T^{\sigma}_{\rho}(G) Theories and Their Hilbert Series

TL;DR

This paper derives explicit formulas for the Higgs and Coulomb branch Hilbert series of a class of 3d N=4 superconformal theories, using Hall-Littlewood polynomials and localization techniques, with applications to classical groups.

Contribution

It provides a new explicit formula for Hilbert series of T^{\sigma}_{ ho}(G) theories, extending previous results to include classical groups and O3 planes.

Findings

Explicit formulas for Higgs and Coulomb branch Hilbert series.

Formulas expressed in terms of Hall-Littlewood polynomials.

Results include cases for orthogonal and symplectic groups.

Abstract

We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3d N=4 superconformal gauge theories T^{\sigma}_{\rho}(G) corresponding to a set of D3 branes ending on NS5 and D5-branes, with or without O3 planes. Here G is a classical group, \sigma is a partition of G and \rho a partition of the dual group G^\vee. In deriving such a formula we make use of the recently discovered formula for the Hilbert series of the quantum Coulomb branch of N=4 superconformal theories. The result can be expressed in terms of a generalization of a class of symmetric functions, the Hall-Littlewood polynomials, and can be interpreted in mathematical language in terms of localization. We mainly consider the case G=SU(N) but some interesting results are also given for orthogonal and symplectic groups.