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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 B^ R\hat{B}_R operators that parametrize the Higgs branch (each contributes τ 2R\tau^{2R} to the index). The Hall-Littlewood index also includes contributions from D R(0,j)D_{R(0,j)} operators (which contribute (−1) 2j+1τ 2(1+R+j)(-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)D_{R(0,j)} operators at genus-0, and so the Hilbert series and Hall-Littlewood index agree.

For genus g>0g\gt0, the gauge symmetry1 cannot be completely Higgsed on the Higgs branch of the theory. For the theory of type J=ADEJ=\text{ADE}, there’s a U(1) rank(J)gU(1)^{\text{rank}(J)g} unbroken at a generic point on the Higgs branch2. Correspondingly, the SCFT contains D R(0,0)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)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 ℤ 2\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.

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