Date: September 18, 2014

Time: 09:00AM - 10:00AM

Location: BLOC 220

Speaker: Prakash Belkale, U N Carolina

  

Title: Gauss-Manin representations of conformal block local systems, III

Abstract:

Conformal blocks give projective local systems on moduli spaces of curves with marked points. One can ask if they are “realizable in geometry”, i.e., as local subsystems of suitable Gauss-Manin local systems of cohomology of families of smooth projective varieties.

I will focus on the genus 0 situation where there is extensive contact with the theory of hyperplane arrangements. In genus zero, conformal blocks are surjected on to by constant vector bundles of classical coinvariants; which carry connections lifting the one on conformal blocks. One would like to have a consistent "integral" Hodge theoretic realization of this entire package. Formulating what one expects for the package is a challenge, and I want to get to that in the lectures. The following will be covered (in an "inverted" order),

1) I will discuss (in genus 0) the proof of Gawedzki et al’s conjecture that Schechtman-Varchenko forms are square integrable (this was proved first for sl(2) by Ramadas). Together with the flatness results of Schechtman-Varchenko, and the work of Ramadas, one obtains the desired realization and a unitary metric on conformal blocks.

2) Characterization of the image of conformal blocks in cohomology (joint with S. Mukhopadhyay)

3) Whether classical coinvariants can be characterized cohomologically (together with an integral structure).

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