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

TL;DR

The paper demonstrates that Hitchin's equation on a punctured Riemann surface provides a UV–IR unified description of 4D N=2 SCFTs obtained from six-dimensional $(0,2)$ $A_N$ theory. Singular Higgs-field residues at punctures distinguish massless theories (nilpotent residues) from mass-deformed ones (semisimple residues), with the moduli spaces of solutions encoding flavor symmetries through nilpotent/semi-simple orbit geometry. The Seiberg–Witten curve is identified as the spectral curve of the Hitchin system, and the flavor data follow from orbit closures and their minimal degenerations, generalizing Gaiotto’s puncture construction to arbitrary $A_{N-1}$ theories using partitions and dual partitions. The framework consistently reproduces known results for SU(2) quivers and extends to SU($N$) with a precise mathematical dictionary linking Higgs-field residues, pole structures, and flavor symmetries, while outlining paths to broader theories ($D_N$, $E_N$) and connections to AGT and 3D dualities.

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.

Figures (20)

• Figure 1: (a) A Type IIA NS5-D4 brane configuration which gives four dimensional $N=2$ superconformal field theory, there are semi-infinite D4 branes on both ends which provide the fundamental hypermultiplets; (b) We use $D6$ branes instead of semi-infinite D4 branes to provide the fundamental hypermultiplets.
• Figure 2: (a) Riemann surface with punctures whose complex structure moduli determines the four dimensional gauge couplings, we take SU(2) with four fundamentals as an example; (b) Seiberg-Witten curve which determines the low energy effective action, in fact we have a family of these curves which are parameterized by Coulomb branch parameters, it may be degenerate for certain parameter.
• Figure 3: A Young tableaux with total of boxes 6, the order of poles of the meromorphic differential $\phi_i$ are: $p_1=1-1=0,p_2=2-1=1,p_3=3-1=2,p_4=4-2=2,p_5=5-2=3,p_6=6-3=3$; the flavor symmetry associated with this puncture is U(1).
• Figure 4: The various weakly coupled limit of SU(2) theory with four fundamental matter. The narrow strip denotes the weakly coupled SU(2) gauge group. The punctures are associated with flavor symmetry $SU(2)$.
• Figure 5: a) The sphere with three full punctures; b)The graph representation of the theory derived from compactification on a).