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

TL;DR

The paper studies the growth of the $1/16$-BPS index in 4d $\mathcal{N}=4$ $U(N)$ SYM and its holographic interpretation via AdS/CFT, focusing on the connection to black hole entropy in AdS$_5\times$S$^5$. It employs a Hamiltonian matrix-integral formulation to analyze finite-$N$ and Cardy-like large-charge regimes, supported by numerical results for $N=2$ through $10$. A key result is a representation-theory proof that, for charges $n\le 2N+1$, the index matches the multi-graviton count, while at larger charges it deviates and exhibits BH-like exponential growth, illustrating a smooth interpolation between graviton-dominated and black-hole regimes. The work also suggests the possibility of an exact finite-$N entropy formula and highlights a deep link between Frobenius duality and holographic state counting.

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.

Figures (4)

• Figure 1: The comparison between the microscopic entropy $\log |d_N(n)|$ and the BH entropy $S_\text{BH}(N,n)$ for $N=2,3,4,10$. The plot for $N=10$ is zoomed in slightly, and clearly shows the initial deviation from the BH curve. The microscopic data for $N=2,3,4$ are computed directly using the gamma function representation. The time taken to calculate $d_N$ increases rapidly with $N$---after initialization, and putting a cutoff at $n=100$, it took 5 ms for $N=2$ and 26 min for $N=4$. (All these calculations were performed using PARI/GP PARI2 on a MacBookPro 2017.) For higher values of $N$ we use the formula \ref{['INpartformula']}. In this method, the computational bottleneck is to produce the characters of the permutation group $S_d$, which leads to the charge cutoff $n \le 2d$. The time taken to calculate the character tables from d=1 to 20 was 4 seconds while the final case dealt with here, namely $d=35$ alone took 20 hours. (All the character tables were computed using GAP GAP4.) Having obtained the characters, calculating the coefficients $d_N$ is quite fast, e.g. the case $N=10$, $n \le 70$ took 14 min using PARI/GP.
• Figure 2: The gravitational black hole entropy $S_\text{BH}=N^2 s(j) = a_2(Nn)^{2/3} + O(n^{1/3})$ for $N=2,3,\dots,10$ and the logarithm of the index of the graviton gas (dashed line).
• Figure 3: The blue (solid) lines are $S_\text{BH}(N,n)/N^{2/3}$ as a function of $n$ (left: $N = 2, \dots 10$, right: $N=2, \dots 100$). The brown (dashed) lines are $\log d_\text{grav}(n)/N^{2/3}$, as a function of $n$. The blue lines converge to the uppermost line as $N \to \infty$ and the brown lines converge to zero as $N \to \infty$. The convergence is seen more clearly for larger $N$ on the right.
• Figure 4: Microscopic data of $d_N(n)$ for $N= 2,3,4$, and of $d_\text{grav}$.