Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0 for arbitrary Lie algebras

TL;DR

This paper proves that vector bundles of conformal blocks for any simple Lie algebra on genus zero moduli spaces have natural unitary metrics preserved by the KZ/Hitchin connection, confirming a conjecture by Gawedzki.

Contribution

It extends the unitarity of conformal blocks to all simple Lie algebras in genus zero, providing explicit geometric metrics preserved by the connection.

Findings

Established unitarity for all simple Lie algebras in genus zero

Constructed explicit geometric unitary metrics

Confirmed Gawedzki's conjecture for genus zero case

Abstract

We prove that the vector bundles of conformal blocks, on suitable moduli spaces of genus zero curves with marked points, for arbitrary simple Lie algebras and arbitrary integral levels, carry unitary metrics of geometric origin which are preserved by the Knizhnik-Zamolodchikov/Hitchin connection (as conjectured by Gawedzki). Our proof builds upon the work of Ramadas who proved this unitarity statement in the case of the Lie algebra sl(2) (and genus 0). We note that unitarity has been proved in all genera (including genus 0) by the combined work of Kirillov and Wenzl in the 90's (their approach does not yield a concrete metric).