# Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms

## TL;DR

This paper develops a theory of intertwining operators among twisted modules of vertex operator algebras with non-commuting automorphisms, establishing conditions for their existence and constructing key isomorphisms.

## Contribution

It introduces and analyzes twisted intertwining operators for non-commuting automorphisms, extending the structure theory of vertex operator algebras.

## Key findings

- Proves that the automorphism associated with a twisted module equals the product of two automorphisms under certain conditions.
- Constructs skew-symmetry and contragredient isomorphisms for twisted intertwining operators.
- Analyzes analytic extensions related to non-commuting automorphisms.

## Abstract

We introduce intertwining operators among twisted modules or twisted intertwining operators associated to not-necessarily-commuting automorphisms of a vertex operator algebra. Let $V$ be a vertex operator algebra and let $g_{1}$, $g_{2}$ and $g_{3}$ be automorphisms of $V$. We prove that for $g_{1}$-, $g_{2}$- and $g_{3}$-twisted $V$-modules $W_{1}$, $W_{2}$ and $W_{3}$, respectively, such that the vertex operator map for $W_{3}$ is injective, if there exists a twisted intertwining operator of type ${W_{3}\choose W_{1}W_{2}}$ such that the images of its component operators span $W_{3}$, then $g_{3}=g_{1}g_{2}$. We also construct what we call the skew-symmetry and contragredient isomorphisms between spaces of twisted intertwining operators among twisted modules of suitable types. The proofs of these results involve careful analysis of the analytic extensions corresponding to the actions of the not-necessarily-commuting automorphisms of the vertex operator algebra.