Even More On Twisted $A_{2n}$ Class-S Theories

TL;DR

This paper advances the understanding of Coulomb branches in twisted $A_{2n}$ class-S theories by defining the automorphism structure, establishing local constraints on invariant polynomials, and constructing explicit Seiberg-Witten curves.

Contribution

It introduces a precise automorphism characterization for twisted punctures, derives local constraints on Higgs field polynomials, and constructs explicit Seiberg-Witten curves for these theories.

Findings

Defined the automorphism of order 4 for twisted punctures.

Established local constraints on Laurent coefficients of invariant polynomials.

Constructed explicit Seiberg-Witten curves for 3-punctured spheres.

Abstract

This paper is a continuation of our investigation into the Coulomb branches of twisted $A_{2n}$ of class-S. In arXiv:2411.17675, we found predictions for the contributions of twisted punctures to the graded dimensions of the Coulomb branch, based on the behaviour under nilpotent Higgsings and S-duality. While surprisingly powerful, these arguments were indirect. Here, we take a different approach: we define precisely the nature of the automorphism under which the twisted punctures are twisted (in particular, it is order-4, not order-2). From that, we find the local constraints satisfied by the Laurent coefficients of the invariant polynomials in the Higgs field, for all twisted punctures in $A_{2n}$, for all n. A crucial role is played by a new (at least, new in physics) order-reversing map on the set of nilpotent orbits in sp(n). Finally, we construct several examples of Seiberg-Witten curves for 3-punctured spheres in these theories.