A G2 unification of the deformed and resolved conifolds

We find general first-order equations for G2 metrics of cohomogeneity one with S3×S3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics B7 and D7, which have weak-coupling limits that are S1 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 B7 and D7 arise as solutions of the same system of first-order equations, with different values of certain integration constants. Additionally, we find a new class of ALC G2 solutions to these first-order equations, which we denote by C7, whose topology is an R2 bundle over T1,1. There are two non-trivial parameters characterising the homogeneous squashing of the T1,1 bolt. Like the previous examples of the B7 and D7 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 C7 metrics gives S1 times a family of Calabi–Yau metrics on a complex line bundle over S2×S2, with an adjustable parameter characterising the relative sizes of the two S2 factors.