Quiver superconformal index and giant gravitons: asymptotics and expansions

TL;DR

This work analyzes the d=4 ${\mathcal N}=1$ superconformal index for toric quivers in the large-$N$ limit, revealing a cycle-factorization of the $R$-charge weighted adjacency matrix $M(t)$ and connecting the combinatorics to saddle-point asymptotics. It establishes Hardy–Ramanujan-type growth for univariate and main-diagonal bivariate coefficients in several families (notably $\hat{A}_{m}$, $Y^{p,p}$, $dP_3$) and identifies polynomial growth in a subset of quivers, accompanied by an effective central charge interpretation. The paper then extends Murthy’s giant graviton expansion to a matrix-coupling setting via the Hubbard–Stratonovich transformation and Borodin–Okounkov formula, enabling finite-$N$ corrections through multi-matrix determinants. Together, these results illuminate state degeneracies and their gravity-dual interpretations, and point to future directions including flavoured indices, broader quiver classes, and rigorous analytic control of tails and bivariate asymptotics.

Abstract

We study asymptotics of the $d=4$, $\mathcal{N}=1$ superconformal index for toric quiver gauge theories. Using graph-theoretic and algebraic factorization techniques, we obtain a cycle expansion for the large-$N$ index in terms of the $R$-charge-weighted adjacency matrix. Applying saddle-point techniques at the on-shell $R$-charges, we determine the asymptotic degeneracy in the univariate specialization for $\hat{A}_{m}$, and along the main diagonal for the bivariate index for $\mathcal{N}=4$ and $\hat{A}_{3}$. In these cases we find $\ln |c_{n}| \sim γn^{\frac{1}{2}}+ β\ln n + α$ (Hardy-Ramanujan type). We also identify polynomial growth for $dP3$, $Y^{3,3}$ and $Y^{p,0}$, and give numerical evidence for $γ$ in further $Y^{p,p}$ examples. Finally, we generalize Murthy's giant graviton expansion via the Hubbard-Stratonovich transformation and Borodin-Okounkov formula to multi-matrix models relevant for quivers.

Figures (12)

• Figure 1: The clover quiver for ${\mathcal{N}}=4$ SYM.
• Figure 2: The periodic tiling and the corresponding dimer with three zig-zag paths for ${\mathcal{N}}=4$ SYM.
• Figure 3: An example of toric duality in the $Y^{4,2}$ quiver.
• Figure 4: Construction of the planar quiver for $Y^{p,q}$.
• Figure 5: The four zig-zag paths for the $Y^{p,q}$ dimer, presented in dotted red and blue, and dashed red and blue, respectively.