Modular functors and CFT correlators via double categories

TL;DR

This paper demonstrates that double categories provide a natural framework for modular functors in conformal field theory, establishing equivalences that lead to universal correlators and field functors being isomorphisms.

Contribution

It introduces a double categorical approach to modular functors and correlators in CFT, showing their equivalences and universal properties using skein theoretic methods.

Findings

Vertical transformations between modular functors are equivalences.

Universal correlators are isomorphisms of vector spaces.

Field functors are equivalences of categories.

Abstract

We point out that double categories provide a natural setting for modular functors obtained by a (bicategorical) string-net construction: The source of the modular functor -- which is now a double functor -- is a symmetric monoidal double category of bordisms, with bordisms as horizontal morphisms and smooth embeddings of manifolds as vertical morphisms. The target of the modular functor is a double category with profunctors and functors as horizontal and vertical morphisms. The correlators and field functors for a conformal field theory based on a pivotal monoidal category $\mathcal C$ can then be understood in the unified setting of a vertical transformation between the modular functors for two pointed pivotal bicategories, the delooping of $\mathcal C$ and the bicategory of $\Delta$-separable symmetric Frobenius algebras in $\mathcal C$. Using skein theoretic methods, we show that this vertical transformation is an equivalence, which implies that field functors are equivalences of categories and that universal correlators are isomorphisms of vector spaces.