Unification of the Deformed and Resolved Conifolds

We find general first-order equations for G 2 metrics of cohomogeneity one with S 3 × 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 B 7 and D 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 B 7 and D 7 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 G 2 solutions to these first-order equations, which we denote by C 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 B 7 and D 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 C 7 metrics gives S 1 times a family of Calabi-Yau metrics on a complex line bundle over S 2 × S 2 , with an adjustable parameter characterising the relative sizes of the two S 2 factors.