PoS(Modave2015)001 http://pos.sissa.it/ Chern-Simons , equivalent to the ) R , 2 ( SL × ) R Wess-Zumino-Witten model, while the second set im- , ) 2 ( R , SL 2 ( SL beyond the semi-classical regime. 3 - dimensional gravity with negative cosmological constant is described at the ) 1 + ∗ 2 ( [email protected] Speaker. Einstein action, to a non-chiral poses constraints on the WZWdiscuss currents the issues that of reduce considering further thetheory latter the describing as AdS action an to effective description Liouville of theory. the dual We conformal field classical level by Liouville theory.two Boundary steps: conditions implement the the first asymptotic set reduction reduces in the These are the lectures notes of theematical course , given at 2015, the Eleventh aimed Modave at SummerWe School review PhD in in candidates details Math- the and result junior ofnamics researchers Coussaert-Henneaux-van Driel of in showing theoretical that the physics. asymptotic dy- ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Eleventh Modave Summer School in Mathematical13-18 Physics, September 2015 Modave, Belgium Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Laura Donnay Asymptotic dynamics of three-dimensional gravity Université Libre de Bruxelles and InternationalULB-Campus Solvay Plaine Institutes CP231 B-1050 Brussels, Belgium E-mail: PoS(Modave2015)001 3 4 6 7 9 2 33 24 24 25 26 26 27 29 29 30 32 23 21 20 20 18 17 10 12 13 13 15 17 Laura Donnay ) 2 , 2 ( SO 1 spacetimes 3 dimensions 1 + 2 0 gravity as a Chern-Simons theory for < The Gauss decomposition Hamiltonian reduction to the Liouville theory At classical level At quantum level: stress tensor and central charge Cardy formula and effective central charge A caveat of the CFT spectrum and Liouville theory Combining the sectors to a non-chiral WZW action Reduction of the action to a sum of two chiral WZW actions Improved action principle Adding the Wess-Zumino term Vielbein and spin connection formalism The Chern-Simons action Λ Some comments on Chern-Simons theories Boundary conditions and phase space Asymptotic symmetry algebra The nonlinear sigma model gravity as a D 8.1 8.2 9.1 9.2 10.1 10.2 7.3 7.2 7.1 6.2 3 4.1 4.2 4.3 4.4 5.1 5.2 6.1 . Liouville and the entropy of the BTZ black hole . Other directions and recent advances . Conventions . Liouville field theory . From the WZW model to Liouville theory . From Chern-Simons to Wess-Zumino-Witten . The three-dimensional black hole . . Asymptotically AdS . A brief introduction to Wess-Zumino-Witten models . Gravity in . Introduction 9 10 11 A 8 7 3 4 5 6 2 Contents 1 Asymptotic dynamics of 3D gravity PoS(Modave2015)001 3 , we 4 . After a 8 , we present Laura Donnay 5 and 7 space is governed 3 the action of gravity in 2 was discovered a long time before 2 /CFT 3 contains a brief introduction on Wess-Zumino- 6 2 spaces is generated by two copies of 3 boundary conditions and compute the associated asymptotic symmetry 3 , using the vielbein and spin connection formalism. In Section ) R , 2 ( SL × ) 1 spacetime dimensions admits black hole solutions [3] whose properties resemble very much R , These notes are organized as follows: After reviewing in Section Three-dimensional provides us with an interesting toy model to investigate The AdS/CFT correspondence permits to formulate questions about in AdS 1 spacetime dimensions and the absence of local degrees of freedom, we present in Section + 2 ( the black hole solution hosted in the case of negative cosmological constant. In Section + Witten (WZW) models. Thenon-chiral WZW two model, steps and of then the to reduction a Liouville of action, the are Chern-Simons detailed action in to, Sections first, a 2 show how the Einstein-Hilbert action canSL be written as a Chern-Simons action for the gauge group 3 the holographic correspondence was formulated:be In considered the as work the [8], precursor which, of accordingsymmetry AdS/CFT correspondence, to algebra Brown Witten, of and can Henneaux asymptotically showed AdS thatwith the non-vanishing central charge, namelydimensions. the At algebra that of moment, local the conformaling, appearance of transformations while a in it central two is charge in nowwas a understood shown classical as [9] context was that a surpris- the crucialby asymptotic ingredient the Liouville dynamics in action, of the a Einstein holographic non-trivialcoincides two-dimensional gravity context. with conformal around field the Soon AdS theory one later, whose found centralbetween it in charge these [8]. theories, showing These in notes detailstein all are gravity the aimed in steps at that asymptotically describing brings AdSconformal the one spaces from field connection three-dimensional to existing theory Ein- the living Liouville atgravity field the in theory boundary three-dimensions; action. of the classical three-dimensionalhas Finding computation gravity to the we would admit dual will Liouville solve present theory quantum as shows an that effective the description dual in theory a certain regime. the Brown-Henneaux AdS algebra in the Chern-Simons formalism. Section diverse aspects of gravityWhile [1, 2] its which, dynamics otherwise, isdimensional would substantially Einstein lie gravity simpler still beyond exhibits than our several phenomena theand current that are one are understanding. still present of poorly in understood, higher its such dimensions in four-dimensional as 2 black analog, hole three- thermodynamics. Remarkably,those Einstein of gravity the four-dimensional black holes,Bekenstein-Hawking as for area instance law. the In fact of orderthe having to an theory admit entropy requires obeying theses the the black presencebehave hole of asymptotically solutions a as in negative Anti-de three-dimensions, cosmological Sitter constantsince (AdS) it [4], space. allows making us This the to makes solutions make thisquantum use to model gravity. of Although even AdS/CFT a more correspondence fully [5] powerful satisfactory totivity quantum ask has version fundamental not of yet questions three-dimensional been about accomplished general [6], rela- in this the program context has of already black shown to hole be physics promising, [7]. specially Asymptotic dynamics of 3D gravity 1. Introduction spaces in terms of its muchAs better a understood matter of dual fact, two-dimensional in conformal three field dimensions, theory the (CFT). relation AdS PoS(Modave2015)001 of (2.5) (2.3) (2.4) (2.1) (2.2) µν g is a three- 0 and locally Laura Donnay . M ) , which is there > νρ B g Λ µσ the possibility of the g 10 − 2), with a metric , 1 νσ , g 0 , B µρ the Ricci tensor. = g ( . ν ) ) + spacetime dimensions, one can al- R , µν . 3 Λ ) 2 1 R µ D 2 ( . = νρ − 0 − in g yields the Einstein equations D µν ) has constant curvature; namely = ( νρ R g µσ ( µν R µν g 2.2 g g µν g 1): det µσ g 0 − − g Λ ≡ ≡ , we discuss in Section √ 1 dimensions is that any solution of the vacuum = νσ 9 c − x g + 3 g 3 + D R d ) is defined up to a boundary term µσ µρ − R g µν M ( 2.1 g D Z νρ Λ 2 1 g − G = ) − − π 1 1 1 µν µρ 16 + 2 R 0 is locally Anti-de Sitter (locally de Sitter if R is the curvature scalar, and µνρσ D . Action ( ≡ ( R 2 < ] νσ µν D g g /` [ Λ g is the cosmological constant, which can be positive, negative, or null, 1 + EH µν Λ − S R 1 spacetime dimensions is defined by the three-dimensional Einstein- νσ = R ≡ + ) with dimensions Λ µρ R 1 g 2.2 , + = 2 components of a symmetric tensor +) of them using diffeomorphism invariance (one removes one d.o.f. per coordinate). 2 / , 3 dimensions has no degrees of freedom (d.o.f.) is counting them explicitly: Out ) + D 1 through components of the metric appear in the Lagrangian with no temporal derivative, they = , µνρσ ` + D − 0). This can be verified by realizing that the full curvature tensor in three dimensions is R D ( D = ( the three-dimensional Newton constant, D Λ G In three dimensions, the Weyl tensor vanishes identically. Pure gravity in 2 Extremizing the action with respect to the metric Physically, this means that on three-dimensional Einstein spacetimes there are no local propa- 1 As a consequence, any solution of Einstein equations ( ways remove Moreover, are therefore no true d.o.f. but Lagrange multipliers. This counting leads therefore to of the 2. Gravity in Hilbert action (where we set the speed of light to gating degrees of freedom: there aregravity no in gravitational waves in this theory. Another way to see that latter to account for the microstates of the BTZ black hole. in order to ensure that the action has a well-defined action principle. flat if totally determined by the Ricci tensor Asymptotic dynamics of 3D gravity brief introduction on Liouville theory, in Section with An important property of general relativityEinstein in equations 2 ( dimensional manifold, and yielding respectively locally de Sitter (dS),be Anti-de concerned Sitter with (AdS), Anti-de or Sitter flatAdS spacetime, spacetimes. radius for Here, which we the will cosmological constant is related to the signature PoS(Modave2015)001 is 1- J + (3.3) (3.2) (3.4) (3.1) . This ) with ϕ ∂ 2.2 and Laura Donnay 0, there exist t ∂ < Λ disappears, leading 0; that is, , + r . 2 ) ) = 2 = dt GJ r ± ) r 4 r r ( and angular ( . N ϕ 2 M ≡ − N / 1 ) + r . (  plays therefore the role of a cosmic ϕ where and an inner Cauchy horizon (when ϕ ` − 2 d r ` ) N ( G ± + ` ; this case corresponds to the so-called + r of mass 2 4 r M r − r M r 2 , = / = + J r | ≤ = 2 ( . It solves the Einstein equation ( J J 2 | π + J − dr r 2 2 2 2 1 r 4 G − ≤ 3 q , )) 16 r ϕ ± 2 − ( 2 r + 1 ` ≤ N 2 2  + is indeed a event horizon, (...). G r ` 0. Relation 8 2 + + ( + r + 2 r = GM and 0 dt 4 = r becomes too large), the horizon at 2  . GM | 1. ∞ 2 ` M J 8 )) | ≡ r + /` − = ( G 1 < N ± r − ( r , the mass and angular momentum read r ± ≡ − , both horizons coincide ) is stationary and axially symmetric, with Killing vectors = r ` < ) 0 (or if r = Λ M 3.1 ( 2 < , 0 N = properties, and this allows for interesting geometrical properties such as non- ds ∞ | M J + | , the BTZ possesses an event horizon at ` < M t global . In terms of − < r | ≤ J ∞ | − The BTZ metric ( In the case For a review, see [10]. Notice that many authors set 8 To everyone’s surprise, Bañados, Teitelboim and Zanelli (BTZ) showed in 1992 that 2 0) at 2 3 6= J When with cosmological constant metric exhibits a (removable) singularity at the points bad choice of system of coordinates), indicatingis the indeed presence a of black even hole, horizons. namely To see that that at the BTZ extremal BTZ. If therefore to a naked singularity at Asymptotic dynamics of 3D gravity A priori, this property mayto look study very four-dimensional disappointing: gravity iflocal How there could d.o.f., is this it no theory turns bebecause at out a of all? that realistic two three-dimensional model However, fundamental despite gravitylocally reasons: the is equivalent absence first, to in of a as fact constant wesolution far curvature will spacetime, by from see it being may later, sterile. differ even from though the This every maximally is spacetime symmetric is where the lapse and shift functions are given by trivial causal structures. The second reason is that, unexpectedly, in the case black hole solutions. 3. The three-dimensional black hole dimensional gravity admits a black holefour-dimensional solution Kerr [3] black that hole. shares manydescribed, The physical in BTZ properties Schwarzschild with black type the hole coordinates, by the metric PoS(Modave2015)001 (3.5) . 3 0, both the Laura Donnay = J . 5 . The vacuum state ` and . 2 M ) ϕ 0 corresponds to naked G d 8 | ≤ 2 ( < J r | / 1 M + 0, 2 − < ≥ 0, separated by a gap from the dr ) . = 2 1 G M = − ϕ 8 M ( d J  2 , the metric reduces to / 2 2 , is thus the same, this is why the BTZ r 1 + r ` G 3 1 r 8 − + + 2 also correspond to naked singularities, in −  1 ) dr r  = 2 2 G r ` 8 + ( M , corresponds to AdS Figure 1: Spectrum of theholes BTZ exist black for hole. Black 2 / + 5 1 2 dt − dt  2 2 2 2 < r ` 0. r ` − = M + r 1 = . The solution with vacuum, since it would imply to go through the regions with 1  0 3 = − J spacetime, and therefore the whole black hole can be expressed ) is locally Anti-de Sitter: every point of the black hole has a = = 3 2 M 0 3.1 = ds J , of the BTZ black hole and AdS G 1 8 , see Figure 4 ) − = G 8 2 M ( / spacetime; namely ds 1 3 = 0 ∆ Since, as we saw above, any solution of pure gravity in three dimensions is locally of constant By asymptotic, here simply mean far away. We will give a more precise definition in Section Very far away from the black hole, namely when 0 and admit to be interpreted as particle-like objects [1]. Therefore, one cannot continuously 4 = curvature, the BTZ solution ( The asymptotic behavior is said to be asymptotically AdS. This is in contrast with the Schwarzschild and Kerr black holes this case with angular excesses around neighborhood isometric to AdS as a collection of patches ofonly by AdS global assembled properties in suggests the that rightof the AdS way. black spacetime The hole by fact metric a that can sub-group BTZTeiteilboim be of and differs its obtained Zanelli from isometry by showed AdS group. identifying in points That [11], is where actually these what identifications Henneaux, were Bañados, given explicitly. Therefore, AdS spacetime is separated frommass gap the of continuous spectrum of thesingularities, BTZ with conical black singularity holes at the by origin. a r These solutions exhibit an angular deficit around deform a black hole state tonaked the singularities. AdS The solutions with Asymptotic dynamics of 3D gravity censorship condition. There is, however, a special case: when horizon and the singularity disappear! At thiscovering point, of) the AdS metric exactly coincides with (the universal PoS(Modave2015)001 law, (3.8) (3.9) (3.6) (3.7) area is a Killing − Laura Donnay r = r ) is that it can be rewritten in terms . , 2.1 dJ GM , 8 ) Ω √ 2 − , , with ` + r + r 2 erg = − ¯ ` A hG r BH 2 + 4 2 − π 6 r r = 2 dS ( = r ¯ h + BH 2 + T = BH r S = q BH , meaning that all observers in this region are unavoidably T = dM erg . r ` ∼ + ergosphere r is the angular velocity at the horizon. Notice that the BTZ also exhibits a ) ` + r the horizon size. That is, three-dimensional black holes also obey the ( / + − component of the BTZ is zero at r region is called r π 00 2 erg = g r = gravity as a gauge theory Ω < A D A crucial property of three-dimensional gravity action ( Finally, it is worth mentioning that, besides pure gravity, the BTZ solution appears in many The r 3 horizon. In fact, even thoughrealistic BTZ (3+1)-dimensional has black no holes: curvature itsesses singularity is at similar the final the properties state origin, also of it atthermodynamical gravitational is quantum properties; collapse quite level. it [12], similar radiates and Indeed, to at pos- a remarkably, Hawking the temperature BTZ exhibits non-trivial 4. other frameworks, such as [13], stringthe theories BTZ [14], turned and out higher-spins to [15]. appearthis Moreover, in showing the the relevance near of horizon this limit black of hole higher-dimensional solution solutions in [16]; more all general of set-ups. Hawking-Page transition at with whose full microscopic understanding isthese one thermodynamical of quantities satisfy the the main first questions law in of black quantum hole gravity. thermodynamics Besides, where of ordinary gauge fields, in suchsimplify a way substantially. that both This the fact structure ofWitten was [18], the discovered action and and by holds equations for Achúcarro of any andextended motion sign to Townsend of supergravity [17], the actions and cosmological [17, 19], constant. latter as The by well validity as of higher-spins this [20, 21]. result can be The dragged along by the rotation ofsphere the region black make hole. the The BTZ existencefeature extremely of that similar BTZ an to shares event with horizon the the and four-dimensional Kerr of Kerr solution an is black ergo- that, hole. in both Another spaces, the surface and has a Bekenstein-Hawking entropy Asymptotic dynamics of 3D gravity which are asymptotically flat. In fact,de there Sitter is in no three-dimensions black (for hole pure asymptotically gravity) flat, [4]. nor asymptotically PoS(Modave2015)001 0, with 6= (4.4) (4.2) (4.3) (4.5) (4.1) µ a g O det , provided − b T Laura Donnay √ b a , the following = 1 a − µ a e de Λ det → , which can be thought a 5 ≡ , we will use an auxiliary . Notice that, for a given T e µ ν µν δ g , transforms as in the following way: = vielbein a ν . abc e b ε µ a d e , or Λ in four dimensions. . d , µ a , c , ) O , ab and c ρ x a ω . e ( R b c a λ ρ c tetrad b a b ν b ν e e 1 δ √ e e = ν b or ∧ − µ a e ab = spin connections = b e cb Λ frame field µ a a η µ a µ e b ) e ω + abc e ω x 7 ε b ( µ ∧ a abc 1 + c e µ vierbein a c ε − ; e a a Λ e e d ω de is a non-singular matrix, with c are equivalent (the transformation is local since it affects ≡ ≡ are called a ) = 2), called + ) 1 µ a , or Palatini formulation of general relativity. This consists ≡ , x e 1 ab − ( µ 1 ab a dreibein , µνρ , µνρ Λ ω 2 T ω µν 0 such that ε ε ( d g ) ; namely . This quantity is introduced because it permits to construct a → = x Minkowski spacetime. 6 SO ( b ba a . Since a µ a D ∈ µ ω a e first-order ω e Λ − 2-form of the connection, = with ab ) and the Levi-Civita in frame components ω x ( µ , the vielbein is therefore defined as torsion a b ν e λ ) can be simply seen as the transformation of a tensor under a change of coordi- . In the tetrad formalism, the role of this connection is played by one-forms dx ) , whose components µ a Γ x b e ( , with 4.1 a a + ) is called the first Cartan structure equation. The second Cartan structure equation is µ 1 b ≡ ω (with a frame index ∂ − a dx is the metric of flat 3 4.3 µ a Λ e ab = µ e ab = ω η D a µ 0 A covariant derivative is made out of the normal derivative plus an affine connection; schemat- We can use the vielbein to define a basis in the space of differential forms. We define the In three dimensions, it receives the name Its existence comes from the fact that the metric tensor can be diagonalized by an orthogonal matrix Relation ( This result is based on the = e 5 6 ab Equation ( the quantity does transform as a vector under local Lorentz transformation, namely ω quantity that transforms as a local Lorentz vector. Indeed, unlike the 2-form ically quantity, called the one-form only the frame indices, while the spacetime indices do not see such transformation). positive eingeinvalue metric, the frame field is nottion unique; indeed, all frame fields related by a local Lorentz transforma- given by where there is an inverse frame field in the following: Instead of working as we usually do with the metric nates described by the matrix Asymptotic dynamics of 3D gravity 4.1 Vielbein and spin connection formalism quantity of as the square root of the metric PoS(Modave2015)001 (4.6) (4.9) (4.7) (4.8) (4.13) (4.12) (4.11) (4.10) Laura Donnay . .   c c e ρ e ∧ ∧ c dx b b b µ e ∧ e ω ν ∧ ∧ ac x ν . . a a 3 c dx , e ω , e ω d c ω A ∧ 3 Λ R − plays the role of a gauge field, notice abc ∧ µ , c ∧ ε abc µνρ b ν − ν ω ω abc ε 3 A ε ] Λ dx ε ω dx + ω + νρ ac − [ µ . ∧ αβ ] µνρ ω ≡ − . bc ω ε µ ≡ − R d dA ω gR R [ e ; ab µν + a c dx ab = − = ∧ ab 1 e ω ) 3! notation (valid only in three dimensions), R µαβ ab R µ a R x R 8 √ ε F ∧ e ( ∧ x ω σ b ) can finally be rewritten as = e ↔ b 3 e ↔ a ν :  e 2 µν 2 1 ab e d ∂ λ a A dual bc 4.9 e R bc 2 ∧ abc dx − = = , 1 2 ω , a R ε 1  = e ab bc ν ω A = abc dx abc M M R ω ε 0 µν abc ε Z + + Z µ ε ab λσ 1 2 2 1 ∧ ∂ ∂ ∂ R R G a G 1 3! edx e ≡ ≡ π 1 π 1 = = a a = = ) = 16 abc 16 D D ω R g ε ω ( − ) reads (we will take care of the boundary term later) ] = ] = . To see that the spin connection µν ab √ ) ω ω x R 2.1 , b ν , 3 e e e [ d [ b a EH EH µ S S ), we notice that ω + 4.2 ), we have a ν e 4.7 µ ∂ ( ρ a e We can of course relate the spin connection with the usual Christofell symbols; the relation is Let us now go back to our three-dimensional Einstein-Hilbert action. In terms of the quantities = µν ρ the curvature tensor, and With this, we notice that the gravity action ( Indeed, using ( In what follows, we will adopt the so-called Asymptotic dynamics of 3D gravity with and, using ( Γ we have defined above, ( the analogy with the Yang-Mills connection PoS(Modave2015)001 (4.18) (4.14) (4.15) (4.16) (4.17) Laura Donnay . existence of , then one has, G . M ) . · ] , · A ( δ ∧ A [ , Tr  M A ∂ . ∧ Z i b A π k 4 ∧ dA is given by A − , ∧ represents a Lie algebra-valued one-form is such that the second term vanishes, we 3 2 0 G a )] A A A + = , h ) takes the form M ∧ ) A ∂ b dA A dG ∧ T 1 invariant bilinear form on the Lie algebra (of the ∧ a 4.14 9 + A − 7 T A ( G +  dA Tr ( = of the Lie algebra of the gauge group Tr dA ∧ 8 A ) to hold everywhere on the manifold ] = M A ≡ mentioned in the previous subsection. Z δ a dA F 2 π T k 4.17 [ 4 ∧ a basis Tr A a [ plays the role of a metric on the Lie algebra, and therefore should , while the gauge field T ] = . M ) Tr A Z b [ for a compact gauge group T M π level a k CS 4 T S , with ( ), a T Tr ] = a is a gauge transformation of the trivial field configuration; in other words, A A ≡ , see for instance [22]. [ 4.14 A D = ab CS S d A . Therefore, a Chern-Simons theory has no true propagating degrees of freedom: δ ). G , and Tr represents a non-degenerate µ is the usual field strength 2-form. These equations imply that, locally, is a constant called dx Chern-Simons action F k µ is chosen such that its value on the boundary pure gauge This is asked in order that all gauge fields haveThis a has kinetic nothing term to in do the with action; the this torsion is always true for semisimple Lie If we write Now that we have at hand the gravity action in terms of the vielbein and the spin connection, A Integrating by parts, the variation of action ( A A 7 8 δ is = algebras. boundaries for the manifold for the first term of ( We then see that be non-degenerate. The existencewhether of the a gauge Chern-Simons action, groupNotice and one that the one wants form can to make it use consider will of admits take, this bilinear such relies form on an to define invariant an non-degenerate inner form. product where A obtain we are ready to prove, astheory announced above, with that a three-dimensional gravity specific isthis kind equivalent to very of a interesting gauge interaction, model calledChern-Simons for in Chern-Simons the 3 theory. readers who We are will not first familiar introduce with it. For an introduction to Asymptotic dynamics of 3D gravity 4.2 The Chern-Simons action If gauge group where which means that it is purely topological.topologies, Indeed, which all prevent the relation physical ( content of the theory is contained in non-trivial A PoS(Modave2015)001 ) 9 1 , 2 ( (4.21) (4.22) (4.20) (4.23) (4.24) (4.19) ISO ) one can 0, Laura Donnay 4.21 < Λ , for . c ) J ,  2 a c , abc ω 2 ω ε d ( . The first term is ) ∧ ∧ 2 b . SO ] = a , ) b e ω 2 b P ( ` 2 ) P , ∧ , b a a , a J = P SO c b e P [ J are the generators of the translations. ω ab = abc d a = ( η abc ε ) G . P ε 3 2 0  + a , b , J b + 2 a c µ P ( c de b ≡ − ) = P e ω b ∧ J SO de ∧ ab + a ] abc , ` J 1 b ε a a A ω e , J P 10 0, since we are interested in Anti-de Sitter spaces. gauge theory relation that we are about to show. a , whose commutation relations are given by ( ∧ µ a ↔ ∧ ) + J ] ] = e a a 2 < b b A ` , 1 e ↔ bc , P ω ω 2 Λ J 2 , 1 ( A ` d a ≡ , + [[ J abc so a [ ∧  µ ), we have used the three-dimensional rewriting ε ab P a Tr living on this Lie algebra as ` A a 1 2 1 e η ) and with the non-degenerate invariant form ( 1 3 3 ) for the gauge group e A  ` 1 4.19 ≡ ` = 1 , a ) = ] = 4.22 2 are raised and lowered with the three-dimensional Minkowski 4.14 b J c , A = 0. P J 1 ] = ( , . In ( , ∧ a > 0 abc ab J A dA ε ( Λ η = ∧ ∧ c A ] = A for [ , [ b b ) J Tr , Tr , 1 a a , 3 2 J 3 [ ( are the usual Lorentz generators, while the SO ab and its inverse gravity as a Chern-Simons theory for J 0 ab 0, or η < = This is not the only one, though; see the comments at the end of this section. We will prove this result for the case What Achúcarro, Townsend and Witten discovered [17, 18] is that three-dimensional gravity Λ 9 Λ This Lie algebra admits the following non-degenerate invariant (symmetric and real) bilinear form One then constructs the gauge field Notice that the Lie algebraconnection; indices this are is identified with crucialEquipped the for with frame the the indices gauge gravity of field the ( vielbein and spin where the write the Chern-Simons action ( while the second term is found to be Asymptotic dynamics of 3D gravity 4.3 In this case, the Lie algebra involved is metric action and equations of motions aregroup. equivalent More to precisely, a their Chern-Simons result theory statesa for that three-dimensional an pure Chern-Simons appropriate gravity theory gauge (Einstein-Hilbert based action) on is the equivalent to gauge group for where the indices PoS(Modave2015)001 . , 2 as 11 /` Γ 1 (4.28) (4.26) (4.27) (4.25) , alge- 12 − ) a P = . Boundary ± Λ Laura Donnay a J ( 2 1 connection . ) ) with ≡ . The breakdown of 0 2 , , ± a in the first and second 2 J  ( 4.13 ] = ¯ c A e − b so , is equal to J ∧ A , ) , b a 2 + a e , T J [ ) 2 ∧ a ( a ω e SO − , abc . We have thus shown that the Chern- ε c /` c 2 a − ` asymptotic symmetry 1 ω e J 3 ∧ abc b = ( ε ] + ¯ . ω A ω [ G ` is semi-simple). Defining ] = a abc 4 11 ε ) − R b , namely depends on the choice of relative orientation of the 2 1 2 J e = , ∧ , , . One can show that the decomposition of the action ± 2 ) k a + − a a ( e J a . Introducing boundary conditions is not the only way to generate T R = [ 2 ) , so 10 ω g a 2  d ( − ω √ sl global = M + , Z a c ` R /` + k a π J e 4 (recall that abc exactly matches the Einstein-Hilbert action ( ) = ( A ε ] = ) R A 2 that, under an appropriate choice of boundary conditions, there is in fact ω , infinite number of degrees of freedom living on the boundary , , 2 ] = 5 2 e ( respectively: [ ( + b ) sl J , CS R SO ⊕ S + , a ) J 2 depends on the identity [ ( R k , sl 2 ( sl ) reads ≈ now being the generators of ) a 2 T Before concluding this section, let us mention a very useful fact, namely the isomorphism The sign of The fact that Einstein gravity is merely a Chern-Simons action reminds us that there is no prop- In other words, the relevant d.o.f. are which we previously called 4.19 , 2 10 11 12 ( bra ( with so conditions are necessary in order tobut ensure the that choice the of action hasare such a extremely conditions well-defined sensitive is variational to principle, not thesymmetry unique. choice on of the In boundary boundary fact, is conditions.gauge the called global In invariance dynamical symmetry at this properties or the context, of the boundaryfreedom. the residual has theory gauge the effect of generating this infinite amount of degrees of chiral copy of agating degree of freedom inabove the that this theory, gauge and theory thus is notions, purely its graviton topological. dynamical in However, content even three-dimensions, though is since there far are from we no being saw local trivial excita- due tonot the existence one, of not boundary two, conditions but an Simons action for We will see in Section the sum of two Chern-Simons actions, each having their connections coordinate basis and the frame basis. global d.o.f., one can also consideronly holonomies a (we finite will number not of study degrees them of here). freedom Notice in however the that theory. holonomies generate Asymptotic dynamics of 3D gravity Therefore, we find that the Chern-Simons action for the group provided that the level acquires the value Thanks to this splitting, one can rewrite the Chern-Simons action for the where we have remembered that PoS(Modave2015)001 ). In (4.29) (4.30) (4.31) (4.32) (4.33) 4.14 Laura Donnay , ab η . 0 action [18], which corresponds ) = , , ] b = ¯ 0 . One can use this new form to P A ) , , b a = e A R P [ b , exotic ( ∧ . e 2 ] b CS ( ¯ A a ∧ S [ sl a ω e ≡ CS ⊕ , 2 ] + S 1 ) ¯ ` A 0 a [ R + ] + , de CS 2 0. More precisely, varying the action with respect A ) = S ab 12 [ ( = b R = sl P − CS a , a ] S a T ¯ ≈ A ⇔ F J [ ( ) 0 ] = ⇔ 0, 2 CS , Γ S = 0 [ 2 = ( E a a = S ¯ , F ] = so F a Γ leads to the torsion free equation ¯ ab [ + F a η a CS − ω F S a ) = F admits a second one, given by b J ) , 2 a , J 2 ( ( so ), 4.21 gives the constant curvature equation Finally, let us mention that Einstein’s are equivalent to the ones in the Before concluding this section, let us make some remarks on Chern-SimonsLet theories us begin that by can noticing that the Chern-Simons description of gravity is valid when the vielbein A second comment regards the definition of the invariant bilinear form appearing in ( a e construct an alternative Chern-Simons action: the so-called Chern-Simons formalism, namely be relevant for their gravity application. is invertible, which is true forof classical view, this solutions is of not gravity. entirely natural. However, fromof This the the is two gauge relevant because, theories, theory despite it point the is identitywhere, not between obvious the besides that actions gravity the and action, Chern-Simons arefunctional one equivalent sum. at has quantum to level Perturbatively, provide closetheory a and to set three-dimensional classical of gravity saddle configurations maystill points, over holds remain which the non-perturbatively valid; [23]. to relation however, Moreover, perform it between toequivalent, the is we claim the have that not to gauge Chern-Simons clear prove and that that gravity theto the theories gauge a transformations are relation and local the Lorentz diffeomorphisms doequations transformation). match of (up motion It are is satisfied, namely shown on-shell. in [18] that this matching occurs only when the that is, can be rewritten as the difference of a chiral and anti-chiral Chern-Simons action. while varying with respect to We thus verify that solving the equationssimpler of than motion solving in Einstein’s the equations. Chern-Simons formalism is considerably 4.4 Some comments on Chern-Simons theories addition to ( consequence of the isomorphism to This action is relevant in the construction of the so-called Topologically [24]. Asymptotic dynamics of 3D gravity then reads to PoS(Modave2015)001 , iS e ) = (5.1) , and F (4.34) , where ∧ M 2 ). There- 1) F , D ( 0 (so that × in a specific Tr Laura Donnay 4.14 = π R R 3 i [23]. Z . We introduce light-cone . π of the spacetime at spatial ∈ as an integral over 2 2 k asymptotically Anti-de Sitter P + S M . , 2 ∂ ϕ . Before that, we need to specify ). P M 4 ∼ , j is the two-dimensional spatial mani- M , see Fig. ϕ 4.14 2 Z corresponds to a topological invariant; dx ϕ i D π , P k t 4 S whose boundary is the three-dimensional dx ) 4 and k is called the Pontryagin form, and is a total x l ) = , M / b r F t ( F i j ∧ is the Chern-Simons Lagrangian of ( ≡ . The boundary γ 13 ∧ ) F τ a ( ϕ + CS boundary conditions . Topological invariant actions exist only in even di- F ∂ 2 L ≡ Tr , with periodicity 13 ab 0 ± 4 dr x d is taken to have the topology of a cylinder ϕ 2 2 τ M r ` ∂ Z ≡ ≡ ( M 2 1 1 = π P k , where x 4 2 , is the genus of the closed manifold = takes discrete values. In fact, one can show that CS r ). g ds = P ± dL P S ∂ S 4.14 = spacetimes 3 P , where being the curvature associated to the four-dimensional gauge field that g with 2 F ϕ − , 2 b ± , one can, after using Stokes’ theorem, rewrite = F τ . This, together with asking that the action is defined modulo 2 x 2 ∧ Z M ≡ a appearing in ( ) is thus a timelike cylinder of coordinates ∈ = ± F gRd ∞ x A 4 n − ab defined on a four-dimensional manifold = where the Chern-Simons theory is defined. Here, we think of a gauge field in four di- d √ M 2 r P ∂ ≡ M S R M P , with π A similar example is the Einstein-Hilbert action in two dimensions, which coincides with the Euler characteristic We adopt Fefferman-Graham coordinate system where the metric is given by ( In this section, we will make more precise what we mean by Finally, let us mention an interesting feature of the Chern-Simons , the level Our three-dimensional manifold 1 2 n 13 2 is parametrized by the time coordinate = π . Generally, a Chern-Simons theory admits to be constructed by starting from a gauge invariant Ξ derivative; more precisely infinity ( Being a topological invariant, 4 which appears in the path integral,be is well defined, single-valued), leads the Chern-Simons to level the has conclusion to that, be for quantized, the namely theory to 5. Asymptotically AdS spacetimes. We will consider a set of metrics which tend to the metric of AdS 5.1 Boundary conditions and phase space k action Asymptotic dynamics of 3D gravity its Lagrangian density is amension, total and derivative that isfour-dimensional the action is reason given why by Chern-Simons actions exist only in odd dimensions. The with fore, if one is left with the three-dimensional Chern-Simons action ( way. Giving such information is actuallycomponents equivalent to at prescribing large fall-off distances, conditions the on so-called what the is metric the boundary of our spacetime. extends the space mensions that is an extensionextension of of the the three-dimensional three-dimensional gauge field gauge is,tion field generically, about non-unique, of topology. and Chern-Simons this In carries theory. fact, informa- the The four-dimensional action R fold parametrized by coordinates coordinates PoS(Modave2015)001 ) G 4 (5.5) (5.2) (5.4) (5.3) ( / for our ) . , ¯ L 0 in global A dr 3 r ` + = M ), where the L 2 Laura Donnay = ω 2 5.1 = ( e , respectively. In . We call asymp- , + M ) x 1  , ( + , − O and − dx ; see Appendix dx ) ) − ab dx ) + + − x ) k x η x x ( − ( ( ¯ L x ) L ( 0 2 r ( i j r ` L g 2 ` r ) is 2 − 2 2 r √ ` − √ 5.2 = + . − − + i j ), ( rdx γ 0 corresponds to the massless BTZ, while + dx dx , 5.1 + ` = dx  ∞ 2 r ¯ 2 − L dx r √ Figure 2: We consider the manifold having the topology of aIts solid boundary cylinder. is taken tocylinder be at the spatial timelike infinity. √ → − dx = ) r 14 = L = = − 1 are Dirichlet boundary conditions with a flat bound- j x 1 ( 4, e ω / . L dx i ) 1 2 with an off-diagonal metric r ` G − b dx 4 ) e − ( , 0 , a = i j ( / e + g ) + ¯ + L ¯ ab L dx dx η = rdx ) − ) , as desired. Then, the torsion free first Cartan structure equation  L + correspond to generic BTZ geometries of mass + = 2 L x x ) ( 2 ¯ ( − L ( 2 ` , ¯ ¯ e 2 L L with boundary conditions ( ds L r = r 2 dr 2 are two single-valued arbitrary functions of 2 2 ` J + ( ` 2 2 /` ) √ 1 r ` √ 1 e + is fixed as 0 x + + − = ) e ( 0 ¯ − 2 2 i j ( − L = g ds dx dx = Λ spaces, in the sense of Brown-Henneaux [8], metrics of the form ( ) on the cylinder located at spacial infinity. ` 2 2 and r 3 2 r √ ) ds 5.2 which satisfies √ − − a x e = ( = L 0 Brown-Henneaux boundary conditions 0 e ω It was shown in [25] that the most general solution (up to trivial diffeomorphisms) to Einstein’s Let us now translate these boundary conditions in the Chern-Simons formalism. We choose a ) determines uniquely the associated spin connections 4.3 ary metric ( These conventions. One can check that reproduces ( equations with with the expansion, closetotically to AdS the the boundary dreibein Asymptotic dynamics of 3D gravity boundary metric where and angular momentum this gauge, one recovers well known geometries when these functions are constant; AdS coordinates is recovered when generic positive values of PoS(Modave2015)001 − a (5.9) (5.6) (5.7) (5.8) ω (5.11) (5.10) = ( . We will ¯ 1 A − , Laura Donnay . b a 0 j j ) = 2 ` . ¯ /` b . √ 15 a ! e    + will be used to further 2 + /` − ) j r a 0 − ii . 0 ( 2 dr dx are chiral. r ω , , r ` dx ) ¯ + 1 b + 1 − d x − − j = ( ( ] 1 ) ¯ L -dependance of the gauge fields ¯ dx − λ dx , A − r r ) ) ¯ , ) b bdb x ) at large ¯ 2 dr 0 − a − − ( x + x + x ¯ ( L − 5.2 ( ( ¯ 1 b ` L + [ -independent; L L . ¯ r 2 − A ¯ ` λ r ` ), we define the off-shell reduced gauge 1 ! √ d ¯ Ab −

2 are    ¯ 5.7 / b = b = 0 ¯ 1 a = ¯ r = , a − connections = = ¯ ¯ a a a ¯ δ 2 ¯ A ¯ a a / 14 1 0 , will reduce the Chern-Simons action to a sum of − 0 15 j r ) , i ) , , ( ,

, + ] as x ! λ (    µ db , db + L ) = 1 1 a + ` r dx − − ( 2 dx a b b ) j b dx 0) is ensured by the fact that + [ r √ ) 0 + µ a + 2 dr + λ = ¯ x + a + ( x ¯ d F − ¯ ( 2 L j Ab = Ab ¯ L = ` = 1 1 a 0 r ` our generators) − − a F b a b /` , + δ j + 1 0 + + r and ¯ = j = ¯ a 2 dr dx a µ dx 2 a r ` ` = √

dx . We impose our boundary conditions in the following way; they come in two 0    a r j are flat (i.e. ¯ = = a = = µ a ¯ A a + a , = − A a A Wess-Zumino-Witten (WZW) actions. The remaining set a 0 = ) ) ) a = R ii i , ( ( r 2 a ( are therefore (with a SL j The phase space is clearly contained in these boundary conditions, with The asymptotic symmetries correspond to the set of gauge transformations The fact that The analysis of asymptotic boundary conditions can also be performed from the geometrical point of view, in terms The corresponding chiral Chern-Simons flat A very useful trick is to notice that one can factorize out the In analogy with the on-shell reduced connections ( ) 14 15 /` a see that the first set of boundary conditions chiral reduce the WZW model to Liouville theory. 5.2 Asymptotic symmetry algebra connections such that sets: e of the metric (or thestudies vielbein the and asymptotic Killing the vectors spin that connection). preserve the In form that of case, the instead metric of ( studying the gauge transformations one Asymptotic dynamics of 3D gravity with Indeed, one can check that the reduced connections by performing the following gauge transformation: PoS(Modave2015)001 ) 2, Z √ ∈ / (5.17) (5.16) (5.13) (5.12) (5.14) (5.15) 1 m λ ` ≡ Laura Donnay ). Writing the Y . ϕ 5.10 d ) − a , . Writing δ , ϕ 17 . , , ¯ λ 2 2 ( ϕ j j ¯ Ld π 1 0 2 ϕ ). Defining the modes ( , , ¯ ¯ 0 Ld λ λ 0 0 im , , Z ¯ + − Y n n e ∂ ∂ π π + + π k 5.13 2 . 2 2 2 , 2 m m 0 ` ` 0 ¯ δ δ Y Y √ √ Z Z − respectively 3 3 3 3 , namely equations ( + − a π ∂ ∂ m m π k − + − k , ¯ 2 2 2 2 1 1 x 1 1 ] = ¯ c c a j j , 12 12 ¯ − λ 1 − − ≡ [ +  ¯ x + + λ ; therefore, the algebra of the canonical gen- ¯ = 0 Q Y Y m } n n ¯ ¯ ¯ − + L δ λ 1 + Y + + Y ∂ ∂ ¯ 2 − 0 transform in the same way as a two-dimensional Q m m that the latter have to be of the form j ¯ Q ∂ L L ¯ 16 L L ¯ , L 2 2 2 1 ) ) 2 16  , Y n n 1 , ; + + ϕ − and identify the components of left and right hand sides along the Q λ ¯ − − L L and 0 ] ϕ ϕ { 2 + − + λ ¯ , d λ m m L ∂ , ∂ ∂ 0 = ¯ ) L Y Ld Ld ) become linear in the deviation of the fields, so that they 2 1 + ¯ we find 2 Y Y + a π  ϕ = = ( = ( Y . a 2 a depend only on − j 2 0 = = + im Q + [ } } } δ ` ¯ 1 0 a 5.14 a Z 1 e n n n , ¯ L L λ Y ¯ ¯ λ λ ¯ ¯ λ π L L L + = λ δ π + δ δ 2 k , , , , L ( 0 ∂ 2 0 0 1 j = m m m π  Z ¯ 2 0 = λ L L L = − 2 ¯ 0 λ ¯ { { { π ` λ + k − i i i Z , 2 = a a a j = = δ π k Y a 2 ≡ ¯ λ λ Q λ m − L = ] = λ λ [ Q δ 2, we find . a √ j / 0 ¯ λ We compute for instance This follows directly from The variation of the canonical generators associated to the asymptotic symmetries spanned by The Poisson brackets fulfill ` 16 17 take a very simple form in the Chern-Simons formalism [26, 27]; one has ≡ ¯ one finds generators can be directly integrated as gauge parameters At this stage, we can already notice that erators can be directly computed from the transformation laws ( λ CFT energy-momentum tensor doesone under can generic already infinitesimal see conformal thatence the transformations, of last a and term, central associated extension. to the Schwarzian derivative, indicates the pres- Asymptotic dynamics of 3D gravity that preserve the asymptotic behavior of the connections where the arbitrary functions One then find that expressions in ( Y PoS(Modave2015)001 ) c n (5.18) ,..., 1 , with = change the i ( ¯ c -dimensional Laura Donnay 24 i φ + m ¯ L ) is often called the infinite → 5.18 m . From the CFT perspec- Virasoro algebra ¯ L 0 , , n c + 24 (recall that the m δ + ) to a new kind of symmetries, ) G m 1 L −  5.17 2 ` → m describes scalar fields m ( ) tends to infinity. Also, notice that the L m and, hence, discrete values of the central . 18 ¯ c ` 12 G G 3 2 5.18 ) is the algebra of local conformal transforma- `/ = and k 0 5.17 , 17 6 n + = ) was first shown in the seminal paper of Brown and m ¯ c δ ) implies that the quantum theory seems to be well defined 1 = 5.17 k c − 2 m ( is also understood from the dual CFT point of view, since the nonlinear sigma model m c c ), the central charge ( 12 G ∼ P . The Virasoro algebra ( ` ) computation that describes a property of the effective action of the quan- 3 ) theory. Algebra ( 2 ). Notice that the standard redefinitions -theorem prohibits the central charge to be continuous [28]. c Z ∈ m asymptotic form consists of a direct sum of two copies of the central charge 3 In this section, we will present an important ingredient in our discussion, the Wess-Zumino- In quantum field theory, a The name comes from the fact that this model appeared for the first time in the description of a spinless meson This shows that the charge algebra associated to the symmetry transformations that preserve Recently, it was shown that the asymptotic symmetries of three-dimensional gravity with It is worth mentioning that in the semi- 18 being the Witten model, which appears as an intermediateLiouville step actions. in the connection between Chern-Simons and 6.1 The nonlinear sigma model tive, the existence of a non-trivialin central the extension is quantum the theory. resulttension of However, a at since conformal the this (or classical derivation Weyl) level anomaly was isa purely quite classical classical, remarkable. bulk finding (AdS In a thetum central context boundary of ex- (CFT AdS/CFT, this is interpreted as quantization of the Chern-Simons level only for discrete values of the dimensionless ratio the so-called symplectic symmetries [30].mations These that symplectic are symmetries defined are everywhere largesuggests gauge in that transfor- the spacetime, dual not two-dimensional only CFTthe in surface is charges not an and necessarily asymptotic algebra only region. are definedof defined symplectic at This on symmetries the is any result boundary, related circle since to located theand in existence is of the most a AdS presympletic likely bulk. form conditioned which The by vanishes appearance the on-shell absence of propagating degrees of6. freedom in A the brief bulk. introduction to Wess-Zumino-Witten models charge. This discretizationZamolodchikov of the AdS Brown-Henneaux boundary conditions can be definedpromoting everywhere in into the this bulk sense of spacetime the [29], two copies of Virasoro algebra ( Henneaux [8] in 1986, andAdS/CFT this correspondence; is notice the that reason for why this this reason result the is central considered charge as ( the precursor of the in three dimensions is tions in two-dimensions; it is the central extension of the Witt algebra and is Asymptotic dynamics of 3D gravity with central elements given by (recall that central terms in the algebra to Brown-Henneaux central charge. PoS(Modave2015)001 , τ = (6.4) (6.3) (6.2) (6.5) (6.1) 0 -model x σ , in which ) x Laura Donnay ( -function (the φ β and the fields are 0. G ) = 1 with coordinates − -dimensional Riemannian Σ n gg ( δ . , i  . We thus see that the equations ) φ g 1 ν − − are the scalar fields ∂ ∂ i g 1 n ( φ − ν µ g M 20 ∂ ∂ g , , ≡ ] , µ µν 0 0 ∂ − G η [ J ) = µν Γ , φ k ) = g + ( involves a particular two-dimensional η J g  + i j , ν − ∂ 18 19 ] + ∂ 1 ∂ Tr ϕ xg g 1 [ − x , which can be derived from D which depends on the fields, this is why the model is − + ± 2 g σ 1 d g ) d S − τ − ( ≡ J φ Σ ν Z ( gg = ≡ Z + + ∂ 2 δ i j ∂ 2 S J 1 ± a 1 g a 1 − x 4 ) implies that the other current has to be conserved as well. . For a two-dimensional manifold 4 g ) x − 6.4 ( ] = g ] = φ ) = g [ 1 [ σ ) do not lead to the independent conservation of the left and right − σ g S S ( 6.2 δ , noted G are the Cartesian coordinates of a flat spacetime. An action for this model is 1), the action of this nonlinear sigma model takes the form [31] D , 0 acquires a scale dependance, leading therefore to a non-vanishing equipped with a metric ,... a has to be semi-simple to ensure the existence of the trace Tr, but can be either compact = 1 n . If one is conserved, ( ν G 0 a dimensionless coupling constant. = , ± M J µ > µ ( . , 2 σ ϕ a µ This theory is conformally invariant only at the classical level. Indeed, under quantization, In order to have two independently conserved currents, it has been observed in [32, 33] that It is important not to mistake the Wess-Zumino-Witten modelNotice with the the useful Wess-Zumino relation model that describes four- The so-called Wess-Zumino-Witten model x 19 20 = 1 where we have defined the currentsof as motion derived from ( quantum theory is in fact asymptotically free).not Furthermore, totally even at satisfactory the since classical it level, this doesholomorphic) theory not is and possess a two conserved right currents (antiholomorphic) thatfactorization part; factorize of this into a is a CFT. the left Indeed, fundamental (or the property equations of of motion holomorphic are which read in light-cone coordinates the coupling x The group or non-compact. called one has to consider, instead, the more involved action dimensional supersymmetric interactions. The former is usually referred to as the Wess-Zumino-Novikov-Witten model. currents 6.2 Adding the Wess-Zumino term with in which the rolematrix of fields the living target on space is played by a semi-simple Lie group given by Asymptotic dynamics of 3D gravity as maps from amanifold flat spacetime to aintrinsically nonlinear. target In manifold. other words,the The the coordinates latter on is a PoS(Modave2015)001 . k = − 0, is (6.8) (6.9) (6.7) (6.6) J (6.11) (6.10) + . ∂ k ) = , leading g Σ − ∂ Laura Donnay 1 − is the extension g 0. For the same ( G + = ∂ extending + J ) read V Wess-Zumino-Novikov- − , and . .  ] ∂ 0 Σ 6.5 G G [ . = ρ .  Γ ∂ ) = 0 1 ), namely k 1 V g − − ∂ + + ∂ gg  ) = 6.10 1 ν (WZW), or g g GG − ∂ ν ν ν g 1 ∂ ∂ ∂ ( 1 1 − 1 − − − being − ∂ , g gg ] ) ) ( µ gg k ), unlike the chiral action we will present G µ . In fact, for the quantum theory to be well- − ∂ 2 µ [ GG − x ∂ 1 Γ a ∂ x µ Γ ( 1 − 2 ∂ as a boundary, µν − − 1 , θ ε + ↔ gg g Σ − )  k 0, one finds the conservation of the current 1 δ 19 3 WZW action, since it does not distinguish between + + G µν ) for our conventions), they become − x x  < µν ( η ) ε  A k yields a two-dimensional functional. Actually, one can g + dG Tr ) + (  1 ν θ Wess-Zumino-Witten g g Tr ∂ − − Tr δ 1 x µνρ = G ∂ ). However, the Wess-Zumino term has the fundamental x 2 − ε ( 1 + 2 g d g  x − d ( g non-chiral 3 6.6 g µ Σ Z Tr d ( ∂ Z → k 2 we are interested in, this issue does not appear. + V V Z Z ∂ g ) µν ) ] = 3 3 1 1 η R k ] = , (see Appendix 2 2 G , which implies g [ 2 a [ ≡ ) a 1 ± ( Γ 2 k ] = 2 , one obtains the x , one finds the conservation of the dual current 2 ) δ G ) SL − ( k [ k WZW / 2 1 Γ . Notice that there are of course several choices for a 2 S ( 1 ( ( / V − / 1 , and with the Wess-Zumino term 1 = − on 21 ) may look surprising since it mixes a nonlinear sigma model in two dimensions 2 = g = a 2 2 6.5 a a is a three-dimensional manifold having (WZNW) action; namely an integer V k Taking Action ( The name is of course not innocent, since we will see that it is indeed related to the Chern-Simons level This action is sometimes called 21 Witten conditions, one can show thatfixed the point. beta-function vanishes [32], representing a conformal invariant therefore to a potential ambiguity indefined, the and definition depending of on the LieHowever, in group the considered, case this of can imply a quantization condition for of the element In light-cone coordinates Therefore, for 0, while for simply Using this result, one can see that the equations of motion derived from ( with the three-dimensional actionproperty ( that its variation under show that its variation is a total derivative, leading to the result (using Stokes’ theorem) Asymptotic dynamics of 3D gravity with where left and right movers (it is symmetric under later on. The solution of the equations of motion derived from ( PoS(Modave2015)001 ) → . In 7.2 g ¯ (7.4) (7.1) (7.2) (7.3) A , (6.12) A .  ] r ) r A , A ) implements Laura Donnay ) has the two ) do not lead τ ϕ ∂ invariance. A [ ) 5.10 − τ 6.10 5.10 − τ A x 2 A ( r G ∂ + ( ) is invariant under × ϕ ) A + , 6.10 x (  ) + A G ϕ . A ∧ ) means that left and right movers 1 r − ∂ A . Therefore, one sees that the global  ] ∧ , − G 6.11 gg ρ ] r A ¯ + A A A [ , 2 3 ∂ ϕ ν . reduces under our boundary conditions to CS ∂ A + 2 ( S [ G τ ≡ − µ π ` /` − A dA A 1 ] + ¯ 16 J 1 3 ∧ A − [ ) + A 20 = + τ =  CS ρ A S ) to a sum of two chiral Wess-Zumino-Witten actions. π k Λ ϕ A Tr 4 , ∂ ν 7.1 g ∂ ] = M ≡ − ¯ µ − A Z , ∂ ϕ κ A 1 π  A k A [ 4 − τ Tr E g ∂ − S ( ρ r ≡ A dx −  ] = : J ∧ A ϕ [ Tr two arbitrary matrices valued in ν , are arbitrary functions. Equation ( ϕ τ CS ± dx , d S ) r Θ τ − ∧ x µ ( , the constant drd − ) dx θ -dimensional gravity with , with G M ) ) M 4 Z 1 − ( Z and κ x κ + ( `/ ) WZW model on the cylinder at spatial infinity. From now on, we will use, instead of − 1 2 − + ) − − ( = x = R ( Θ k , g ] = + 2 ) invariance of the sigma model has been promoted to a local θ A ( [ + At this stage, we have a small issue to solve: our boundary conditions ( Notice that we have changed the overall sign for later convenience. Chern-Simons theories have been shown to reduce to Wess-Zumino-Witten theories on the Again, let us emphasize the advantage of the Chern-Simons formulation: Instead of working 22 G x SL CS ( 22 S × + the the level The independent conservation of the two currents implies that action ( with describing this section, we want to showthe explicitly reduction how of the the first Chern-Simons set action of ( boundary conditions ( 7.1 Improved action principle to a well-defined action principle.explicitly in To coordinates see that, let us first rewrite our Chern-Simons action ( boundary [34,35]. In particular, we will see explicitly in this section that the Chern-Simons theory with a second order action in terms of the metric, we work with two flat gauge connections conserved currents Θ G 7. From Chern-Simons to Wess-Zumino-Witten Asymptotic dynamics of 3D gravity where do not interfere between each others. One checks that the model described by ( PoS(Modave2015)001 = of τ ϕ r (7.6) (7.9) (7.7) (7.8) (7.5) A F (7.10) ,  ). ϕ ϕ A ¯ A τ Laura Donnay A =  τ ¯ Tr A , ϕ  ϕ d 2 ϕ A ¯ τ  A d , , ]) = +  r  r τ M ϕ A 2 ϕ ϕ ∂ , ¯ A A A F Z ϕ  τ δ A κ τ A ¯ Tr , A 2 +  + [ ϕ , − + 2 ϕ d ϕ  r τ ϕ A A ˙ 2 ϕ A  r d A ¯ ), we realize that the component A ϕ ∂ δ Tr A M τ + − ∂ ϕ 7.6 A r − 2 ϕ Z  d A A ϕ τ κ  ϕ ˙ Tr d A ∂ r − ϕ ( Tr ] A M τ d ¯  A ϕ ∂ τ A [ 21 Z d ) is given by on the boundary imply 2 d Tr τ CS κ + d are required to vanish on the boundary. Therefore, in S + ϕ M 7.9 ¯ A r − ∂ d + ˙ − ] M A Z τ ¯ ] = A ∂ A ϕ [ κ A Z 0 [ A 2 drd κ CS CS − = and S − S M ) + ϕ − − Z ≡ ˙ A = = A A ] r κ I I A A [ −  EOM + S Tr . The last boundary contribution being irrelevant, one can reabsorb it E ] = τ S = ( ϕ ). Looking at the last term of ( ∂ A [ d E ≡ τ S is the curvature two-form associated to the connection. However, this action ] 7.6 CS ¯ δ A S A , 0 (recall that drd A ∧ [ A M S ] = Z ¯ + A κ , given by ( A − dA [ ] vanish because of periodicity), one has, using Stokes’ theorem, S A = [ δ ϕ ] = F A CS [ S The chiral part of the improved action ( Now that we have an action with a well-defined variational principle, we are ready to reduce ), one finds CS S 7.1 the Chern-Simons to the WZWdo model. a similar First, computation we for will the focus anti-chiral on sector the to finally chiral compose sector, the and full then WZW we action. will 7.2 Reduction of the action to a sum of two chiral WZW actions where satisfies does not have a well-defined( action principle. Indeed, computing the variation of the total action which is not zero on-shellorder when to have a well-definedaction variational principle, one must add the following surface term to the which is such that the improved action the connection merely plays the0. role of Therefore, a Lagrange assuming multiplier, no implementing holonomies the constraint (i.e. no holes in the spatial section), one can solve this with where the dot stands for in the definition of thecoordinates action. is given Therefore, by the final form for the Chern-Simons action in terms of Asymptotic dynamics of 3D gravity Integrating by parts the secondones on and fifth terms, and keeping only the radial boundary terms (the PoS(Modave2015)001 given (7.12) (7.13) (7.15) (7.18) (7.17) (7.11) (7.16) (7.14) . ] 23 . ˙ h ] r 0. We are ˙ h 0 . ∂ 0 0 ] g = ˙ Laura Donnay g g h 1 r 1 1 − ∂ − − M 0 g g ∂ g 1 . | ˙ 1 gg 1 − ˙  h 1 − 0 is locally − 3 − h ) g hh 0) , g 1 r = 1 + ∂ )) − = dG A 1 − ˙ h h h 1 r 0 1 ϕ − ∧ r − ∂ hh g + h − 1 F r ), the boundary term simply 1 G A h h ∂ − ( − − r  ˙ + + ∂ g + ˙ ˙ hh 1 gg 1 7.12 0 0 Tr 1 ˙ − − gh g g dA − 1 1 1 1, g 0 ˙ M g , hh − − − , 0 g 1 Z  ≡ ) g g g . g 1 2 − 1 1 1 ) 1 ) and ( 3 ϕ κ ) − ϕ − − − τ , − g g rt hh r , h h g 1 0, which allows to factorize the general r r + ϕ ε ( ˙ hh 1 ( ∂ − ∂  − = 7.11 h 1 [ h − 0 1 = g i − t ) h + g hh − ( r Tr 0 1 r ∂ g ϕ A 1 hh g − ∂ − ( , ϕ r 1 ϕ −  ˙ g h τ d ∂ 0 − − ∂ 1 ( , with the prime standing for the derivative with [ τ 22 g g gg Tr g − h , 1 0 1 Tr ϕ h 24 ϕ g − − ( ∂ G 1 d ϕ drd r g i 1 ) = 1 ∂ − τ d ˙ ∂ ). Plugging ( hh − ), one finds g 1 − 1 ϕ M d τ g 1 , Z − −  r − hh ) = M , κ G 7.10 r h drd 7.14 ∂ hh τ Tr 0, as a first class constraint, generates gauge transformations. G ∂ r ( Z + τ 1 = ∂ ϕ = M = [ ∂ G  i κ − d Z 1 3 h ϕ A ). ϕ τ ) − Tr − r κ , we have (using Stokes’ theorem again) A d F − ϕ 5.6 0 = dG d ˙ M M g GG 1  τ 1 ∂ ϕ ∂ and − 3 Z − ∂ ) G 1 h g κ group element. Indeed, the solution to ( r drd 1 −  ∂ dG ) − 0 on 1 G 1 M ( R ] = Tr − − Z hh r , = A h r ∂ 2 [ G κ ˙ M ∂ h ( ( Z =  CS − S r . The condition SL 3 κ ] = Tr A A [ . We will assume that, as it happens for the solutions of interest, dG M 1 ] = ϕ CS Z is an − A S [ κ G G 3 1 CS = S Since we do not consider holonomies, we assume that this will also hold globally. In general,In one fact, should this consider consists of a gauge fixing condition only in an off-shell formulation, since this relation is obviously A 24 23 the more general case withwants holonomies, to which describe appear three-dimensional as black additional holes. zero-modes A that treatment one of should holonomies take wassatisfied into considered for account in the if [36, on-shell one 37]. connection ( This implies while the three-dimensional term gives explicitly (using the constraint Integrating by parts the third term in ( Asymptotic dynamics of 3D gravity constraint as solution into respect to now ready to reduce the chiralreads action ( and recalling that where by One can partially fix the gauge by imposing Finally, realizing that the last two terms are nothing but On the other hand, one can see that, with the convention PoS(Modave2015)001 (7.22) (7.23) (7.21) (7.24) (7.19) (7.20) direction. ) reduces to Laura Donnay , which is the + ) x ), can be written + 7.10 x . ( M  k . ] ∂ ) 2 ϕ ] ¯ ¯ τ G A G ( [ [  f Γ Γ . 0 on ] Tr κ κ = G ϕ = . In light-cone coordinates [ + + g d moves along the g ˙ ¯ h  Γ  τ ), one can use the Hamiltonian g . Indeed, the equations of mo- ) ) κ ¯ d g g , g ; this is the reason for the name ϕ  ϕ + g M ∂ 2 ϕ 6.10 ∂  (with ∂ 1 . ¯ 1 A ] g Z ) − ¯  − g t ¯ − g [ , g κ ∂ r Tr 1 ( + − + ¯ h ϕ − ¯ . g i g ) t d L WZW t r 25 ˙ ¯ ∂ S τ gg ∂ ϕ A 1 1 , d ϕ ϕ t − − − ∂ ¯ ( ¯ ] A g 1 ¯ g M g ( g ( − group element ∂ [ ¯ g − g 23 g Z ϕ  ϕ ϕ ˙ ¯ ) = κ ∂ direction ∂ A 1 1 ϕ r R WZW Tr + , − ¯ − S A r ] + x ¯ ϕ g g , h ¯  . A t d  [ ) ( τ ] = Tr ¯ 0, whose solution is given by − Tr ¯ Tr G d CS A x ϕ , S , ϕ ( ϕ ¯ d ¯ M ; it is an abuse of notation to mean that k A right-moving G d d ) i [ ) = ≡ τ ) ∂ , 0 τ , τ ] ∂ + S ] ] Z τ g ¯ d x 1 g ¯ d ( g A 1 ( [ ¯ κ [ f [ − drd g − 2 M ¯ S M G g ∂ ∂ = M ( Z Z L WZW Z = R WZW − g ] = κ S i κ S ∂ κ A ¯ A [ ≡ − S ≡ ] = ] = ¯ A A [ 0 by [ ] = S ¯ S 0 imply ¯ ) reads A [ Wess-Zumino action action for the group element = denotes a WZW action for a left-moving element ¯ S φ ] r ) = ¯ ¯ 7.19 g 0 F [ ¯ g ,( 1 chiral ϕ − ¯ g L WZW ± ( S + τ The only difference with the chiral action being the sign of the two-dimensional term, one Therefore, combining left and right sectors, we have shown that the total Chern-Simons action A right mover is often denoted Similarly, the anti-chiral action In order to recover the standard (non-chiral) WZW action ( Therefore, we have shown that the Chern-Simons action for the chiral copy ( ∂ = 25 ± finds easily that is given by as where tion equation for an element moving along the after solving form, since the chiral and anti-chiral actions are linear and of first order in time derivative. We 7.3 Combining the sectors to a non-chiral WZW action Asymptotic dynamics of 3D gravity This is a This first order action describes“chiral”, a which means thatequations the of action motion distinguishes are between left and right movers. Indeed, the x PoS(Modave2015)001 0 < (8.1) (7.25) (7.28) (7.26) (7.27) Λ .  3 Laura Donnay ) dK , 1  − 3 ) K (  dK 1 Tr . − . k  K M 1 ( Z  − , 3 κ Tr . In terms of the latter, the action ! dgg − Π 1 1 , M ¯  g − Z ) ¯ 1 0 Y ϕ g 2 ¯ 3 κ ∂ g ) and 0 1 . However, only the first set of boundary d k ) k − − 1 ¯ ! g R  ; we define as well − , to k φ Tr ¯ k G − 1 2 2 − g 1 0 ( ¯ g − M ∂ − 1 + ( e 1 1 ∂ SL − 2 G Z − . We assume that the decomposition holds globally Y j 24 φ and ¯ 2 r 1 2 Π 0 − gg kk = ( g √ e ] ϕ + e 1 2 ¯ K ∂ 2 G ∂ j [ and 1 , 1 WZW action for an element − ¯ φ Γ g − − ! ϕ e ˙ ¯ ) 1 k g k , 0 1 − 2 R X u ] + − g ,  X j by using its equation of motion, one finally gets k 2 1 0 1 2 G ≡ − [ ( ≡ √ Tr Π Π Γ

e  k Π ϕ SL − = = d Tr τ K d ϕ ] = d K M τ [ ∂ d Γ Z M κ ∂ Z ] = κ k ) was used so far. In this section, we will see how the use of the second set [ S ] = are fields that depend on 5.10 Π ) in local form upon performing a Gauss decomposition of the form φ , , k Y [ , S 7.28 X ) reads In order to perform the reduction at the level of the action, it is useful to express the WZW So far, we have shown that the asymptotic dynamics of three-dimensional gravity with Notice that the above change of variables above is not well-defined for the zero modes [38]. 7.24 where conditions ( (for subtleties in the presence of global obstructions, see [55]). The Gauss decomposition allows further reduces the WZWtwo-dimensional model conformal to invariant field eventually theory getJoseph whose Liouville Liouville origin in can field the be 19th theory. century. tracedLiouville We back will theories Liouville give to in a theory the the brief last work is introduction section. of to a the classical and quantum 8.1 The Gauss decomposition action ( and observe that which is the standard non-chiral is described by the (non-chiral) WZW action for 8. From the WZW model to Liouville theory As a consequence, the equivalencenot of valid the in that sum sector. of two chiral models with the non-chiral theory is Asymptotic dynamics of 3D gravity combine left and right movers as We are allowed to change the( variables from Eliminating the auxiliary variable PoS(Modave2015)001 . , 0 + j a ) (8.3) (8.7) (8.6) (8.8) (8.2) (8.4) (8.5) since + 26 = . x (  1 M ) L − ∂ ` | Y 2 kk + K ∂ + √ Laura Donnay ∂ X by + − − k 2 ∂ j = . 0 + ) + ¯ Y Y + J − − 1 , , j ∂ ∂ . − ) ¯ X  X  a . + /` + ): Y 1 Y ¯ γ ∂ 2 ∂ g = , which has not to be confused with + − ¯ ( g ∂ d k ∂ k 1 √ φ . 0, we obtain a simple relation − 1 + X − 0 − − X − ¯ , g a . Y β ∂ e ¯ = g − 0 1 k 1 ∂ = ( + 2 = = ∂ − φ ∂ − + = φ k − 2 = + ¯ + + k ¯ a − X ¯ J − kk a a e + 2 ) has for effect to set the WZW currents − φ a e − , , a = J = ∂ 2 ∂ + ) ( α dg − ii − − = + φ ∂ 1 ( J 5.10 φ∂ a − , . − + φ ¯ = − , e J g , ) + ∂ 2 a 2 αβγ ¯ i ` M J ) reduces to ( = 2 √ ϕ ∂ ) on the gauge fields set the currents of the WZW ` φ ∂ ) as a two-dimensional integral using the relation ϕ d a ϕ ε 25 √ − d = , τ d 7.28 ∂ τ 1 d − τ = , ] 5.10 1 2 ¯ d 7.28 a 1 k 0 ] M a  − + k M ∂ drd ∂ ∂ 1 Z ϕ k − kk Z = − d ∂ − + 1 ) can be rewritten equivalently by replacing = k 3 a τ ∂ , one finds = ) − 1 3 d 0 = ) k − j  − k k ) a 7.28 dK M J − 1 ∂ dK = [ = [ − /` ∂ 1 Z − 2 1 0 1 − + − ¯ = κ K J J − √ ( K 2 − ( kk J Tr = + ( + Tr , we deduce (recall that 1 3 symbols appearing denote the group element ∂ 2 k have been pull-backed on j k 1 red − M − S φ 0 Z k , 3 1 2 + Y  , exactly cancel in the trace. Therefore, one finds 1 − j X ¯ h ) Tr , − h ϕ x ( d ¯ , the WZW currents, and the gauge fields: L τ k ` d 2 M √ ∂ In this section, all the Let us begin by considering the left and right moving WZW currents. They are given by The second set of boundary conditions ( Z = 26 − ¯ between The two-dimensional integral in ( Then, using the first set of boundary conditions Implementing the second set of boundary conditions a Using the definition of We thus see that the second set of boundary conditions ( model to constants. This is the[39–41]. well-known Hamiltonian reduction of the WZW model to Liouville 8.2 Hamiltonian reduction to the Liouville theory the level, which is a constant labeled by the same letter. Asymptotic dynamics of 3D gravity to rewrite the three-dimensional integral in ( Therefore, one finds (keeping only the radial boundary term) all factors of One can then combine all terms and find that ( where all fields PoS(Modave2015)001 . 3 m ¯ L (8.9) (8.10) (8.11) and themselves. m Y L Laura Donnay 3 dimensions . Also, notice and ) = X φ fields introduced ( D Y , . exp X ) . ,  2 and, then, finds that the 2 1 ) φ τ τ φ

/` rather than φ 2 ( 4  e X ( ) the dual conformal theory of − ϕ = ∂ X has a constant scalar curvature exp is d − 2 , Ω τ 2 ∂ ` µν and ` d 1 Y g Y + . − M Ω + + φ φ ∂ ∂ = − R Y − ≡ ∂ + κ Y ` ∂ φ ∂ µν + 2 X g ∂ + ( − φ ∂ φ − 1 2 = − e . But, before anticipating this discussion, let us e  to any (positive) value. 2 Y  26 2 ϕ ϕ /` d d , τ π , d ` 1 2 ) is equivalent, in terms of the 0 φ M Z ) contributes as 2 = + ∂ 8.8 κ ∂ Z X 2 ` 0? Actually not. The reduction we have presented is a classical − κ 2 8.10 − ∂ 2 < φ = − red Λ such that a new metric ˜ e X ] = S φ Ω [ ≡ impr S Liouville S by a constant, one can set 2 φ 27 is very tempting. Does it mean that Liouville theory 5 This boundary term comes from the fact that the constraints restrict The Liouville differential equation was introduced in the 19th century in the context of the We have therefore shown that the boundary dynamics of AdS gravity in Before solving these constraints in the action, one has to be sure that the latter has a well- ) as follows , does it exist a function 27 ? The answer turns out to be yes. To see this, one first defines µν 8.5 ˜ After inserting the constraints, we are left with the Liouville action R for Riemann surfaces [42–44].that This can be is rephrased a as classical the problem followingg of question: mathematics In a two-dimensional space equipped with a metric above, to the set Notice that the boundary termthat by in shifting ( is described by the two-dimensionaland, Liouville action. therefore, Liouville has theory associated is a two conformal mutually field commuting theory sets of Virasoro generators three-dimensional gravity with computation, working at theconditions; level to of establish the aunderstanding actions correspondence at the through between quantum the level. the Nevertheless, prescription the twoand connection of Liouville theories between Einstein theory specific one theory on on boundary would AdS thean also boundary effective need we theory have of a described, the full briefly shows holographic introduce that dual the the Liouville CFT latter field theory theory for is, the at readers least, who are not familiar with it. 9. Liouville field theory 9.1 At classical level Asymptotic dynamics of 3D gravity to constants. The set of constraints ( Then, identifying these generators within the Section ones appearing in the asymptotic analysis carried out defined variational principle. To achieve( so, one needs to add an improvement term to the action PoS(Modave2015)001 ; ), λ 9.2 ≡ (9.2) (9.4) (9.5) (9.1) (9.3) 2 b πµ ) is actually Laura Donnay 9.2 and 8 , b , /   φ φ ), which is consistent ϕ 2 b e = 2 . e 2 λ ϕ 8.11  πµ φ ) + 2 4 2 e b + 2 λ which satisfies ( + ϕ φ + 1 R ( ) R b φ φ / R 1 , + . ) + 0 + φ φ b φ b = b  corrections correspond here to corrections of φ φ∂ 2 . 2 ¯ h + ( a φ∂ (the sign is chosen for latter convenience) leads e ϕ a ∂ − ϕ ∂ λ λ b R ab 27 ( ab − 28 g + is a dimensionless positive parameter which controls φ g . The curvature scalar of this two-dimensional space, ϕ∂  2 φ b a  | ab − ∂  g e g g . Finding a solution | 2 ab 0 (indeed, √ = g p − x ˜  2 x R → | R 2 d g 2 d in the Lagrangian. 2 | b C R Z Z p | ) corresponds to a non-free scalar field theory formulated on a two- x g Liouville action | 2 = = d metric is given by 9.4 q p S x g Z 2 2 b d π 1 π 4 Liouville equation to an arbitrary constant Liouville R 4 . This shifting is a symmetry of the classical theory as it merely generates a S ˜ can be set to 1 without loss of generality by shifting the field as follows ) in the limit R = ∝ µ ≡ ) was interpreted by Polyakov as the equation of motion of the quantum field q µ 9.3 cl S log S 9.2 1 − ) b 2 ( is an arbitrary positive constant and ). − 2 µ b ϕ See for instance App. D in [45] Before closing this section, let us present some quantum properties of Liouville action. As an The value of One can consistently recover the classical Liouville action by setting 28 → , couples non-minimally (linearly) with the field the quantum effects. Action ( which is the so-called where which reduces to ( to the nonlinear equation order Setting the curvature On the cylinder (or on the torus),ville one action can coincides, always up set the to second field term redefinition, to with zero. the In one this case, obtained the in Liou- ( the action above then becomes dimensional manifold doted with a metric R total derivative term exact conformal field theory, the quantum Liouville action is giving an answer to the uniformizationLater, problem. equation ( which can be derived from the with the fact that the metric at infinity is the9.2 flat metric At on quantum the level: cylinder. stress tensor and central charge ϕ theory that appears inunder string Weyl theory rescaling. when The one classical studies equation how of the motion path of integral Liouville measure field transforms theory would be ( Asymptotic dynamics of 3D gravity curvature associated to the ˜ PoS(Modave2015)001 (9.8) (9.6) (9.9) (9.7) (9.13) (9.10) (9.11) (9.12) and satisfies Laura Donnay . One then reads from | ..., , namely that the theory 2 continuous ... L z c + ) + − 2 ) 1 z ) 2 z ( 2 | z . z α ( 2 , , − V T ϕ 1  ∂ , are given by [46] ϕ . z z 2 2log . The operator product expansion ∂ 2 ) 1 b z ( R 2 ∂ ∂ T α − z ) ) + ∈ + b b . , 1 ; which can be easily computed starting = − b s / CFT, one can show that the holomorphic 2 2 / 2 contribution to 1 ) and ) i 1  ) z 1  , z ) 2 b ) D 2 ) ( z α 2 zz 1 4 ¯ z z + / h + = z ( ( T 1 / with ( b − b − 1 1 ≥ T z αφ 1 ϕ ) + ( + b 1 2 2 2 , z ) 2 + ( e s + ( b ( 1 z O ( 2 z ( + is 2 28 ) ( 6 + α + b  + 2 ϕ V ) = 4 +  h z 2 ) ∂ϕ  b ( ) 1 ∂ϕ 1 2 ( α 2 2 1 b 2 z α  to the conformal anomaly (see [47] for more details). z / − = V = L + ) − + ∆ − L − c 2 1 = b 2 b ∆ c b 1 z = z ( /  zz ( z = 1 T z 4 1 is given in terms of the momentum by ( ) that there is a T α ) = O ∆ ≡ ≡ = ) = 2 9.8 2 z ∆ T T z ( ( T α ) V 1 ) z 1 ( z T ( T . One can compute the central charge of Liouville by computing the operator ¯ the complex coordinates as usual in 2 z ∂ ) z , = z ( ¯ ∂ , z ∂ ) and using the free field correlator = 9.6 ∂ Defining The spectrum of primary operators in Liouville field theory is represented by the exponential ) the value of the central charge 9.7 ( Remarkably, one notices from ( product expansion of two stress-tensorMore operators precisely, one and, has from this, one can read the central charge. with which create primary statesbetween of the stress-tensor the and theory these vertex with operators momentum is where the conformal dimension and analogously for the antiholomorphic component.theory On correspond the to other momenta hand, normalizable states in the Therefore, the spectrum of normalizable states of Liouville field theory is vertex operators Asymptotic dynamics of 3D gravity where the ellipses stand for termsfrom that ( are regular at presents a classical contribution and antiholomorphic components of the stress tensor, PoS(Modave2015)001 of about (10.1) (9.14) (9.15) elliptic parabolic conjugacy , and under ) knows Laura Donnay ∆ R 0. , 2 = ( f SL )) around singularities. z ( × T 29 ) + R is given by 2 , ∂ 2 ∆ ( ( 0) correspond to the 0) belonging to the SL hyperbolic conjugacy class < = , J M ) 0 = , at large , 6= ∆ ) 0 and the minimum eigenvalue ( 1 ) M ∆ ρ ( G = − 24 # 8 ρ L ( ∆ c / cyl 1 . ∆ 6 = L − 2 eff c 24 c  29 ≈ 1 b , denoted r 0 of the so-called Hill equation ∆ 0) correspond to the π + f ∆ 2 b > "  M 1 4 exp between the value = ≈ | ≤ 0 ) ∆ /` ∆ gap ( J | ρ ; the massless BTZ black hole ( ) R , 2 ( , such that its partition function on the torus is modular invariant; then, the c SL asymptotic symmetry manages to account for three-dimensional the black hole 3 of . It happens that all these solutions can actually be gathered in Liouville theory , the degeneracy of states in the limit of large conformal dimension 2 , the particle-like solutions (for example, ) R , More precisely, they are related to the solutions Before concluding this section, let us make the following comment: As mentioned in the pre- In this section, we will discuss how the conformal field theory description appearing in the In a CFT 2 29 ( vious sections, through the reduction fromthe Einstein contribution gravity of to holonomies. Liouville This theory we candescribe, did be for not seen instance, consider as a the limitation BTZ since solution. holonomies are However, important it to turns out that Liouville theory where the continuum starts. Anlimit, observation where that the will central be charge relevant becomes later large, is this that, gaps in reads the semi-classical the holonomies. This is because the holonomiesspond of to the the Chern-Simons different gauge BTZ connections that geometries corre- can be classified in terms of Asymptotic dynamics of 3D gravity This means that the theory has a classes, and the latter arethe closely related BTZ to black the holes classicalSL solutions (namely of Liouville field theory. While conjugacy class certain assumptions, is givenwith by central the charge Cardy formuladegeneracy of [48]. states of Namely, conformal dimension let a (chiral part of a) CFT boundary can be usedby to reviewing reproduce how, the by BTZ meansthrough black of the hole a AdS entropy. Cardy formula, More the precisely, conformal we field will theory begin structure appearing 10. Liouville and the entropy of the BTZ black hole entropy. Then, we willspectra. discuss some issues about Liouville field theory and the BTZ10.1 black Cardy hole formula and effective central charge conjugacy class by studying the monodromies of theWe classical will solutions not of discuss the the field details equation of this in these notes. PoS(Modave2015)001 , 3 2 is ∆ ∆ (10.5) (10.3) (10.4) (10.6) (10.2) Laura Donnay central charge 1. For different , the asymptotic 5  c effective  , ∆ cyl ¯ ∆ 6 , 24 is the conformal dimension eff ) ¯ / c ) black holes. ) is the J c c is given by two copies of Virasoro s − 2 − ), which can be written π 10.4 ∆ M /` 2 ` 1 ( = ). + . 10.1 − 1 2 is associated to whether the theory has a cyl cyl 3.8 BH = . = ∆ 0 S ∆ (and small c 6 . Λ ∆ eff ` ∆ cyl = G 6= c 3 ¯ ∆ 2 24 is large but not necessarily much smaller than + ) one assumes that the conformal dimension eff r c r − G 30 = , c π 4 π c ¯ ) c 2 2 . In other words, we have J 10.1 = black hole ( = G = + 3 c eff ) = c ¯ M on the sphere, ∆ ` , CFT 0 ( S ∆ L 1 2 ( ρ = log cyl finite. Notice that this is not in contradiction with the semi-classical ), ∆ ≡ c dimensional gravity with 3.4 − is an effective central charge, defined by ) CFT 1 S eff c + 2 ( 24 is related to the conformal mapping between the Riemann sphere and the cylinder, which / c , and . This is related to the fact that, when deriving the asymptotic growth of states at − c 30 ) since one can always consider large mass black holes while considering the AdS c ) as the effective central charge, and the fact that the BTZ conserved charges are given point of view. Indeed, applying Cardy formula ( is the lowest eigenvalue of 2 0 , one resorts to the saddle point approximation, which selects the state of lower conformal 10.3 ∆ ∆ Let us now discuss a subtlety in relation to the Liouville field theory spectrum. Let us begin In the derivation of the Cardy formula ( The shifting In [7], Strominger made use of this result to reproduce the entropy of the BTZ black hole from rather than 30 eff by noticing that what actually enters in the Cardy formula ( 10.2 A caveat of the CFT spectrum and Liouville theory produces a non-vanishing Schwarzian derivative in the stress-tensor transformation law. one finds, using relations ( large and the central charge limit (large radius much larger than thelimits Planck in length relation to the Cardysee formula, [49]; in see which also [23] for a computation with small c large dimension. In other words, the possibility of the CFT by using ( This is a remarkableexactly result! reproduces the It entropy of manifestly the shows AdS that the Cardy formula of the boundary CFT Asymptotic dynamics of 3D gravity where on the cylinder symmetry group of algebra with central charges This result is notably relevantshow for how the the Cardy applications formula to canwith three-dimensional be the gravity. used observation to In of compute fact, Brown the one and entropy of can Henneaux BTZ that, black as holes. we This reviewed starts in Section PoS(Modave2015)001 ), = M ` 9.13 (10.7) (10.8) geom- (10.10) 3 Laura Donnay 1, and this would = 0 between AdS ) also agrees with the ∆ of Liouville action are 0 b ∆ 24 9.15 − L ) one finds that for the BTZ 1. c ; (10.9) different from zero. Therefore, =  10.5 J 0 k eff to be the Brown-Henneaux central . ∆ / c ). In fact, apart from the normaliz- = 1 L eff ), one finds that, in the semi-classical c 24 c cyl ≈ ¯ ∆ 2 9.8 10.10 b − ≈ − . ` , cyl G G 3 2 ` 2 ∆ 16 1 − , reads = , k − 31 G k 6 = = M  ` 2 ≈ ` cyl b L ), which requires = ¯ ∆ c cyl = ¯ 10.6 ∆ cyl of the WZW action and the parameter + ∆ ) cyl G ∆ 4 ( `/ space, which is the natural vacuum of the theory, corresponds to = 3 k gravity in more details. 0, which reads ), Liouville effective central charge would be 3 = J 9.14 and ) G 8 ). This suggests to include, apart from the normalizable states that account for the black The reduction from a WZW model to Liouville can be performed at the quantum level: this This issue has been discussed in the literature [50–52], where it has been proposed that this is ( / 1 9.15 according to ( approximation, gap in the spectrum ofblack BTZ holes black one has holes. More precisely, from ( We see therefore that the mass gap of the BTZ spectrum (namely the gap hole spectrum, other states in Liouville theory, such as ( when formulated on the sphere has a minimum eigenvalue of which in the semi-classical approximation That is, the centraladdition, charge one of observes Liouville that, coincides in the exactly same with limit, the the one gap in of the Brown-Henneaux. spectrum ( In etry and the massless BTZ( black hole) coincides with the gap in the spectrum of Liouville theory is the so-called (Drinfeld-Sokolov) Hamiltonianone finds reduction that [39, the 54, level 55]. Through this procedure, an indication that the descriptionconsidered of only thermodynamics as in an effective terms description. oftheory In Liouville and other field Liouville words, theory the theory, different which shouldtaken steps be were as connecting a proven Einstein proof to that the hold theories atquantum involved are regimes the equivalent of beyond level the both of semi-classical theories limit. the canissues In actions, be fact, that the should notably make not different. difficult be Liouvilleclassical theory to approximation. exhibits believe indeed that Nevertheless, many one it canand could still see describe how insist far three-dimensional with it the gravity bringsto Liouville beyond us. the effective With the one description this of aim, AdS let us discuss the Liouville theory spectrum in relation Asymptotic dynamics of 3D gravity gap or not. This is exactly what occurs in Liouville theory which, as wecharge mentioned in in order to ( reproduce the black hole entropy. related as follows: on the other hand, AdS − Since the central charge of Liouville theory is given by ( be in conflict with the derivation of ( PoS(Modave2015)001 n < 0 are 1 (10.11) (10.12) (10.13) = = r 1 become m > Laura Donnay s living at the 1 corresponds 2 m = n has been discussed 3 = ). A natural question ) with m 10.10 10.11 , 0 = 24 as in ( cyl / ¯ ∆ c , , 0 b + geometries also exhibit interesting super- m = cyl ≈ − 2 J ∆ − 1 cyl , , ¯ ∆ + L 2 G 32 = n c 12 n ; that is, 8 b 2 ) − 2 2 cyl n − spacetimes provide new potential CFT b 1 ∆ 3 4 = 0. Notice also that the state with ( = ≈ − / M ) α → ` 2 cyl n b ¯ ∆ − + 1 0, namely exact angular excesses ( cyl ), in the semi-classical limit theses states must correspond to gravity = ≈ ∆ ¯ ¯ ∆ ∆ 9.11 = = . These ∆ ∆ π ), in Liouville field theory there exists another set of interesting operators to be two positive integer numbers. Notice that the states ( . These states play an important role in the discussion [55] and can be shown to 0 n 9.12 > Z ). According to ( and ∈ n m The asymptotic symmetries and dynamics of three-dimensional gravity is highly sensitive to Also, the scope of these lecture notes was limited to the case of pure gravity, but the study of Before concluding these notes, let us mention an important direction that these notes have 10.11 with is to find what geometries correspondin to ( the other degenerate states,solutions namely with those charges with the choice of boundaryHenneaux conditions. boundary conditions, In where theseyears, the lectures, there metric have we been at many have the worksthe focused boundary generalizing boundary on or has metric modifying the no is these seminal dynamics. allowed boundarywhile conditions. to Brown- in In be In [60, 61] the [58], dynamical; the last in boundaryThese [59], metric new chiral is notions boundary in of conditions a asymptotically were conformal AdS given, gauge and light-cone gauge, respectively. explored [50, 53]. Thesemomenta are [46] the so-called degenerate operators, which correspond to values of represent the special cases of negative mass geometries whose angular excesses around 11. Other directions and recent advances asymptotic symmetries and dynamics of three-dimensional AdShere spaces similar can to the be one addressed extended[65, 66]. to broader set-ups such as supergravity [19, 36]not and explored, since, higher-spin motivated theories byHowever, AdS/CFT, over we focused the on last the years,case case significant of of effort Anti-de non-AdS has Sitter spacetimes. been spacetimes. a made In vanishing to particular, cosmological extend holographic constant holographic properties have toolsrich of asymptotic been to dynamics gravitational investigated. the can theories be In found with at this null flat infinity: holography the symmetry perspective, algebra a of asymptotically flat Asymptotic dynamics of 3D gravity able states ( irrelevant in the semi-classical limit or equivalently, with integer multiples of 2 recently [57]. boundary [62–64]. symmetric properties [56], and their relevance in higher-spin theories in AdS exactly to the vacuum PoS(Modave2015)001 (11.1) Laura Donnay , ! 1 − 1, and the tangent space 1 0 0 . One can also connect ∞

= 1 2 → 012 , , ` ε = 0 0 , , 2 n n j + + m m δ δ 3 3 2). , m , m . 1 2 1 , c 12 c 12 ! 0    + + = n 0 0 1 0 n b + + , , respectively. This algebra can be obtained from

m 0 1 0 1 0 0 0 0 1 m a m 33 ) by writing the latter in terms of the generators P ( J J 2 ) )    1 n n ab √ = η 5.17 − − and 2 1 = , ab m m m 1 and then taking the limit 0 j P η WZW model and the Hamiltonian reduction reduces further ) ) is an infinite-dimensional algebra made out of supertrans- ) = = = ( = ( ) m b j 1 } } − } , a n n ¯ n L j 11.1 2 P J P , ( ( , , , − m m m ! J P J m , Tr { { { L c i i i j 0 1 0 0 2, is off-diagonal, given by = ( abc , algebra [67, 68]

ε 1 3 m . Algebra ( 2 , J invariant Liouville theory [38, 72] (generalized afterward adding higher-spin 1 0 G √ , 3 ] = / b = 3 j /` = ) a , 0 = a m j j 2 − [ c ¯ L generators, we take , with 0, + ) ab m R = , η L 1 2 ( c I would like to thank all the organizers of this great edition 2015 of the Modave Summer Our conventions are such that the Levy-Civita symbol fulfills sl = ( m metric As which satisfy School, as well as all theshared participants during for their the questions, lectures. comments andGonzález for interesting I enlightening discussions also discussions we on want these to topics.Belgium. thank L.D. is Glenn a Barnich, FRIA research Gaston fellow of Giribet, the and FNRS Hernán A. A. Conventions the (two copies of the) Virasoro algebra ( the theory to a flat chiral boson action which can be finally relatedAcknowledgments to a Liouville theory. with through a well-defined flat-space limittimes: [69] the the phase limit space of of theerties asymptotically BTZ AdS can black and be holes flat are understood space- cosmological fromthat solutions a the whose holographic dual thermodynamical dynamics perspective prop- of [70, three-dimensional 71].shown asymptotically to flat Finally, spacetimes be let at a us null BMS mention infinity has the been fact fields [73] and [74]),one goes through first a through reduction a chiral very iso similar to the one presented here; lations and superrotations generated by P Asymptotic dynamics of 3D gravity spacetimes is the BMS PoS(Modave2015)001 , ϕ ϕ = ± 9812 1002 1 (1993) τ x , (1994) Phys. Phys. , = Phys. τ , 72 D47 ± JHEP JHEP , = x (1986) Laura Donnay 0 x , 104 (2000) 3758 arXiv:1103.4304 1 black hole”, (1995) 2961 Phys. 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