# On fusion rules and intertwining operators for the Weyl vertex algebra

## TL;DR

This paper studies the fusion rules and intertwining operators in the Weyl vertex algebra, confirming a conjecture and explicitly constructing key operators, while relating modules to affine Lie superalgebra representations.

## Contribution

It provides a detailed description of fusion rules for the Weyl vertex algebra and constructs explicit intertwining operators, confirming a conjecture based on the Verlinde algebra.

## Key findings

- Confirmed the conjecture on fusion rules for the Weyl vertex algebra
- Explicitly constructed intertwining operators for fusion rules
- Related Weyl vertex algebra modules to affine Lie superalgebra modules

## Abstract

In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde algebra. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{gl(1 \vert 1)}$.