AdS black holes and finite $N$ indices

We study the index of 4d $\mathcal{N}=4$ Yang-Mills theory with $U(N)$ gauge group, focusing on the physics of the dual Bogomolny-Prasad-Sommerfield black holes in ${\mathrm{AdS}}_5\times S^5$. Certain aspects of these black holes can be studied from finite $N$ indices with reasonably large $N^2$. We make numerical studies of the index for $N\le 6$, by expanding it up to reasonably high orders in the fugacity. The entropy of the index agrees very well with the Bekenstein-Hawking entropy of the dual black holes, say at $N^2=25$ or 36. Our data clarifies and supports the recent ideas which allowed analytic studies of these black holes from the index, such as the complex saddle points of the Legendre transformation and the oscillating signs in the index. In particular, the complex saddle points naturally explain the $\frac{1}{N}$-subleading oscillating patterns of the index. We also illustrate the universality of our ideas by studying a model given by the inverse of the MacMahon function.

Figure 1

Plots of $\log |{\mathrm{\Omega }}_j|$ for $U(5)$ and $U(6)$ indices. The colors of the points encode the sign of $(-1{)}^j{\mathrm{\Omega }}_j$: red being positive and blue being negative. $\mathrm{Re}(S(j))$ computed from the black hole entropy function is the Bekenstein-Hawking entropy, given by the curve drawn with a solid black line. (a) $N=5$ and (b) $N=6$.

Figure 2

Same plots as Fig. 1, with the extra red/blue curves for $\mathrm{Re}(S(j))+\log |\cos [\mathrm{Im}(S(j))+\pi j+\eta ]|$. A subleading constant $\eta$ is empirically tuned to $\eta \approx -1$ to minimize the overall off-phase behaviors. The red and blue colors of the curves denote $\cos [\mathrm{Im}(S(j))+\pi j+\eta ]\gtrless 0$, respectively. (a) $N=5$ and (b) $N=6$.

Figure 3

Two plots of $\log |{\mathrm{\Omega }}_j|$ for the MacMahon function. Red/blue colors denote the positive/negative signs of ${\mathrm{\Omega }}_j$.

Figure 4

$N=2$

Figure 5

$N=3$

Figure 6

$N=4$