Enumerative geometry via the moduli space of super Riemann surfaces

Abstract: Mumford initiated the calculation of many algebraic topological invariants over the moduli space of Riemann surfaces in the 1980s, and Witten related these invariants to two dimensional gravity in the 1990s. This viewpoint led Wittento a conjecture, proven by Kontsevich, that a generating function for intersection numbers on the moduli space of curves is a tau function of the KdV hierarchy, now known as the Kontsevich-Witten tau function, which allowed their evaluation. In 2004, Mirzakhaniproduced another proof of Witten's conjecture via the study of Weil-Petersson volumes of the moduli space using hyperbolic geometry. In this lecture I will describe a new collection of integrals over the moduli space of Riemann surfaces whose generating functionis a tau function of the KdV hierarchy, known as the Brezin-Gross-Witten tau function. I will sketch a proof of this result that uses an analogue of Mirzakhani's argument applied to the moduli space of super Riemann surfaces - defined by replacing the fieldof complex numbers with a Grassman algebra - which uses recent work of Stanford and Witten. This appearance of the moduli space of super Riemann surfaces to solve a problem over the classical moduli space is deep and surprising.

HEP - theorymathematical physicsalgebraic geometrycombinatoricsexactly solvable and integrable systems

Audience: researchers in the discipline

IBS-CGP Mathematical Physics Seminar

Series comments: Registration is required at cgp.ibs.re.kr/activities/talkregistration