# The infinitesimal moduli space of heterotic $G_2$ systems

## TL;DR

This paper characterizes the infinitesimal moduli space of heterotic G2 systems using a covariant derivative operator, revealing a cohomological structure that generalizes Atiyah's deformation analysis to heterotic string compactifications.

## Contribution

It introduces a covariant exterior derivative $\\cal D$ for heterotic G2 systems, establishing a cohomological framework for their infinitesimal deformations, extending previous deformation theories.

## Key findings

- The operator \$\cal D\$ encodes heterotic G2 geometry and satisfies \$\check{\cal D}^2=0$.
- The moduli space is given by the cohomology group \$\check H^1_{\check{\cal D}}(\cal Q)\$.
- Results are applicable to all orders in the \$\alpha'\$ expansion.

## Abstract

Heterotic string compactifications on integrable $G_2$ structure manifolds $Y$ with instanton bundles $(V,A), (TY,\tilde{\theta})$ yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative $\cal D$ and show that it is equivalent to a heterotic $G_2$ system encoding the geometry of the heterotic string compactifications. This operator $\cal D$ acts on a bundle ${\cal Q}=T^*Y\oplus{\rm End}(V)\oplus{\rm End}(TY)$ and satisfies a nilpotency condition $\check{\cal D}^2=0$, for an appropriate projection of $\cal D$. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group $\check H^1_{\check{\cal D}}(\cal Q)$. We comment on the similarities and differences of our result with Atiyah's well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the $\alpha'$ expansion.