The Super Period Matrix with Ramond Punctures in the supergravity formulation

TL;DR

This paper extends Witten’s recent construction of the super period matrix with Ramond punctures to the supergravity formulation, providing new insights into supermoduli variations and degenerations.

Contribution

It offers a supergravity-based construction of the super period matrix with Ramond punctures, complementing Witten’s holomorphic approach and analyzing its deformation behavior.

Findings

Constructed the super period matrix in supergravity formulation.

Derived variations of the super period matrix with respect to supermoduli.

Illustrated degeneration of the super period matrix with Ramond punctures.

Abstract

In a very recent preprint, Witten showed how to construct a $g|r \, \times \, g|r$ super period matrix for super Riemann surfaces of genus $g$ with $2r$ Ramond punctures, which is symmetric in the ${\bf Z}_2$ graded sense. He also showed how it can be applied to analyze supersymmetry breaking in string compactifications which are supersymmetric at tree-level. Witten's construction is in the purely holomorphic formulation of super Riemann surfaces. In this paper, a construction is given in the formulation of two-dimensional supergravity. The variations of the super period matrix with respect to supermoduli deformations are also given, as well as an explicit illustration of how the super period matrix with two Ramond punctures would emerge from a degeneration of the super period matrix without punctures in higher genus.