The ‘Harder-Narasimhan trace’ and unitarity of the KZ/Hitchin connection: genus 0. (English) Zbl 1167.32011

Summary: Let a reductive group \(G\) act on a projective variety \(X_+\), and suppose given a lift of the action to an ample line bundle \(\widehat \theta\). By definition, all \(G\)-invariant sections of \(\widehat \theta\) vanish on the nonsemistable locus \(X_+^{\text{nss}}\). Taking an appropriate normal derivative defines a map \(H^0(X_+,\widehat\theta)^G \rightarrow H^0(S_{\mu},V_{\mu})^G\), where \(V_{\mu}\) 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 the “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.


32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
14D20 Algebraic moduli problems, moduli of vector bundles
14L30 Group actions on varieties or schemes (quotients)
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