Topological recursion, discrete surfaces and cohomological field theories

Elba Garcia Failde (Sorbonne U, Paris)

Nov 17. 2021, 10:15 — 11:15

For some decades, deep connections have been forming among enumerative geometry, complex geometry, intersection theory of the moduli space of curves and integrable systems. In 1990, Witten formulated his celebrated conjecture that predicts that the generating series of intersection numbers of psi-classes is a tau function of the KdV hierarchy, which was first proved by Kontsevich making use of a cell decomposition of a combinatorial model of the moduli space of curves by means of certain ribbon graphs which are Feynman graphs of a cubic hermitian matrix model with an external field. In 2007, Chekhov, Eynard and Orantin introduced a procedure that associates a family of differentials to a Riemann surface with some extra data, which we call spectral curve. This tool naturally fits in numerous algebro-geometric contexts, helping build relations among them. In the Witten—Kontsevich case, the Airy curve allows to build the connection with the 4 mentioned areas. After an introduction to topological recursion in general and the Witten—Kontsevich case in particular, I will introduce more general structures which help organising the intersection theory of the moduli space of curves: cohomological field theories. I will relate them to topological recursion in general and I will describe a recent work with R. Belliard, S. Charbonnier and B. Eynard, in which we extend both this relation and the Witten—Kontsevich instance to intersection numbers with Witten’s r-spin class, allowing us to complete the connections to the 4 mentioned areas in the context of Witten’s generalised conjecture for r>1.

Further Information
ESI Boltzmann Lecture Hall
Slides Wulkenhaar
Associated Event:
Higher Structures Emerging from Renormalisation (Graduate School)
Pierre Clavier (U of Haut-Alsace)
Kurusch Ebrahimi-Fard (NTNU, Trondheim)
Peter K. Friz (TU Berlin)
Harald Grosse (U of Vienna)
Dominique Manchon (U Clerment Auvergne)
Sylvie Paycha (U of Potsdam)
Sylke Pfeiffer (U of Potsdam)