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 p µ −2φ µν SSU(4) = d x −g R − 2@µφ∂ φ − e FµνF : (1) Z 4 p 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.