Hitchin Equation, Singularity, and N=2 Superconformal Field Theories

TL;DR

This paper demonstrates that Hitchin's equations encode both the UV and IR physics of 4D N=2 superconformal theories derived from 6D theories on punctured Riemann surfaces, linking geometric singularities to physical properties.

Contribution

It establishes a detailed correspondence between Hitchin's equations with singularities and the structure of 4D N=2 SCFTs, including flavor symmetries and mass deformations, extending Gaiotto's framework.

Findings

Hitchin's equations describe the UV theory of 4D N=2 SCFTs.

Singular solutions relate to flavor symmetries via boundary conditions.

Seiberg-Witten curves are identified as spectral curves of Hitchin's systems.

Abstract

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 $A_N$ $(0,2)$ theory on a punctured Riemann surface. We study the singular solution to Hitchin's equation and the Higgs field of solutions has a simple pole at the punctures; We show that the massless theory is associated with Higgs field whose residual 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 the mass-deformed theory the residual 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 the 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.