Hitchin equation, singularity, and N = 2 superconformal field theories

We argue that Hitchin’s equation determines not only the low energy effective theory but also describes the UV theory of four dimensional N = 2 superconformal field theories when we compactify six dimensional AN (0, 2) theory on a punctured Riemann surface. We study singular solutions to Hitchin’s equation and the Highs field of equation has a simple pole at the punctures; We show that the massless theory is associated with Higgs field whose residue is a nilpotent element; We identify the flavor symmetry associated with the puncture by studying the singularity of closure of the moduli space of solutions with the appropriate boundary conditions. For mass-deformed theory the residue of the Higgs field is a semi-simple element, we identify the semi-simple element by arguing that the moduli space of solutions of mass-deformed theory must be a deformation of the closure of the moduli space of massless theory. We also study the Seiberg-Witten curve by identifying it as the spectral curve of the Hitchin’s system. The results are all in agreement with Gaiotto’s results derived from studying the Seiberg-Witten curve of four dimensional quiver gauge theory.