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We analyze M theory fivebrane in order to study the moduli space of vacua ofN=1 supersymmetric $Sp(N_c)$ gauge theories with $N_f$ flavors in fourdimensions. We show how the N=2 Higgs branch can be encoded in M theory bystudying the orientifold which plays a crucial role in our work. When all thequark masses are the same, the surface of the M theory spacetime representing anontrivial ${\bf S^1}$ bundle over ${\bf R^3}$ develops $A_{N_f-1}$ typesingularities at two points where D6 branes are located. Furthermore, byturning off the masses, two singular points on the surface collide and produce$A_{2N_f-1}$ type singularity. The sum of the multiplicities of rational curveson the resolved surface gives the dimension of N=2 Higgs branch which agreeswith the counting from the brane configuration picture of type IIA stringtheory. By rotating M theory fivebranes we get the strongly coupled dynamics ofN=1 theory and describe the vacuum expectation values of the meson fieldparameterizing Higgs branch which are in complete agreement with the fieldtheory results. Finally, we take the limit where the mass of adjoint chiralmultiplet goes to infinity and compare with field theory results. For massivecase, we comment on some relations with recent work which deals with N=1duality in the context of M theory.Comment: 25 pages, Latex:We clarified the discussion of Higgs and Coulomb  branch on page 14. This final version will appear in Phys. Rev.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">Sp</mml:mi><mml:mrow><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:math>gauge theories and the M theory fivebrane

Sp(Nc)gauge theories and the M theory fivebrane