July 15, 2022

HL ≠ HS

There’s a nice new paper by Kang et al, who point out something about class-S theories that should be well-known, but isn’t.

In the (untwisted) theories of class-S, the Hall-Littlewood index, at genus-0, coincides with the Hilbert Series of the Higgs branch. The Hilbert series counts the $\hat{B}_R$ operators that parametrize the Higgs branch (each contributes $\tau^{2R}$ to the index). The Hall-Littlewood index also includes contributions from $D_{R(0,j)}$ operators (which contribute $(-1)^{2j+1}\tau^{2(1+R+j)}$ to the index). But, for the untwisted theories of class-S, there is a folk-theorem that there are no $D_{R(0,j)}$ operators at genus-0, and so the Hilbert series and Hall-Littlewood index agree.

For genus $g\gt0$, the gauge symmetry1 cannot be completely Higgsed on the Higgs branch of the theory. For the theory of type $J=\text{ADE}$, there’s a $U(1)^{\text{rank}(J)g}$ unbroken at a generic point on the Higgs branch2. Correspondingly, the SCFT contains $D_{R(0,0)}$ multiplets which, when you move out onto the Higgs branch and flow to the IR, flow to the $D_{0(0,0)}$ multiplets3 of the free theory.

What Kang et al point out is that the same is true at genus-0, when you include enough $\mathbb{Z}_2$-twisted punctures. They do this by explicitly calculating the Hall-Littlewood index in a series of examples.

But it’s nice to have a class of examples where that hard work is unnecessary.