Quiver theories and formulae for Slodowy slices of classical algebras

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 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 S ∩ N of 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 S ∩ 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 3 d mirror symmetry within the class of SCFTs known as T σ ρ ( G ) theories. We also give simple group theoretic formulae for the Hilbert series of Slodowy slices; these draw directly on the S U ( 2 ) embedding into G of the associated nilpotent orbit, and the Hilbert series of the nilpotent cone.