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

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

2011-01-31

arxiv.org/abs/1101.5846v3

Mathematics

Algebraic Geometry

Several references were added. Some speculations were removed (Remarks 3.9 and 9.1 in v1). A section on descent of metrics fro

Scientific paper

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).

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