Twisted logarithmic modules of vertex algebras

TL;DR

This paper introduces a new class of twisted modules for vertex algebras involving logarithmic fields, extending the theory to non-semisimple automorphisms with applications in conformal field theory and Gromov-Witten theory.

Contribution

It develops the foundational theory of twisted logarithmic modules for vertex algebras, including identities and formulas, broadening the scope of vertex algebra representations.

Findings

Derived Borcherds identity for twisted logarithmic modules

Established commutator formulas for these modules

Analyzed examples involving affine and Heisenberg vertex algebras

Abstract

Motivated by logarithmic conformal field theory and Gromov-Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.