Hermitian geometry on the big phase space

Ian Strachan (University of Glasgow)

Tuesday 16th October, 2012 15:00-16:00 325


In the early '90s Witten, by studying intersection theory on the moduli space of curves, defined a Topological Quantum Field Theory (TQFT) on an infinite dimensional manifold known as the big phase space. By restricting this theory to the so-called small phase space one obtains a finite dimensional object known as a Frobenius manifold. While one talks of metrics and connections on a Frobenius manifold everything is actually holomorphic and not real. However, again in the early '90s, Cecotti and Vafa showed how one can construct an Hermitian geometry on a Frobenius manifold in such a way that the Hermitian and holomorphic structures are compatible. This leads to an additional set of (integrable) equations known as the tt*-equations. The aim of this talk is to link these two ideas together and define an Hermitian metric on the infinite dimensional big phase space, compatible with the holomorphic structures arising from the TQFT.

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