On Z_2-twisted representation of vertex operator superalgebras and the Ising model SVOA.pdf

a r X i v : m a t h / 0 2 0 3 0 8 6 v 4 [ m a t h .Q A ] 1 8 M a r 2 0 0 2 On Z2-twisted representation of vertex operator superalgebras and the Ising model SVOA Hiroshi Yamauchi Graduate School of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan e-mail: hirocci@math.tsukuba.ac.jp Abstract We investigate a general theory of the Z2-twisted representations of vertex op- erator superalgebras. Certain one-to-one correspondence theorems are established. We also give an explicit realization of the Ising model SVOA and its Z2-twisted modules. As an application, we obtain the Gerald Ho?hn’s Babymonster SVOA VB? and its Z2-twisted module VB ? tw from the moonshine VOA V ? by cutting off the Ising models. It is also shown in this paper that Aut(VB?) is finite. 1 Introduction In the theory of vertex operator algebras (VOAs), we sometimes notice that it makes the theory simpler to use some representations of vertex operator superalgebras (SVOAs for short) instead of VOAs. For example, as one can see in [M1]-[M3], in the representation theory of the Ising model VOA L(1 2 , 0) it seems more natural that we should treat the theory in the view point of the SVOA L(1 2 , 0) ⊕ L(1 2 , 1 2 ), which is the most interested object in this paper. For the theory of VOAs, we have many remarkable results on the fundamental representation theory, so-called Zhu’s theory (cf. [Z] [FZ] [Li2] [DLM1] [DLM2] [Y]) and some of them are extended to that for SVOAs (cf. [KW]). The Zhu algebra A(V ) is an associative algebra associated to every VOA V . It is well-known that there exists a one-to-one correspondence between the category of irreducible V -modules and that of irreducible A(V )-modules (cf. [Z], [DLM1]). Since every SVOA has a canonical involution, we can think of the Z2-twisted representations of SVOAs and the above theory should be naturally extended to Z2-twisted representations for SVOAs. To start the investigation of SVOAs, the Ising model SVOA L(1 2 , 0) ⊕ L(1 2 , 1 2