Riemann Surfaces: Vector Bundles, Physics, and Dynamics.

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  • Shehryar Sikander, Denmark
We construct quantum representation of a subgroup of the mapping class group of a genus two surface. Our construction relies on realizing this subgroup as the orbifold fundamental group of a Teichmueller curve, pulling back the Hitchin connection to this Tecihmueller curve, and computing the monodromy with respect to the pulled back connection. The formula for the representation includes a series with coefficients as iterated integrals. This series is closely related to the cyclotomic version of the Drinfel'd associator.

The geodesic flow in the unit the tangent bundle of this Teichmueller curve is ergodic. We construct a cocycle on this ergodic dynamical system by using the parallel transport of the pulled back Hitchin connection and study its properties. We conjecture that in level one, this cocycle coincides with the Kontsevich-Zorich cocycle.
Original languageEnglish
PublisherCentre for Quantum Geometry of Moduli Spaces, Aarhus University
Number of pages114
Publication statusPublished - 2014

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