The Hitchin and Knizhnik-Zamolodchikov connections are projectively equivalent in the genus zero case

TL;DR

This paper proves that in genus zero with at least three marked points, the Knizhnik-Zamolodchikov and Hitchin connections are projectively equivalent, linking conformal field theory and geometric quantisation.

Contribution

It establishes the projective equivalence of the KZ and Hitchin connections in genus zero, providing a new understanding of their relationship in conformal field theory.

Findings

The KZ and Hitchin connections are projectively equivalent in genus zero.

The isomorphism intertwines these connections up to a scalar-valued one-form.

Constructs a projectively unique and flat Hitchin connection using an auxiliary metaplectic correction.

Abstract

This paper establishes the projective equivalence between the Knizhnik-Zamolodchikov connection and the Hitchin connection in genus 0 with at least 3 marked points. The Knizhnik-Zamolodchikov connection is defined on the sheaf of conformal blocks in the Tsuchiya-Ueno-Yamada model of conformal field theory. The Hitchin connection is defined on the Verlinde bundle via geometric quantisation of the moduli space of flat connections. Pauly's isomorphism establishes the equivalence of these two vector bundles. The main theorem of this paper is that the isomorphism intertwines these two connections up to a scalar-valued one-form. In addition, this theorem is used to construct a Hitchin connection through an auxiliary metaplectic correction. As a corollary of the main theorem, this construction of the Hitchin connection is projectively unique and projectively flat.