Three-dimensional vacua of $G_2$ flux compactifications

Abstract

We study supersymmetric vacua of three-dimensional $\mathcal{N}=1$ effective theories from type IIA, type IIB/type I, and heterotic compactifications on $G_2$-structure spaces. Superpotential homogeneity and the physical scalar metric yield universal vacuum relations and axion stabilization constraints without specifying the full moduli-space metric. We express AdS${}_3$ scalar masses through $\Delta$, the scaling dimension of the corresponding operator in the dual two-dimensional conformal field theory. In massive type IIA AdS${}_3$ vacua, we derive the bound ${\Delta }_{\max }\ge (20+4\sqrt{67})/7\approx 7.5$ and an exact dimension ${\Delta }_b=6$ for $B_2$ axions on the torsion-free branch. Type IIB type I and heterotic AdS${}_3$ vacua contain a universal volume-dilaton mode with ${\Delta }_{\varphi s}=4$. On the singlet-torsion type IIB branch, we also obtain the volume-dilaton mode with ${\Delta }_{\varphi s}=20/7$ and ${\Delta }_b=16/7$. Every supersymmetric Minkowski vacuum in the retained classical closed-string theories possesses an exact volume-dilaton flat direction. We characterize physical large-flux limits after tadpole constraints and axionic identifications, and identify torsion obstructions to isotropic scale separation.

40 pages