The 4d superconformal index near roots of unity and 3d Chern-Simons theory

TL;DR

The paper analyzes the 4d N=1 superconformal index on S^3 × S^1 in Cardy-like limits as q approaches roots of unity, revealing an all-order expansion in ˜τ = mτ + n with terms in powers k ∈ { -2, -1, 0, 1 }. The authors show that the leading singular pieces are controlled by background Chern-Simons couplings, while the constant term receives a dynamical contribution from a 3d Chern-Simons theory on S^3/ℤ_m, with linear terms tied to the supersymmetric Casimir energy; this structure is derived both from direct asymptotics of the 4d index and from dimensional reduction to a 3d 𝒩=2 theory with supersymmetric CS terms. In the rational-tau regime, the analysis extends to orbifold geometries, connecting the 4d master index to 3d CS partition functions on S^3/ℤ_m and lens-space sectors, including nontrivial holonomy contributions. Overall, the work provides a coherent 4d→3d EFT explanation for the exponential growth of the index near roots of unity and elucidates how background CS data encode the dominant growth, with potential implications for black hole entropy counting and holographic interpretations.

Abstract

We consider the $S^3\times S^1$ superconformal index $\mathcal{I}(τ)$ of 4d $\mathcal{N}=1$ gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential $τ$ coupled to a combination of angular momenta on $S^3$ and the $U(1)$ R-charge. We develop the all-order asymptotic expansion of the index as $q = e^{2 πi τ}$ approaches a root of unity, i.e. as $\widetilde τ\equiv m τ+ n \to 0$, with $m,n$ relatively prime integers. The asymptotic expansion of $\log\mathcal{I}(τ)$ has terms of the form $\widetilde τ^k$, $k = -2, -1, 0, 1$. We determine the coefficients of the $k=-2,-1,1$ terms from the gauge theory data, and provide evidence that the $k=0$ term is determined by the Chern-Simons partition function on $S^3/\mathbb{Z}_m$. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the $S^1$. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the $\widetilde τ\to 0$ expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index.

Figures (2)

• Figure 1: The catastrophic behavior of $V^Q(u_{ij})$, drawn over the range $mu_{ij}\in(-1,1)$, for $\mathrm{arg}\widetilde{\tau}>\frac{\pi}{2}$. The control parameter $m\xi$ determines the M or W type behavior.
• Figure 2: The functions $C'^{-3}\overline{B}_3(C'\Delta)$ with $C'=1,\cdots,13$, for $0\le\Delta\le1$. For $0<\Delta<1/2$ the blue curve corresponding to the fully-deconfined phase takes over. The take-over of the orange curve signifies the partially-deconfined 2-center phase in the corresponding region ($1/2<\Delta\lesssim .72$), and so on.