Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 1

Special Issue on Super Riemann Surfaces and String Theory

Guest Editor: Edward Witten

Superstring perturbation theory via super Riemann surfaces: an overview

Pages: 517 – 607

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n1.a4


Edward Witten (School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A.)


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.

The arXiv version of this article was originally entitled “More on Superstring Perturbation Theory.”

Research supported in part by NSF Grant PHY-0969448. I thank K. Becker, P. Deligne, R. Donagi, G. Moore, N. Seiberg, and H. Verlinde for useful advice and comments.

Received 6 September 2018

Published 2 April 2019