Higgs branch, Coulomb branch, and Hall-Littlewood index

Abstract

The Higgs branch of 4D $\mathcal{N}=2$ superconformal field theories can be analyzed via the Hilbert series of the Higgs branch or, in special cases, by computing the Hall-Littlewood index. For any class $\mathcal{S}$ theory corresponding to a genus-zero Riemann surface, they are conjectured to be identical. We present several families of counterexamples. We find that for any class $\mathcal{S}$ theory with four or more ${\mathbb{Z}}_{2}$-twisted punctures, they do not match. We construct 3D mirrors for such theories and analyze their Coulomb branch Hilbert series to compute the Higgs branch Hilbert series of the 4D theory. We further construct $a=c$ theories in class $\mathcal{S}$ using the twisted punctures, and these theories, which includes the ${\stackrel{^}{D}}_{4}(SU(2n+1))$ theories, have Hall-Littlewood index different from the Hilbert series of the Higgs branch. We conjecture that this is the case for all $a=c$ theories with nonempty Higgs branch, including $\mathcal{N}\ensuremath{\ge}3$ superconformal field theories.