Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories

TL;DR

The paper develops a Hitchin-system framework for 4D $N=2$ gauge theories with asymptotic freedom by introducing irregular singularities on the Hitchin moduli space. It shows how irregular data, classified for $SU(2)$ and generalized to $SU(N)$ quivers, encode dynamical scales and UV couplings and reproduce the Seiberg-Witten curves via the spectral curve $\det(x-\Phi(z))=0$. The main contributions include explicit irregular Higgs-field forms, dimension counts for the local moduli spaces, and the demonstration that UV parameters can be captured by irregular data rather than solely by complex structure moduli. This work bridges Hitchin-system techniques with $N=2$ gauge theory, suggesting links to AGT with irregular states and pointing toward broader applications in dualities and geometric Langlands. It opens several directions, including extensions to other gauge groups and a deeper exploration of degeneration limits and dual frames across non-conformal quivers.

Abstract

In this paper, we study irregular singular solution to Hitchin's equation and use it to describe four dimensional $N=2$ asymptotically free gauge theories. For $SU(2)$ $A$ type quiver, two kinds of irregular singularities besides one regular singularity are needed for the solution of Hitchin's equation; We then classify irregular singularities needed for the general $SU(N)$ $A$ type quiver.

Figures (6)

• Figure 1: a) Brane configuration of $SU(2)$ theory with four fundamentals; b)Brane configuration of $SU(2)$ theory with three fundamentals.
• Figure 2: a) Brane configuration of $SU(2)$ theory with two fundamentals; b) Another Brane configuration of $SU(2)$ theory with two fundamentals.
• Figure 3: a) Brane configuration of $SU(2)$ theory with one fundamentals; b) Another Brane configuration of pure $SU(2)$ theory
• Figure 4: a) Brane configuration of conformal $SU(2)$ quivers; b) A brane configuration with non-conformal gauge group
• Figure 5: a) A nonconformal $SU(2)$ quiver with three gauge groups; b) The degeneration limit of the Riemann sphere corresponding to quiver a), the regular singularity is represented by cross, and the irregular singularity is represented by box; c) Another degeneration limit of the same Riemann sphere; d) After complete degeneration, we get a new $SU(2)$ theories with two irregular singularity and a simple singularity.