XXXII Workshop on Geometric Methods in Physics 30.06-6.07.2013

Sergey Shadrin

Matrix models topological recursion and Givental theory

Topological recursion for matrix models has appeared in the papers of Chekhov, Eynard, and Orantin, and it serves both as a very powerful computational tool and as a way to defines the correlation forms of a matrix model in a mathematically rigorous way.

We explain a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve. Under some conditions we can identify the n-point functions computed from spectral curve with the Givental formula for the ancestor formal Gromov-Witten potential.

Using this identification one can deduce various relations of combinatorial problems to the intersection theory of the moduli spaces of curves. Examples include the Norbury-Scott conjecture on a particular spectral curve for the Gromov-Witten theory of $CP^1$, a new way to derive the ELSV formula for Hurwitz numbers, and a mathematical physics proof of the 2006 conjecture of Zvonkine that relates Hurwitz numbers with completed cycles to the intersection theory of the moduli spaces of $r$-spin structure.

The talk will follow my recent works with Dunin-Barkowski, Orantin, Spitz, and Zvonkine.

Event sponsored by:
Belgian Science Policy Office     PAI          University
of Bialystok
University of Bialystok

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