Modular functors from conformal blocks of rational vertex operator algebras

TL;DR

This paper proves that conformal blocks of strongly rational vertex operator algebras form a modular functor, leading to a 3D topological field theory with a rich categorical structure.

Contribution

It establishes the modular functor property for conformal blocks of rational vertex operator algebras, connecting algebraic and topological structures.

Findings

Conformal blocks form vector bundles over moduli spaces.

The category of modules inherits a ribbon Grothendieck-Verdier structure.

The modular functor extends to a 3D topological field theory.

Abstract

For a vertex operator algebra $V$, one may naturally define spaces of conformal blocks following a construction of Frenkel-Ben-Zvi generalized by Damiolini-Gibney-Tarasca. If $V$ is strongly rational, these spaces of conformal blocks form vector bundles over a suitable moduli space of algebraic curves. In this article, we establish, under the same assumptions, the widely expected topological result that the spaces of conformal blocks produce a modular functor, i.e. a modular algebra over an extension of the surface operad. This entails that the category $\mathcal{C}_V$ of admissible $V$-modules inherits from the topology of genus zero surfaces a ribbon Grothendieck-Verdier structure that leads even to the structure of a modular fusion category whose structure comes directly from the spaces of conformal blocks of $V$. As a direct consequence, we prove that the modular functor from conformal blocks extends to a three-dimensional topological field theory and comes with a description in terms of factorization homology.