The Super Period Matrix With Ramond Punctures

TL;DR

This paper extends the super period matrix concept to super Riemann surfaces with Ramond punctures, providing a new mathematical framework and applying it to analyze genus 2 vacuum amplitudes in supersymmetric string theory.

Contribution

It introduces a generalized super period matrix for super Riemann surfaces with Ramond punctures, expanding the classical theory and enabling new insights into string theory amplitudes.

Findings

The generalized period matrix is symmetric in the Z_2-graded sense.

The genus 2 vacuum amplitude vanishes pointwise after summing over spin structures.

The result explains previous explicit computations by D'Hoker and Phong.

Abstract

We generalize the super period matrix of a super Riemann surface to the case that Ramond punctures are present. For a super Riemann surface of genus g with 2r Ramond punctures, we define, modulo certain choices that generalize those in the classical theory (and assuming a certain generic condition is satisfied), a g|r x g|r period matrix that is symmetric in the Z_2-graded sense. As an application, we analyze the genus 2 vacuum amplitude in string theory compactifications to four dimensions that are supersymmetric at tree level. We find an explanation for a result that has been found in orbifold examples in explicit computations by D'Hoker and Phong: with their integration procedure, the genus 2 vacuum amplitude always vanishes "pointwise" after summing over spin structures, and hence is given entirely by a boundary contribution.