Getting superstring amplitudes by degenerating Riemann surfaces

TL;DR

This paper demonstrates how to derive superstring amplitudes via degenerations of Riemann surfaces, providing new insights into higher-genus computations and the structure of the superstring measure.

Contribution

It introduces a method to obtain superstring amplitudes through Riemann surface degenerations and derives consistency relations at any genus, including new corrections for genus higher than three.

Findings

Computed two-point functions at genus two, three, and four.

Found that the two-point function vanishes at genus two and three after GSO projection.

Identified discrepancies at genus four suggesting the need for measure corrections.

Abstract

We explicitly show how the chiral superstring amplitudes can be obtained through factorisation of the higher genus chiral measure induced by suitable degenerations of Riemann surfaces. This powerful tool also allows to derive, at any genera, consistency relations involving the amplitudes and the measure. A key point concerns the choice of the local coordinate at the node on degenerate Riemann surfaces that greatly simplifies the computations. As a first application, starting from recent ansaetze for the chiral measure up to genus five, we compute the chiral two-point function for massless Neveu-Schwarz states at genus two, three and four. For genus higher than three, these computations include some new corrections to the conjectural formulae appeared so far in the literature. After GSO projection, the two-point function vanishes at genus two and three, as expected from space-time supersymmetry arguments, but not at genus four. This suggests that the ansatz for the superstring measure should be corrected for genus higher than four.