Quiver superconformal index and giant gravitons: asymptotics and expansions

A bstract We study asymptotics of the d = 4, 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 $$ {\hat{A}}_m $$ A ̂ m , and along the main diagonal for the bivariate index for 𝒩 = 4 and $$ {\hat{A}}_3 $$ A ̂ 3 . In these cases we find $$ \ln \left|{c}_n\right|\sim \gamma {n}^{\frac{1}{2}}+\beta \ln n+\alpha $$ ln c n ∼ γ n 1 2 + β ln n + α (Hardy-Ramanujan type). We also identify polynomial growth for dP 3, 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.