Sp(N_c) gauge theories and the M theory flvebrane

We analyze an M theory fivebrane in order to study the moduli space of vacua of N = 1 supersymmetric Sp(N_c) gauge theories with N_f flavors in four dimensions. We show how the N = 2 Higgs branch can be encoded in M theory by studying the orientifold which plays a crucial role in our work. When all the quark masses are the same, the surface of the M theory spacetime representing a nontrivial S¹ bundle over R³ develops A_Nf-1 type singularities at two points where D6 branes are located. Furthermore, by turning off the masses, two singular points on the surface collide and produce an A_2Nf-1 type singularity. The sum of the multiplicities of rational curves on the resolved surface gives the dimension of an N = 1 Higgs branch which agrees with the counting from the brane configuration picture of type IIA string theory. By rotating M theory fivebranes we get the strongly coupled dynamics of N = 1 theory and describe the vacuum expectation values of the meson field parametrizing the Higgs branch which are in complete agreement with the field theory results. Finally, we take the limit where the mass of an adjoint chiral multiplet goes to infinity and compare it with field theory results. For the massive case, we comment on some relations with recent work which deals with the N = 1 duality in the context of M theory.