Calabi-Yau 3-folds from 2-folds

We consider type IIA string theory on a Calabi-Yau 2-fold with D6-branes wrapping 2-cycles in the 2-fold. We find a complete set of conditions on the supergravity solution for any given wrapped brane configuration in terms of SU(2) structures. We reduce the problem of finding a supergravity solution for the wrapped branes to finding a harmonic function on R x CY2. We then lift this solution to eleven dimensions as a product of R(4,1) and a Calabi-Yau 3-fold. We show how the metric on the 3-fold is determined in terms of the wrapped brane solution. We write down the distinguished (3, 0) form and the Kaehler form of the 3-fold in terms of structures defined on the base 2D complex manifold. We discuss the topology of the 3-fold in terms of the D6-branes and the underlying 2-fold. We show that in addition to the non-trivial cycles inherited from the underlying 2-fold there are N - 1 new 2-cycles. We construct closed (1, 1) forms corresponding to these new cycles. We also display some explicit examples. One of our examples is that of D6-branes wrapping the 2-cycle in an A1 ALE space, the resulting 3-fold has h(1,1) = N, where N is the number of D6-branes