Bootstrapping the superconformal index with surface defects

TL;DR

The paper develops a physical bootstrap for the N=2 superconformal index of class S theories by interpreting certain poles in flavor fugacities as IR endpoints with BPS surface defects. It introduces RG constructions that realize surface defects via vortices and FI parameters, and shows that residues are encoded by Ruijsenaars-Schneider difference operators acting on the IR index, connecting to a 2d TQFT and Macdonald-type structures. For A1 theories, residues generate elliptic RS operators, whose eigenfunctions diagonalize the index and enable a complete bootstrap, with Macdonald limits reproducing known results; these ideas extend to higher rank A_{N-1} theories and admit a consistent 3d reduction to Wilson-line data in mirror descriptions. The framework unifies 4d indices, 2d defect insertions, and integrable systems, providing a practical route to compute indices with surface defects across broad classes of theories and offering multiple avenues for mathematical and physical exploration.

Abstract

The analytic properties of the N = 2 superconformal index are given a physical interpretation in terms of certain BPS surface defects, which arise as the IR limit of supersymmetric vortices. The residue of the index at a pole in flavor fugacity is interpreted as the index of a superconformal field theory without this flavor symmetry, but endowed with an additional surface defect. The residue can be efficiently extracted by acting on the index with a difference operator of Ruijsenaars-Schneider type. By imposing the associativity constraints of S-duality, we are then able to evaluate the index of all generalized quiver theories of type A, for generic values of the three superconformal fugacities, with or without surface defects.

Figures (7)

• Figure 1: Our first RG example. On the left we indicate schematically the quiver for the SCFT ${\cal T}_{IR}$, and on the right the quiver for ${\cal T}_{UV}$. The dashed lines represent $SU(N)$ gauge groups.
• Figure 2: Our second RG example.
• Figure 3: The setup of figure 2 for the $A_1$ generalized quivers. The SCFT ${\cal T}_{IR}$, associated to a Riemann surface ${\cal C}$ with $s$ punctures, is coupled to the SCFT of free trifundamental half-hyper, associated to a sphere with three punctures. The two SCFTs are coupled by choosing an $SU(2)$ flavor symmetry of each theory, and weakly gauging a diagonal $SU(2)$ subgroup. The resulting SCFT is denoted by ${\cal T}_{UV}$, and corresponds to the degeneration limit of a surface with $s+1$ punctures shown in the figure.
• Figure 4: By generalized S-duality, the order in which one extracts the residues in fugacities $a$ and $b$ is expected to be immaterial, indeed the three different decompositions of the surface shown in the figure are topologically equivalent. This implies $[{\frak S}_{(r,s)} \, ,{\frak S}_{(r',s')}] = 0$.
• Figure 5: Three different but equivalent ways to introduce a surface defect in a tube. First by coupling the two edges of the tube to a free hypermultiplet and computing a $U(1)_f$ residue of the index. Then by acting with the relevant difference operator on either of the edges and then gauging. The equivalence of these different procedures implies that the index is self-adjoint under the natural measure.