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Riemann-Hilbert problems on Riemann Surfaces: Tyurin data, non-abelian Cauchy kernels and the Goldman bracket

Marco Bertola
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Wednesday, March 18, 2020 - 14:30
In this talk I will start with  a review of Riemann—Hilbert problems for matrices on Riemann surfaces. 
They can be rephrases as (stable) vector bundles and we will review the (very simple)  notion of moduli of (stable) vector bundles  expressed by Tyurin in late ‘60s. 
For each (stable) vector bundle $\mathcal  E$ of rank $n$ and degree $ng$ we can associate a matrix kernel (the non—Abelian Cauchy kernel) which is written explicitly in terms of the Tyurin data and the ordinary third kind differentials. 
Using the Cauchy kernel (and additional datum of divisor of degree $g$) we construct a map from the cotangent bundle (of the moduli space)  to the space of representations of the fundamental group of the surface. 
The final goal is to show that the complex-analytic canonical symplectic structure on the cotangent bundle is mapped to the symplectic structure introduced by Goldman on the space of representations. 
The main tool is the “Malgrange-Fay” form, expressing the tautological form  on the cotangent bundle; it has also an interesting algebro—geometric relevance because this form has a pole along the non-Abelian Theta divisor whose residue is interpreted as dimension of an appropriate space. 
Joint work (in progress) with C. Norton, G. Ruzza.

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