ar X iv : h ep - t h / 07 02 13 5 v 1 1 6 Fe b 20 07 Calabi - Yau 3 - folds from 2 - folds

We consider type IIA string theory on a Calabi-Yau 2-fold with D6branes 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×CY2. We then lift this solution to 11-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 Kähler form of the 3-fold in terms of structures defined on the base 2-d complex manifold. We discuss the topology of the 3-fold in terms of the D6-branes and the underlying 2fold. 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. BCCUNY-HEP /07-01 hep-th/0702135 ∗email: Ansar [email protected]