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Institut de Mathématiques de Marseille, UMR 7373
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Operators appearing in Teichmuller theory and conformal field theory

14h30 - 15h30
horaire FRUMAM

Aix-Marseille Université - Site St Charles
3, place Victor Hugo - case 39
13331 MARSEILLE Cedex 03

This talk will describe the singular integral operators that appear in Teichmuller theory, including the Hilbert transform on the circle and in the complex plane. These can be used to solve problems of gluing and uniformisation of multiply connected domains. They also can be used to understand the Szego projections onto the Hardy spaces of these domains. These provide elements of a universal fermionic semigroup, which can be "quantised" through Hilbert-Schmidt operators. Composition of operators corresponds to gluing of domains. It leads to a theory on bordered Riemann surfaces with spin structure, understood in terms of half-order differentials and the Arf invariant. The building blocks are annuli and trinions. Annuli result in a holomorphic semigroup, due to Graeme Segal and Neretin, that provides a complexification of the diffeomorphism group of the circle. Trinions or two-holed discs encode completely the structure of the vertex operator algebra of a single complex fermion.