A G_2 Unification of the Deformed and Resolved Conifolds

TL;DR

The paper addresses unifying deformed and resolved conifolds within seven-dimensional G2 holonomy by deriving the most general first-order system for cohomogeneity-one metrics with $S^3×S^3$ principal orbits. It introduces a five-function first-order framework with an algebraic constraint, plus two constants $p$ and $q$ from the associative 3-form, that contains the known $B_7$ and $D_7$ families as special cases and reveals a new two-parameter class $\widetilde{{{\Bbb C}}}_7$ with a $T^{1,1}$ bolt. The weak-coupling limits reproduce $S^1$-times Calabi–Yau metrics on appropriate line bundles, while the GH limits connect to the deformed and resolved conifolds; the formalism also yields smooth KK reductions to Type IIA with finite dilaton. This work thus provides a supersymmetric path between Calabi–Yau geometries within M-theory and expands the landscape of noncompact G2 manifolds with potential phenomenological applications.

Abstract

We find general first-order equations for G_2 metrics of cohomogeneity one with S^3\times S^3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics \bB_7 and \bD_7, which have weak-coupling limits that are S^1 times the deformed conifold and the resolved conifold respectively. Our more general first-order equations provide a supersymmetric unification of the two Calabi-Yau manifolds, since the metrics \bB_7 and \bD_7 arise as solutions of the {\it same} system of first-order equations, with different values of certain integration constants. Additionally, we find a new class of ALC G_2 solutions to these first-order equations, which we denote by \wtd\bC_7, whose topology is an \R^2 bundle over T^{1,1}. There are two non-trivial parameters characterising the homogeneous squashing of the T^{1,1} bolt. Like the previous examples of the \bB_7 and \bD_7 ALC metrics, here too there is a U(1) isometry for which the circle has everywhere finite and non-zero length. The weak-coupling limit of the \wtd\bC_7 metrics gives S^1 times a family of Calabi-Yau metrics on a complex line bundle over S^2\times S^2, with an adjustable parameter characterising the relative sizes of the two S^2 factors.