### 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 symmetry^{1} 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 branch^{2}. 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 $D_{0(0,0)}$ multiplets^{3} 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.

Consider the $J=D_N$ theory. The punctures in the $\mathbb{Z}_2$-twisted sector are labeled by nilpotent orbits in $\mathfrak{g}=\mathfrak{sp}(N-1)$. The twisted full puncture is $[1^{2(N-1)}]$ and the twisted simple puncture is $[2(N-1)]$. Consider a 4-punctured sphere with 2 twisted full punctures and two twisted simple punctures. The manifest $\mathfrak{sp}(N-1)_{2N}\times \mathfrak{sp}(N-1)_{2N}$ symmetry is enhanced to $\mathfrak{sp}(2N-2)_{2N}$. In a certain S-duality frame, this is a Lagrangian field theory: $SO(2N)$ with $2(N-1)$ hypermultiplets in the vector representation. That matter content is insufficient to Higgs the $SO(2N)$ completely. At a generic point of the Higgs branch, there’s an $SO(2)=U(1)$ unbroken.

We can construct the corresponding $D_{R(0,0)}$ operator, where $R=N-1$. Organize the scalars in the hypermultiplets into complex scalars $\phi^i_a,\tilde{\phi}^i_{\tilde{a}}$, where $i=1,\dots,2N$ is an $SO(2N)$ vector index, and $a,\tilde{a}=1,\dots, 2(N-1)$ span the $4(N-1)$-dimensional defining representation of $Sp(2N-2)$. Let $\Phi^{i j}=-\Phi^{j i}$ be the complex scalar in the adjoint of $SO(2N)$. Then the superconformal primary of the $D_{R(0,j)}$ multiplet with $R=N-1,\; j=0$ is

which we see is in the traceless, completely anti-symmetric rank-$(2N-2)$ tensor representation of $Sp(2N-2)$ (the representation with Dynkin labels $(0,\dots,0,1)$). This has $\Delta=1+2R+j=2N-1$ and contributes $-\tau^{2N}\chi(0,\dots,0,1)$ to the Hall-Littlewood index.

The above statement takes a little bit of work. At zero gauge coupling, the formula $\Delta=2N-1$ obviously holds. We need to worry that, at finite gauge coupling this operator recombines with other operators to form a long superconformal multiplet (whose conformal dimension is not fixed). The relevant recombination formula is $A^{2R+1}_{R-1,0(0,0)} = \hat{C}_{R-1(0,0)}\oplus D_{R(0,0)}\oplus \overline{D}_{R(0,0)}\oplus \hat{B}_{R+1}$ where we denote a long multiplet by $A^\Delta_{R,r(j_1,j_2)}$. One can check that the free theory has no candidate $\hat{B}_N$ operator transforming in the appropriate representation of the flavour symmetry. So (1) necessarily remains in a short superconformal multiplet and $\Delta$ is independent of the gauge coupling.

Similarly, you can replace one of the twisted full punctures with $[2,1^{2N-4}]$. The resulting SCFT has a Lagrangian description $SO(2N-1)$ as $SO(2N-1)+(2N-3)(V) + (N-1)(1)$. Again, this matter content leaves an unbroken $U(1)$ at a generic point on the Higgs branch. The $D_{R(0,0)}$ multiplet (for $R=(2N-3)/2$) in the traceless rank-$(2N-3)$ completely anti-symmetric tensor representation of $Sp(2N-3)$, constructed by the analogue of (1), has $\Delta=2N-2$ and contributes $-\tau^{2N-1}\chi(0,\dots,0,1)$ to the Hall-Littlewood index.

These examples were rather special, in that they had an S-duality frame in which they were Lagrangian field theories. Generically that won’t be the case. But there’s no reason to expect that theories, with an S-duality frame in which they are Lagrangian, should be distinguished in this regard. And, indeed, Kang *et al* find that the presence of $D_{R(0,0)}$ operators in the spectrum persists in examples with no Lagrangian field theory realization.

^{1} For genus-$g$ and $n$ punctures, the class-S theory can be presented (in multiple ways) as a “gauge theory” with $(3g-3+n)$ simple factors in the gauge group. This statement has to be modified slightly in the presence of “atypical” punctures in the twisted theory.

^{2} Here, I’m taking “Higgs branch” to mean the branch on which the gauge symmetry is maximally-Higgsed.

^{3} The superconformal primary of the $D_{0(0,0)}$ multiplet is the complex scalar in the free $\mathcal{N}=2$ vector multiplet. Its superconformal descendents include the photino and the imaginary-self-dual part of the field strength.