Supersymmetric Deformations of G_2 Manifolds from Higher-Order Corrections to String and M-Theory

TL;DR

The paper shows that higher-order string and M-theory corrections, specifically $ obreak\alpha'^3$ $R^4$-type terms, deformation of Ricci-flat $G_2$ metrics can be treated systematically by expressing corrections through the leading-order calibrating 3-form $c_{ijk}$ and an associated $Z^{mn}$ tensor. It derives explicit first-order corrected Einstein and Killing spinor equations and proves that supersymmetry can be preserved to order $\alpha'^3$ via a modified Killing spinor condition, with the modification encoded in a spinor–multiplication term $Q_i$. The authors then apply the framework to several cohomogeneity-one $G_2$ geometries—$S^3\times S^3$, $\mathbb{C}P^3$, and $SU(3)/U(1)^2$-based spaces—obtaining concrete integro-differential solutions for the corrected metric components and, in special cases, closed-form expressions; they also discuss the M-theory lift and show that the corrected internal space deformations are compatible with 11D dynamics and KK reduction to 10D. Overall, the work provides explicit, SUSY-preserving deformations of a broad class of $G_2$ compactifications and clarifies how higher-derivative corrections modify the geometry while retaining a controlled, supersymmetric structure.

Abstract

The equations of 10 or 11 dimensional supergravity admit supersymmetric compactifications on 7-manifolds of $G_2$ holonomy, but these supergravity vacua are deformed away from special holonomy by the higher-order corrections of string or M-theory. We find simple expressions for the first-order corrections to the Einstein and Killing spinor equations in terms of the calibrating 3-form of the leading-order G_2-holonomy background. We thus obtain, and solve explicitly, systems of first-order equations describing the corrected metrics for most of the known classes of cohomogeneity-one 7-metrics with G_2 structures