On the definition of Heisenberg category

TL;DR

This paper refines the definition of the Heisenberg category across various central charges, removing unnecessary relations and clarifying its structure, including cyclotomic quotients.

Contribution

It simplifies and clarifies the definitions of the Heisenberg category for different central charges, removing redundant relations and unifying the framework.

Findings

Redundant cyclicity relations are unnecessary for central charge -1.

The definition for negative central charges is streamlined by removing extra relations.

Central charge zero corresponds to the affine oriented Brauer category.

Abstract

We revisit the definition of the Heisenberg category of central charge k. For central charge -1, this category was introduced originally by Khovanov, but with some additional cyclicity relations which we show here are unnecessary. For other negative central charges, the definition is due to Mackaay and Savage, also with some redundant relations, while central charge zero recovers the affine oriented Brauer category of Brundan, Comes, Nash and Reynolds. We also discuss cyclotomic quotients.