Twisted representations of vertex operator algebras

Abstract

Let V be a vertex operator algebra and g an automorphism of order T . We construct a sequence of associative algebras A g,n (V ) with n ∈ 1 T Z nonnegative such that A g,n (V ) is a quotient of A g,n+1/T (V ) and a pair of functors between the category of A g,n (V )-modules which are not A g,n−1/T (V )-modules and the category of admissible V -modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is g-rational if and only if all A g,n (V ) are finite-dimensional semisimple algebras.

Logarithmic extensions of minimal models: Characters and modular transformations

“…, N m ) ⊂ W generated by the modes W i [n i ] with n i N i be finite-dimensional in any module from W. In particular, this means that the category W contains a finite number of irreducible representations, and hence a finite number of projective modules (cf. [33,27]). …”

Shifted vertex operator algebras

“…That is, it is regular and has a unique simple module. By Theorem 4.1, we can conclude that V L,h is also a holomorphic vertex operator algebra whenever h ∈ L. Remark: By results from [18] and [7], we know that if V is holomorphic and C 2 -cofinite, then we necessarily have 8|c. For each integer r ≥ d, let (L r , h r ) be a pair consisting of a self-dual, even lattice L r of rank 24r +c and an element h r ∈ L r which satisfies (h r , h r ) = 2r.…”

“…Since one knows ( [7], Proposition 12.5) that V L is C 2 -cofinite, then so too are the vertex operator algebras V L,h . This completes the proof of the Theorem.…”

“…We are particularly interested in the case in which the vertex operator algebra V is nevertheless well-behaved from the representation-theoretic point-of-view. By this we generally have in mind that V is rational in the sense of [7], that is, the module category V − Mod is semisimple. Closely related to this is the condition that V is regular in the sense of [6].…”

A new rational conformal field theory extension of the fully degenerate W1+∞(m)

“…It was first introduced in [35,24] on RCFTs characterized as the tensor product of m copies of a given RCFT. The subject was further developed in [26,25,17,7,4,21,5], and it was also studied by using the approach of the m-reduction technique in [37,13,14]. A more complete classification has been recently presented in [2], where the orbifold construction is applied in the more general framework of the lattice vertex algebras (which are not simply tensor products of RCFTs).…”