Black holes, wormholes, and AdS/CFT correspondence

Prieslei Estefânio Dominik Goulart Santos

Supervisor: Prof. Dr. Victor de Oliveira Rivelles

Abstract The AdS/CFT correspondence, or gauge/gravity duality, is a relation between a D-dimensional at space conformal eld theory and a (D+1)-dimensional grav- itational theory. Black holes play an important role in the duality, and give a description of strongly interacting theories at nite and zero temperature. In this research project we propose to study black holes in dimensions D ≤ 4, the count- ing of their microstates, and applications in the context of gauge/gravity duality and condensed matter, and also in the Sachdev-Ye-Kitaev model. We also propose to study some aspects of wormholes, as well as their relation with gauge/gravity duality.

São Paulo, Agosto de 2017 1 Introduction: Black holes and AdS/CFT

It is commonly believed that string theory is the best candidate for a unied theory of Nature. The matter and gauge particles correspond to dierent modes of vibration of a fundamental string, so the standard model of particle physics and gravitational phenomena can be described by one of the so many vacua of the theory. By itself string theory is very interesting and promising. It turns out that Maldacena has proposed in 1997 a way to use string theory as a mathematical tool to handle dicult nonper- turbative problems [1] (see [2] for a book). In his work he cojectured that there is a duality between the N = 4 Super-Yang-Mills theory, which is a superconformal theory 5 in at four-dimensional space, and string theory in AdS5×S space. As the eective eld theory on the string theory side is gravity in anti-de-Sitter spacetime, and on the

eld theory side is conformal eld theory, the duality takes the name of AdS5/CFT4. The duality is shown to work in dierent dimensions, with the gravity theory always having one more dimension than the conformal eld theory. This extrapolation goes further, conjecturing that, in general, gauge theories have a gravity dual, justifying why this is sometimes called gauge/gravity duality. In the last few years, a more phenomenological approach of gauge/gravity duality has been used to handle problems in condensed matter, going by the name AdS/CMT (see [3] for a review). The philosophy behind it is that we can extract the desired properties of some condensed matter systems by using some gravity theory. The physics of both sides of the duality must match, so one can in principle use gravity to make predictions or explain phenomena for condensed matter systems in certain limits when the eld theory description is untractable or even unknown. It turns out that condensed matter systems are at nite temperature, which requires the gravity theory to have also a well-dened notion of temperature. This is achieved by the introduction of a black hole on the gravity side. It is undeniable that black holes have always been a hot topic in theoretical physics. The detection of gravitational waves emitted by the colision of two black holes in a coalescing binary system, made by the Laser Interferometer Gravitational-Wave Obser- vatory (LIGO) in 2015, did not leave space for questioning their existence [4]. From a theoretical point of view, black holes are considered the perfect laboratory to test any theory of quantum gravity. Because they behave as thermodynamical objects, the area of the event horizon has a large entropy associated to it, which matches the counting of their inner microstates, at least in the string theory context [5]. Quantum processes taking place near the event horizon lead the black hole to evaporate [6], changing its temperature. It turns out that, even in the most symmetric cases, quantum gravity in four dimensions is a very challlenging subject. The simplest charged known solution in four dimensions is the Reissner-Nordström black hole. A special feature about this solution is that it can achieve a zero temperature limit, and in this case we say that the black hole is extremal. A more general feature about most of the charged extremal black holes is that their geometry in the near D−2 horizon region is AdS2 × S , where D is the number of dimensions of the spacetime. The one-dimensional conformal eld theory dual to the gravity theory in AdS2 space is just quantum mechanics. One can use gauge/gravity duality in order to study the

1 properties of this quantum mechanical system. Moreover, four-(or higher-)dimensional black holes can be reduced eectively to black holes in three or two dimensions, where the same conceptual issues arise, but turn to be simpler. So, one can gain important insights about quantum gravity and black holes in higher dimensions by combining the study of black holes in lower dimensions [7, 8] with gauge/gravity duality. In this research project we propose to study black holes in D ≤ 4, as well as its relation to string theory and gauge/gravity duality.

2 Goals

In this section, we describe in more details the problems we intend to investigate during the postdoctoral period.

2.1 Black holes solutions for supergravity in D = 4 The maximally supersymmetric gravity theory in four dimensions, and in anti-de- Sitter spacetime, is N = 8, SO(8) gauged supergravity. The work of nding the most general non-extremal solution with the full charge content is far from complete. The knowledge of explicit non-extremal black hole solutions is of fundamental importance, for instance, in order to address their microscopic properties. Moreover, thermal states of a boundary conformal eld theory are dual to non-extremal black holes in AdS space- times, which are solutions to supergravity theories with a non-trivial scalar potential, namely gauged supergravities. One can fully understand the topic of phase transitions only if the full non-extremal solution is known. It turns out that the presence of cou- plings among the elds and a non-trivial scalar potential makes the work of nding analytical solutions a very dicult task. Although some explicit (and complicated) non-extremal four-dimensional solutions were found, it is understandable why much of the progress over the past years was restricted to the treatment of extremal solutions. Consider for instance the so-called Einstein-Maxwell-dilaton system, which corre- sponds to the bosonic part of an N = 4 supergravity theory. The theory corresponds to an Abelian truncation of N = 8 supergravity, and it comes in two versions. One is the SU(4) version, which contains only one Abelian gauge eld, and the other is the SO(4) version, sometimes called U(1)2 supergravity. The elds of the SU(4) version are the metric gµν, the dilaton φ, and the gauge eld Aµ, whereas the SO(4) version contains one more gauge eld, Bµ. The actions are written as Z 4 √ µ −2φ µν
SSU(4) = d x −g R − 2∂µφ∂ φ − e FµνF . (1) Z 4 √ µ −2φ µν 2φ µν
SSO(4) = d x −g R − 2∂µφ∂ φ − e FµνF − e GµνG . (2)

We take unities in which (16πGN ) ≡ 1, and the eld strengths are written as Fµν = ∂µAν − ∂νAµ, and Gµν = ∂µBν − ∂νBµ. When Fµν is just electric, and Gµν is just magnetic, the theories are equivalent, meaning that their equations of motion can be

2 mapped into each other. The most general dyonic1 black hole solution for such theories were found analytically in [9], and it has pretty much the same metric structure as the Reissner-Nordström solution, with two horizons. The simplest potential that can be added to the theory is a constant potential V = 2Λ, where Λ is the cosmological con- stant. Planar black hole solution for this theory in asymptotically Lifshitz spacetimes were found in [10, 11]. Apart from not satisfying the relativistic symmetries, the dilaton eld in this case diverges at the boundary, which is an issue for holography. One goal is to attempt to nd analytically the non-extremal black hole solution in AdS spacetime which respects the relativistic symmetries and with regular dilaton at the boundary. This is actually an ambitious task, but it is motivated by the experience of the applicant. In particular, the applicant "rederived" the non- extremal black hole solution for the Einstein-Maxwell-dilaton theory in absence of a potential [12], writing it in terms of the integration constants. He showed that there is a freedom of choosing independent and dependent physical parameters for the theory, and solved old puzzles related to the dilaton eld on the horizon of the extremal black hole. He also showed that one can construct massless black holes and naked singularities for this theory [13], and used the massless solution to construct Einstein-Rosen bridges satisfying the null energy condition. The consequences of nding such a solution in AdS spacetimes will be of great relevance, since, as we mentioned, this will give us insight about the form of the non-extremal black hole of N = 8 gauged supergravity. The Eintein-Maxwell-dilaton theory is also interesting for another reason: The dila- ton eld is attracted to a xed point on the horizon of the extremal black hole, inde- pendently of its boundary conditions at innity. This phenomena is called the attractor mechanism. It is believed that the attractor points of the dilaton eld correspond to the critical points of the Beta function of the dual conformal eld theory. This is of par- ticular interest for this research project, since the applicant is already familiar with the subject, which is one of the topics of his PhD thesis. Specically, the applicant used the Sen's entropy function method to compute [14] analytically the extremal dyonic black hole entropy for the N = 8, SO(8) gauged supergravity. The best candidate for the dual three dimensional theory is the ABJM model (due to Aharony, Bergman, Jaeris and Maldacena), which is an N = 8 superconformal Chern-Simons theory. Recently, the counting of microstates of the twisted and mass-deformed ABJM theory was per- formed [15, 16] by evaluating and extremizing its twisted index, and this reproduced the

Bekenstein-Hawking entropy for AdS4 black holes in N = 2 gauged supergravity. We intend to perform the counting of microstates of dyonic black holes in AdS4 in the same spirit as was proposed in [15, 16] and check whether the result matches with the Bekenstein-Hawking entropy, computed by the applicant in [14]. This will be a consistency check of both the method and the applicant's result.

2.2 Charged wormholes in AdS spacetimes Wormholes are solutions to General Relativity which correspond to shortcuts that connect two dierent regions of the same spacetime, or a bridge that connects two

1A dyonic black hole is a black hole that contains electric and magnetic charges at the same time.

3 regions of dierent spacetimes. One of the rst bridge constructions dates back to Einstein and Rosen [17], who used the Schwarzschild solution in an attempt to give a geometrical description of elementary particles which excludes singularities. A simple coordinate transformation brings the Schwarzschild metric into the form of a bridge, with two dierent slices connected by a surface called "throat". In general, bridges constructed from black hole solutions can not be traversed even for light, rst due to its dynamics, and second because of the presence of spacetime singularities. In 1988, Morris and Thorne [18] introduced the concept of a traversable wormhole. These are solutions that contain no singularities in the whole spacetime, and more importantly, it allows light (or even a human being) to cross it. It turns out that the matter content necessary for the construction of such solutions is of the "exotic" type, which means that it does not satisfy the null energy condition. Since then, the Einstein-Rosen bridges and the traversable wormholes, which are both referred to as just wormholes in modern language, were considered only mere curiosities and would not be considered physically acceptable objects. One way to violate the energy conditions is by the introduction of "phantom" scalar elds in the theory (called also "ghost" elds in other contexts). Current cosmological observations [19, 20] suggest the existence of a uid with negative pressure, so, this phantom scalar is the natural candidate for modelling such uids. It turns out that a ghost scalar can give rise to wormhole solutions. In fact, the applicant presented very recently [21] an analytical charged traversable wormhole solution for the Einstein- Maxwell-phantom-dilaton theory, whose asymptotic regions are two Minkowski space- times. It is well-known that anti-de Sitter space does not satisfy the null energy condition, so, a natural direction is to attempt to construct analyti- cal charged wormhole solutions whose asymptotic regions are anti-de Sitter spaces. We will analyse whether it is necessary or not to introduce phantom elds in order to obtain traversable wormhole solution with anti-de Sitter asymptotics. Hopefully, this will generalize his solution. This solution will be very interesting in the context of holography, since Gao, Jaeris, and Wall recently showed [22] that adding certain interactions that couple the two boundaries of the eternal AdS-Schwarzschild space renders a traversable wormhole. The so-called ther- moeld double state is a state in a Hilbert space of two copies of the original system. When these two copies are coupled, it is conjectured that their gravity dual is the AdS- Schwarzschild wormhole [23] with the two boundaries coupled. This will serve as a guide to give a physical interpretation in the context of AdS/CFT for the charged traversable wormhole solution with AdS asymptotics.

2.3 Transport coecients and quantum phase transitions in AdS/CMT The computation of transport coecients for strongly interacting systems with a net charge density ρ is a very challenging task. In order to obtain nite results for the thermoelectric conductivities, a mechanism for breaking translation invariance must be introduced. In applications of gauge/gravity duality to condensed matter systems, or just AdS/CMT, the breaking of translation invariance is achieved by the introduction

4 of elds with specic proles that do the the job. This was the case, for instance, of massive gravity theories [24, 25, 26], lattice models [27, 28, 29], and linear axions [30]. As was stated in the introduction, we need to introduce a black hole with temperature T in the bulk in order to describe condensed matter systems. A novel technique to compute holographic thermoelectric conductivities for dyonic black holes was recently developed [31, 32, 33]. By dening radially independent boundary currents, one can study linear perturbation about the black holes induced by applied electric elds and thermal gradients, and then extract analytically the thermoelectric conductivities in terms of only horizon data. With the knowledge of the full solution, one can then compute analytically the conductivities in terms of the charges of the black hole. In collaboration with Erdmenger, Fernandéz, and Witkowski [34], the applicant adapted the Sen's entropy function formalism for black holes with planar horizons, and then computed all the horizon data for Einstein-Maxwell-dilaton systems in the presence of a scalar potential. Inserting such horizon data in the expressions for the conductivities, explicit results written in terms of the charges of the black hole were obtained. This method works only for extremal black holes, and it is useful because one does not need to know the full extremal solution to obtain the horizon data. A natural direction is to understand what the attractor mechanism and these analytical results have to teach us about strongly coupled systems at zero temperature. In this sense, it will be interesting to understand the relation among the attractor mechanism, explicit expressions for conductivities, and the topic of quantum phase transitions (which occur at T = 0). One interesting direction is to investigate the behavior of all the con- ductivities as a function of the temperature for an Einstein-Maxwell-dilaton theory with a specic potential. In order to do so, we will use the Gao and Zhang planar black hole solution presented in reference [35], which makes use of a combination of three Liouville potentials. This will allow us to test analytically, at some good approximation, some numerical predictions of reference [36].

2.4 The Sachdev-Ye-Kitaev model and dilaton-gravity in two and three dimensions

The Sachdev-Ye-Kitaev model [37, 38] is a 0 + 1-dimensional quantum mechani- cal model, and consists of N-Majorana fermions with q-interactions. For q = 4, the Hamiltonian is written as

1 X H = J ψaψbψcψd, (3) 4! abcd a,b,c,d with a, b, c, d = 1, ..., N. This model presents three very important features: i) it is solvable at strong coupling; ii) it is maximally chaotic; iii) it has emergent conformal symmetry. The rst point is extremally important, since we do not know many models which are solvable at strong coupling. This happens because we can sum all Feymann diagrams at large N, being able to compute correlation functions. For the second

5 point, it is known that Lyapunov exponent quanties the amount of chaos. For black holes [39], it was shown that the Sachdev-Ye-Kitaev achieves the maximal allowed Lyapunov exponent[37]. For the third point it was shown that two point functions present conformal symmetry at low energies [37, 40, 41, 42]. This emergent conformal invariance suggests that the theory has a holographic gravity dual. Although this model has been of great interest recently, it is still unclear what is its correct gravity dual. It was argued [43, 44, 45] that this must be the Jackiw- Teilteboim dilaton-gravity model with a cosmological constant [46], which was studied with more details in [47]. Another suggestion is that it is Liouville theory [48]. It is worth emphasizing that AdS2 holography with constant and non-constant dilaton elds was recently studied [49] by Grumiller and collaborators. It was also shown that the spectrum of the Sachdev-Ye-Kitaev model can be interpreted as the spectrum of a three dimensional scalar coupled to gravity [50]. The important point is that dilaton-gravity in two or three dimensions captures many features of the model, but so far, it is just believed that four-dimensional models can be capable of capturing such features. Some eective four-dimensional supergravity theories give rise to charged black holes, which have a well-dened extremal or near-extremal limit, whose near-horizon geometry is 2 AdS2 × S . So, extremal or near-extremal black holes are strong candidates to be the gravity dual of the Sachdev-Ye-Kitaev. We propose to investigate how a four- dimensional theory of this kind, i.e. the Einstein-Maxwell-dilaton theory, is related to the Sachdev-Ye-Kitaev model. We propose to investigate more closely the relation between the physical observables of the Sachdev-Ye- Kitaev model and those of extremal or near-extremal black holes of Eintein- Maxwell-dilaton theory. Notice that this goal can also be generalized to consider more general theories, which will include other elds such as axions, for instance.

3 Work plan and timeline

• From December, 2017 to March, 2018: In order to be updated with the developments of the topic, during the rst four months the applicant will study the relevant references about the Sachdev-Ye- Kitaev model and the proposed gravity duals. Being updated with the references will be important, since this is a rapid developing eld;

• From March, 2018 to December, 2018: Then, the applicant will focus on the proposed problem, and attack in parallel the problems proposed in sections 2.1, 2.2, and 2.4;

• From January, 2019 to October, 2020: During this period the applicant will deal also with the problems proposed in section 2.2, and new problems that might arise as a consequence of all his research.

6 4 Proposal for BEPE

During the postdoctoral period the applicant intends to apply for a BEPE grant to visit an international center of excelence. He will present a new research project aligned with the interests of a supervisor who can possibly collaborate with him.

5 Collaboration between USP and IFT/UNESP/ICTP- SAIFR

The city of São Paulo holds the two best institutes for investigation in string theory and holography in the country, USP and IFT/UNESP/ICTP-SAIFR. It is also good to emphasize that the applicant will be moving from the institution where he will get his doctoral degree, IFT/UNESP, to USP. Changing institution will force him to collaborate with a dierent group, and this will be of great importance for his carreer as a researcher. On the other hand, staying in the same city will still be fruitful, since the IFT holds the ICTP-SAIFR, which organizes every year many international events of the highest level, bringing world class researchers to visit and teach the new generation of researchers. The group at USP develops research in dierent branches of string theory and holog- 5 raphy, such as strings in AdS5 × S space, Wilson-loops in AdS/CF T , entanglement entropy, etc. These topics are aligned with the applicant's interests, and this shows a great potential of collaboration not only with the supervisor of the project, but also with other members of the group.

6 Conclusions

In this research project we proposed to study several aspects of black holes, worm- holes, and AdS/CFT. Specically, we proposed to study black hole solutions of su- pergravity theories, and check whether the counting of microstates proposed recently matches the Bekenstein-Hawking entropy for N = 8, SO(8) gauged supergravity. In the context of applications to condensed matter, we proposed to investigate in more details the relation between the attractor mechanism for black holes and strongly cou- pled at zero temperature. We also proposed to study the Sachdev-Ye-Kitaev model, and make progress towards understanding how the four-dimensional Eintein-Maxwell- dilaton theory and its black hole solution capture all the interesting features of the model.

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