Supergeometry: introduction and some applications
When
February 8, 2019 | 2:30 - 3.30 p.m.
Where
Campbell Hall 443
Speaker
Dr. Theodore Voronov, University of Manchester
Abstract
Supergeometry owes its name to supersymmetry, which is a (theoretical) symmetry mixing bosons and fermions in physics. Supersymmetric models appeared in 1970s, but "super" ideas have much deeper roots both in mathematics and physics e.g. in the works of Hermann Graßmann and Élie Cartan. The pioneer of supermathematics was Felix Alexandrovich Berezin. His discovery of what we now call "Berezin integral" was used by Faddeev and Popov for quantization of gauge fields even before the official birth of supermathematics.
Supergeometry provides powerful tools for "ordinary" mathematics and mathematical physics, and has led to spectacular applications such as analytic proof of the Atiyah-Singer index theorem based on supermanifold quantization. It possesses a unifying power, making it possible to see e.g. Clifford algebra and differential operators as basically the same thing. Recent applications of supergeometry are related with symmetry structures "up to homotopy." This includes the proof of the existence of deformation quantization of Poisson manifolds due to Kontsevich.
I will give an introduction to the main ideas of supergeometry with examples (including my own work). No prior knowledge will be assumed. I hope to touch on some of my recent works, such as volumes of classical supermanifolds (originated from my counterexample to a conjecture by Witten), differential operators on the superline, and (time permitting) an entirely new idea of "microformal geometry."