Black Holes in 4D $\mathcal{N}=4$ Super-Yang-Mills Field Theory

Black-hole solutions to general relativity carry a thermodynamic entropy, discovered by Bekenstein and Hawking to be proportional to the area of the event horizon, at leading order in the semiclassical expansion. In a theory of quantum gravity, black holes must constitute ensembles of quantum microstates whose large number accounts for the entropy. We study this issue in the context of gravity with a negative cosmological constant. We exploit the most basic example of the holographic description of gravity ($\mathrm{AdS}/\mathrm{CFT}$): type IIB string theory on ${\mathrm{AdS}}_5\times S^5$, equivalent to maximally supersymmetric Yang-Mills theory in four dimensions. We thus resolve a long-standing question: Does the four-dimensional $\mathcal{N}=4$ $\mathrm{SU}(N)$ Super-Yang-Mills theory on $S^3$ at large $N$ contain enough states to account for the entropy of rotating electrically charged supersymmetric black holes in 5D anti–de Sitter space? Our answer is positive. By reconsidering the large $N$ limit of the superconformal index, using the so-called Bethe-ansatz formulation, we find an exponentially large contribution which exactly reproduces the Bekenstein-Hawking entropy of the black holes. Besides, the large $N$ limit exhibits a complicated structure, with many competing exponential contributions and Stokes lines, hinting at new physics. Our method opens the way toward a quantitative study of quantum properties of black holes in anti–de Sitter space.

Popular Summary

One of the great challenges of modern theoretical physics is to reconcile Einstein’s theory of gravitation with the principles of quantum mechanics. This is particularly difficult for black holes, which must be featureless according to general relativity and yet should have enormous entropy to comply with the laws of thermodynamics. Hope for reconciling these pillars of modern physics comes from the so-called anti–de Sitter/conformal field theory ($\mathrm{AdS}/\mathrm{CFT}$) correspondence, which relies on principles of holography to provide a consistent quantum theory of gravity in terms of an ordinary quantum field theory in one less dimension. Here, we analyze a particular class of black hole in the most basic example of $\mathrm{AdS}/\mathrm{CFT}$ and find that it arises from an ensemble of quantum microstates whose number reproduces the required contribution to entropy.

More precisely, we analyze extremal and supersymmetric charged and rotating black holes in the prime example of $\mathrm{AdS}/\mathrm{CFT}$: type IIB string theory in anti–de Sitter space in five dimensions. We identify such black holes, known as Kerr-Newman black holes, with particular ensembles of supersymmetric states in the boundary field theory. Counting the states is a daunting task, which we overcome using modern nonperturbative techniques such as localization. The counting returns the sought-after entropy.

Our result sets the foundational stage to study quantum properties of black holes in a controlled framework. It opens the way to the thrilling possibility of computing nonperturbative quantum corrections to the entropy of black holes and, more generally, to their thermodynamic and statistical properties.

Figure 1

In yellow is highlighted the domain, Eq. (56), in the complex $\mathrm{\Delta }$ plane. The right boundary is the line $\gamma$, passing through 0 and $\tau$. The left boundary is the line $\gamma -1$, passing through $-1$ and $\tau -1$. The dashed lines are other elements of $\gamma +\mathbb{Z}$.

Figure 2

The upper-left plot shows the values of $\mathrm{I}\mathrm{m}\mathrm{\Theta }(\mathrm{\Delta };\tau +r)$ as a function of $r$, for a sample value of $\tau$ inside the semicircle (130). The red dot corresponds to $r=0$, which is the dominant contribution in this case. The upper-right plot shows $\mathrm{I}\mathrm{m}\mathrm{\Theta }(\mathrm{\Delta };\tau +r)$ for $\tau$ inside the semicircle (131). The green dot corresponds to $r=-1$, which is the dominant contribution in this case. The lower plot shows the values of $\mathrm{I}\mathrm{m}\mathrm{\Theta }(\mathrm{\Delta };\tau +r)$ for $\tau$ outside the two semicircles, where there is no dominant contribution.

Figure 3

Stokes lines in the $\tau$ plane. The red semicircle corresponds to the domain of analyticity where the $r=0$ contribution dominates the index. The green semicircle instead corresponds to the domain where the $r=-1$ contribution dominates. In the remaining region we do not have a dominant contribution. The blue and orange lines are the critical points of the entropy function for BPS black holes, in the two possible formulations.

Figure 4

Stokes lines in the $q$ plane, where the variable is $q^{1/3}$. The notation is the same as in Fig. 3. The dashed line indicates the subspace where both fugacities $q$ and $y$ are real and the computation of Ref. [30], which reproduced the index of multigraviton states, applies.