Ron Donagi, University of Pennsylvania
Super Riemann surfaces and foundations of superstring perturbation theory
Superstring perturbation theory expresses scattering amplitudes as integrals over the moduli of punctured super Riemann surfaces. The moduli space of super Riemann surfaces has many analogies with the moduli space of ordinary Riemann surfaces. For example, there is a super analog of the Deligne-Mumford compactification, and it is important in understanding the qualitative properties of superstring scattering amplitudes. There is also a superanalog of the Mumford isomorphism between certain line bundles over moduli space, and it plays a role analogous to the role that the ordinary Mumford isomorphism plays in bosonic string theory.
Integration over supermoduli space gives a powerful framework for understanding important properties of string theory such as spacetime supersymmetry. But in practice the understanding of superstring perturbation theory via super Riemann surfaces has been relatively little-developed, beyond elementary examples in genus 0 and 1. Only in the early 2000s did DHoker and Phong succeed in determining some genus 2 amplitudes explicitly, by exploiting the super period matrix of a super Riemann surface. Likewise there has been only limited mathematical work on super Riemann surfaces in recent years, one recent result being that despite the analogies between them, super moduli space cannot be reduced to the ordinary moduli space in the sense that there is no holomorphic projection from super moduli space to its underlying reduced space.
Main topics will include: supermanifolds, supersymmetry, super Riemann surfaces, moduli spaces, Deligne-Mumford compactifications, novel issues related to Ramond punctures, super Mumford isomorphisms, super period matrices, and how these all fit into perturbative superstring theory: the measure on super moduli space, the Neveu-Schwarz and Ramond sectors, and other topics including the spontaneous breaking of supersymmetry at 1 loop.