22 jun 2016 -- 14:30
Aula di Consiglio, Dip.Matematica, Università "La Sapienza", Roma
Abstract.
Super Riemann surfaces exhibit many of the familiar features of ordinary Riemann surfaces, and some novelties. They have moduli spaces and Deligne-Mumford compactifications. One can integrate and construct measures on moduli spaces. The punctures one can insert come in two varieties: Ramond and Neveu-Schwarz. I will survey some of the expected and unexpected features, including some aspects of non splitness: for genus g>4, the moduli space of super Riemann surfaces 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. When we examine the Deligne-Mumford compactification of moduli space, and especially the Ramond boundary divisors, we find that the interesting new phenomena start already in genus one. This is interpreted as the mechanism that allows supersymmetry to remain unbroken at tree level in certain models of superstring perturbation theory, but to be spontaneously broken at one loop.