Superstring perturbation theory via super Riemann surfaces: an overview

This article is devoted to an overview of superstring perturbation theory from the point of view of super Riemann surfaces. We aim to elucidate some of the subtleties of superstring perturbation that caused difficulty in the early literature, focusing on a concrete example—the $SO(32)$ heterotic string compactified on a Calabi–Yau manifold, with the spin connection embedded in the gauge group. This model is known to be a significant test case for superstring perturbation theory. Supersymmetry is spontaneously broken at 1-loop order, and to treat correctly the supersymmetry-breaking effects that arise at 1- and 2-loop order requires a precise formulation of the procedure for integration over supermoduli space. In this paper, we aim as much as possible for an informal explanation, though at some points we provide more detailed explanations that can be omitted on first reading.