T ρ σ ( G) theories and their Hilbert series

摘要

A bstract We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3 d N = 4 $$ mathcal{N}=4 $$ superconformal gauge theories T ~( ρ )_( σ ) ( 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, σ is a partition of G and ρ a partition of the dual group G _(∨). 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 $$ mathcal{N}=4 $$ 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.