The `Harder-Narasimhan trace’ and unitarity of the KZ/Hitchin connection: genus 0

Abstract

Let a reductive group $G$ act on a projective variety $\mathcal{X}_+$, and suppose given a lift of the action to an ample line bundle $\hat{\theta}$. By definition, all $G$-invariant sections of $\hat{\theta}$ vanish on the nonsemistable locus $\mathcal{X}_+^{nss}$. Taking an appropriate normal derivative defines a map $H^0(\mathcal{X}_+,\hat{\theta})^G \to H^0(S_{\mu},\mathcal{V}_{\mu})^G$, where $\mathcal{V}_{\mu}$ is a $G-$vector bundle on a $G-$variety $S_{\mu}$. We call this the Harder-Narasimhan trace. Applying this to the Geometric Invariant Theory construction of the moduli space of parabolic bundles on a curve, we discover generalisations of “Coulomb-gas representations”, which map conformal blocks to hypergeometric local systems. In this paper we prove the unitarity of the KZ/Hitchin connection (in the genus zero, rank two, case) by proving that the above map lands in a unitary factor of the hypergeometric system. (An ingredient in the proof is a lower bound on the degree of polynomials vanishing on partial diagonals.) This elucidates the work of K. Gawedzki.