From multi-gravitons to Black holes: The role of complex saddles

By applying the Atiyah-Bott-Berline-Vergne equivariant integration formula upon double dimensional integrals, we find a way to compute the matrix integral representations of $4d$ $\mathcal{N}=1$ superconformal indices. The final formula allows us to easily extract analytic results in the large-rank expansion of certain theories. As an example, we compute the leading one-loop corrections to the effective action of the known complex saddles in those theories. For a superconformal index of $SU(N)$ $\mathcal{N}=4$ SYM, we use the equivariant integration formula and the Picard-Lefschetz method to show that at large enough values of $N$, only two, among the known complex saddles, dominate the counting of large operators i.e. of operators with charges of order $N^2$. Contributions from other known complex saddles are present, but we show they are exponentially suppressed in that range of charges; the smaller the charges the less suppressed they are, and eventually, to count small operators i.e. operators with charges smaller than $N^{2/3}$, like multi-gravitons, they can not be neglected.