Hodge structures on conformal blocks

TL;DR

This paper establishes the existence and uniqueness of complex Hodge structures on modular functors, linking them to Frobenius algebras and providing explicit formulas for certain cases, advancing the mathematical understanding of conformal blocks.

Contribution

It introduces a novel framework connecting Hodge structures on modular functors with Frobenius algebras and cohomological field theories, including explicit formulas for specific cases.

Findings

Proved existence and uniqueness of Hodge structures on modular functors.

Connected Hodge numbers to Frobenius algebra structures.

Derived explicit formulas for Hodge numbers in certain SU(2) cases.

Abstract

We prove existence and uniqueness of complex Hodge structures on modular functors. The proof is based on the non-Abelian Hodge correspondence and Ocneanu rigidity. Given a modular functor, we explain how its Hodge numbers fit into a Frobenius algebra and the Chern characters of its Hodge decompositions into a new cohomological field theory (CohFT). In the case of $\mathrm{SU}(2)$ modular functors of level $2$ times an odd number, we give explicit formulas for all Hodge numbers, in any genus $g$.