G(2) Holonomy Spaces from Invariant Three-Forms

TL;DR

This work develops a practical, invariant-3-form framework to construct torsion-free $G_2$ holonomy metrics, enabling systematic generation of non-compact $G_2$ spaces by enforcing $d\Phi=0$ and $d*_{\Phi}\Phi=0$. It extends known $SU(2)^3$ symmetric AC manifolds to new $SU(2)^2\times U(1)$-symmetric geometries that realize M-theory lifts of IIA backgrounds with wrapped D6 branes and RR flux on conifold cycles, including a novel branch with a finite-size $U(1)$ fiber. The authors derive both analytic and numerical solutions, derive the corresponding IIA reductions, and demonstrate the framework’s capacity to model geometric transitions such as conifold transitions and flux-induced vacua. This approach offers a compact, flexible toolkit for constructing and exploring $G_2$ manifolds and their string-theoretic duals, with potential applications to flux compactifications and gauge/gravity dualities.

Abstract

We construct several new G(2) holonomy metrics that play an important role in recent studies of geometrical transitions in compactifications of M-theory to four dimensions. In type IIA string theory these metrics correspond to D6 branes wrapped on the three-cycle of the deformed conifold and the resolved conifold with two-form RR flux on the blown-up two-sphere, which are related by a conifold transition. We also study a G(2) metric that is related in type IIA to the line bundle over S^2 x S^2 with RR two-form flux. Our approach exploits systematically the definition of torsion-free G(2) structures in terms of three-forms which are closed and co-closed. Besides being an elegant formalism this turns out to be a practical tool to construct G(2) holonomy metrics.