$G$-crossed braided zesting

TL;DR

This paper introduces a method called $G$-crossed braided zesting to construct new $G$-crossed braided fusion categories from existing ones using cohomological data, enhancing the understanding of their structure.

Contribution

The authors develop an explicit cohomological construction for $G$-crossed braided zesting, extending previous methods for fusion categories and linking to homotopy-based descriptions.

Findings

Explicit construction of $G$-crossed braided zestings.

Characterization of $G$-extensions arising from zestings.

Connection between zestings and homotopy-based $G$-extensions.

Abstract

For a finite group $G$, a $G$-crossed braided fusion category is $G$-graded fusion category with additional structures, namely a $G$-action and a $G$-braiding. We develop the notion of $G$-crossed braided zesting: an explicit method for constructing new $G$-crossed braided fusion categories from a given one by means of cohomological data associated with the invertible objects in the category and grading group $G$. This is achieved by adapting a similar construction for (braided) fusion categories recently described by the authors. All $G$-crossed braided zestings of a given category $\mathcal{C}$ are $G$-extensions of their trivial component and can be interpreted in terms of the homotopy-based description of Etingof, Nikshych and Ostrik. In particular, we explicitly describe which $G$-extensions correspond to $G$-crossed braided zestings.