A geometric unification of dualities

We study the dynamics of a large class of A = I quiver theories, geometrically realized by type IIB D-brane probes wrapping cycles of local Calabi-Yau three-folds. These include V = 2 (affine) A-D-E quiver theories deformed by superpotential terms, as well as chiral A = I quiver theories obtained in the presence of vanishing 4-cycles inside a Calabi-Yau. We consider the various possible geometric transitions of the 3-fold and show that they correspond to Seiberg-like dualities (represented by Weyl reflections in the A-D-E case or "mutations" of bundles in the case of vanishing 4-cycles) or large N dualities involving gaugino condensates (generalized conifold transitions). Also duality cascades are naturally realized in these classes of theories, and are related to the affine Weyl group symmetry in the A-D-E case. (C) 2002 Elsevier Science B.V. All rights reserved.