# NGK and HLZ: fusion for physicists and mathematicians

## TL;DR

This paper compares two approaches to fusion products in conformal field theory and vertex operator algebras, clarifying their relationship and providing illustrative examples for better understanding.

## Contribution

It elucidates the duality between NGK and HLZ fusion constructions and adapts NGK algorithms within the HLZ framework, enhancing conceptual clarity.

## Key findings

- NGK coproducts are dual to HLZ P(w)-compatibility.
- NGK fusion algorithm can be adapted to HLZ setting.
- Explicit examples illustrate the correspondence between approaches.

## Abstract

In this expository note, we compare the fusion product of conformal field theory, as defined by Gaberdiel and used in the Nahm-Gaberdiel-Kausch (NGK) algorithm, with the $P(w)$-tensor product of vertex operator algebra modules, as defined by Huang, Lepowsky and Zhang (HLZ). We explain how the equality of the two "coproducts" derived by NGK is essentially dual to the $P(w)$-compatibility condition of HLZ and how the algorithm of NGK for computing fusion products may be adapted to the setting of HLZ. We provide explicit calculations and instructive examples to illustrate both approaches. This document does not provide precise descriptions of all statements, it is intended more as a gentle starting point for the appreciation of the depth of the theory on both sides.