International Conference on Strings, M-Theory and Quantum Gravity Centro Stefano Franscini, Monte Verita, Ascona, 27 July 2010 Aspects of the BMS/CFT correspondence Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes
Overview Classical gravitational aspects of AdS3/CFT2 correspondence 4d flat case, null infinity: asymptotic symmetries 3d flat case, null infinity: BMS3/CFT1 correspondence 4d flat case, null infinity: solution space work done in collaboration with C. Troessaert
AdS3/CFT2 Asymptotic symmetries Fefferman-Graham ansatz g µν = l 2 r 2 0 0 g AB Λ= 1 l 2 g AB = r 2 γ AB (x C )+O(1) r t, φ 2d metric γ AB conformal to flat metric on the cylinder γ AB = e 2ϕ η AB η AB dx A dx B = dτ 2 + dφ 2, τ = t l, ϕ = ϕ(xa ) asymptotic symmetries L ξ g rr =0=L ξ g ra, L ξ g AB = O(1), general solution determined by conformal Killing vector ξ r = 1 2 ψr, ξ A = Y A + I A, I A = l2 2 Bψ r dr r Y A of η AB g AB = l2 4r 2 γ AB B ψ + O(r 4 ). ψ = D A Y A
AdS3/CFT2 Asymptotic symmetries metric dependence ξ µ = ξ µ (x, g) δ g ξ 1 g µν = L ξ1 g µν modified bracket [ξ 1, ξ 2 ] µ M =[ξ 1, ξ 2 ] µ δ g ξ 1 ξ µ 2 + δg ξ 2 ξ µ 1 linear representation of conformal algebra [ξ 1, ξ 2 ] r M = 1 2 ψr, [ξ 1, ξ 2 ] A M = Y A + I A, Y A =[Y 1,Y 2 ] A, ψ = DA Y A light-cone coordinates x ± = τ ± φ, 2 ± = τ ± φ, γ ABdx A dx B = e 2ϕ dx + dx Y ± (x ± ) ± = n Z c n ±l ± n, l ± n = ie inx± ± [l ± m,l ± n ]=(m n)l ± m, [l ± m,l n ]=0 include Weyl rescalings of boundary metric L ξ g rr =0=L ξ g ra, L ξ g AB =2ωg AB + O(1) direct sum with abelian algebra of Weyl rescalings ( Y,ω) = [(Y 1, ω 1 ), (Y 2, ω 2 )] Y A = Y B 1 B Y A 2 Y B 2 B Y A 1, ω =0
AdS3/CFT2 Solution space existence of general solution integration constants Ξ ++ = Ξ ++ (x + ), Ξ = Ξ (x ) when ϕ =0 g AB dx A dx B = (r 2 + l4 r 2 Ξ ++Ξ )dx + dx + l 2 Ξ ++ (dx + ) 2 + l 2 Ξ (dx ) 2, BTZ black hole Ξ ±± =2G(M ± J l ) AdS3 space M = 1,J =0 8G general solution ϕ = 0 g AB dx A dx B = e 2ϕ r 2 +2γ + r 2 e 2ϕ (γ + 2 + γ ++ γ ) dx + dx + +γ ++ (1 r 2 e 2ϕ γ + )(dx + ) 2 + γ (1 r 2 e 2ϕ γ + )(dx ) 2, γ ±± = l 2 Ξ ±± (x ± )+ 2 ±ϕ ( ± ϕ) 2 γ + = l 2 + ϕ
AdS3/CFT2 Conformal properties asymptotic symmetries transform solutions into solutions g AB = g AB (x, Ξ,ϕ) g AB (x, δξ, δϕ) =L ξ g AB conformal transformation properties δ Y +,Y,ωΞ ±± = Y ± ± Ξ ±± +2 ± Y ± Ξ ±± 1 2 3 ±Y ± δ Y +,Y,ωϕ = ω asymptotically AdS3 gravity: dual to conformal boundary theory
AdS3/CFT2 Charge algebra Hamiltonian approach Q ξ surface charge generators, Dirac algebra centrally extended charge representation of conformal algebra Brown & Henneaux 1986 Q ξ [g ḡ, ḡ] = 1 8πG 2π 0 dφ (Y + Ξ ++ + Y Ξ ) Q ξ1 [L ξ2 g, ḡ] Q [ξ1,ξ 2 ] M [g ḡ, ḡ]+k ξ1,ξ 2, K ξ1,ξ 2 = Q ξ1 [L ξ2 ḡ, ḡ] = 1 8πG 2π 0 dφ ( φ Y τ 1 2 φy φ 2 φy τ 2 2 φy φ 1 ) modes Strominger 1998: combine with Cardy formula to provide a microscopic derivation of the Bekenstein-Hawking entropy of BTZ black hole
BMS4/CFT2 Asymptotically flat spacetimes BMS ansatz g µν = Minkowski u = t r η µν = e2β V r + g CDU C U D e 2β g BC U C e 2β 0 0 g AC U C 0 g AB u 1 1 0 0 1 0 0 0 0 0 r 2 0 0 0 0 r 2 sin 2 θ g AB dx A dx B = r 2 γ AB dx A dx B + O(r) r x A = θ, φ ζ, ζ Sachs: unit sphere γ AB = e 2ϕ 0γ AB 0 γ AB dx A dx B = dθ 2 + sin 2 θdφ 2 Riemann sphere ζ = e iφ cot θ 2, γ ABdx A dx B = e 2 eϕ dζd ζ dθ 2 +sin 2 θdφ 2 = P 2 dζd ζ, P(ζ, ζ) = 1 2 (1 + ζ ζ), ϕ = ϕ ln P determinant condition fall-off conditions det g AB = r4 4 e4 eϕ β = O(r 2 ), U A = O(r 2 ), V/r = 1 2 R + O(r 1 )
BMS4/CFT2 Asymptotic symmetries asymptotic symmetries general solution L ξ g rr =0, L ξ g ra =0, L ξ g AB g AB =0, L ξ g ur = O(r 2 ), L ξ g ua = O(1), L ξ g AB = O(r), L ξ g uu = O(r 1 ) ξ u = f, f = f ϕ + 1 ξ A = Y A + I A, I A = f,b dr (e 2β 2 ψ f = eϕ T + 1 g AB 2 ), r ξ r = 1 2 r( D A ξ A f,b U B +2f u ϕ), ψ = D A Y A u 0 du e ϕ ψ, Y A = Y A (x B ) T = T (x B ) conformal Killing vectors of the sphere generators for supertranslations spacetime vectors with modified bracket form linear representation of bms 4 algebra [(Y 1,T 1 ), (Y 2,T 2 )] = ( Y, T ) Y A = Y1 B B Y2 A Y1 B B Y2 A Sachs 1962, T = Y1 A A T 2 Y2 A A T 1 + 1 2 (T 1 A Y2 A T 2 A Y1 A ) standard GR choice: restrict to globally well-defined transformations SL(2, C)/Z 2 SO(3, 1) Y A generators of Lorentz algebra
BMS4/CFT2 New proposal CFT choice : allow for meromorphic functions on the Riemann sphere solution to conformal Killing equation Y ζ = Y ζ (ζ), Y ζ = Y ζ( ζ) generators l n = ζ n+1 ζ, ln = ζ n+1 ζ, n Z T m,n = ζ m ζn, m, n Z commutation relations [l m,l n ]=(m n)l m+n, [ l m, l n ]=(m n) l m+n, [l m, l n ]=0, [l l,t m,n ]=( l +1 2 m)t m+l,n, [ l l,t m,n ]=( l +1 2 n)t m,n+l. Poincaré subalgebra l 1,l 0,l 1, l 1, l 0, l 1, T 0,0,T 1,0,T 0,1,T 1,1,
BMS3/CFT1 ansatz for asymptotically flat metrics g µν = Asymptotic symmetries e 2β Vr 1 + r 2 e 2ϕ U 2 e 2β r 2 e 2ϕ U e 2β 0 0 r 2 e 2ϕ U 0 r 2 e 2ϕ Minkowski spacetime ds 2 = du 2 2dudr + r 2 dφ 2 u = t r fall-off conditions β = O(r 1 ), U = O(r 2 ) V = 2r 2 u ϕ + O(r) asymptotic symmetries L ξ g rr =0=L ξ g rφ, L ξ g φφ =0, L ξ g ur = O(r 1 ), L ξ g uφ = O(1), L ξ g uu = O(1) ξ u = f, ξ φ = Y + I, I = e 2ϕ φ f dr r 2 e 2β = 1 r r e 2ϕ φ f + O(r 2 ), ξ r = r φ ξ φ φ fu + ξ φ φ ϕ + f u ϕ, u f = f u ϕ + Y φ ϕ + φ Y f = e ϕ T + u 0 du e ϕ ( φ Y + Y φ ϕ) solution involves 2 arbitrary functions on the circle spacetime vector form faithful representation of Y = Y (φ), T = T (φ) bms 3 algebra [(Y 1,T 1 ), (Y 2,T 2 )] = ( Y, T ) Y = Y 1 φ Y 2 (1 2), T = Y1 φ T 2 + T 1 φ Y 2 (1 2)
BMS3/CFT1 Solution space and conformal properties general solution parametrized by Θ = Θ(φ), Ξ = Ξ(φ) s uφ = e ϕ Ξ + u 0 ds 2 = s uu du 2 2dudr +2s uφ dudφ + r 2 e 2ϕ dφ 2, s uu = e 2ϕ Θ ( φ ϕ) 2 +2 2 φϕ 2r u ϕ, du e ϕ 1 2 φθ φ ϕ[θ ( φ ϕ) 2 +3 2 φϕ]+ 3 φϕ. bms 3 transformation properties δ Y,T Θ = Y φ Θ +2 φ Y Θ 2 3 φy, δ Y,T Ξ = Y φ Ξ +2 φ Y Ξ + 1 2 T φθ + φ T Θ 3 φt, covariant charges Q ξ [g ḡ, ḡ] 1 16πG K ξ1,ξ 2 = 1 8πG 2π 0 dφ 2π 0 dφ (ΘT +2ΞY ) φ Y 1 (T 2 + φt 2 2 ) φ Y 2 (T 1 + φt 2 1 )
BMS3/CFT1 Charge algebra modes Y (θ) 1 copy of Wit algebra acting on i 1 iso(2, 1) charge algebra: relation to AdS 3 similar to contraction between so(2, 2) iso(2, 1) L ± m = 1 2 ( lp ±m ± J ±m ) l collaboration with G. Compère
BMS4/CFT2 solution space ansatz g AB = r 2 γ AB + rc AB + D AB + 1 4 γ ABC C DC D C + o(r ) determinant condition C A A =0=D A A Sachs: power series and D AB =0 guarantees absence of log terms equations of motion imply β = β(g AB ) U A = 1 2 r 2 DB C BA 2 3 r 3 (ln r + 1 3 ) D B D BA 1 2 CA B D C C CB + N A + o(r 3 ε ), angular momentum aspect N A (u, x A ) u dependence fixed log terms also absent when D ζζ = d(ζ), D ζ ζ = d( ζ), D ζ ζ =0.
BMS4/CFT2 Solution space V r = 1 2 R + r 1 2M + o(r 1 ) mass aspect M(u, x A ) u dependence fixed news tensor u C AB (u, x A ) only arbitray function of u general solution: 4 arbitrary functions of 3 variables & 3 arbitrary functions of 2 variables g AB (u 0,r,x A ) u C AB (u, x A ) M(u 0,x A ) N A (u 0,x A ) for simplicity ϕ =0 γ AB dx A dx B = dζd ζ C ζζ = c, C ζ ζ = c, C ζ ζ =0 redefinitions M = M 2 c 2 c Ñ ζ = 1 12 [2N ζ +7 c c +3c c] evolution equations u M = ċ c 3 u Ñ ζ = M 2 3 c ( c +3 c )ċ
BMS4/CFT2 Conformal properties bms4 transformations δc = fċ + Y A A c +( 3 2 Y 1 2 Ȳ )c 2 2 f δd = Y A A d +2 Y d f = T + 1 2 uψ δċ = f c + Y A A ċ +2 Y ċ 3 Y δ M = fċ c + Y A A M + 3 2 ψ M + c 3 Y + c 3 Ȳ +4 2 2 T δñ ζ = Y A A Ñ ζ +( Y +2 Ȳ )Ñ ζ + 1 (ψ d) 3 f( M +2 2 c + cċ) f M 3 +2 3 c +( c +3 c )ċ Interpretation and consequences: work in progress
BMS4/CFT2 Conclusions and perspectives 4d gravity is dual to some conformal field theory classifiy (non)-central extensions; study representation theory of bms4 to be done: surface charge algebra non extremal Kerr/CFT correspondence? angular momentum problem in GR: Lorentz = bms4(old)/supertranslations versus bms4(new)/supertranslations = Virasoro
References Asymptotically flat spacetimes & symmetries
References Gravitational AdS3/CFT2 & Kerr/CFT
References
References Holography at null infinity in 3 & 4 dimensions
References This work based on
Quantum Coulomb solution Toy model to understand BH entropy quantum understanding of BH entropy: what microstates are responsible? physical toy model for BH: electromagnetic Coulomb solution quantize EM field in the presence of external source Q action S Q = d 4 x [ 1 4 F µνf µν j µ A µ ] j µ = δ µ 0 Qδ3 (x ) modification of constraint associated with Gauss law φ Q 2 πi, i +j 0 =0
Quantum Coulomb solution BFV-BRST quantization standard BFV-BRST operator quantization in Hilbert space with indefinite norm modified BRST charge Ω Q = d 3 kc ( k)[a( k) q( k)] + [a ( k) q( k)]c( k) q( k)= Q (2π) 3/2 2k 3/2 null oscillators a( k)=a 3 ( k)+a 0 ( k) b( k)= 1 2 (a 3( k) a 0 ( k)) [â( k), ˆb ( k )] = δ 3 ( k k ) standard vacuum no longer BRST invariant! shifted oscillators a Q ( k)=a( k) q( k), a Q = a ( k) q( k) new vacuum in the presence of source â Q ( k) 0 Q =0
Quantum Coulomb solution Coherent state of unphysical photons solution in terms of old vacuum 0 Q = k exp q( k)ˆb ( k) 0 =exp d 3 kq( k)ˆb ( k) 0 null oscillator Q 0 0 Q = 0 0 =1 expectation values of electric and magnetic fields Q 0 ˆπ i (x) 0 Q = Qxi 4πr 3 Q 0 ˆ A 0 Q =0 quantum mechanical understanding of Coulomb solution only possible in Gupta-Bleuler/ BRST quantization BH problem: one needs to count coherent states of unphysical degrees of freedom