Quiver Theories and Formulae for Slodowy Slices of Classical Algebras

TL;DR

This paper uses supersymmetric quiver gauge theories to compute and analyze the algebraic and geometric properties of Slodowy slices associated with classical Lie algebras, revealing new insights into their structure and dualities.

Contribution

It introduces methods to compute Hilbert series and algebraic descriptions of Slodowy slices using quiver gauge theories, extending to classical algebras up to rank 4 and exploring dualities.

Findings

Refined Hilbert series for classical algebras up to rank 4 and A5.

Descriptions of generators and relations of the chiral ring as algebraic varieties.

Identification of dual quiver theories with Coulomb branches corresponding to intersections of Slodowy slices and nilpotent cones.

Abstract

We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra $\mathfrak g$. We analyse classes of quiver theories, with Classical gauge and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the nilpotent cone $\mathcal S\cap \mathcal N$ of $\mathfrak{g}$. We calculate refined Hilbert series for Classical algebras up to rank $4$ (and $A_5$), and find descriptions of their representation matrix generators as algebraic varieties encoding the relations of the chiral ring. We also analyse a class of dual quiver theories, whose Coulomb branches are intersections $\mathcal S\cap \mathcal N$; such dual quiver theories exist for the Slodowy slices of $A$ algebras, but are limited to a subset of the Slodowy slices of $BCD$ algebras. The analysis opens new questions about the extent of $3d$ mirror symmetry within the class of SCFTs known as $T_\sigma^\rho(G)$ theories. We also give simple group theoretic formulae for the Hilbert series of Slodowy slices; these draw directly on the $SU(2)$ embedding into $G$ of the associated nilpotent orbit, and the Hilbert series of the nilpotent cone.