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ST4 Lectures on Assorted Topics in Ads3/CFT2

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ST4 Lectures on Assorted Topics in Ads3/CFT2 4 ST Lectures on Assorted Topics in AdS3/CFT2 Pinaki Banerjee˚ ˚International Centre for Theoretical Sciences Survey No. 151, Shivakote, Hesaraghatta Hobli, Bengaluru - 560 089, India. E-mail: [email protected] Abstract: These lecture notes can broadly be divided in three parts : overview, basics and recent developments. In the first part gauge/gravity duality is reviewed with motiva- tions, “derivation” and the prescription. Second part deals with basic of AdS3 and CFT2 and the related thermodynamics. I also discuss Cardy formula, BTZ black hole entropy and Hawking-Page transition in this section. Finally the last part is based on some selected current activities in this area which include conformal blocks, geodesic Witten diagrams and Cardy-like formula for three point functions in AdS3/CFT2 context. These notes are based on lectures delivered at Student Talks on Trending Topics in Theory (ST4) 2018, held at the National institute for Science Education and Research, Bhubaneshwar. Contents 1 Introduction1 1.1 Why bother?1 1.2 Brief outline3 2 Lightning review of AdS/CFT4 2.1 Dualities in physics5 2.1.1 Dualities in quantum field theory5 2.1.2 Dualities in string theory8 2.2 Strings, D-branes & Holography 10 2.2.1 String theories 10 2.2.2 D-branes 12 2.2.3 Why do we expect AdS/CFT duality? 13 2.2.4 The decoupling argument 17 2.2.5 The dictionary of parameters 24 2.2.6 Generalization to finite temperature and density 26 2.2.7 The GKPW prescription 26 3 AdS3 and its thermodynamics 31 3.1 AdS3 and its ‘excitations’ 33 3.1.1 Global AdS 34 3.1.2 Poincaré patch 36 3.1.3 Conical defect 36 3.1.4 Thermal AdS3 37 3.1.5 BTZ black hole 38 3.2 Hawking-Page transition 39 3.2.1 Crtitical Temperature 41 3.2.2 Order of the Transition 42 3.2.3 Is BTZ stable? 42 3.3 Brown-Henneaux in a nutshell 42 3.3.1 Why are boundary conditions important? 42 3.3.2 Asymptotic symmetries 44 4 CFT2 and its thermodynamics 47 4.1 ABC of CFT2 47 4.2 CFT on a torus 53 4.3 Modular invariance 56 4.4 Derivation of Cardy formula 57 4.5 BTZ from Cardy 61 – i – 5 2D blocks, 3-point functions and AdS3 61 5.1 Local observables in CFT 61 5.1.1 2-point function 61 5.1.2 3-point function 62 5.1.3 4-point function 62 5.2 Heavy-light conformal blocks at large c 64 5.2.1 Monodromy method 66 5.2.2 Geodesic Witten diagrams 69 5.2.3 Entanglement entropy : an application 70 5.3 A Cardy formula for three-point coefficients 72 5.3.1 Torus 1-point function 73 5.3.2 Modular invariance 73 5.3.3 Asymptotic formula 74 5.3.4 Dual gravity interpretation 76 A Some useful tricks & results 79 A.1 Lagrange multipliers 79 A.2 ‘Solvable’ path integrals 80 A.3 Gaussian is special 80 B Maximally symmetric spaces 81 B.1 Killing vectors 81 B.2 Maximal symmetry 82 C “ 4 super Yang-Mills 83 N C.1 The Lagrangian 83 C.2 Dimensional reduction of “ 1 SYM 84 N C.3 Symmetries 85 C.4 Phases 85 1 Introduction 1.1 Why bother? AdS/CFT is very important, if not the most important ingredient of present day theoretical high energy physics research. I’ll briefly review that duality below. In case some of you are not at all familiar with it, here is the basic picture. It relates certain gravitational theories to certain field theories (without gravity). When I say relates I actually mean equates i.e. they are two completely different but equivalent descriptions. AdS “ CFT (1.1) – 1 – Here AdS is an abbreviation of Anti-de Sitter space which is solution to Einstein equa- tions with negative cosmological constant and CFT is Conformal Field Theory which is a field theory with extra symmetries known as conformal symmetries. The equation (1.1) is considered one of the most beautiful ‘equations’ of theoretical physics. In what follows I will try to motivate why AdS3/CFT2 is particularly interesting. There will be some remarks that you probably find difficult to digest if you are not familiar with AdS/CFT or/and AdS and CFT - but please don’t panic! I shall review (though briefly) it before starting the actual content of these lectures. Before I discuss about why AdS3/CFT2 is particularly interesting for the sake of com- pleteness let me tell you about gauge/gravity duality in other spacetime dimensions. The duality was originally discovered by Maldacena in the context of AdS5/CFT4 where he conjectured that maximally supersymmetric Yang-Mills theory in 4 dimensions was dual to 5 type IIB string theory in AdS5 ˆS . This is the most well understood version of the duality till date. There have been a vigorous development in this context, both from the gauge the- ory side and the gravitational side, just after it was proposed. Then there is examples from even higher dimensional context e.g. AdS7/CFT6 where it is believed that six dimensional conformal gauge theory which lives on the world volume of stack of M5 branes is actually 4 dual to string theory in AdS7 ˆ S . This field theory is quite “mysterious” and AdS/CFT is one possible way to understand this. In lower dimensions too there are proposed examples of similar duality - particularly popular these days is AdS2/CFT1. AdS2 space is ubiquitous d´2 in BH physics. Near horizon limit of any extremal BH is AdS2 ˆM . This was one of the major reasons string theorists were particularly interested in this duality. But the recent fuzz about it is mostly due to the SYK model1. There are different other examples as well but let’s focus on the topic of this set of lectures i.e. AdS3/CFT2. AdS3 3D gravity is easier! No propagating DOF and locally always AdS (for EH with • 3 negative cosmological constant). Since three-dimensional Einstein gravity has no local degrees of freedom, it is not a realistic model of the world where gravitational waves do exist2! Then why should we study such a “unrealistic” model? Because different things get simplified in 3D and we can learn several aspects of the theory even ignoring GW3. Calculations are often easy due to mere few number of spacetime directions. • AdS can be seen as a group manifold. This feature plays important role while • 3 studying string propagation in this background. 1See Pranjal’s lectures in ST4 2017 for more details. 2Yes, I am referring to LIGO detections [1,2]. In fact, Jahanur and Aneesh are going to talk about related aspects of gravity in de Sitter universe where we actually live in. 3One can always add matter to EH action to get dynamics i.e. contain local degrees of freedom. See TMG, NMG. Those theories are interesting by their own. We don’t discuss them in these lectures. – 2 – Higher spin gauge fields (like gravity) also don’t have any propagating degrees of free- • dom. Little group is trivial. HS tower can be truncated to spins s “ 2; 3;:::N in AdS3. But the most important feature is, even after all those simplifications, 3D gravity is still nontrivial enough to be interesting. It has BH solutions. BH entropy computa- tion, Info loss, String propagation, etc. CFT2 Conformal field theory is QFT which enjoys “extra” symmetries including Poincare in- • variance. Poincare transformations preserve the distance or length. Roughly speaking if we relax the condition more and consider theories which remain unchanged under any angle preserving transformations (Poincare transformations will only be a subset). pd`2qpd`1q For d dimensional CFTs the symmetry group is SOpd; 2q which has 2 genera- tors. But d “ 2 case is special, it enjoys Virasoro symmetry which has infinite number of symmetry generators and therefore very constraining. One can directly compute correlators even without writing down the Lagrangian of the theory! CFT2’s are most well studied among all their higher dimensional (or even the lower dimensional) siblings. Infinite number of relevant operators unlike higher dimensions. So there can be arbi- • trary number of directions one can flow to get new fixed points (i.e. CFTs). For Euclidean 2D CFT one can use well developed tools of complex analysis (partic- • ularly analyticity properties) which is extremely useful in exploring the theory. There is modular invariance. We don’t know analogous property for higher dimen- • sional CFTs yet. It plays an important role in 2D CFTs. We will get to see its power in next lecture. In 2D for c ă 1 there are unitary minimal models - an infinite family of discrete models • which are unitary and have finite numbers of “primary”. These can be mapped to different statistical mechanical models like Ising, Potts etc. and one can write down all the correlators of these theories without even mentioning their Lagrangians! Hope I could convince you how AdS3 and CFT2 are special and particularly interesting (I’ll talk more about them tomorrow). So by now we all are motivated enough to start learning/discussing about AdS3/CFT2. Before that here is the plan of my lectures. 1.2 Brief outline Lecture I : Why bother? (done!) • Outline of the lectures (going on!) Lightening review of AdS/CFT – 3 – Lecture II : AdS and its thermodynamics • 3 CFT2 and its thermodynamics Lecture III : Cardy formula and BTZ black hole entropy • Lecture IV : Conformal blocks (cont.) • Geodesic Witten diagrams, EE computation Cardy-like formula for three point functions Here is a list of some interesting topics in AdS3/CFT2 which I shall not cover.
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