rise to a 9m+ 3-dimensional modular invariant space. Acknowledgments The second author was partially supported by NSF grant DMS-0802962. References [1] Abe T., Buhl G., Dong C., Rationality, regularity, and C2-cofiniteness, Trans. Amer. Math. Soc. 356 (2004), 3391–3402, math.QA/0204021. [2] Adamović D., Classification of irreducible modules of certain subalgebras of free boson vertex algebra, J. Algebra 270 (2003), 115–132, math.QA/0207155. [3] Adamović D., Milas A., Logarithmic intertwining operators and W(2, 2p − 1)-algebras, J. Math. Phys. 48(2007), 073503, 20 pages, math.QA/0702081. [4] Adamović D., Milas A., On the triplet vertex algebra W(p), Adv. Math. 217 (2008), 2664–2699, arXiv:0707.1857. [5] Adamović D., Milas A., The N = 1 triplet vertex operator superalgebras, Comm. Math. Phys., to appear, arXiv:0712.0379. [6] Adamović D., Milas A., An analogue of modular BPZ-equations in logarithmic (super)conformal field theory, submitted. [7] Adamović D., Milas A., Lattice construction of logarithmic modules, in preparation. [8] Arakawa T., Representation theory of W-algebras, II: Ramond twisted representations, arXiv:0802.1564. [9] Bakalov B., Kac V., Twisted modules over lattice vertex algebras, in Lie Theory and Its Applications in Physics V, World Sci. Publ., River Edge, NJ, 2004, 3–26, math.QA/0402315. [10] Dong C., Twisted modules for vertex algebras associated with even lattices, J. Algebra 165 (1994), 91–112. [11] Dong C., Zhao Z., Modularity in orbifold theory for vertex operator superalgebras, Comm. Math. Phys. 260 (2005), 227–256, math.QA/0411524. [12] Dörrzapf M., Highest weight representations of the N = 1 Ramond algebra, Nuclear Phys. B 595 (2001), 605–653. [13] Feingold A.J., Frenkel I.B., Ries J.F.X., Spinor construction of vertex operator algebras, triality, and E (1) 8 , Contemporary Mathematics, Vol. 121, American Mathematical Society, Providence, RI, 1991. [14] Feingold A.J., Ries J.F.X., Weiner M., Spinor construction of the c = 1 2 minimal model, in Moonshine, the Monster and Related Topics (South Hadley, MA, 1994), Editors C. Dong and G. Mason, Contemporary Mathematics, Vol. 193, American Mathematical Society, Providence, RI, 1996, 45–92, hep-th/9501114. [15] Feigin B.L., Găınutdinov A.M., Semikhatov A.M., Tipunin I.Yu., Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center, Comm. Math. Phys. 265 (2006), 47–93, hep-th/0504093. http://arxiv.org/abs/math.QA/0204021 http://arxiv.org/abs/math.QA/0207155 http://arxiv.org/abs/math.QA/0702081 http://arxiv.org/abs/0707.1857 http://arxiv.org/abs/0712.0379 http://arxiv.org/abs/0802.1564 http://arxiv.org/abs/math.QA/0402315 http://arxiv.org/abs/math.QA/0411524 http://arxiv.org/abs/hep-th/9501114 http://arxiv.org/abs/hep-th/0504093 24 D. Adamović and A. Milas [16] Frenkel I.B., Lepowsky J., Meurman A., Vertex operator algebras and the monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Inc., Boston, MA, 1988. [17] Huang Y.-Z., Milas A., Intertwining operator superalgebras and vertex tensor categories for superconformal algebras. I, Commun. Contemp. Math. 4 (2002), 327–355, math.QA/9909039. [18] Iohara K., Koga Y., Fusion algebras for N = 1 superconformal field theories through coinvariants. II. N = 1 super-Virasoro-symmetry, J. Lie Theory 11 (2001), 305–337. [19] Iohara K., Koga Y., Representation theory of the Neveu–Schwarz and Ramond algebras. I. Verma modules, Adv. Math. 178 (2003), 1–65. [20] Iohara K., Koga Y., Representation theory of the Neveu–Schwarz and Ramond algebras. II. Fock modules, Ann. Inst. Fourier (Grenoble) 53 (2003), 1755–1818. [21] Kac V.G., Vertex algebras for beginners, 2nd ed., University Lecture Series, Vol. 10, American Mathematical Society, Providence, RI, 1998. [22] Kac V., Wakimoto M., Unitarizable highest weight representations of the Virasoro, Neveu–Schwarz and Ramond algebras, in Conformal Groups and Related Symmetries: Physical Results and Mathematical Back- ground (Clausthal-Zellerfeld, 1985), Lecture Notes in Phys., Vol. 261, Springer, Berlin, 1986, 345–371. [23] Kac V., Wakimoto M., Quantum reduction in the twisted case, in Infinite Dimensional Algebras and Quan- tum Integrable Systems, Progr. Math., Vol. 237, Birkhäuser, Basel, 2005, 89–131, math-ph/0404049. [24] Li H., Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, in Moonshine, the Monster, and Related Topics (South Hadley, MA, 1994), Editors C. Dong and G. Ma- son, Contemporary Mathematics, Vol. 193, American Mathematical Society, Providence, RI, 1996, 203–236, q-alg/9504022. [25] Meurman A., Rocha-Caridi A., Highest weight representations of the Neveu–Schwarz and Ramond algebras, Comm. Math. Phys. 107 (1986), 263–294. [26] Milas A., Fusion rings for degenerate minimal models, J. Algebra 254 (2002), 300–335, math.QA/0003225. [27] Milas A., Characters, supercharacters and Weber modular functions, J. Reine Angew. Math. 608 (2007), 35–64. [28] Miyamoto M., Modular invariance of vertex operator algebras satisfying C2-cofiniteness, Duke Math. J. 122 (2004), 51–91, math.QA/020910. [29] Xu X., Introduction to vertex operator superalgebras and their modules, Mathematics and Its Applications, Vol. 456, Kluwer Academic Publishers, Dordrecht, 1998. [30] Zhu Y.-C., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237–302. http://arxiv.org/abs/math.QA/9909039 http://arxiv.org/abs/math-ph/0404049 http://arxiv.org/abs/q-alg/9504022 http://arxiv.org/abs/math.QA/0003225 http://arxiv.org/abs/math.QA/020910 1 Introduction 2 Preliminaries 3 Highest weight representations of the Ramond algebra 3.1 Intertwining operators among twisted modules 4 -twisted modules for the super singlet algebra SM(1) 5 The N=1 Ramond module structure of twisted V_L \otimes F-modules 6 The -twisted SW(m)-modules 7 Modular properties of characters of \sigma-twisted SW(m)-modules 7.1 Modular differential equations for \sigma-twisted SW(m) characters 8 Conclusion References

The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector

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