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Supergravity can't be the whole story

(Or at least, it can't produce the full black hole entropy)

Today, there is a kind of Big Tuesday on hep-th. And I am not the only one who believes that the first three hep-th papers on the arXiv are the three most interesting papers. Greetings to SlavaM. ;-)

This triumvirate includes two F-theory uplift papers - that were probably published on the same day by an accident (although the authors have obviously communicated with each other) - and one paper comparing SUGRA-calculated black hole entropies with the full quantum gravity result.

We will begin with the latter:

Jan de Boer & 3 co-authors: A bound on the entropy of supergravity
To appreciate what they're doing, we must begin with some history.

Black holes with explained microscopic entropies

In 1974, Stephen Hawking has figured out that black holes were not quite black.

Because the particles can't be quite confined inside the black hole walls (recall quantum tunneling) or because different Hamiltonian-like generators have different ground states (related by the Bogoliubov transformation) or because the virtual pair production may become real near the event horizon or because superluminal trajectories do contribute to the path integral, he was able to show that the particles leaving the black hole match a black body curve with the temperature determined by the "gravitational acceleration" at the event horizon, using classical terms - a quantity that is inversely proportional to the black hole radius.

Max Planck was able to deduce the spectrum emitted by all black bodies 100+ years ago - so it shouldn't be shocking that a black hole is a black body, too, especially after John Wheeler gave it the nice name. ;-)

Completely off-topic video clip by Lucie Vondráčková.

This conclusion only depended on long-distance physics of gravity, i.e. on its semiclassical approximation. However, if something has a temperature, the equations "E=T.dS" and "E=mc^2" can be applied to find out that the black hole also has a nonzero entropy. Hawking calculated it by these "indirect" thermodynamic tricks and found out that the entropy was proportional to the area of the event horizon (in "G=1" Planck units, over four), which confirmed a prophecy by Jacob Bekenstein who was able to guess the right basic dictionary between the geometry of black holes and thermodynamics before any semiclassical calculation was made.

However, thermodynamics is not really "fundamental". Its conclusions must be derivable from statistical physics. More concretely, the entropy should arise as the logarithm of the number of quantum microstates (or the volume of a phase space, if you use a classical approximation). Black holes have the maximum entropy among all localized (or bound) objects of a fixed mass but where does the huge number of microstates come from?

For 20 years, all known calculations that led to the right result were heuristic, vague, and unreliable. People had to wait until January 1996 when Strominger and Vafa published their groundbreaking calculation of the microstates of a black hole in 4+1 (large) dimensions, embedded into type IIB string theory where the supersymmetric black hole (that is macroscopic and qualitatively analogous to the four-dimensional Schwarzschild black holes) can be constructed out of many D5-branes, D1-branes, and some units of momenta.

Needless to say, their completely independent and geometrically "non-manifest" calculation led to the very same result as the Hawking-Bekenstein geometric calculation. String theory passed this very difficult test. The work of Strominger and Vafa was followed by 1,300+ papers, many of which have verified more complicated black holes (and black rings), near-extremal ones, non-extremal ones, higher-order corrections to the entropy (that can be mapped to Wald's formula in supergravity or its generalizations), and lots of other things. Whenever there was a well-defined test, string theory confirmed the geometric picture.

Maldacena-Strominger-Witten black holes

Today, the list of successes contains many four-dimensional black holes, including the nearly extremal Kerr black holes observed in the telescopes. But it took some time before people could generalize the Strominger-Vafa methods to other objects in the "upper" stringy landscape. The first successful black hole test of a hole localized in 3+1 large spacetime dimensions was published by Maldacena, Strominger, and Witten (MSW) in 1997, a few weeks before Maldacena's epochal discovery of the AdS holography.

MSW considered a black holes in type IIA string theory on a Calabi-Yau three-fold, the same manifold that is used in the realistic heterotic compactifications. Nice and large black holes are obtained if you put D0-branes on the manifold and wrap some D4-branes on some four-cycles in the six-manifold.

As they already have known from an M-theoretical pioneering insight by Witten, this configuration may also be described as M-theory on a circle times a Calabi-Yau manifold. The D4-branes become M5-branes wrapped on the circle (and the four-cycles) while the D0-branes become units of momenta along the circle. So the black hole is made out of some M5-branes with some stuff running on them.

M5-branes wrapped on the four-cycles give you points - but the circle remains macroscopic in the near-horizon description. Consequently, there exists a holographic description in terms of a string (M5-brane minus four dimensions) wrapped on a circle. The AdS3 near-horizon geometry of this string has a dual, holographic, description in terms of a conformal field theory, CFT2.

Because of the chronological subtlety mentioned above, the MSW paper actually couldn't use the proper AdS/CFT terminology: it appeared before this revolution! Still, it contained pretty much all the relevant formulae. And as expected, the microstates in the CFT2 exactly come in the right number to match the entropy of the four-dimensional black hole.

String theorists like to use the Calabi-Yau manifolds for many other reasons so this new kind of a black hole, technically distinct from the Strominger-Vafa black hole, became popular, too.

Fuzzballs and LLM

But you could still prefer to count the microstates in some variables of gravity or supergravity, instead of some a priori non-geometric (or at least non-back-reacting) branes in string theory. Is it possible? Well, the most geometric methods to count microstates in string theory are known as "bubbles in AdS space" (LLM) or "fuzzballs" (Samir Mathur).

The main idea is actually the same in both cases (and in many other backgrounds): see a modern review of fuzzballs. The microstates are visualized as some smooth configurations in supergravity. The smoothness means that there is no singularity, no event horizon, and therefore no entropy: they're entropy-less. But there are many of them and this large number may account for the entropy. The "empty" black hole interior only emerges if you average over many of these states.

LLM were able to create nice smooth solutions out of a two-dimensional map with arbitrary black and white spots: near the "base" of the map, the geometry converges to a "four-ball times three-sphere" or "three-sphere times four-ball", depending on the (non)color (black or white): both of these geometries are regular. Samir Mathur was able to imprint the information about any shape of a string into a smooth supergravity solution. His solution generated in this way contains some singularities but if you work a little bit, you may check that all of them are just coordinate singularities, just like in the LLM case.

But those successes mostly applied to black holes with a very high number of supercharges - as much as 16 - and these black holes cannot have a macroscopic area of the event horizon as long as you stick to Einstein's equations without corrections. Can the construction be generalized to more realistic black holes with macroscopic horizon areas? Are their microstates still encoded in the supergravity degrees of freedom?

The paper today brings a very powerful body of evidence that the answer is No. The massive states of string/M-theory become essential for more realistic black holes.

SUGRA only contains some states

The maximum N=8 supergravity in four dimensions (with 32 supercharges) was a nice unifying idea in the late 1970s and early 1980s that created the right mood for the first superstring revolution: superstrings extend the low-energy physics of supergravity to arbitrarily high center-of-mass energies.

Despite its beauties, the N=8 supergravity was a sick theory. Its supersymmetry could erase some infinities but it also made the theory too unrealistic, unable to contain the "ugly" particle species that we know from observations and that are compatible at most with N=1 supersymmetry. If one broke SUSY, the finiteness went away. Moreover, it was believed that even with those 32 supercharges, the theory would be non-renormalizable: the lethal divergences would just appear at a higher number of loops.

In the recent years, an increasing body of evidence suggests that the theory is actually finite to all orders of perturbation theory. However, it is also obvious that this consistency is not extended non-perturbatively. We have explained, e.g. in Two roads from N=8 SUGRA to string theory or Dixon's puzzles about N=8 SUGRA, that SUGRA must therefore be viewed as a low-energy launching pad for string theory. Whatever basic problem of SUGRA you will try to solve (while keeping its attracting features) will lead you to superstring theory.

The today's mostly Dutch paper (either by ethnicity, or by affiliation of the authors) escalates these arguments about the incompleteness of SUGRA to the context of black hole entropy counting. Is supergravity able to account for all degrees of freedom of the black holes? Is the question well-defined at all?

A strong argument that it is well-defined is that the authors actually present not one but two SUGRA calculations of the black hole entropy. They look pretty different from each other. Recall that they're essentially looking at the MSW black hole whose Bekenstein-Hawking thermodynamic (macroscopic) entropy was previously correctly matched with the full, stringy calculation involving CFTs.

The Dutchmen want to calculate the microscopic entropy, but only its supergravity part, and they do so in two ways:
  1. One of them counts states (or phase space) of multi-centered solutions with D0-branes attached to D6-brane and anti-D6-brane pairs. The brane-anti-brane pair may be called a "dipole" and the D0-branes add a "halo" to it.
  2. The other counts supergravitational excitations of the near-horizon AdS3 geometry.
You may expect what I am going to say: these two a priori very different computations match. That's a pretty strong circumstantial evidence supporting the opinion that the "supergravity part of the entropy" is a pretty well-defined notion, at least at the leading order, and that either of these two calculations gives them its magnitude.

And needless to say, this entropy is smaller than the full back hole entropy that was previous calculated either by the Bekenstein-Hawking thermodynamic tricks, or from the full stringy near-horizon CFT by Cardy's formula: the summarized formulae are added in the appendix of this blog article. So supergravity doesn't account for all the entropy of the black holes with large event horizons. There may still exist a calculation that is morally analogous to supergravity and includes all the essential higher-energetic string-theoretical stuff. But this calculation simply can't be "just supergravity".

I am inclined to believe that some construction of the microstates or fuzzballs (and these words are irritating mostly because several younger workers in the field assign them with slightly incorrect interpretations, even though they are pretty familiar with the technology) will exist for the black holes with many charges and large event horizons - but they will have excited "stringy" states, not just the massless (supergravity) states. Somebody should construct them, with some string field theory, added states of branes, or something else. For example, the Rindler space in string theory or M-theory could actually have a conceptually simple description in terms of smooth solutions that involve SUGRA plus something else.

And that's the memo.

Appendix: formulae for the entropy

The specific MSW-like black hole they study may be obtained from "q0" D0-branes and "p" D4-branes wrapped on a four-cycle. It matters what cycle you're considering: this information is encoded in the homology "p_A", but only the triple self-intersection of the D4-branes will influence the entropy. This triple intersection determines the central charge of the CFT,
c = dABC pA pB pC = 6I.
So "I" stands for the triple self-intersection divided by six.

If "q0" is negative and if its absolute value is much greater than "I", which is already much greater than one, the full stringy result for the entropy is
S = 2 pi (-q0)1/2 I1/2
which agrees with the Bekenstein-Hawking entropy when translated to the "astrophysical" variables. In the same limit, their new SUGRA calculation gives
S = [3/8 zeta(3)]1/3 (-q0)1/3 I1/3
You see that exponents of both "q0" and "I" were reduced. They actually have a more accurate formula that also holds when "I" and "q0" are comparable, and my "much greater than" inequality may be replaced by a finer "ordinary" inequality. They also have found another formula valid when "q0" is rather close to "I/4". In that regime, the entropy gets even smaller because the power of "I" is replaced by the same power of a (smaller, in this regime) "N" which is defined as the difference "I/4 - q0".

These formulae and limits are derived in two "supergravity only" ways and they're smaller than the Bekenstein-Hawking = stringy result, showing that supergravity doesn't contain all the degrees of freedom (unless you believe that Bekenstein, Hawking, string theorists, mathematics, and Nature are victims of halucinations of seeing the same number of states that don't exist haha: if you believe so, you will be permanently banned once you repeat this belief in the comments).

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