The superconformal index of N=1 class S fixed points

TL;DR

The authors analyze four-dimensional $\mathcal{N}=1$ class $\mathcal{S}$ fixed points arising from M5-branes on holomorphic curves in local Calabi–Yau three-folds and compute their superconformal index. They show the index organizes into a two-dimensional topological quantum field theory, generalizing the known $\mathcal{N}=2$ case, and express indices of all fixed points in terms of the $\mathcal{N}=2$ index. For accessible rank-one theories, they establish IR equivalence across different UV constructions and demonstrate dualities via rank-one crossing symmetry, with a broader universal TQFT description that extends to inaccessible fixed points. In Hall–Littlewood/Coulomb limits, the index counts relevant operators in a way that geometrically matches the Morse index of $SU(2)$ Yang–Mills connections on the UV curve, suggesting a deep link between class $\mathcal{S}$ dualities and two-dimensional gauge theory.

Abstract

We investigate the superconformal index of four-dimensional N=1 superconformal field theories that arise on coincident M5 branes wrapping a holomorphic curve in a local Calabi-Yau three-fold. The structure of the index is very similar to that which appears in the special case preserving N=2 supersymmetry. We first compute the index for the fixed points that admit a known four-dimensional ultraviolet description and prove infrared equivalence at the level of the index for all such constructions. These results suggest a formulation of the index as a two-dimensional topological quantum field theory that generalizes the one that computes the N=2 index. The TQFT structure leads to an expression for the index of all class S fixed points in terms of the index of the N=2 theories. Calculations of spectral data using the index suggests a connection between these families of fixed points and the mathematics of SU(2) Yang-Mills theory on the wrapped curve.

Figures (4)

• Figure 1: Two generalized quivers that flow to the same SCFT. Unshaded versus shaded triangles corresponds to $T_N^+$ constituents versus $T_N^-$.
• Figure 2: Representatives constructions of all genus four generalized quivers with $|z|<1$.
• Figure 3: The two local moves that allow arbitrary modifications of a refined pants decomposition within a fixed topological class, demonstrated on "renormalized" trinions (indicated by semi-circular external legs). The first figure denotes crossing symmetry at an $\mathcal{N}=2$ gluing, while the second shows the crossing symmetry at an $\mathcal{N}=1$ gluing.
• Figure 4: [left] A propagator is obtained from the trinion and cap amplitudes. This determines the cap amplitude to be a simple inverse of the trinion amplitude in the diagonal basis. [right] Gluing a cap and trinion with a different choice of levels leads to a propagator that shifts the overall level of the cobordism.