A modular functor from state sums for finite tensor categories and their bimodules

TL;DR

This paper develops a state-sum based modular functor for finite tensor categories and bimodules, enabling explicit computations of surface invariants and mapping class group representations without requiring semisimplicity or pivotal structures.

Contribution

It introduces a novel state-sum construction for a modular functor valued in finite categories, accommodating boundaries, defects, and non-semisimple categories.

Findings

Constructed a modular functor on 2-framed bordisms.

Enabled explicit computation of surface functors.

Represented mapping class groups on surfaces.

Abstract

We construct a modular functor which takes its values in the monoidal bicategory of finite categories, left exact functors and natural transformations. The modular functor is defined on bordisms that are 2-framed. Accordingly we do not need to require that the finite categories appearing in our construction are semisimple, nor that the finite tensor categories that are assigned to two-dimensional strata are endowed with a pivotal structure. Our prescription can be understood as a state-sum construction. The state-sum variables are assigned to one-dimensional strata and take values in bimodule categories over finite tensor categories, whereby we also account for the presence of boundaries and defects. Our construction allows us to explicitly compute functors associated to surfaces and representations of mapping class groups acting on them.