The growth of the $\frac{1}{16}$-BPS index in 4d $\mathcal{N}=4$ SYM

Abstract

We study the Hamiltonian index of $\frac{1}{16}$-BPS operators in 4d $\mathcal{N}=4$ U(N) super Yang-Mills (SYM) theory numerically for $N=2,\dots ,10$. We show that the large-charge asymptotics agree with analytic results in the Cardy-like limit, as consistent with the entropy of supersymmetric black hole in the dual AdS. The numerics also agree with the large-N analytic result, thus providing hints towards an exact formula for the index. We then prove, using ideas from representation theory, that for values of charges (quantized in integer units) less than 2(N+1) the index agrees precisely with the multi-graviton index, and then begins to deviate for larger charges. Thus the U(N) SYM index interpolates between multi-graviton values at small charge and black hole growth at large charges.

18 pages, 4 figures, 2 tables; v2: typos corrected, statements sharpened

References in corpus (7)

• Nonperturbative black hole entropy and Kloosterman sums
• Supersymmetric Index from Black Hole Entropy
• Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula
• Counting BPS operators in N=4 SYM
• Comments on 1/16 BPS Quantum States and Classical Configurations
• Free Fermions and Thermal AdS/CFT
• Supergravitons from one loop perturbative N=4 SYM

Cited by in corpus (10)

• A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index
• $SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index
• Superconformal index of low-rank gauge theories via the Bethe Ansatz
• Logarithmic Corrections to the Entropy of Rotating Black Holes and Black Strings in AdS${}_5$
• Spin Matrix Theory in near $\frac{1}{8}$-BPS corners of $\mathcal{N}=4$ super-Yang-Mills
• From multi-gravitons to Black holes: The role of complex saddles
• Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index
• The Yang-Mills duals of small AdS black holes
• The Giant Graviton Expansion
• Symmetry structure of the interactions in near-BPS corners of $\mathcal{N}=4$ super-Yang-Mills