Real-Time, Finite-Temperature AdS/CFT

Real-Time, Finite-Temperature AdS/CFT p. 1/34 Real-Time, Finite-Temperature AdS/CFT Chaolun Wu Physics Department, University of Virginia April 28, 2010

Real-Time, Finite-Temperature AdS/CFT p. 2/34 Real-Time, Finite-Temperature AdS/CFT Work done with Peter Arnold, Eddy Barnes & Diana Vaman e-print: arxiv:1004.1179 [hep-th]

Real-Time, Finite-Temperature AdS/CFT p. 3/34? Question? How to calculate n-point correlators like T O(x 1 )O(x 2 )O(x 3 ) in a Strongly Coupled, Finite Temperature field theory? Strong Coupling: AdS/CFT duality Finite Temperature: AdS-Schwarzschild/Thermal CFT duality QCD (Quark-Gluon Plasma) when g is large but slow-changing = N = 4, SU(N c ) Supersymmetric Yang-Mills (SYM) Field Theory

Real-Time, Finite-Temperature AdS/CFT p. 4/34 Outline A Brief Review CFT: N = 4 SU(N c ) Super-YM Field Theory AdS: Type IIB Supergravity in AdS 5 Space T = 0 Duality: AdS/CFT T = 0 Duality: AdS-Schwarzschild/Thermal CFT Scalar in AdS-Schwarzchild Bulk-to-Boundary Propagators 2-Point Functions 3-Point Functions related to Schwinger-Keldysh Formalism Summary & Outlook

Real-Time, Finite-Temperature AdS/CFT p. 5/34 { L = tr N = 4SU(N c ) Super-YM Field Theory 6 D μ φ i D μ φ i i=1 4 I=1 i ψ I γ μ D μ ψ I 1 2g 2 Y M d = 4 flat Minkowskian space Poincaré invariance (rotation and boost + translation) Symmetry group SO(1, 3) Maximal supersymmetry } F μν F μν + Bosons: φ i (x) (i = 1..6), A μ (x) Fermions: ψ I α(x) (I = 1..4) R-symmetry: mix same spin fields Symmetry group SO(6) β(g YM ) = 0 at quantum level Scale invariance Conformal field theory Symmetry group SO(2, 4) (together with Poincaré) Global symmetry: SO(2, 4) SO(6)

Real-Time, Finite-Temperature AdS/CFT p. 6/34 AdS 5 S 5 Geometry SO(6) = S 5 (5-d sphere) Embedding: X 2 1 +X 2 2 +X 2 3 +X 2 4 +X 2 5 +X 2 6 = R 2 Metric: ds 2 = R 2 dω 2 5 SO(2, 4) =?

Real-Time, Finite-Temperature AdS/CFT p. 7/34 AdS 5 S 5 Geometry SO(2,4) = AdS 5 (5-d Anti-de Sitter space) Embedding: X 2 1 X 2 0 +X 2 1 +X 2 2 +X 2 3 +X 2 4 = R 2 Metric: ds 2 = r2 R 2( dt2 +d x 2 )+ R2 r 2 dr2 = R2 z 2 ( dt2 +d x 2 +dz 2 ) Curvature: negative constant 2 Poincaré patches 1 boundary in each Poincaré patch: at r = or z = 0

Real-Time, Finite-Temperature AdS/CFT p. 8/34 S = 1 4πα + 1 4πα Type IIB Superstring Theory d 2 ξ { } γ [γ mn G μν (X)+ε mn B μν (X)] m X μ (ξ) n X ν (ξ)+α R (2) γ Φ(X) Σ d 2 ξ [ ] γg μν (X) ψ+(ξ)d z μ ψ+(ξ)+ψ ν (ξ)d μ z ψ (ξ) ν + Σ String 1 spatial dimension d = 2 worldsheet: ξ m = (τ,σ), (m = 0,1) Worldsheet metric: γ mn (ξ) Bosonic field: embedding space d = 10 spacetime: X μ = (t, x,r,...), (μ = 0..9) Target space metric: G μν (X) Fermionic field: ψ μ ±(ξ) Coupling constant: g s, string length: α = l 2 s AdS 5 S 5 is the backgound

Real-Time, Finite-Temperature AdS/CFT p. 9/34 Type IIB Supergravity (SUGRA): d = 10 S = 1 2κ 2 10 d 10 X } Ge {R Φ (10) +4 μ Φ(X) μ Φ(X) + where κ 10 = 8π 7 2g s l 4 s Low Energy (Point) limit of string: massive string modes decouple, only massless mode survives. l s R 0 d = 10 spacetime: X μ = (t, x,r,...), (μ = 0..9) Fields: metric G μν (X), scalar Φ(X),... G μν (X) = AdS 5 S 5 is a classical solution Classical limit: suppress quantum fluctuations g s 0

Real-Time, Finite-Temperature AdS/CFT p. 10/34 Classical Type IIB Supergravity: d = 5 Kaluza-Klein reduction from d = 10 to d = 5, eg: integrate over S 5 Φ(X) = φ(t, x,r)y lm... (Ω 5 ) Eigenvalue of Y lm... looks like a mass term in d = 5 5-d effective action S = N2 8π 2 R 3 d 5 x { g R (5) 1 2 S 5 = Δ(Δ 4) m 2 ( μ φ μ φ+m 2 φ 2) } R2 8 Fa μνf aμν + This is the starting point of an actual AdS calculation.

Real-Time, Finite-Temperature AdS/CFT p. 11/34 String/Gauge Duality 2 quite different theories: String/SUGRA vs. CFT Same global symmetry groups Prerequisite of forming a duality Duality: Mapping of everything Map of (coupling) constants Map of String fields CFT operators Map of String action CFT generating functional Map of correlation functions

Real-Time, Finite-Temperature AdS/CFT p. 12/34 AdS/CFT Duality: (Coupling) Constants Quantum IIB String in AdS 5 S 5 (g s,l s ) N = 4 SU(N c ) SYM (CFT) (N c,λ=g 2 YM N c) g s 0 l s R 0 4πg s = λ N c ( ) 4 R = λ λ N c λ l s Weakly coupled IIB SUGRA in AdS 5 Strongly coupled large N c Quantum CFT

Real-Time, Finite-Temperature AdS/CFT p. 13/34 AdS/CFT Duality: Fields / Operators AdS Fields CFT Operators φ L A a μ J a μ g μν T μν. m Δ

Real-Time, Finite-Temperature AdS/CFT p. 14/34 AdS/CFT Duality: Action & Correlators AdS (classical) CFT (strongly coupled) lim z 0 φ(x, z) z 2 Δ 2 = φ(x) φ(x) = J(x) J(x) O(x) S cl [ φ] = d 5 x g( φ) 2 + Z[J] = Dφe is+i d 4 xj(x)o(x) e is cl[ φ] = Z[J] δ n S cl [ φ] δ φ(x 1 )δ φ(x 2 )... φ 0 O(x 1 )O(x 2 )... δ n lnz[j] δj(x 1 )δj(x 2 )... J 0

Real-Time, Finite-Temperature AdS/CFT p. 15/34 AdS-Schwarzschild/CFT Duality: T = 0 Weakly coupled IIB SUGRA in AdS Schwarzschild 5 T H [Witten] Strongly coupled Quantum CFT at Finite Temperature T AdS 5 : AdS Schwarzschild 5 : ds 2 = r2 R 2( dt2 +d x 2 )+ R2 r 2 dr2 ds 2 = r2 R 2( fdt2 +d x 2 )+ R2 r 2 f dr2 f = 1 ( r 0 ) 4 r - Blackening function T H = r 0 - Hawking temperature πr 2

Real-Time, Finite-Temperature AdS/CFT p. 16/34 AdS-Sch/CFT Duality: Procedure & Complications 1. Solve EOM in AdS-Schwarzschild background ( m 2 )φ(t, x,r) = 0 with = 1 g μ gg μν ν Find 2 eigenfunctions AdS: 2 poles at r = 0 and r = = Bessel eq. AdS-Sch: 4 poles, 2 extra from f at r 2 = ±r0 2 =? Superposition by B.C. s =? 2. Calculate classical on-shell action S cl [ φ] = d 5 x g { μ φ μ φ+m 2 φ 2} global structure: 4 quadrants + 2 timelike boundaries, so d 5 x =? 3. Take functional derivative of S cl [ φ]

Real-Time, Finite-Temperature AdS/CFT p. 17/34 ODE with 4 Regular Poles: Heun s Equation d 2 φ(z) dz 2 + ( γ z + δ z 1 + ε ) dφ(z) z a dz 4 poles: z = 0, 1, a, + αβz q z(z 1)(z a) φ(z) = 0 Regularity at z = : ε = α+β γ δ +1 Solution: Heun s function in z [0, min{1, a }) Hl(a,q,α,β,γ,δ;z) = k=0 c k z k = 1+ q z + γa Recursion relation: c k+1 =...c k +...c k 1 2 independent solutions (eigenfunctions): φ 1 (z) = Hl(a,q,α,β,γ,δ;z) φ 2 (z) = z 1 γ Hl(a,q (γ 1)(aδ +ε),β γ +1,α γ +1,2 γ,δ;z) { or φ 3 (z) = lim φ 1 (z) c } 2(γ, ) γ 1 γ +1 φ 2(z)

Real-Time, Finite-Temperature AdS/CFT p. 18/34 { Solving Scalar EOM in AdS-Schwarzschild ( m 2 )φ(t, x,r) = 0 with = 1 g μ gg μν ν F w.r.t. (t, x) φ(t, x,r) φ(ω, k;u) ω= E 2πT, k= p 2πT u = ( r 0 ) 2 r f = 1 u 2 boundary: u = 0 horizon: u = 1 2 u 2 + u2 +1 u(u 2 1) u + ω 2 u(u 2 1) + k 2 2 u(u 2 1) + m 2 R 2 4u 2 (u 2 1) } φ(ω, k;u) = 0 φ(ω, k;u) = F(ω, k;u) φ(ω, k) F(ω, k;u) = 1 u 0 u 2 Δ 2 F(u): bulk-to-boundary propagator φ(ω, k): boundary field

Real-Time, Finite-Temperature AdS/CFT p. 19/34 AdS-Schwarzschild: Bulk-to-Boundary Propagators (1) i.e., 2 independent solutions of the EOM: [Starinets] F(ω, k;u) = N ω, k u 2 Δ 2 (1+u) ω 2 (1 u) iω 2 Hl(a,q,α,β,γ,δ;1 u) with α = β = ( 2 Δ 2 γ = 1 iω δ = 3 Δ a = 2 ( ) q = 4 2Δ+ Δ2 4 ) + 1 i 2 ω + [ 1 2 +( ) ] Δ 7 2 i ω i 2 ω2 ω 2 + k 2 The other is F (ω, k;u).

Real-Time, Finite-Temperature AdS/CFT p. 20/34 What isf(ω, k;u)? F(ω, k;u) F(ω +iε, k;u): = Retarded propagator analytic in UHP of ω Near-horizon u 1 behavior inside the F.T. F(ω, k;u) (1 u) iω 2 e iet F(ω, R2 ie(t+ 4r ln k;u) e r r 0 0 r ) 0 V iω = F(ω, incoming wave in R k;u) is outgoing wave in L F(ω, k;u) has supports only inside the future light-cone = Retarded propagator

Real-Time, Finite-Temperature AdS/CFT p. 21/34 Global Structure of AdS-Schwarzschild Poincaré (local) ds 2 = r2 f R 2 dt2 + R2 r 2 f dr2 Coordinate transformation: t K = βe αr2 2r r 0 sinhαt in R r K = βe αr2 2r r 0 coshαt t K = βe αr2 2r r 0 sinhαt in L r K = βe αr2 2r r 0 coshαt r = arctan ( ) ( r r 0 arctanh r0 ) r t K = V +U r K = V U Kruskal (global) ds 2 = e Λ(t K,r K ) ( dt 2 K +drk) 2 AdS-Sch Penrose diagram F U=0 L R P V=0

Real-Time, Finite-Temperature AdS/CFT p. 22/34 Prescriptions & Boundary Conditions Defined in Kruskal (global) coordinates Action integral: only in L & R quadrants, minus R L [Frolov & Martinez] Continuity across the horizon: [Unruh] f(u,v = 0) f(u = 0,V) Frequency selection: [Son & Herzog] Physical particles R: incoming L: outgoing

Real-Time, Finite-Temperature AdS/CFT p. 23/34 AdS-Schwarzschild: Bulk-to-Boundary Propagators (2) Bulk field (globally defined): φ R (t, x,r) in R φ(t K, x K,r k ) = φ L (t, x,r) in L d 4 { } p φ R (t, x,r) = f RR (ω, k;u) φ R (ω, k)+f RL (ω, k;u) φ L (ω, k) φ L (t, x,r) = (2π) 4eipx d 4 { } p f (2π) 4eipx LR (ω, k;u) φ R (ω, k)+f LL (ω, k;u) φ L (ω, k) Bulk-to-boundary propagators: Feynman & Wightman f RR (ω, k;u) = e2πω e 2πω 1 F(ω, k;u) 1 e 2πω 1 F (ω, k;u) f RL (ω, [ k;u) = eπω e 2πω 1 F(ω, k;u)+f (ω, ] k;u) f LR (ω, [ k;u) = eπω e 2πω 1 F(ω, k;u) F (ω, ] k;u) f LL (ω, k;u) = 1 e 2πω 1 F(ω, k;u)+ e2πω e 2πω 1 F (ω, k;u) where e 2πω = e E/T - Boltzman factor [Son & Herzog]

Real-Time, Finite-Temperature AdS/CFT p. 24/34 S[φ] = Structure of On-Shell Action d 5 x gl[φ] R L d 5 x gl[φ] L 0 = μ φ μ φ+m 2 φ 2, L pert = λ 3! φ3 Substitute 0 th order EOM solution into the action S cl [φ] = d 4 x gg uu φ u φ + λ dud 4 x gφ 3 R L u 0 3! R L S cl [ φ] = d 4 p 1 d 4 p 2 δ (4) (p 1 +p 2 ) f ij1 (p 1 ;u) gg uu u f ij2 (p 2 ;u) (p 1 ) φ j2 (p 2 ) ij u 0 φj1 + λ 1 du g d 4 p 1 d 4 p 2 d 4 p 3 δ (4) (p 1 +p 2 +p 3 ) f ij1 (p 1 ;u)f ij2 (p 2 ;u) 3! 0 ij f ij3 (p 3 ;u) φ j1 (p 1 ) φ j2 (p 2 ) φ j3 (p 3 ) where i,j = R,L Next: to take functional derivative w.r.t. φ.

Real-Time, Finite-Temperature AdS/CFT p. 25/34 AdS-Schwarzschid: 2-Point Functions Define: G ij (p 1,p 2 ) = ( ) i+j δ 2 S cl [ φ] δ φ i (p 1 )δ φ j (p 2 ) φ 0 where i,j = 1,2 (1 R, 2 L) 2-point functions: F(u) F(p;u) = F(ω, k;u) [Son & Herzog] [ G 11 (p) = N2 8π 2 R gg uu e 2πω 3 e 2πω 1 F(u) uf(u) 1 e 2πω 1 F (u) u F (u)] G 12 (p) = N2 8π 2 R gg uu e πω 3 e 2πω 1 [ F(u) uf(u)+f (u) u F (u)] u 0 u 0 G 21 (p) = G 12 (p) G 22 (p) = N2 8π 2 R gg uu 3 [ 1 e 2πω 1 F(u) uf(u) e2πω e 2πω 1 F (u) u F (u)] u 0 Same structure as the propagators: Retarded vs. Feynman & Wightman?

Real-Time, Finite-Temperature AdS/CFT p. 26/34 AdS-Schwarzschid: 3-Point Functions Define: G ijk (p 1,p 2,p 3 ) = ( ) i+j+k δ 3 S cl [ φ] δ φ i (p 1 )δ φ j (p 2 )δ φ k (p 3 ) φ 0 where i,j,k = 1,2; (1 R, 2 L) 3-point functions: G ijk (p 1,p 2,p 3 ) = ( ) i+j+k N 2 16π 2 R 3λδ(4) (p 1 +p 2 p 3 ) du g 0 [ ] f Ri (p 1 ;u)f Rj (p 2 ;u)f Rk (p 3 ;u) f Li (p 1 ;u)f Lj (p 2 ;u)f Lk (p 3 ;u) 1 1 = ( ) i+j+k N 2 16π 2 R 3λδ(4) (p 1 +p 2 p 3 ) du g 0 [ n ( e 2πω 1,e 2πω 2,e ) ] 2πω 3 F(p 1 ;u)f(p 2 ;u)f(p 3 ;u)+ Tree-level diagrams & Circling rules

Real-Time, Finite-Temperature AdS/CFT p. 27/34 Schwinger-Keldysh Formalism for Field Theory at T = 0 Real-time formalism Complex t-contour C, doubler fields S = dtl C e β ˆH it ˆH e Generating function Z[J 1,J 2 ] = Dφe is+i d 4 xj 1 O 1 i d 4 xj 2 O 2 n-point functions G (n) ab... ({x i}) = ( i) n 1 δ n lnz[j 1,J 2 ] δj a (x 1 )δj b (x 2 )... (a,b... = 1,2)

Real-Time, Finite-Temperature AdS/CFT p. 28/34 Mapping AdS-Sch Structure to Schwinger-Keldysh L & R structure S-K contour 2 boundaries 2 real time paths R = 1, L = 2 Fields: right real, left doubler φ R = J 1, φl = J 2 φ R = O 1, φ L = O 2 n-point function n n matrices match, e.g.: G RLRR... = G 1211... Hawking temperature imaginary contour period σ = β 2

Real-Time, Finite-Temperature AdS/CFT p. 29/34 2-Point Function: Matrix & Causal Schwinger-Keldysh 2-point function marix has the following properties: KMS (Kubo-Martin-Schwinger) relation G 11(p) = G 22 (p), G 12(p) = G 21 (p) Largest time identity (πω = E 2T ) G 11 e πω G 12 e πω G 21 +G 22 = 0 Feynman, Wightman & Causal G 11 (p) = G F (p), G 12 (p) = e πω G (p), G R (p) = G 11 e πω G 12 G 11 (p) = RG R (p)+icoth(πω)ig R (p) G 12 (p) = icsch(πω)ig R (p) G R (p) = N2 gg uu 8π 2 R 3 F(ω, k;u) u F(ω, k;u) u 0

Real-Time, Finite-Temperature AdS/CFT p. 30/34 3-Point Function: Matrix & Causal Schwinger-Keldysh 3-point function marix has the following properties: KMS relation G 111(p) = G 222 (p), G 121(p) = G 212 (p) Largest time identity (p 3 = p 1 +p 2 ) G 111 e πω 1 G 211 e πω 2 G 121 +e πω 3 G 221 +e πω 1 G 122 +e πω 2 G 212 e πω 3 G 112 G 222 = 0 Feynman & Wightman G 111 ({x}) = G F ({x}) = T e β ˆHO(x 1 )O(x 2 )O(x 3 ), Causal [Kobes] G R ({p}) = G 111 e πω 1 G 211 e πω 2 G 121 +e πω 3 G 221 G R ({x}) = θ(t 3 > t 2 > t 1 ) e β ˆH[[O(x 3 ),O(x 2 )],O(x 1 )] θ(t 3 > t 1 > t 2 ) e β ˆH[[O(x 3 ),O(x 1 )],O(x 2 )]

Real-Time, Finite-Temperature AdS/CFT p. 31/34 Causal 3-Point Function G R ({p}) = G 111 e πω 1 G 211 e πω 2 G 121 +e πω 3 G 221 =? G 111 ( ( G 121 e ω 2 π ( (? G e (ω 1 + ω 2 ) π 221 G211 e ω π 1

Real-Time, Finite-Temperature AdS/CFT p. 32/34 Causal 3-Point Function G R ({p}) = G 111 e πω 1 G 211 e πω 2 G 121 +e πω 3 G 221 ( ( 1 2 G 111 G 121 e ω 2 π λ ( ( 3 G e (ω 1 + ω 2 ) π 221 G211 e ω π 1 1 G R ({p}) = λn2 16π 2 R 3δ(4) (p 1 +p 2 p 3 ) du gf(p 1 ;u)f(p 2 ;u)f(p 3 ;u) 0 G R ({x}) λ dud 4 y gf(x 1 y;u)f(x 2 y;u)f (x 3 y;u)

Real-Time, Finite-Temperature AdS/CFT p. 33/34 Conclusions & Outlook Conclusions Strongly-coupled FT Weakly-coupled gravity Finite-temperature Black Hole in gravity Mapping of everything: built & checked 2-pt & 3-pt functions: calculated (in terms of Heun s function) Causal correlators: very simple structures in gravity dual Future Work More realistic calculations (such as T μν ) Non-relativistic, scale-invariant FT gravity dual

Real-Time, Finite-Temperature AdS/CFT p. 34/34