Quiver theories and formulae for nilpotent orbits of Exceptional algebras

Abstract

Abstract:We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different nilpotent orbits and propose candidates for the constructions of some non-normal nilpotent orbits of Exceptional algebras.

Discrete gauging in Coulomb branches of three dimensional $$ \mathcal{N}=4 $$ supersymmetric gauge theories

“…Thus, the free part of the Coulomb branch is [10]: C f.g. = H 3 , which is generated by the fundamental representation of Sp (3), denoted by [1,0,0]. Hence, the global symmetry has two constituent parts: 21) such that G global,free = Sp(3).…”

“…The central node is balanced for a special case n 1 = 4, which is indicated by the radial color gradient of the node. The theory in figure 21 has a SU(2) n 1 global symmetry which enhances 21 to SO(8) for k 1 = 4.…”

Non-Abelian mirror symmetry beyond the chiral ring

“…58) so these correlators respect the full S 3 symmetry. 21 Specifically, the Higgs branch computation gives (for…”

“…It turns out that there is no contradiction. By writing sh(s) 2 ch(s/2) 2n = 1 ch(s/2) 2n−4 − 4 ch(s/2) 2n−2 (5.63) 21 One can go on to define additional composite operators. For example, U 2 C (defined as a shift of U C U C ) can mix with V C and V 2 C can mix with U C , which is consistent with the S 3 symmetry.…”

“…A particular class of operators in the SCFT is that of chiral ring operators, which are half-BPS and whose vacuum expectation values give rise to the Coulomb branch, Higgs branch, and mixed branches. The matching of the chiral rings (equivalently, the moduli spaces) [17][18][19][20][21][22] provides a first check for mirror duality proposals beyond anomaly matching.…”