Quiver superconformal index and giant gravitons: asymptotics and expansions

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 ${\widehat{A}}_m$, and along the main diagonal for the bivariate index for $\mathcal{N}=4$ and ${\widehat{A}}_3$. In these cases we find $\ln \mathrm{\mid }{c}_n\mathrm{\mid }\sim \gamma n^{\frac{1}{2}}+\beta \ln n+\alpha$ (Hardy-Ramanujan type). We also identify polynomial growth for $dP3$, $Y^{3,3}$ and $Y^{p,0}$, and give numerical evidence for $\gamma$ 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.