The physical moduli of heterotic G_2 string compactifications

TL;DR

This paper develops a refined operator framework to characterize the physical moduli space of heterotic G_2 string compactifications, linking deformations of the spin connection to G_2 manifold moduli and ensuring consistency with known cases.

Contribution

It introduces an operator $reve D$ that captures the physical moduli of heterotic G_2 compactifications, eliminating spurious degrees of freedom and aligning with established results in special cases.

Findings

The operator $reve D$ accurately describes the physical moduli space.

The proposed moduli space metric reduces to known $SU(3)$ metrics.

Kernels of $reve D$ and its adjoint correspond to F- and D-term conditions.

Abstract

In previous works, an operator was developed for heterotic compactifications on $\mathbb{R}^{2,1}\times G_2$ and $AdS_3 \times G_2$, which preserves $N=1$ $d=3$ supersymmetry and whose kernel is related to the moduli of the compactification. The operator is described in terms of non-physical spurious degrees of freedom, specifically, deformations of a connection on the tangent bundle. In this paper, we eliminate these spurious degrees of freedom by linking deformations of the spin connection to the moduli of the $G_2$ manifold $Y$. This results in an operator $\check D$ that captures the physical moduli space of the $G_2$ heterotic string theory. When $Y=X\times S^1$, with $X$ an $SU(3)$ manifold, we show $\check D$ produces results that align with existing literature. This allows us to propose a $G_2$ moduli space metric. We check that this metric reduces to the $SU(3)$ moduli metric constructed in the literature. We then define an adjoint operator ${\check D}^\dag$. We show the $G_2$ moduli correspond to the intersection of the kernels of $\check D$ and ${\check D}^\dag$. These kernels reduce to the $SU(3)$ F-terms and D-terms respectively on $X\times S^1$. This gives two non-trivial consistency checks of our proposed moduli space metric. Working perturbatively in $\alpha'$, we also demonstrate that the heterotic $G_2$ moduli problem can be characterised in terms of a double extension of ordinary bundles, just like in the $SU(3)$ case.