The superconformal index of theories of class $\cal S$

TL;DR

The article surveys the superconformal index for 4d ${\cal N}=2$ class ${\cal S}$ theories and exhibits its reformulation as a 2d topological field theory on the UV curve ${\cal C}$. It develops a bootstrap program that exploits generalized S-duality, Higgsing, and dimensional reduction to completely determine the index for A-type theories, and connects special limits (Schur, Macdonald, Hall–Littlewood, Coulomb) to well-known mathematical structures like $q$-deformed YM and Macdonald polynomials. A central theme is the relation between 4d indices and 2d chiral algebras, via the Schur index, and the interpretation of difference operators as surface defects tied to punctures on ${\cal C}$. The work highlights deep links among protected spectra, dualities, integrable models (Ruijsenaars–Schneider), and higher-dimensional/topological structures, providing a powerful framework for exact results in non-Lagrangian theories. It also outlines important open questions, including extensions to other gauge groups, irregular punctures, and broader partition functions on nontrivial manifolds.

Abstract

This is the 8th article in the collection of reviews "Exact results in N=2 supersymmetric gauge theories", ed. J. Teschner. The article reviews the superconformal index. It is often simpler to calculate than instanton partition functions, but nevertheless allows one to perform many nontrivial checks of conjectured dualities. It turns out to admit a representation in terms of a new type of topological field theory associated to the Riemann surfaces $C$ parameterising the class $\cal S$ theories.

Figures (5)

• Figure 1: Two different pair of pants decompositions corresponding to two different S-duality frames of the field theory. In the two duality frames the minimal puncture labeled by $a$ sits in a pair-of-pants with a different maximal puncture. The index computed in the two frames should give the same result.
• Figure 2: One can determine the structure constants $C_\lambda$ of the $A_{n-1}$ theories by studying the theory associated to a sphere with two maximal and $n-1$ minimal punctures (top picture). In one duality frame (middle picture) this is given by a $T_n$ theory, involving $C_\lambda$, coupled to a "superconformal tail" quiver. In another duality frame (bottom picture) this is given by a linear quiver with an $SU(n)^{n-2}$ gauge group, where each $SU(n)$ is coupled to $2n$ hypermultiplets. For $n=3$, the equivalence of the two frames is the celebrated Argyres-Seiberg duality, whose consequences for the index of $T_3$ ($\equiv$ the $E_6$ SCFT) have already been explored in section \ref{['E6']}.
• Figure 3: The difference operators ${\frak S}_{(r,s)}$, which compute residues and introduce surface defects, can be visualized as special punctures on the UV curve. The action of ${\frak S}_{(r,s)}$ on a flavor fugacity is interpreted as the collision of the special puncture with a flavor puncture. We can act on different punctures and obtain the same result for the index (top and middle pictures). We can also define the action of ${\frak S}_{(r,s)}$ on a long tube (bottom picture), by cutting open a cylinder, acting on one of the open punctures and gluing the surface back. S-duality guarantees that this is a well-defined procedure. In this way we can introduce the special punctures ${\frak S}_{(r,s)}$ on a UV curve with no flavor punctures at all.
• Figure 4: On the left we have an example of a star-shaped quiver mirror of the $A_3$ theory corresponding to a sphere with four punctures, two of which are maximal and one is minimal. On the right the quiver theories for ${\frak T}_{(1,1,1,1)}[\mathfrak{su}(4)]$ corresponding to the maximal puncture and ${\frak T}_{(3,1)}[\mathfrak{su}(4)]$ corresponding to minimal puncture are depicted.
• Figure 5: The Schur index of a guage theory is given by an integral over fugacities ${\bf z}$ taking value in a torus with modular parameter $\tau$. After modular transformation, $\tau \to -\frac{1}{\tau}$, the index is written as an integral over the dual cycle.