${\mathbbm {C}}$ C -Graded vertex algebras and conformal flow

Abstract

We consider $\mathbbm {C}$C-graded vertex algebras, which are vertex algebras V with a $\mathbbm {C}$C-grading such that V is an admissible V-module generated by “lowest weight vectors.” We show that such vertex algebras have a “good” representation theory in the sense that there is a Zhu algebra A(V) and a bijection between simple admissible V-modules and simple A(V)-modules. We also consider pseudo vertex operator algebras (PVOAs), which are $\mathbbm {C}$C-graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L(0); essentially, these are VOAs that lack any semisimplicity or integrality assumptions on L(0). As a motivating example, we show that deformation of the conformal structure (conformal flow) of a strongly regular VOA (e.g., a lattice theory, or Wess-Zumino-Witten model) is a path in a space whose points are PVOAs.