Quantum aspects of heterotic $G_2$ systems
Xenia de la Ossa, Magdalena Larfors, Matthew Magill, Eirik E. Svanes
TL;DR
This work develops a perturbative, holography-inspired framework for heterotic string compactifications on 7-manifolds with integrable G2 structure, recasting the moduli problem as critical points of a real superpotential $W$. By computing first- and second-order variations of $W$, the authors relate $\delta W$ to the BPS equations and $\tfrac{1}{2}(\delta^2 W)$ to infinitesimal moduli via a nilpotent differential $\mathbb{D}$ and a shifted symplectic pairing, yielding a bicomplex that encodes gauge and geometric data. They prove nilpotency and ellipticity of the projected differential $\check{\mathbb{D}}$, construct a corresponding bicomplex, and analyze infinitesimal moduli $H^0_{\check{\mathbb{D}}}$ and obstructions $H^1_{\check{\mathbb{D}}}$, finding an index-zero moduli problem with Serre-like dualities. Quantum aspects are addressed by interpreting $\tfrac{1}{2}\delta^2 W$ as a quadratic action and computing the absolute value of the one-loop partition function, which factorizes into determinants associated with Abelian and non-Abelian instanton sectors after a diagonalization of the complex. The results connect heterotic G2 deformation theory to elliptic double complexes and generalized geometry, suggesting potential topological invariants and guiding future BV quantization studies and comparisons with topological G2 string invariants.
Abstract
Compactifications of the heterotic string, to first order in the $α'$ 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, $δW$, reproduces the BPS conditions for the system, and furthermore shows that $δ^2 W=0$ 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 $δ^2 W$ 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.