Superconformal Index, BPS Monodromy and Chiral Algebras

TL;DR

The paper establishes a deep link between specialized 4d ${ m N}=2$ superconformal indices and traces of powers of Kontsevich–Soibelman BPS-monodromy operators, ${ m Tr}\,{ m M}(q)^N$, on the Coulomb-branch BPS spectrum. It generalizes the Schur-limit connection (N=-1) to arbitrary integer N, showing that each specialization corresponds to a family of 2d chiral algebras ${\cal A}_N$ with central charges tied to the 4d conformal data; for Lagrangian theories these algebras are constructed explicitly in the extreme weak-coupling limit, and for Argyres–Douglas theories they yield interesting 2d algebras via monodromy. The work also connects these traces to elliptic genera of S^2-twisted compactifications, and develops a comprehensive framework—via Moyal products, modular properties, and adjoint line-operator insertions—for computing and interpreting ${\rm Tr}\,{ m M}(q)^N$ across a broad class of theories. Collectively, it provides both concrete examples and a broad methodological bridge between 4d SCFT invariants and 2d chiral-algebra/data, with rich implications for wall-crossing and dualities.

Abstract

We show that specializations of the 4d $\mathcal{N}=2$ superconformal index labeled by an integer $N$ is given by $\textrm{Tr}\,{\cal M}^N$ where ${\cal M}$ is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras $\mathcal{A}_{N}$. This generalizes the recent results for the $N=-1$ case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on $S^2\times T^2$ where we turn on $\frac{1}{2} N$ units of $U(1)_r$ flux on $S^2$.

Figures (4)

• Figure 1: Schematic matter content of the 2d gauge theory obtained by twisted reduction on $S^2$. ${\cal G}$ denotes the gauge group and ${\cal F}$ denotes the flavor group.
• Figure 2: The $Q(r,s)\equiv Q(s,r)$ quiver $(r+s)>0$. The corresponding $\mathcal{N}=2$ theory has one--dimensional Coulomb branch and rank $(r+s)$ flavor symmetry group $G_F$ with the Dynkin graph in the lower graph.
• Figure 3: BPS quiver
• Figure 4: The bi--graph of the ideal triangulation of the punctured disk with $r$ marks on the boundary having a self--folded triangle inside an internal 2--gon.