C-Graded Vertex Algebras and Conformal Flow

TL;DR

This paper introduces C-graded vertex algebras and pseudo vertex operator algebras, establishing their representation theory and exploring conformal flow deformations in strongly regular VOAs.

Contribution

It develops the theory of C-graded vertex algebras and pseudo VOAs, including the construction of Zhu algebras and the concept of conformal flow deformations.

Findings

C-graded vertex algebras have a well-behaved representation theory with a bijection between modules.

Pseudo vertex operator algebras generalize VOAs without semisimplicity assumptions.

Deformation of conformal structures in strongly regular VOAs forms a continuous space of PVOAs.

Abstract

We consider C-graded vertex algebras, which are vertex algebras V with a 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, which are 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 (eg a lattice theory, or WZW model) is a path in a space whose points are PVOAs.