Mirror symmetry and exact solution of 4D $N = 2$ gauge theories: I

Using geometric engineering in the context of type II strings, we obtain exact solutions for the moduli space of the Coulomb branch of all N=2 gauge theories in four dimensions involving products of SU gauge groups with arbitrary number of bi-fundamental matter for chosen pairs, as well as an arbitrary number of fundamental matter for each factor. Asymptotic freedom restricts the possibilities to SU groups with bi-fundamental matter chosen according to ADE or affine ADE Dynkin diagrams. Many of the results can be derived in an elementary way using the self-mirror property of K3. We find that in certain cases the solution of the Coulomb branch for N=2 gauge theories is given in terms of a three dimensional complex manifold rather than a Riemann surface. We also study new stringy strong coupling fixed points arising from the compactification of higher dimensional theories with tensionless strings and consider applications to three dimensional N=4 theories.