Large-$N$ limit of chiral quiver theories: free energy, supersymmetric indices and operator counting

We find the first analytic solution to the large-$N$ saddle point equations for the $S^3$ free energy of a chiral $\mathcal N=2$ Chern-Simons-matter theory with an M-theory dual. Focusing on the chiral quiver dual to AdS$_4\times Q^{1,1,1}/\mathbb Z_{\rm k}$, we show how the long-range forces of the matrix model cancel at large $N$, resolving a long-standing tension with the expected $N^{3/2}$ scaling. The resulting free energy precisely matches the holographic prediction for all assignments of trial R-charges. Our results extend to the superconformal and topologically twisted indices on $S^2\times S^1$, whose large-$N$ and Cardy limit is determined by the same saddle. The solution agrees with the predictions of the operator counting method, giving strong evidence for this conjecture beyond non-chiral quivers. We support the analytic solution with a finite-$N$ numerical analysis for multiple assignments of R-charges, extending recent numerical results obtained for this quiver.