Non compact (2+1)-TQFTs from non-semisimple spherical categories

TL;DR

This paper develops a new framework for non-compact (2+1)-TQFTs using non-semisimple spherical categories, extending existing theories and introducing admissible skein modules and chromatic categories.

Contribution

It introduces the concept of chromatic categories and constructs associated non-compact (2+1)-TQFTs, broadening the scope beyond semisimple cases.

Findings

Defined admissible skein modules for non-semisimple categories

Constructed non-compact (2+1)-TQFTs from chromatic categories

Proved spherical tensor categories are chromatic categories

Abstract

This paper contains three related groupings of results. First, we consider a new notion of an admissible skein module of a surface associated to an ideal in a (non-semisimple) pivotal category. Second, we introduce the notion of a chromatic category and associate to such a category a finite dimensional non-compact (2+1)-TQFT by assigning admissible skein modules to closed oriented surfaces and using Juh\'asz's presentation of cobordisms. The resulting TQFT extends to a genuine one if and only if the chromatic category is semisimple with nonzero dimension (recovering then the Turaev-Viro TQFT). The third grouping of results concerns sided chromatic maps in finite tensor categories. In particular, we prove that every spherical tensor category (in the sense of Etingof, Douglas et al.) is a chromatic category (and so can be used to define a non-compact (2+1)-TQFT).