Black Holes, Matrix Models and Large D - QG-Paris2017.pdf · Black Holes, Matrix Models and Large D Frank FERRARI Université Libre de Bruxelles International Solvay Institutes Quantum

Black Holes, Matrix Models and Large D

Frank FERRARI

Université Libre de Bruxelles International Solvay Institutes

Quantum Gravity in Paris IHP, Paris, March 23th, 2017

Plan of the talk

1. Building Quantum Models for Black Holes from Holography

2. Basic Properties of Black Holes From the Bulk Description

3. Sachdev-Ye-Kitaev

4. An interesting development: large D in General Relativity (Emparan et al.)

5. The new large D limit of planar diagrams: towards an exactly solvable quantum black hole (Ferrari 2017).

The holographic correspondence teach us that bulk gravitational physics in d dimensions is an emergent phenomenon. The fundamental description is a standard quantum mechanical system with no gravity, living in d’<d dimensions.

Building Quantum Models for Black Holes from Holography

Bulk space-times containing a black hole at Hawking temperature T are described in this way by an ordinary (meaning: well-defined from standard quantum mechanics without gravity) quantum system in the canonical ensemble at temperature T.

This is an extraordinarily non-trivial statement, in particular because black holes have many puzzling properties that are not usually associated with standard quantum systems, as we shall review.


Clearly, only rather special and non-trivial quantum theories can do the job.

As it will become clear, the required models must describe some kind of many-body system in some thermodynamical limit (large number of degrees of freedom).

Moreover, black hole physics and more generally an emergent space-time description can only be found in the fully strongly coupled quantum regime of these models.

Note also that SUSY cannot help because of the finite temperature (it may help only for some extremal black holes, which have T=0 but can still have a large entropy).

In résumé, a quantum black hole must be some kind of strongly interacting many-body system with a very large number of degrees of freedom.


Solving the problem requires:

1. To guess what precisely these models could be (understood).

2. To solve them at strong coupling (we are making nice progress right now).

3. To understand the dictionary with the bulk physics (mainly remains to be done).

Overall an extremely difficult but very exciting research programme which has kept a large community of people busy for many many years.

The great news is that we may, after all, be rather close to complete this programme at least in a class of interesting examples.


1. To guess what precisely these models could be.

To solve this first point, string theory has been an invaluable tool. All our basic intuition really comes from D-brane constructions, in the spirit of the original argument by Maldacena for the holographic correspondence.

The outcome, that we shall briefly explain below, is that one needs to consider gauged quantum mechanical models of NXN matrices in the large N (planar) limit and at strong coupling.

We thus see that the problem will be somehow similar and related to the problem of solving QCD. A startling illustration of the unification of ideas in theoretical physics and also a clue to the difficulty of the problem.


1. To guess what precisely these models could be.

A stack of N Dp-branes is described microscopically by NXN matrix degrees of freedom

Ai↵ j , 0 ↵ p

1 i, j N

i

j

gauging

open strings closed stringsXij tr (X · · ·X)

U(N)G

Newton

⇠ 1

N2

Xiµ j , 1 µ D = d� p� 1


Two salient examples:

1. p=3, D3-branes, N=4 super Yang-Mills

Xµ , 1 µ 6 = 10� 3� 1

2. p=0, D0-branes, BFSS matrix quantum mechanics

Xµ , 1 µ 9 = 10� 0� 1

Typical interaction terms are

V (X) = �N� tr⇥Xµ, X⌫

⇤2trXµXµX⌫X⌫ , trXµX⌫XµX⌫

as in


ia) The continuous spectrum

ib) The quasi-normal behaviour

ic) Unitarity violation

ii) The chaotic behaviour

iiia) The emergent space-time: probes

iiib) The emergent space-time: operators

iiic) The emergent space-time: entanglement entropy

etc…


ia) The continuous spectrum

Puzzling in the quantum mechanical description, where the spectrum is always discrete!

Intuition: any frequency gap can be infinitely redshifted near the horizon.

Math: the potential in tortoise coordinate is becoming flat when . r⇤ r⇤ ! �1

However, we expect the typical gap to be due to the high degeneracy at zero coupling.

⇠ e�↵N2

0.02 0.04 0.06 0.08 0.10

5

10

15

20

The discreteness of the spectrum is a non-perturbative quantum gravity effect.

(with A. Bilal, C. De Lacroix and Tatsuo Azeyanagi)


ib) The quasi-normal behaviour

Intuition: any small perturbation in the black hole background will eventually falls into the horizon and thus should decay.

Math: wave operators are not self-adjoint because of the boundary condition at the horizon; the eigenvalues can have a negative imaginary part.


ibc) The quasi-normal behaviour and loss of unitarity

In quantum mechanics, the consequence of a small perturbation is governed by two-point functions via linear response theory,

⇢(!) =1� e��!

Z

X

p,q

e��EpApqBqp�(! + Ep � Eq)

The quasi-normal behaviour is inconsistent with unitarity and a discrete spectrum.

The large N limit should thus yield a non-unitary model with a continuous spectrum. At large N, the Poincaré recurrences are pushed away to infinity.

⌦[A(t), B]

↵�=

Z +1

�1⇢(!)e�i!td!


ii) The chaotic behaviour

Intuition: very small perturbations at early times can be infinitely blueshifted and thus have drastic effects. This is chaos. (Shenker, Stanford)

In the quantum mechanics, this is governed by out-of-time-ordered four point functions which can grow exponentially in time.

The associated Lyapunov exponent for black holes is universal

⌦[A(t), B]2

↵�⇠ 1

N2e2�Lt

�L =2⇡

�

and is supposed to be an upper bound over a large class of QM models for which it can be defined.


iii) The emergent space-time

Big question: how do we see the emergent space-time from the point fo view of the quantum mechanics?

Possible strategies:

1) Use D-probes, for example D-particles. These can always be defined in matrix theories and the geometry can be read off in full details from their effective action.

2) Use correlation functions of high dimension operators to compute geodesic lengths.

3) Use entanglement entropy to compute areas of embedded surfaces à la Ryu-Takayanagi.


iii) The emergent space-time

Eventually, one would like to understand the full holographic dictionary.

The most puzzling questions concern the interior of the black hole, if it exists (i.e. there is no firewall).

In the interior, time emerges. To make this precise is one of the main conceptual challenge in the field. There is also a deep link with cosmology.


Consider the following Hamiltonian:

H =1

4!Jijkl �i�j�k�l

hhJ2ii ⇠ 1

N3

where the are Dirac gamma matrices in N=2n dimensions. �i

The couplings J are chosen randomly from a Gaussian distribution with

The model is simple to treat because interesting quantities, like the free energy or the correlation functions, are self-averaging at large N. One can thus average over the disorder to compute at large N.


When we do so, it is not too difficult to realize that the large N limit is dominated by so-called melon Feynman diagrams.

For example, a typical melon for the two-point function is

These diagrams have a recursive structure and can be resummed exactly from a Schwinger-Dyson equation.

For four-point functions, one gets similar ladder diagrams which can be resummed as well.


Another interesting aspect of the model is that it is very easy to implement the finite N version on a computer. Finite N (i.e. “non-perturbative quantum gravity”) effects can thus be studied.

The result of these calculations is extremely surprising and interesting. One finds that the model shares some of the non-trivial features expected for quantum black holes.

This includes the appearance of a continuous spectrum at large N, the quasi-normal behaviour, the loss of unitarity and the chaotic behaviour with the correct Lyapunov exponent for black holes!�L = �/2⇡


On the other hand, the bulk interpretation is much more mysterious. The model does not have any natural string theory interpretation. It might be related to higher spins in the bulk, since it looks like a vector model. But in vector models, you do not have black holes…

The use of disorder is another problem. It is OK at large N for self-averaging quantities, but otherwise averaging over disorder does not seem to make sense (what one gets is then not a quantum theory).

So any study of the finite N effects along these lines seem rather dubious, at least to me.


The conclusion is that the reason why SYK is good, at least on some of the important aspects, seems quite mysterious…

The remaining of the talk will be devoted to provide an answer to this question.

We shall enlarge enormously the range of models that can be studied and make the link with string theory precise.

4. Large D in General Relativity

Let us now briefly review a seemingly totally unrelated development, initiated by Emparan, Suzuki and Tanabe in 2014.

The intuition is that at large d, the gravitational field falls off very quickly. A black hole gravitational field is thus limited to a very small region around the horizon.

These authors proposed to study general relativity, in particular the black hole space-times in general relativity, by looking at the limit of very large space-time dimension d.

It turns out that this idea allows to simplify drastically some analytical calculations, while keeping the main qualitative features of black hole physics.

The authors were in particular able to compute the quasi-normal spectrum in the large d expansion in a variety of cases, and to study the dynamics of multi black hole solutions in a membrane approach.

4. Large D in General Relativity

Let us now briefly review a seemingly totally unrelated development, initiated by Emparan, Suzuki and Tanabe in 2014.

The intuition is that at large d, the gravitational field falls off very quickly. A black hole gravitational field is thus limited to a very small region around the horizon.

These authors proposed to study general relativity, in particular the black hole space-times in general relativity, by looking at the limit of very large space-time dimension d.

It turns out that this idea allows to simplify drastically some analytical calculation, while keeping the main qualitative features of black hole physics.

The authors were in particular able to compute the quasi-normal spectrum in the large d expansion in a variety of cases, and to study the dynamics of multi black hole solutions in a membrane approach.

4-5. Large D in General Relativity and in Matrix Quantum Mechanics

Xiµ j , 1 µ D = d� p� 1

I won’t into more details about this approach, because I simply want to use it as a motivation for the following observation.

One is naturally led to the idea of studying the large D limit of matrix quantum mechanics!

5. The new large D limit of planar diagrams

Let us thus consider O(D) invariant matrix quantum mechanics of the form

L = ND⇣tr�X†

µXµ +m2X†µXµ

��

X

B

tBIB(X)⌘

IB = tr�Xµ1X

†µ2Xµ3X

†µ4Xµ5 · · ·X†

µ2s

�where the interaction terms are

For example, take the interaction potential to be

t1 tr�XµXµX⌫X⌫

�+ t2 tr

�XµX⌫XµX⌫

�

t1 t2


L = ND⇣tr�X†

µXµ +m2X†µXµ

��

X

B

tBIB(X)⌘

IB = tr�Xµ1X

†µ2Xµ3X

†µ4Xµ5 · · ·X†

µ2s

�

One can then consider large D at fixed couplings t. This is the usual “vector model like” large D limit. In effect, we simply have vectors of matrices. We get a double large N and large D expansion of the form

F =X

(g,`)2N2

fg,`N2�2gD1�`

At leading order, only the vertex with no crossing contribute.


There is an interesting interplay between the large N and the large D limit. For example, if only the vertex with no crossing is included (or similar vertices),

F =X

(g,`)2N2

fg,`N2�2gD1�`

one can show that , which is an interesting non-renormalization theorem in the large D expansion.

` � g

However, this limit is too simple: the physics is similar to vector model physics, and there is no black hole. We have thrown away to many diagrams.


t1 tr�XµXµX⌫X⌫

�+ t2 tr

�XµX⌫XµX⌫

�

t1 t2

The question is: can we do the limit in a more interesting way, trying to keep more diagrams?

This would mean that, instead of maintaining the ts fixed, we try to enhance them by letting them scale as a power of D:

t = D�� , � > 0 ?


On first sight, this seems inconsistent. Indeed, with the enhanced couplings, one can typically build diagrams with arbitrarily high powers of D. ’t Hooft-like scaling is very delicate and naively cannot be changed without either eliminating all diagrams (trivial) or having a diverging limit…

However, this is where a miracle occurs. It turns out that, to any vertex B, one can associate a genus g(B) such that the large D limit with

tB = Dg(B)�B , �B fixed

makes sense!


I have no time to prove this statement here. The idea is to generalise recent developments in tensor models (led by Adrian Tanasa and collaborators). The nice thing is that these tools can be adapted to matrix models, and to large D!

Maybe this is not too surprising: tensor models deal for example with three-index objects and have a or symmetry. U(N)3 O(N)3

Our objects also have three indices:

1 i, j NXiµ j , 1 µ D = d� p� 1

The symmetry is or for complex or Hermitian matrices.

U(N)2 ⇥O(D) U(N)⇥O(D)

Hermitian matrices seem incompatible with tensor models, but it turns out that the tensor model technology continues apply for planar diagrams, which are the physically most relevant diagrams.



1 i, j NXiµ j , 1 µ D = d� p� 1

U(N)2 ⇥O(D) U(N)⇥O(D)

Conceptually, it is crucial to distinguish between D and N to make the link with string theory: U(N) is a gauge symmetry whereas O(D) is a global symmetry!

A very strong statement is also that these results apply to Hermitian matrices, where the symmetry is reduced. This is an unexpected extension of tensor model technology, which relies heavily in having a distinct symmetry group attached to each tensor index.


l is a new index associated to any Feynman diagram.

In the new scaling, we can prove that the large D limit of the sum over diagrams of any fixed genus g exist:

F =X

g�0

N2�2gFg

Fg =X

`�0

D1+g�`/2Fg,`

The crucial point is that the exponent of the power of D that can show up in a diagram is bounded above for any given genus.

Of course, the new large D limit does not commute with the large N limit: one first take large N, and then large D. The other order does not make sense.


For example, the genera associated with the two basic quartic vertices are

The vertex with crossings can be enhanced! One then gets many more Feynman diagrams at leading order!

g(B)=0, no enhancement.

g(B)=1/2, enhancement!


It turns out that the leading large D planar diagrams are melons :

The link with SYK is made: we’ll get here all the nice features of SYK, plus many extra bonuses because we are working in “string-inspired” matrix models.

This obviously opens up a large realm of possibilities: it seems that we have a new approximation scheme for matrix theories, which has the two superficially incompatible properties:

1) it is amenable to analytic studies, because melons are tractable 2) it captures the essential physics contained in planar diagrams.

Thank you for your attention!

Black Holes, Matrix Models and Large D - QG-Paris2017.pdf · Black Holes, Matrix Models and Large D Frank FERRARI Université Libre de Bruxelles International Solvay Institutes Quantum

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