Boundary contributions to three loop superstring amplitudes

TL;DR

This paper investigates boundary contributions to three-loop superstring amplitudes, showing that under certain conditions, these contributions vanish, simplifying the calculation of the vacuum amplitude in type II superstring theory.

Contribution

It demonstrates that boundary contributions to the three-loop vacuum amplitude vanish in type II superstring theory with unbroken supersymmetry, using boundary divisor analysis.

Findings

Boundary contributions to three-loop amplitude vanish.

Holomorphic projection issues addressed in higher genus.

Boundary divisors enable measure integration despite projection problems.

Abstract

In type II superstring theory, the vacuum amplitude at a given loop order $g$ can receive contributions from the boundary of the compactified, genus $g$ supermoduli space of curves $\overline{\mathfrak M}_g$. These contributions capture the long distance or infrared behaviour of the amplitude. The boundary parametrises degenerations of genus $g$ super Riemann surfaces. A holomorphic projection of the supermoduli space onto its reduced space would then provide a way to integrate the holomorphic, superstring measure and thereby give the superstring vacuum amplitude at $g$-loop order. However, such a projection does not generally exist over the bulk of the supermoduli spaces in higher genera. Nevertheless, certain boundary divisors in $\partial\overline{\mathfrak M}_g$ may holomorphically map onto a bosonic space upon composition with universal morphisms, thereby enabling an integration of the holomorphic, superstring measure here. Making use of ansatz factorisations of the superstring measure near the boundary, our analysis shows that the boundary contributions to the three loop vacuum amplitude will vanish in closed oriented type II superstring theory with unbroken spacetime supersymmetry.