Quiver Maps, Nilpotent Orbits and Special Pieces of Nilcones

TL;DR

This paper develops a new framework linking 3d $ abla=4$ quiver gauge theories to the geometry of nilpotent orbits and special pieces of nilcones, introducing a quiver map that partially resolves duality obstructions.

Contribution

It introduces a novel quiver map involving symmetric group actions that relates magnetic and electric quivers, aiding the study of nilpotent orbit intersections and dualities.

Findings

New quivers for intersections within Exceptional nilcones.

A map between magnetic and electric quivers involving symmetric group actions.

Partial resolution of duality obstructions in quiver theories.

Abstract

This paper explores 3d $\mathcal{N}=4$ quiver gauge theories whose moduli spaces represent nilpotent orbits, S\l odowy slices or, more generally, S\l odowy intersections, which span the Special Pieces of nilcones of Classical or Exceptional algebras. We introduce a map between magnetic and electric quivers containing symmetric group actions, such as wreathings (or loops), bouquets, and/or non-simply laced foldings, which can be related to symmetric subgroups of Lusztig's canonical quotient groups for Special Pieces. The map on quivers induces a map on nilpotent orbits that partially resolves the obstruction to quiver dualities presented by the non-involutive nature of the Lusztig Spaltenstein and Barbasch Vogan maps. We use Coulomb and Higgs branch quiver methods complemented by localisation formulae. Some new quivers for intersections within Exceptional nilcones are presented.