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July 15, 2022


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 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.

Consider the J=D NJ=D_N theory. The punctures in the 2\mathbb{Z}_2-twisted sector are labeled by nilpotent orbits in 𝔤=𝔰𝔭(N1)\mathfrak{g}=\mathfrak{sp}(N-1). The twisted full puncture is [1 2(N1)][1^{2(N-1)}] and the twisted simple puncture is [2(N1)][2(N-1)]. Consider a 4-punctured sphere with 2 twisted full punctures and two twisted simple punctures. The manifest 𝔰𝔭(N1) 2N×𝔰𝔭(N1) 2N\mathfrak{sp}(N-1)_{2N}\times \mathfrak{sp}(N-1)_{2N} symmetry is enhanced to 𝔰𝔭(2N2) 2N\mathfrak{sp}(2N-2)_{2N}. In a certain S-duality frame, this is a Lagrangian field theory: SO(2N)SO(2N) with 2(N1)2(N-1) hypermultiplets in the vector representation. That matter content is insufficient to Higgs the SO(2N)SO(2N) completely. At a generic point of the Higgs branch, there’s an SO(2)=U(1)SO(2)=U(1) unbroken.

We can construct the corresponding D R(0,0)D_{R(0,0)} operator, where R=N1R=N-1. Organize the scalars in the hypermultiplets into complex scalars ϕ a i,ϕ˜ a˜ i\phi^i_a,\tilde{\phi}^i_{\tilde{a}}, where i=1,,2Ni=1,\dots,2N is an SO(2N)SO(2N) vector index, and a,a˜=1,,2(N1)a,\tilde{a}=1,\dots, 2(N-1) span the 4(N1)4(N-1)-dimensional defining representation of Sp(2N2)Sp(2N-2). Let Φ ij=Φ ji\Phi^{i j}=-\Phi^{j i} be the complex scalar in the adjoint of SO(2N)SO(2N). Then the superconformal primary of the D R(0,j)D_{R(0,j)} multiplet with R=N1,j=0R=N-1,\; j=0 is

(1)ϵ i 1i 2i 2Nϕ a 1 i 1ϕ a 2 i 2ϕ a 2N2 i 2N2Φ i 2N1i 2N\epsilon_{i_1 i_2\dots i_{2N}}\phi^{i_1}_{a_1}\phi^{i_2}_{a_2}\dots\phi^{i_{2N-2}}_{a_{2N-2}}\Phi^{i_{2N-1}i_{2N}}

which we see is in the traceless, completely anti-symmetric rank-(2N2)(2N-2) tensor representation of Sp(2N2)Sp(2N-2) (the representation with Dynkin labels (0,,0,1)(0,\dots,0,1)). This has Δ=1+2R+j=2N1\Delta=1+2R+j=2N-1 and contributes τ 2Nχ(0,,0,1)-\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 Δ=2N1\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 R1,0(0,0) 2R+1=C^ R1(0,0)D R(0,0)D¯ R(0,0)B^ R+1 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 R,r(j 1,j 2) ΔA^\Delta_{R,r(j_1,j_2)}. One can check that the free theory has no candidate B^ N\hat{B}_N operator transforming in the appropriate representation of the flavour symmetry. So (1) necessarily remains in a short superconformal multiplet and Δ\Delta is independent of the gauge coupling.

Similarly, you can replace one of the twisted full punctures with [2,1 2N4][2,1^{2N-4}]. The resulting SCFT has a Lagrangian description Layer 1 SO ( 2 N 1 ) SO(2N-1) [ 1 2 N ] [1^{2N}] ( N 2 ) ( V ) (N-2)(V) ( N 1 ) ( V ) + ( N 1 ) ( 1 ) (N-1)(V)+(N-1)(1) ( [ 1 2 N ] , SO ( 2 N 1 ) ) \bigl([1^{2N}],Spin(2N-1)\bigr) [ 2 ( N 1 ) ] [2(N-1)] z 1 z_1 [ 2 ( N 1 ) ] [2(N-1)] z 2 z_2 [ 1 2 ( N 1 ) ] [1^{2(N-1)}] z 3 z_3 [ 2 , 1 2 ( N 2 ) ] [2,1^{2(N-2)}] z 4 z_4 \begin{svg}<svg width="434" height="165" xmlns="" xmlns:svg="" xmlns:se="" xmlns:math="" se:nonce="89522"> <g> <title>Layer 1</title> <ellipse ry="70" rx="100" id="svg_89522_8" cy="71" cx="101" stroke-linecap="null" stroke-linejoin="null" stroke-dasharray="null" stroke-width="2" stroke="#000000" fill="#ffeeee"/> <circle fill="#ffeeee" stroke="#000000" stroke-width="2" stroke-dasharray="null" stroke-linejoin="null" stroke-linecap="null" cx="362.5" cy="71" r="70" id="svg_89522_65"/> <foreignObject x="202.5" y="46.5" id="svg_89522_66" font-size="16" width="88" height="24"> <math display="inline" xmlns=""> <semantics> <mrow> 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</marker> </defs> </svg>\end{svg} as SO(2N1)+(2N3)(V)+(N1)(1)SO(2N-1)+(2N-3)(V) + (N-1)(1). Again, this matter content leaves an unbroken U(1)U(1) at a generic point on the Higgs branch. The D R(0,0)D_{R(0,0)} multiplet (for R=(2N3)/2R=(2N-3)/2) in the traceless rank-(2N3)(2N-3) completely anti-symmetric tensor representation of Sp(2N3)Sp(2N-3), constructed by the analogue of (1), has Δ=2N2\Delta=2N-2 and contributes τ 2N1χ(0,,0,1)-\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)D_{R(0,0)} operators in the spectrum persists in examples with no Lagrangian field theory realization.

1 For genus-gg and nn punctures, the class-S theory can be presented (in multiple ways) as a “gauge theory” with (3g3+n)(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)D_{0(0,0)} multiplet is the complex scalar in the free 𝒩=2\mathcal{N}=2 vector multiplet. Its superconformal descendents include the photino and the imaginary-self-dual part of the field strength.

Posted by distler at July 15, 2022 12:34 PM

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