Real-normalized differentials and the elliptic Calogero-Moser system

Abstract

In our recent works we have used meromorphic differentials on Riemann surfaces all of whose periods are real to study the geometry of the moduli spaces of Riemann surfaces. In this paper we survey the relevant constructions and show how they are related to and motivated by the spectral theory of the elliptic Calogero-Moser integrable system.

v2: this version combines a survey and section 5 of version 1 of the preprint; other parts of the original preprint will appear elsewhere. Published in the proceedings of Abel conference 2013