Quantum aspects of heterotic $G_2$ systems

TL;DR

This paper studies the mathematical structure of heterotic string compactifications on manifolds with $G_2$ structure, revealing new cohomological tools to analyze moduli and computing the one-loop partition function.

Contribution

It introduces a bicomplex framework for heterotic $G_2$ systems and relates moduli and obstructions to cohomology groups, advancing understanding of string compactifications.

Findings

Identified a nilpotent differential and symplectic pairing for the system.

Constructed a double complex to analyze infinitesimal moduli.

Computed the one-loop partition function as a product of gauge theory contributions.

Abstract

Compactifications of the heterotic string, to first order in the $\alpha'$ expansion, on manifolds with integrable $G_2$ structure give rise to three-dimensional ${\cal N} = 1$ supergravity theories that admit Minkowski and AdS ground states. As shown in arXiv:1904.01027, such vacua correspond to critical loci of a real superpotential $W$. We perform a perturbative study around a supersymmetric vacuum of the theory, which confirms that the first order variation of the superpotential, $\delta W$, reproduces the BPS conditions for the system, and furthermore shows that $\delta^2 W=0$ gives the equations for infinitesimal moduli. This allows us to identify a nilpotent differential, and a symplectic pairing, which we use to construct a bicomplex, or a double complex, for the heterotic $G_2$ system. Using this complex, we determine infinitesimal moduli and their obstructions in terms of related cohomology groups. Finally, by interpreting $\delta^2 W$ as an action, we compute the one-loop partition function of the heterotic $G_2$ system and show it can be decomposed into a product of one-loop partition functions of Abelian and non-Abelian instanton gauge theories.