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

Abstract

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.

42 pp + appendices

References in corpus (11)

• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• $\mathcal{N}=1$ supersymmetric indices and the four-dimensional A-model
• Supersymmetric multi-charge AdS_5 black holes
• Localization on Hopf surfaces
• Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula
• Counting BPS operators in N=4 SYM
• Delayed Deconfinement and the Hawking-Page Transition
• Supersymmetric counterterms from new minimal supergravity
• Supergravitons from one loop perturbative N=4 SYM
• The growth of the $\frac{1}{16}$-BPS index in 4d $\mathcal{N}=4$ SYM
• Thermodynamics of Near BPS Black Holes in AdS${}_4$ and AdS${}_7$

Cited by in corpus (5)

• EFT and the SUSY Index on the 2nd Sheet
• $SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index
• Superconformal index of low-rank gauge theories via the Bethe Ansatz
• The large-$N$ limit of 4d superconformal indices for general BPS charges
• Expanding on the Cardy-like limit of the superconformal index of 4d $\mathcal{N}=1$ ABCD SCFTs