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

Understand

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.