Approaching Argyres-Douglas theories

TL;DR

This paper uses Seiberg-Witten theory to analyze the structure of Argyres-Douglas points in $ ext{SU}(N)$ super-Yang-Mills, deriving series expansions, locating walls of marginal stability, and exploring the properties of the intrinsic K"ahler potential.

Contribution

It provides a convergent series expansion for Seiberg-Witten periods near Argyres-Douglas points and investigates the positivity and convexity of the intrinsic K"ahler potential at these superconformal fixed points.

Findings

Series expansion for periods near Argyres-Douglas points

Location of walls of marginal stability for $ ext{SU}(3)$

Intrinsic K"ahler potential is positive definite and convex when only operators with $ ext{Δ}>1$ acquire VEVs

Abstract

The Seiberg-Witten solution to four-dimensional $\mathcal{N}=2$ super-Yang-Mills theory with gauge group $\text{SU}(N)$ and without hypermultiplets is used to investigate the neighborhood of the maximal Argyres-Douglas points of type $(\mathfrak{a}_1,\mathfrak{a}_{N-1})$. A convergent series expansion for the Seiberg-Witten periods near the Argyres-Douglas points is obtained by analytic continuation of the series expansion around the $\mathbb{Z}_{2N}$ symmetric point derived in arXiv:2208.11502. Along with direct integration of the Picard-Fuchs equations for the periods, the expansion is used to determine the location of the walls of marginal stability for $\text{SU}(3)$. The intrinsic periods and K\"ahler potential of the $(\mathfrak{a}_1,\mathfrak{a}_{N-1})$ superconformal fixed point are computed by letting the strong coupling scale tend to infinity. We conjecture that the resulting intrinsic K\"ahler potential is positive definite and convex, with a unique minimum at the Argyres-Douglas point, provided only intrinsic Coulomb branch operators with unitary scaling dimensions $\Delta>1$ acquire a vacuum expectation value, and provide both analytical and numerical evidence in support of this conjecture. In all the low rank examples considered here, it is found that turning on moduli dual to $\Delta \leq 1$ operators spoils the positivity and convexity of the intrinsic K\"ahler potential.