Poisson brackets for some Coulomb branches

Abstract

Abstract We construct Poisson bracket relations between the operators which generate the chiral ring of the Coulomb branch of certain 3d$$ \mathcal{N} $$ N = 4 quiver gauge theories. In the case where the Coulomb branch is a free space, ADE Klein singularity, or the minimal A2 nilpotent orbit, we explicitly compute the Poisson brackets between the generators using either inherited properties of the abstract Coulomb branch variety, or the expected charges of these operators under the global symmetry (known through use of the monopole formula). We also conjecture Poisson brackets for Higgs branches that originate from 6d theories with tensionless strings or 5d theories with massless instantons for which the HWG is known, based on representation theoretic and operator content constraints known from the study of their magnetic quiver.