The decoupling of moduli about the standard embedding

TL;DR

This paper analyzes the cohomology of a differential complex related to heterotic string theory moduli at the standard embedding, demonstrating a decomposition into a direct sum and suggesting a perfect obstruction theory.

Contribution

It explicitly demonstrates the decomposition of cohomology groups at the standard embedding, which was previously assumed but not shown, and relates this to the Euler characteristic and obstruction theory.

Findings

Cohomology groups decompose into a direct sum at the standard embedding.

Euler characteristic of the complex is zero under certain conditions.

Supports the existence of a perfect obstruction theory for heterotic moduli.

Abstract

We study the cohomology of an elliptic differential complex arising from the infinitesimal moduli of heterotic string theory. We compute these cohomology groups at the standard embedding, and show that they decompose into a direct sum of cohomologies. While this is often assumed in the literature, it had not been explicitly demonstrated. Given a stable gauge bundle over a complex threefold with trivial canonical bundle and no holomorphic vector fields, we also show that the Euler characteristic of this differential complex is zero. This points towards a perfect obstruction theory for the heterotic moduli problem, at least for the most physically relevant compactifications.