Central Charges and Vacuum Moduli of 2d $\mathcal{N}=(0,4)$ Theories from Class $\mathcal{S}$

TL;DR

This work studies 2d $\mathcal{N}=(0,4)$ theories arising from topologically twisted reductions of 4d $\mathcal{N}=2$ class $\mathcal{S}$ theories on Riemann surfaces, focusing on infrared central charges and vacuum moduli spaces. By analyzing two Higgs-branch components—the special and twisted Higgs branches—via Hilbert-series techniques, the authors propose conjectural formulas for the IR right-moving central charge $c_R$ and demonstrate their validity for $G=SU(2)$ through explicit Lagrangian calculations and moduli-space counting. The study highlights the limitations of naive anomaly-based central-charge computations in the presence of unbroken Abelian gauge sectors and emergent IR R-symmetries, and shows how IR dynamics, particularly Higgsing patterns, determine the correct $c_R$. The results provide a framework for understanding IR R-symmetries and central charges in a broad class of reduced class $\mathcal{S}$ theories, with clear directions for extending to higher-rank groups, punctured surfaces, and potential 2d TQFT structures.

Abstract

We investigate 2d $\mathcal{N}=(0,4)$ supersymmetric theories obtained from a topologically-twisted reduction of 4d $\mathcal{N}=2$ class $\mathcal{S}$ theories on a Riemann surface. This study addresses subtle aspects of central charges, unbroken gauge groups, and emergent superconformal R-symmetries of these theories. Focusing on infrared vacuum structures, we propose conjectural formulas for the central charges. For theories with the gauge group $SU(2)$, we use a Lagrangian description to analyze the vacuum moduli spaces. In particular, we examine two distinct branches -- the special Higgs branch and the twisted Higgs branch -- by computing their Hilbert series, and find agreement with the proposed central charge formulas.

Figures (12)

• Figure 1: For a puncture labeled by $\rho=[4,3,1]$ when $N=8$, label the boxes in $\rho^{\mathsf{T}}$ by $k=1,\cdots,8$ as on the right, and the corresponding $h_k$ are $(1,1,1,2,2,3,3,4)$, and consequently $n_v(\rho)=199,~ n_h(\rho)=204$.
• Figure 2: The $\mathcal{N}=(0,4)$ reduction of the 4d $\mathcal{N}=2$ multiplets on a Riemann surface of genus $g_2$. For simplicity, in the remaining part, we will use a single solid blue line to represent $g_2$ twisted hypermultiplets $(\Sigma_j,\tilde{\Sigma}_j)$, and a single dashed blue line to represent $g_2$ Fermi multiplets $\Gamma_j$.
• Figure 3: The tadpole frame of the case with generic $(g_1,n)$. The gauge nodes are of two types: one are on the loops of the quiver, and the others are not. The special Higgs branch requires the scalars of the twisted hypermultiplets $\left(\Sigma_j^{\text{off-loop}},\tilde{\Sigma}_j^{\text{off-loop}}\right)$ connecting to the nodes of the latter type must vanish.
• Figure 4: Gluing two quivers. The gluing will introduce 3 new $J$-term equations of $q^1,q^2$, and modify the $J$-term and $E$-term equations on both sides.
• Figure 5: The basic block with hypermultiplets in fundamental representation of both the gauge and flavor $SU(2)$'s.