More On Superstring Perturbation Theory: An Overview Of Superstring Perturbation Theory Via Super Riemann Surfaces

TL;DR

The paper analyzes how superstring perturbation theory, formulated on super Riemann surfaces, handles subtleties in integrating over supermoduli, using the SO(32) heterotic string on a Calabi–Yau manifold as a concrete testbed. It dissects three tightly linked phenomena: one-loop mass splittings in charged chiral multiplets arising from a D-term; two-loop vacuum energy corrections governed by boundary contributions at infinity; and the supersymmetric Ward identities that reveal a Goldstone fermion when SUSY is spontaneously broken by loop effects. The authors develop a unified viewpoint that regularizes conditionally convergent integrals through appropriate gluing-parameter formalisms, clarifies the role of the super period matrix, and shows how V_D controls both one- and two-loop effects in this class of models. The work advances understanding of when and how perturbative string theory yields SUSY breaking, and furnishes a framework that supports broader conclusions about integration over supermoduli in loop amplitudes with anomalous U(1) sectors. Overall, it demonstrates that the nontrivial infrared structure of supermoduli space is essential for predicting mass splittings, vacuum energies, and symmetry-breaking phenomena in realistic string vacua.

Abstract

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.

Figures (14)

• Figure 1: The mass shift of a massless particle $\rho$ can be computed slightly off-shell by treating $\rho$ as a resonance in a scattering amplitude with four external particles. This process is affected by the one-loop mass shift of $\rho$, but now the $\rho$ particle whose mass is shifted is generically off-shell, giving a sound framework for the $k^2/k^2$ computation.
• Figure 2: A process (a) in which two punctures on a Riemann surface $\Sigma$ -- here of genus 1 -- approach each other is equivalent conformally to a process (b) in which $\Sigma$ splits into two components $\Sigma_\ell$ and $\Sigma_r$, connected by a narrow neck, with one of them a genus 0 surface that contains the two punctures.
• Figure 3: A generalization of fig. \ref{['Short']} in which a surface $\Sigma$ splits into a pair of components $\Sigma_\ell$ and $\Sigma_r$ of arbitrary genus, joined via a narrow neck. In the example shown, $\Sigma_\ell$ and $\Sigma_r$ are both genus 1 surfaces with punctures.
• Figure 4: A Riemann surface or super Riemann surface can undergo either a separating degeneration as in (a) or a nonseparating one as in (b). In each case the degeneration involves the collapse of a narrow neck, labeled by the arrow. The singular configurations that arise when the neck collapses are sketched in fig. \ref{['Limiting']}.
• Figure 5: Collapse of the narrow necks in fig. \ref{['Narrow']} leads to these limiting configurations. The singularities depicted here are known as ordinary double points. The local picture is that two branches meet at a common point. The fact that the only singularities that occur in the Deligne-Mumford compactification are ordinary double point singularities (which have a long distance or infrared interpretation in spacetime) is the reason that there are no ultraviolet divergences in superstring perturbation theory.