Geometry, D-branes and N=1 duality in four dimensions with product gauge groups

We study N = 1 dualities in four dimensional supersymmetric gauge theories by D 6-branes wrapping around S-cycles of Calabi-Yau threefolds in type IIA string theory. We consider the models involving SU(N-c1) X SU((c2)) product gauge group without or with adjoint matter in terms of geometrical realization of the configuration of D 6-branes wrapped 3-cycles. We also find simple geometric descriptions for the triple product gauge group SU(N-c1) X SU(N-c2) X SU(N-c3) interpreted recently by Brodie and Hanany in the context of D brane configurations together with NS 5-branes. Their introduction of semi infinite D 4-branes appears naturally by looking at the flavor group in the dual theory. We generalize to product of arbitrary number of gauge groups. (C) 1997 Elsevier Science B.V.