Categorified Open Topological Field Theories

TL;DR

This paper classifies 2D categorified open topological field theories using pivotal Grothendieck-Verdier categories, linking them to conformal blocks, modular structures, and mapping class group representations, expanding the theoretical framework beyond rigid categories.

Contribution

It introduces a classification of 2D categorified open topological field theories via pivotal Grothendieck-Verdier categories and develops new descriptions of conformal blocks and mapping class group representations.

Findings

Classification of theories using pivotal Grothendieck-Verdier categories

Description of conformal blocks via cyclic associative operad

Construction of mapping class group representations from non-rigid categories

Abstract

In this short note, we classify linear categorified open topological field theories in dimension two by pivotal Grothendieck-Verdier categories, a type of monoidal category equipped with a weak, not necessarily rigid duality. In combination with recently developed string-net techniques, this leads to a new description of the spaces of conformal blocks of Drinfeld centers $Z(\mathcal{C})$ of pivotal finite tensor categories $\mathcal{C}$ in terms of the modular envelope of the cyclic associative operad. If $\mathcal{C}$ is unimodular, we prove that the space of conformal blocks inherits the structure of a module over the algebra of class functions of $\mathcal{C}$ for every free boundary component. As a further application, we prove that the sewing along a boundary circle for the modular functor for $Z(\mathcal{C})$ can be decomposed into a sewing procedure along an interval and the application of the partial trace. Finally, we construct mapping class group representations from Grothendieck-Verdier categories that are not necessarily rigid and make precise how these generalize existing constructions.