Branes, geometry and N=1 duality with product gauge groups of SO and Sp

We study N = 1 dualities in four-dimensional supersymmetric gauge theories as the worldvolume theory of D4 branes with one compact direction in type I IA string theory. We generalize the previous work for SO(N-c1) x Sp (N-c2) with the superpotential W = Tr X-4 to the case of W = Tr X4(k+1) in terms o f brane configuration. We conjecture that the new dualities for the product gauge groups of SO(N-c1) x Sp(N-c2) x SO(N-c3), SO(N-c1) x Sp(N-c2) x SO(N -c3) x Sp(N-c4) and higher multiple product gauge groups can be obtained by reversing the ordering of NS5 branes and D6 branes while preserving the li nking numbers. We also describe the above dualities in terms of wrapping D6 branes around 3-cycles of Calabi-Yau threefolds in type IIA string theory. The theory with adjoint matter can be regarded as taking multiple copies o f NS5 brane in the configuration of brane or geometric approaches. (C) 1999 Elsevier Science B.V. All rights reserved.