Members’ Seminar

This will be a gentle intro, aimed at a general mathematical audience, to supergeometry: supermanifolds, super Riemann surfaces, super moduli, etc. As time permits, we will discuss various aspects of supergeometry, including deformation theory and the related obstruction, Atiyah, and super-Atiyah classes. The goal is to apply this to the geometry of the moduli space of super Riemann surfaces. There one can prove that for genus greater than or equal to 5, this moduli space is not projected (and in particular is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathematically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own.