| 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 Witten to 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 which
allowed their evaluation. In 2004, Mirzakhani produced 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 different collection of integrals over the moduli space of
Riemann surfaces whose generating function is a tau function of the
KdV hierarchy. 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 field of complex numbers
with a Grassmann algebra - which uses recent work of Stanford and
Witten. This appearance of the moduli space of super Riemann surfaces
to potentially solve a problem over the classical moduli space is deep
and surprising.
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