A G2 unification of the deformed and resolved conifolds

We find general first-order equations for G(2) metrics of cohomogeneity one with S-3 x S-3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotic ally 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, solutions to these first-order equations, which we denote by (C) over tilde7(,) 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 bolts. 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) over tilde (7) metrics gives S-1 times a family of Calabi-Yau metrics on a complex line bundle over S-2 x S-2. with an adjustable parameter characterising the relative sizes of the two S-2 factors. (C) 2002 Published by Elsevier Science B.V.