Open modular functors from non-finite tensor categories

TL;DR

This paper constructs mapping class group representations from non-finite tensor categories, inspired by vertex operator algebra modules, and shows they have a 3D origin and compatibility with gluing, extending tools for conformal blocks.

Contribution

It introduces a framework for modular functors from non-finite tensor categories, connecting them to 3D topological quantum field theories and conformal blocks.

Findings

Mapping class group representations are compatible with gluing along intervals.

These representations admit a factorization homology description.

The framework extends tools for conformal blocks to non-finite, rigid categories.

Abstract

We show that a compact rigid balanced braided monoidal category with enough compact projective objects gives rise to a system of mapping class group representations compatible with the gluing along marked intervals. A motivation to consider non-finite tensor categories is the theory of vertex operator algebras where such categories arise as categories of modules. The mapping class group representations presented in this article admit a factorization homology description. In other words, they are of three-dimensional origin and hence obey a holographic principle. A compact projective symmetric Frobenius algebra endows the representations with a pointing that is mapping class group invariant and compatible with the gluing along intervals. This shows that, at least to some extent, many of the tools for the construction and study of spaces of conformal blocks and correlators remain available in a non-finite, but rigid setting.