The superconformal index and an elliptic algebra of surface defects

TL;DR

The paper constructs an elliptic algebra of difference operators $G_R$ that act on the 4d $\mathcal{N}=2$ superconformal index to insert surface defects labeled by all irreps of $su(N)$. In the rank-1 (SU(2)) case the operators reduce to the Hamiltonians of the elliptic Ruijsenaars-Schneider system, and in the Macdonald limit their structure constants become $(q,t)$-deformed Littlewood-Richardson coefficients, yielding a Verlinde-like algebra tied to refined Chern-Simons theory. Dimensional reduction to 3d identifies the defects with Wilson loops on the central node of the $T(SU(N))$-type star quivers, while mirror symmetry relates these to 3d line defects; non-minuscule representations exhibit mixing with lower-rank operators, requiring a basis change. The 4d–3d–2d network is further enriched by S-duality domain walls, which relate Wilson and ’t Hooft loops via Verlinde-type operators in Toda/CFT, consistent with AGT-type dualities. Overall, the work provides a cohesive framework connecting elliptic algebras of surface defects in 4d, Macdonald/Verlinde structures in 2d TQFTs, and dual presentations in 3d and Toda CFT, with concrete checks in multiple limits and dimensions.

Abstract

In this paper we continue the study of the superconformal index of four-dimensional $\mathcal{N}=2$ theories of class $\mathcal{S}$ in the presence of surface defects. Our main result is the construction of an algebra of difference operators, whose elements are labeled by irreducible representations of $A_{N-1}$. For the fully antisymmetric tensor representations these difference operators are the Hamiltonians of the elliptic Ruijsenaars-Schneider system. The structure constants of the algebra are elliptic generalizations of the Littlewood-Richardson coefficients. In the Macdonald limit, we identify the difference operators with local operators in the two-dimensional TQFT interpretation of the superconformal index. We also study the dimensional reduction to difference operators acting on the three-sphere partition function, where they characterize supersymmetric defects supported on a circle, and show that they are transformed to supersymmetric Wilson loops under mirror symmetry. Finally, we compare to the difference operators that create 't Hooft loops in the four-dimensional $\mathcal{N}=2^*$ theory on a four-sphere by embedding the three-dimensional theory as an S-duality domain wall.

Figures (12)

• Figure 1: Schematic illustration of the renormalization group flow $\mathcal{T}_{UV} \to \mathcal{T}_{IR}$ that can be used to introduce surface defects. The white dots represent full punctures with $SU(N)$ symmetry while the black dot is a simple puncture with $U(1)$ symmetry. The red dot represents a codimension-four defect engineering a surface defect in four dimensions.
• Figure 2: Sequence of dualities that maps the four-dimensional $\mathcal{T}_N$ theory (upper-left) to the three-dimensional star-shaped quiver theory (lower-right).
• Figure 3: The left picture illustrates the Riemann surface $C$ corresponding to a theory $\mathcal{T}_{UV}$, which is obtained by coupling the theory $\mathcal{T}_{IR}$ to a bifundamental field. An RG flow, that is initiated by turning on a Higgs vev for the bifundamental scalar, relates the theory $\mathcal{T}_{UV}$ to the original theory $\mathcal{T}_{IR}$ with a surface defect $G_r$. This is illustrated on the right.
• Figure 4: The left picture illustrates the Riemann surface $C$ corresponding to the theory $\mathcal{T}_{UV}^{'}$, which is obtained by coupling the theory $\mathcal{T}_{IR}$ to two bifundamental fields. An RG flow, that is initiated by turning on Higgs vevs for both bifundamental scalars, relates the theory $\mathcal{T}_{\rm UV}^{'}$ to the original theory $\mathcal{T}_{\rm IR}$ with two surface defects $G_{r_1}$ and $G_{r_2}$. This is illustrated on the right.
• Figure 5: Linear quiver description of $\mathcal{N}=2$ superconformal QCD. The flavor symmetry group of each set of $N$ hypers is enhanced to $U(N)$. This splits into an $SU(N)$ plus a diagonal $U(1)$ flavor symmetry group.