Lectures on gauge-gravity duality Annamaria Sinkovics Department of Applied Mathematics and Theoretical Physics Cambridge University Tihany, 25 August 2009
1. Review of AdS/CFT i. D-branes: open and closed string picture ii. AdS/CFT conjecture iii. generating function and correlators II. Extensions and applications i. Finite T and thermal aspects ii. Wilson-loop iii. Confinement-deconfinement transition and QCD iv. Non-relativistic CFTs
Open-closed duality open closed open loop closed propagator b.c. on open strings: σ=π σ Polyakov action S = 1 4πα δs = 1 2πα M =0,..., D 1 d 2 σ a x M a x M d 2 σ a x M a δx M σ=0 τ σ x M δx M σ =0, π =0
D-branes in open string picture b.c. Neumann Dirichlet σ x M =0 δx M =0 x M = c M = fixed Dp-branes: σ x µ =0 x i = c i µ =0,..., p i = p +1,..., D 1 D0-brane: particle D1-brane: string D2-brane: membrane... D9-brane: space-filling brane superstrings: carries RR-charge IIA-strings: even Dp-branes IIB-strings: odd Dp-branes
D-brane action DB1 action S DBI = T p d p+1 ξe ϕ det(γ ab +2πα F ab + B ab ) Induced metric γ ab = xm ξ a ϕ : dilaton x N ξ b g MN a =0... p B : B-field Low-energy expansion: α 0 ( S = (2πα ) 2 T p d D+1 ξ e ϕ 1 4 F abf ab }{{} YM theory on brane + 1 2 aφ i a φ i +... )
D-branes from closed strings massive charged objects in II string theory S d 10 x g look for solutions ( e 2ϕ ( R + 4( ϕ) 2) F p+2 = N S 8 p 2 (8 p)! F 2 p+2 ) 3-brane solution ds 2 = H 1/2 dx µ dx µ + H 1/2 (dr 2 + r 2 dω 2 s) H = 1 + R4 r 4, D-brane and p-brane are same objects R4 =4πg s α 2 N
Near-horizon limit r 0 AdS5 S 5 ds 2 = r2 R 2 dx µ dx µ + R 2 dr2 r 2 + R2 dω 2 5 AdS/CFT: AdS5 x S 5 N =4 SYM II string theory on AdS5 S 5
Parameters g S closed string theory on AdS5 : string coupling l S : string scale R : radius of AdS5 and S 5 SUGRA valid: R 4 =4πg s α 2 N string pert. theory valid: N λ =4πg S N = R4 α 2 = R l S λ 1 g S 1 N 1 g s 1/N open SYM : rank of gauge group : coupling const. g YM g 2 YM =4πg S ( R l S α 1/ λ λ = g 2 YMN ) 4 strongly coupled field theory large N field theory
Generating function string theory on AdS5 S 5 N =4 SYM theory S(g µν,a (4), ϕ,...) Φ(x, r) boundary coupling: d 4 x Φ(x)O(x) CFT operators on boundary O(x) Poincare coordinates: generating function: Z gauge = ds 2 = R2 z 2 (dx µ dx µ + dz 2 ) boundary: z 0 R e d 4 x Φ(x)O(x) = Z string (Φ(x, z) z=0 = Φ(x)) CFT
SUGRA regime Z string (Φ) =e I SUGRA(Φ) eg. field in AdS5 operator in CFT d 4 x g(g µν T µν + A µ J µ + ϕf µν F µν +... ) match of symmetries bosonic SO(2,4) SO(6) (supergroup PSU(2,4,4)) massless spectrum of string theory BPS ops in gauge theory massive string modes non-bps-operators R/l s (g 2 YMN) 1/4
Correlators - 1 eg. massive scalar field in AdS I(φ) = d 4 x dz g( µ φ µ φ + m 2 φ 2 ) ( 2 m 2 )φ =0 ds 2 = R2 z 2 (dx µdx µ + dz 2 ) 1 ( ) z 3 µ µ 1 φ + z z 3 zφ m 2 R2 z 5 φ =0 z 0 φ z a a(a 4) m 2 R 2 =0 m 2 R 2 = a(a 4)
Correlators - 2 m 2 = a(a 4) a ± =2± 4+m 2 R 2 a + = 4, a =4 near boundary z 0 z 4 dominates renormalized b.c.: φ(x, ɛ) =ɛ 4 φ 0 (x) coordinate rescaling source in generating function x λx, z λz φ(x, z) : no change φ 0 (x) : dim 4, O : = 2 + 4+m 2 R 2
Two-point functions Conformal invariance: Bulk to boundary propagator: O (x)o (x ) cδ K (z, x, x ) (x y) 2 ( 2 x m 2 )K (z, x, x )=0 in bulk at boundary, K (z, x, x ) z 4 δ(x x ) z 0 K (z, x, x )=c z (z 2 +(x x ) 2 ) (m=0) 2 AdS = 1 ( ) z 3 µ µ 1 + z z 2 m2 R 2 z 5
Solution of Laplace equation 1 φ(x, z) = d 4 x K (z, x, x )φ 0 (x ) generating function: Z gauge = φ(x, z) =c φ(x, ɛ) =ɛ 4 φ 0 (x) d 4 x R e d 4 x φ 0 (x)o(x) CFT z (z 2 +(x x ) 2 ) φ 0(x ) renormalized b.c. = Z string ( φ(ɛ,x)=ɛ 4 φ 0 (x) ) lim ɛ 0 δ 2 δφ 0 (x)δφ 0 (x ) : O(x)O(x ) =...
Solution of Laplace equation 1I S = = = Z string = e S SUGRA = lim ɛ 0 d 5 x g µ φ µ φ b.c.φ0 (x) d 5 x µ ( gφ µ φ) φ µ ( g µ φ) d 5 x µ ( gφ µ φ)+ d 4 x ( gφ z φ) z=ɛ e.om g = R 5 z 5 g zz = z2 R 2
Two-point function from SUGRA S = lim d 4 x 1 ɛ 0 ɛ 3 φ(x, ɛ) zφ(x, z) z=ɛ lim φ(x, z) ɛ 0 =ɛ4 φ 0 (x) φ(x, z) =c d 4 x z [z 2 +(x x ) 2 ] φ 0(x ) S = c d 4 xd 4 x 1 (x x ) 2 φ 0(x)φ 0 (x ) O(x)O(x ) 1 (x x ) 2 as expected from CFT
Higher correlators S = ( φ) 2 + m 2 φ 2 + bφ 3 Diagrammatic representation (Witten diagrams) O(x 1 ) O(x 2 ) int. vertex φ 3 = O(x 3 ) d 5 x K(x 2, x, z) : bulk to boundary propagator d 4 x 1 d 4 x 2 d 4 x 3 K 1 (x 1, x, z) K 2 (x 2, x, z)k 3 (x 3, x, z)φ 0 (x 1 )φ 0 (x 2 )φ 0 (x 3 ) O(x 1 )O(x 2 )O(x 3 ) = dz = d 4 xk 1 (x 1, x, z)k 2 (x 2, x, z)k 3 (x 3, x, z) z 5
Higher correlators - 2 O(x 1 )O(x 2 )O(x 3 ) c 1 2 3 (g 5,N) x 1 x 2 1+ 2 3 x1 x 3 1+ 3 2 x2 x 3 2+ 3 1 in agreement with CFT Non-renormalization theorem: c 1 2 3 (g S,N) AdS = c 1 2 3 (g 2 YM,N) SYM Compare SYM : SUGRA AdS : g YM 1 N, λ
Finite T AdS/CFT black hole physics phase structure of SYM theories QCD, hydrodynamics, condensed matter... SYM: (periodic imaginary time) S 1 R 3 T = 1 2πR gravity: near-extremal 3-brane solution ds 2 = H 1/2 (r)( f(r)dt 2 + dx i dx i ) +H 1/2 (r)(f 1 (r)dr 2 + r 2 dω 2 s) i =1, 2, 3 H(r) = 1 + R4 r 4 R 4 =4πg S α 2 N f(r) = 1 r4 0 r 4
Black 3-brane Decoupling limit: ds 2 = r R r2 R 2 ( + R2 r 2 ( 1 r4 0 r 4 ( 1 r4 0 r 4 Exercise: compute Hawking T ) 1 dt 2 + dx i dx i ) ) 1 dr 2 + R 2 dω 2 5 horizon at r = r 0 Find coord. singularity at horizon. Removable with suitable choice of coordinates (i.e. only if Euclidean time is periodic). β = 1 T = πr2 r 0 T H = r 0 πr 2
Bekenstein-Hawking entropy ds 2 = r2 R 2 ( ( 1 r4 0 r 4 ) dt 2 + dx i dx i ) + R2 r 2 ( 1 r4 0 r 4 ) 1 dr 2 A = ( r0 R volume of S 5 ) 3 {}}{ V3 R 5 π 3 = T 3 R 8 V 3 π 6 +R 2 dω 2 5 spatial volume of D3-branes T = r 0 πr 2 S BH = A 4G 10 holography G 10 : 10D Newton constant
YM entropy S BH = cf. A = π2 4G 10 2 N 2 V 3 T 3 R 4 =4πNg s α 2 G 10 =8π 6 g 2 sα 4 Agreement up to a factor 4/3! S corr = 2π2 3 f(λ)n 2 V 3 T 3 leading corrections: f(λ) = 1 3 2π 2 λ +... field theory: f(λ) = 4 3 + 45 32 S = 2π2 α corrections: α 3 R 4 +... ζ(3) λ 3/2 +... 3 N 2 V 3 T 3 λ λ diagrammatic methods in pert. finite T field theory leading α correction in SUGRA
Wilson loop ( ( )) W (C) = Tr P exp i C A path ordering contour order parameter for confining-deconfining phase transition confining phase: W (C) exp( σa C ) area law computes quark-antiquark potential ex. show this taking a rectangular loop T L V (L) = lim T log W (C)
D-brane picture }{{} N D-branes 1 D-brane U(N + 1) U(N) U(1) open massive string states massive quarks in gauge theory non-dynamical external quark geometric picture: string worldsheet in AdS
Wilson loop in AdS/CFT S NG = 1 2πα string chooses minimal area in flat 5d spacetime, surface of minimal area with boundary C would lie in boundary dτ dσ det(g MN a x M a x N ) AdS metric ds 2 = r2 R 2 dx µ dx µ + R 2 dr2 r 2 string stretching in AdS energetically favored quark-antiquark potential can be computed from geometry
Exercise: compute the quark-antiquark potential for a rectangular loop 1.write NG-action for AdS-space 2.extremize to find minimal area 3.regularize area by a cutoff near boundary 4.find energy of two separated quark λ E = V (L) L QCD string fundametal string in AdS!
Compactification confining example: ds 2 AdS 5 = R2 z 2 N =4 (1 z4 z 4 0 ) SYM on R 2,1 S 1 R 0 dt 2 + dx i dx i + dz 2 ( 1 z4 z0 4 ) double Wick rotation t = iy y = y +2πR 0 x 3 = it ds 2 = R2 z 2 [ dt 2 + dx 2 1 + dx 2 2 + (1 z4 z 4 0 ) dy 2 + dz2 1 z4 z 4 0 ] boundary: R 2,1 S 1 as required z 0 =2R 0
Mass gap ds 2 = R2 z 2 space terminates warp factor [ dt 2 + dx 2 1 + dx 2 2 + z z 0 w(z) w(z 0 )= R z 0 (1 z4 z 4 0 ) dy 2 + dz2 1 z4 z 4 0 AdS-soliton ] UV/IR correspondence x µ YM = 1 w(z) xµ proper E YM = w(z)e proper = R z 0 E proper UV in field theory IR in gravity theory area law and mass gap
Finite T finite T: YM theory on R 2 S 1 β S 1 R 0 gravity theory: ds 2 s = R2 z 2 2. black brane ds 2 bb = R2 z 2 [ [ ( 1 z4 1. AdS-soliton dτ 2 + dx 2 1 + dx 2 2 + z 4 0 (1 z4 z 4 0 ) dy 2 + dz2 1 z4 z 4 0 ) dτ 2 + dx 21 + dx 22 + dy 2 + dz2 (1 z4 z 4 0 ] ) ] But Lorentzian different! τ y β 2πR 0
Confinement-deconfinement transition P.I. sum over geometries with same boundary dominated by lowest action (infinite volume) phase transition: β c =2πR 0 (T c =1/2πR 0 ) (Hawking-Page phase transition) field theory: confinement-deconfinement phase transition T < T c : T > T c : AdS-soliton, confined, N-independent spectrum Black-brane, deconfined, O(N 2 ) states not real QCD: KK-modes do not decouple Λ QCD 1 R 0
AdS/CFT and QCD advantage: can examine QCD-like theories (but not QCD...) minus: qualitative geometric interpretation of Wilson-loop confinement-deconfinement transition easy to compute at finite T cf. hydrodynamics and quark-gluon plasma N, λ : handles strong coupling, large N region exact solutions, integrability weakly coupled gauge theory highly curved string background at weak coupling, KK-modes important
Non-relativistic CFTs invariant under Galilean transformation invariant under non-relativistic scale invariance many non-relativistic CFTs govern physical systems in condensed matter physics e.g. fermions at unitarity gravity duals? holographic dictonary?
Symmetry algebra rotations {M ij } translations {P i } Galilean boost {K i } time translations {H} dilatations {D} dynamical exponent : t λ z t, x λ x [D, M ij ]=0, [D, P i ]=ip i, [D, H] =izh [D, K i ]=i(1 z)k i [D, N] =i(2 z)n [P i,k j ]= δ ij N N : z =2: Schrodinger-algebra number operator (eg. fermion number) special conformal transformations C [D, C] = 2iC [H, C] = id [D, N] = 0 i, j =1... d
Dual geometry ( ds 2 = R 2 dt2 r 2z + dxi dx i +2dξdt r 2 + dr2 r 2 ) i =1... d scaling x λx t λ z t ξ λ 2 z ξ Galilean boost if ξ = ξ + 1 2 (2 v x v2 t), x = x vt N = i ξ ex. Take a scalar field on this background. Find relevant scaling dimension and two point function in NR boundary theory.
String theory embedding to quantize N, ξ ξ +2πL ξ DLCQ Null Melvin twist: sequence of boost, T-dualities, and twist extremal D3-brane solution (σ σ + αdy) Schrodinger geometry, d=2, z=2 dual field theory is N =4 SYM twisted by an R-charge SU(4) SU(3) U(1) non-extremal D3-brane solution finite T black hole, asymptotically Schrodinger thermodynamics, shear viscosity
Summary AdS/CFT based on open-closed duality holography large N expansion range of applicability from fundamental string questions to condensed matter systems...