Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes

TL;DR

This paper explores the cohomological structure of chiral superstring amplitudes on genus 2 surfaces, demonstrating that they can always be represented holomorphically through a novel hybrid cohomology framework.

Contribution

It introduces a hybrid cohomology theory that unifies de Rham and Dolbeault cohomologies, providing a constructive method to obtain holomorphic representatives of superstring amplitudes.

Findings

Hybrid cohomology classes admit holomorphic representatives.

A recursive construction using monodromies is developed.

New classification of kinematic invariants and Green's functions is provided.

Abstract

The relation between superholomorphicity and holomorphicity of chiral superstring N-point amplitudes for NS bosons on a genus 2 Riemann surface is shown to be encoded in a hybrid cohomology theory, incorporating elements of both de Rham and Dolbeault cohomologies. A constructive algorithm is provided which shows that, for arbitrary N and for each fixed even spin structure, the hybrid cohomology classes of the chiral amplitudes of the N-point function on a surface of genus 2 always admit a holomorphic representative. Three key ingredients in the derivation are a classification of all kinematic invariants for the N-point function, a new type of 3-point Green's function, and a recursive construction by monodromies of certain sections of vector bundles over the moduli space of Riemann surfaces, holomorphic in all but exactly one or two insertion points.