Aspects of Four Dimensional N = 2 Field Theory

New four dimensional N = 2 field theories can be engineered from compactifying six dimensional (2, 0) superconformal field theory on a punctured Riemann surface. Hitchin?s equation is defined on this Riemann surface and the fields in Hitchin?s equation are singular at the punctures. Four dimensional theory is entirely determined by the data at the punctures. Theory without lagrangian description can also be constructed in this way.

We first construct new four dimensional generalized superconformal quiver gauge theory by putting regular singularity at the puncture. The algorithm of calculating weakly coupled gauge group in any duality frame is developed. The asymptotical free theory and Argyres-Douglas field theory can also be constructed using six dimensional method. This requires introducing irregular singularity of Hithcin?s equation.

Compactify four dimensional theory down to three dimensions, the corresponding N = 4 theory has the interesting mirror symmetry. The mirror theory for the generalized superconformal quiver gauge theory can be derived using the data at the puncture too. Motivated by this construction, we study other three dimensional theories deformed from the above theory and find their mirrors.

The surprising relation of above four dimensional gauge theory and two dimensional conformal field theory may have some deep implications. The S-duality of four dimensional theory and the crossing symmetry and modular invariance of two dimensional theory are naturally related.