Associative algebras and intertwining operators

TL;DR

This paper establishes a correspondence between logarithmic intertwining operators among modules of a vertex operator algebra and homomorphism spaces over certain associative algebras, extending the algebraic understanding of module interactions.

Contribution

It introduces new associative algebra structures and demonstrates their use in characterizing intertwining operators for generalized modules of vertex operator algebras.

Findings

Isomorphism between intertwining operators and hom spaces over $A^{ abla}(V)$

Extension of algebraic framework to lower-bounded generalized modules

Characterization of intertwining operators via $A^{N}(V)$-bimodule homomorphisms

Abstract

Let $V$ be a vertex operator algebra and $A^{\infty}(V)$ and $A^{N}(V)$ for $N\in \mathbb{N}$ the associative algebras introduced by the author in [H5]. For a lower-bounded generalized $V$-module $W$, we give $W$ a structure of graded $A^{\infty}(V)$-module and we introduce an $A^{\infty}(V)$-bimodule $A^{\infty}(W)$ and an $A^{N}(V)$-bimodule $A^{N}(W)$. We prove that the space of (logarithmic) intertwining operators of type $\binom{W_{3}}{W_{1}W_{2}}$ for lower-bounded generalized $V$-modules $W_{1}$, $W_{2}$ and $W_{3}$ is isomorphic to the space $\hom_{A^{\infty}(V)}(A^{\infty}(W_{1})\otimes_{A^{\infty}(V)}W_{2}, W_{3})$. Assuming that $W_{2}$ and $W_{3}'$ are equivalent to certain universal lower-bounded generalized $V$-modules generated by their $A^{N}(V)$-submodules consisting of elements of levels less than or equal to $N\in \mathbb{N}$, we also prove that the space of (logarithmic) intertwining operators of type $\binom{W_{3}}{W_{1}W_{2}}$ is isomorphic to the space of $\hom_{A^{N}(V)}(A^{N}(W_{1})\otimes_{A^{N}(V)}\Omega_{N}^{0}(W_{2}), \Omega_{N}^{0}(W_{3}))$.