The M theory lift of two O6− planes and four D6 branes

We solve for the effective actions on the Coulomb branches of a class of N = 2 supersymmetric theories by finding the complex structure of an M5 brane in an appropriate background hyperkahler geometry corresponding to the lift of two O6 orientifolds and four D6 branes to M theory. The resulting Seiberg-Witten curves are of finite genus, unlike other solutions proposed in the literature. The simplest theories in this class are the scale invariant Sp(k) theory with one antisymmetric and four fundamental hypermultiplets and the SU(k) theory with two antisymmetric and four fundamental hypermultiplets. Infinite classes of related theories are obtained by adding extra SU(k) factors with bifundamental matter and by turning on masses to flow down to various asymptotically free theories. The N = 4 supersymmetric SU(k) theory can be embedded in these asymptotically free theories, allowing a derivation of a subgroup of its S duality group as an exact equivalence of quantum field theories. ∗On leave from Newman Laboratory, Cornell University, Ithaca NY 14853