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

Understand

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.