Abstract
There has been a recent progress in understanding the chiral ring of 3d $\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques. We find a simple formula of the Hilbert series with background magnetic charges for $T_\rho(G)$ theories in terms of Hall-Littlewood polynomials. Here $G$ is a classical group and $\rho$ is a certain partition related to the dual group of $G$. The Hilbert series for vanishing background magnetic charges show that Coulomb branches of $T_\rho(G)$ theories are complete intersections. We also demonstrate that mirror symmetry maps background magnetic charges to baryonic charges.
Highlights
Identifying the chiral ring and moduli space on the Coulomb branch of an N = 4 supersymmetric gauge theory in 2 + 1 dimensions has been a long standing problem
In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques
We find a simple formula of the Hilbert series with background magnetic charges for Tρ(G) theories in terms of Hall-Littlewood polynomials
Summary
In this paper we discuss the general properties of the Hilbert series with background fluxes and we provide computations for a class of simple theories, the three-dimensional superconformal field theories known as Tρ(G) [2] The latter are linear quiver theories with non-decreasing ranks associated with a partition ρ and a flavor symmetry G and were defined in terms of boundary conditions for 4d N = 4 SYM with gauge group G [2]. Given a classical group G and a corresponding partition ρ of the dual group, the Coulomb branch Hilbert series of Tρ(G) with background monopole fluxes for the flavor symmetry G can be written in terms of Hall-Littlewood polynomials
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