Supersymmetric M3-branes and G_2 Manifolds

TL;DR

This work generalizes the classic complete $G_2$ holonomy metric on $\mathbb{R}^4\times S^3$ to a two-parameter family dependent on $\lambda$, with regular geometries at $\lambda\in\{-1,0,+1\}$ and an $S_3$ triality relating these cases. It then constructs explicit supersymmetric M3-brane solutions in $D=11$ supergravity, where the transverse space is a deformation of these $G_2$ metrics, obtained from first-order equations derived from a superpotential and linked to Killing-spinor integrability. The paper also develops additional deformed $G_2$-holonomy backgrounds, including a known metric by Brandhuber et al., and shows that corresponding M3-branes remain supersymmetric. Together, these results deepen the connection between $G_2$ holonomy geometry, M-theory brane constructions, and type IIA reductions to conifold geometries, with implications for dual gauge theories and non-perturbative dynamics in four dimensions.

Abstract

We obtain a generalisation of the original complete Ricci-flat metric of G_2 holonomy on R^4\times S^3 to a family with a non-trivial parameter λ. For generic λthe solution is singular, but it is regular when λ={-1,0,+1}. The case λ=0 corresponds to the original G_2 metric, and λ={-1,1} are related to this by an S_3 automorphism of the SU(2)^3 isometry group that acts on the S^3\times S^3 principal orbits. We then construct explicit supersymmetric M3-brane solutions in D=11 supergravity, where the transverse space is a deformation of this class of G_2 metrics. These are solutions of a system of first-order differential equations coming from a superpotential. We also find M3-branes in the deformed backgrounds of new G_2-holonomy metrics that include one found by A. Brandhuber, J. Gomis, S. Gubser and S. Gukov, and show that they also are supersymmetric.