Mirror symmetries for brane configurations and branes at singularities

We study local mirror symmetry on non-compact Calabi-Yau manifolds (conifol d type of singularities) in the presence of D3 brane probes. Using an inter mediate brane setup of NS 5-branes 'probed' by D-branes, we can explicitly T-dualize three isometry directions to relate a non-compact Calabi-Yau mani fold to its local mirror. Both intervals and boxes of NS 5-branes appear as brane setups. Going from one to the other is equivalent to performing a co nifold transition in the dual geometry. One result of our investigation is that the brane box constructions, as they have been discussed so far, shoul d be modified. (C) 2000 Elsevier Science B.V. All rights reserved.