Exploring the Strong-Coupling Region of $SU(N)$ Seiberg-Witten Theory

TL;DR

This paper develops a series expansion for Seiberg-Witten periods in SU(N) gauge theory near a symmetric point, analyzes the Kähler potential's structure, and investigates stability walls, enhancing understanding of the theory's strong-coupling regime.

Contribution

It provides a new exact series expansion for Seiberg-Witten periods in SU(N) theories and explores the global structure of the Kähler potential and stability walls.

Findings

Kähler potential is convex with a unique minimum at the symmetric point

Series expansion generalizes previous results for N=2 and N=3

Evidence of the relation between stability walls and vanishing Kähler potential surface

Abstract

We consider the Seiberg-Witten solution of pure $\mathcal{N} =2$ gauge theory in four dimensions, with gauge group $SU(N)$. A simple exact series expansion for the dependence of the $2 (N-1)$ Seiberg-Witten periods $a_I(u), a_{DI}(u)$ on the $N-1$ Coulomb-branch moduli $u_n$ is obtained around the $\mathbb{Z}_{2N}$-symmetric point of the Coulomb branch, where all $u_n$ vanish. This generalizes earlier results for $N=2$ in terms of hypergeometric functions, and for $N=3$ in terms of Appell functions. Using these and other analytical results, combined with numerical computations, we explore the global structure of the K\"ahler potential $K = \frac{1}{2\pi} \sum_I \text{Im}(\bar a_I a_{DI})$, which is single valued on the Coulomb branch. Evidence is presented that $K$ is a convex function, with a unique minimum at the $\mathbb{Z}_{2N}$-symmetric point. Finally, we explore candidate walls of marginal stability in the vicinity of this point, and their relation to the surface of vanishing K\"ahler potential.