Introduction to the AdS/CFT correspondence

Horatiu Nastase IFT-UNESP

Strings at Dunes, Natal July 2016

1 Plan:

•Lecture 1. Conformal field theories, gauge theories and nonperturbative issues

•Lecture 2. Strings and Anti-de Sitter space

•Lecture 3. The AdS/CFT map and gauge/gravity duality

2 Lecture 1

Conformal field theories, gauge theories

and nonperturbative issues

3 1. Conformal field theories: relativistic . ′ •Classical scale invariance: xµ → xµ = λxµ.

•QM scale invariance → usually conformal invariance. β fct. =0.

•Conformal transformation of flat space: x′(x) such that 2 ′ ′µ −2 µ ds = dxµdx = [Ω(x)] dxµdx . •Is an invariance of flat space under transformations.

•Conformal invariance is not general coord. invariance → in 2d, ↔ ∃ of generalization that is diff. and Weyl invariant ∫ ∫ 2 2 d z(∂µϕ) = dzdz∂ϕ¯ ∂ϕ¯ inv. ∫ ∫ d2z m2ϕ2 = dzdz¯ m2ϕ2

4 •So: conformal transf. = generalization of scale transf. of flat

.space that change distance between points by local factor.

•Infinitesimal: ′ xµ = xµ + vµ(x); Ω(x) = 1 − σν(x) 1 ∂µvν + ∂νvµ = 2σνδµν ⇒ σν = ∂ · v d •d = 2 Euclidean: ds2 = dzdz¯ ⇒ Most general solution is holo- morphic transformation z′ = f(z), ′ ′ ′ ′ − ds2 = dz dz¯ = f (z)f¯(¯z)dzdz¯ ≡ [Ω(z, z¯)] 2dzdz¯ •d > 2: most general solution is 2 vµ(x) = aµ + ωµνxν + λxµ + bµx − 2xµb · x σν(x) = λ − 2b · x

5 •Pµ ↔ aµ,Jµν ↔ ωµν: ISO(d − 1, 1) (Poincar´e). Also D ↔ λ dilatation; Kµ ↔ bµ special conformal . •Form generators of SO(2, d)   ¯ ¯ Jµν Jµ,d+1 Jµ,d+2  ¯  JMN = −Jν,d+1 0 D  ¯ −Jν,d+2 −D 0 Kµ − Pµ Kµ + Pµ J¯ = , J¯ = , J¯ = 0. µ,d+1 2 µ,d+2 2 d+1,d+2 •Symmetry group of AdSd+1 : CFT on Minkd = gravity in AdSd+1?

′ 2 2 •Obs.: Inversion I : xµ = xµ/x ⇒ Ω(x) = x is conformal also. I & rotation & translation ⇒ all finite conformal: e.g. xµ → λxµ and special conformal xµ + bµx2 µ → x 2 2 1 + 2xνbν + b x

6 d=2 conformal fields and correlators

•. → ′ Covariant GR tensor: under zi zi(z), ⃗z = (z1, z2), ′j ′j ′ ∂z 1 ∂z n Ti ...i (z1, z2) = Tj ...j (z1, z2) ... 1 n 1 n ∂zi1 ∂zin is generalized to primary field (tensor operator) of CFT, of dimensions (h, ˜h) ( ) ( ) ′ h ′ ˜h (h,˜h) ′ dz dz¯ ϕ (z, z¯) ≡ T ¯ ¯ = T z...zz...z z...zz...¯ z¯ dz dz¯ •Operator product expansion (OPE) → in any QFT, ∑ k Oi(xi)Oj(xj) = C ij(xi − xj)Ok(xj) k •In CFT, conformal invariance gives for n-point correlators ( ) Cδ ∑ Ck x + x ⟨O (x )O (x ) = ij ⇒ ⟨O (x )O (x )...⟩ = ij ⟨O i j ...⟩ i i j j 2∆ i i j j ∆ +∆ −∆ k |xi − xj| i |xi − xj| i j k 2 k

7 •Know ALL OPEs → can solve CFT for correlators. •Symmetry algebra is infinite dimensional. Energy momentum tensor (not primary!) decomposes as . ∑ ∑ Lm L˜m Tzz(z) = , T˜ (¯z) = m+2 z¯z¯ ¯m+2 m∈Z z m∈Z z Then we have the Virasoro algebra c 3 [Lm,Ln] = (m − n)L + (m − m)δ − m+n 12 m, n † and idem for L˜m. Here Lm = L−m and c=central charge. •{L0,L1,L−1} form a closed subalgebra: Sl(2, C).

•Representations: given by ”highest weight state” |h⟩. In CFT, h ∃ operator-state correspondence: |h⟩ = limz→0 ϕ (z)|0⟩: primary field. L−n: gives ”descendants”:

L0|h⟩ = h|h⟩; Ln|h⟩ = 0,L0(L−n|h⟩) = (h + n)(L−n|h⟩)

8 d > 2 conformal fields and correlators •Eigenfcts. of D with eigenv.−i∆ (like L0 and ”energy” of state)

′ ∆ . ϕ(x) → ϕ (x) = λ ϕ(λx). † Then Kµ ∼ Ln, Pµ ∼ L−n: a and a : create representations

[D,Pµ] = −iPµ ⇒ D(Pµ|ϕ) = −i(∆ + 1)(Pµϕ) [D,Kµ] = +iKµ ⇒ D(Kµ|ϕ) = −i(∆ − 1)(Kµϕ) •Inversion generates conf. transf. Defined by orthog. matrix ′µ ∂x ′ 2 Rµν(x) = Ω(x) ⇒ for x = xµ/x , ∂xν µ 2xµxν Rµν(x) ≡ Iµν(x) = δµν − x2 •For correlators, 3 point functions of scalars. C ⟨O (x)O (y)O (z)⟩ = ijk i j k ∆ +∆ −∆ ∆ +∆ −∆ ∆ +∆ −∆ |x − y| i j k |y − z| j k i |z − x| k i j amd currents → analyze properties under inversion ab δ Iµν(x − y) ⟨Ja(x)Jb(y)⟩ = C µ ν |x − y|2(d−1)

9 •QCD at high energies ≃ conformal. . •Condensed matter systems: Euclidean CFT appears near criti- cal point. Correlation length → ∞. ( )−α ( )−γ ( )−ν T − Tc T − Tc m T − Tc C ∝ ; χm ∝ ; ξ ∝ → ∞ Tc Tc Tc •Near Tc : conformal field theory: correlation functions defined by ∑ lat lat lat −1 lat lat −βH({s}) ⟨ϕ1 (r1)ϕ2 (r2)...ϕn (rn)⟩ = Z ϕ1 (r1)...ϕn (rn)⟩e {s} and take scaling limit a → 0 with ξ fixed, such that   ∏n  1  lat lat lat ⟨ϕ1(r1)...ϕn(rn)⟩ = lim ⟨ϕ1 (r1)ϕ2 (r2)...ϕn (rn)⟩ a→0 ∆i i=1 a •So: critical point: relativistic. 10 2. Conformal field theories: non-relativistic

•Condensed matter → ∃ also nonrelativistic scaling near ”Lifshitz . points”: t → λzt, ⃗x → λ⃗x, z= dynamical critical exponent. e.g. Lifshitz field theory with z = 2, ∫ d 2 2 2 2 L = d xdt[(∂tϕ) − k (∇⃗ ϕ) ] •Lifshitz algebra generated by Poincar´egenerators i j H = −i∂t; Pi = −i∂i; Mij = −i(x ∂j − x ∂i) and generator of scaling transformations i D = −i(zt∂t + x ∂i) •Algebra is

[D,H] = izH, [D,Pi] = iPi, [D,Mij] = 0, [Pi,Pj] = 0, i − j [Mij,Pk] = i(δkPj δkPi), [Mij,Mkl] = i[δikMjl − δjkMil − δilMjk + δjlMik]

11 •Larger algebra: conformal Galilean algebra, e.g. cold atoms and fermions at unitarity.

•. ∃ ”Galilean boosts” (nonrelativistic version of boosts) and con- served rest mass (particle number) N.

•Represent them by introducing direction ξ. i D = −i(zt∂t + x ∂i + (2 − z)ξ∂ξ) i Ki = −i(x ∂ξ − t∂i); N = −i∂ξ and the rest, same. Then, algebra = one before, plus

[D,Ki] = (1 − z)iKi [D,N] = (2 − z)iN, [D,H] = ziH, [D,Pi] = iPi [Ki,Pj] = iδijN, [Ki,H] = −iPi, [Ki,Mij] = i(δjkKi − δikKj) and rest zero. For z = 2, ∃ special conformal generator C for Schr¨odingeralgebra

[C,D] = +2iC, [C,H] = −iD, [C,Pi] = −iKi, [C,Mij] = [C,Ki] = 0.

12 3. Gauge theory

a •∃ gauge field Aµ, with field strength . 1 µ ν µ a F = dA + gA ∧ A; F = Fµνdx ∧ dx ,A = Aµdx ,Aµ = A Ta 2 µ c where [Ta,Tb] = fab Tc and gauge invariance a a a a b c δAµ = (Dµϵ) = ∂µϵ + gf bcAµϵ •The field strength transforms covariantly under a finite transf.

a U(x) = egλ (x)Ta, ϵa → λa ′ −1 Fµν = U (x)FµνU(x) •Couple to fermions and scalars. In Euclidean space, ∫ E E 4 ¯ ∗ µ S = SA + d x[ψ(D/ + m)ψ + (Dµϕ) D ϕ] µ where D/ ≡ Dµγ and − ⇒ a a Dµ = ∂µ ieAµ (Dµ)ijδij∂µ + g(TR)ijAµ(x).

13 •Green’s functions (correlation functions) from partition function = generating functional (in Euclidean space) ∫ ∫ −S [ϕ]+i ddxJ(x)ϕ(x) ZE[J] = Dϕe E . by

δ δ (E) (E) Gn (x1, ..., xn) = ... Z [J] ∫δJ(x1) δJ(xn) J=0 −S [ϕ] = Dϕe E ϕ(x1)...ϕ(xn) •Can be generalized to existence of composite operators, e.g. gauge invariant operators in gauge theory O(x), ∫ ∫ −S + ddxO(x)J(x) ZO[J] = Dϕe E giving correlation functions

δ δ ⟨O O ⟩ (x1)... (xn) = ... ZO[J] ∫δJ(x1) δJ(xn) J=0 −S [Φ] = Dϕe E O(x1)...O(xn)

14 •Noether theorem: (global) symmetry ↔ current L a,µ ∂ a i j j = (T ) jϕ ∂(∂µϕ) . µ,a ′ Classically, ∂µj = 0. Quantum mechanically, if Dϕ = Dϕ , ∫ µ a −SE[ϕ] µ a ⟨∂ jµ⟩ = 0 = Dϕe ∂ jµ(x) = 0 •If Dϕ ≠ Dϕ′ → ∃ anomaly.

•Chiral anomaly: ψ(x) → eiαγ5ψ(x), ψ¯(x) → eiαγ5ψ¯(x), 5 ¯ jµ = ψγµγ5ψ ⇒ e 1 ⟨∂µj5⟩ = (2d) ϵµνF ext µ 4π 2 µν e2 1 = (4d) ϵµνρσF extF ext 16π2 2 µν ρσ • i a a i j ⇒ Global nonabelian: δψ = ϵ (T ) jψ

a i a 1 + γ5 j j = ψ¯ γµ(T ) ψ . µ ij 2

15 4. Nonperturbative issues .

•Strong coupling: difficult. In particular, correlators (n-point functions). Also for currents n ∫ ∫ δ − µ ⟨ a1µa( ) anµn( )⟩ = D(fields) SE+ j Aµ j x1 ...j xn a1 an e δAµ1...δAµn(xn) •QCD: conformal at E ≫ ΛQCD. Also, toy model, N = 4 SYM a aI a[IJ] (4 susies): {Aµ, ψ ,X , I,J = 1, ..., 4. SU(4) = SO(6) global symmetry. Is exactly conformal (β = 0). From KK dimensional reduction of N = 1 SYM, ∫ [ ] 1 1 S = d10xTr − F F MN − ¯λΓM D λ 4 MN 2 M •CFT easy to define in Euclidean space. But Minkowski? Sub- tle.

16 •Instantons: nonperturbartive Euclidean solution. ∫ ∫ [ ] . 1 1 1 S = d4x(F a )2 = d4x F a ∗ F aµν + (F a − ∗F a )2 4g2 µν 4g2 µν 8g2 µν µν a a •Only in Euclidean space Fµν = ∗Fµν has real solutions, and 2 2 then S = Sinst. = 8π /g n. Solution a 2 η (x − x )ν a = µν i Aµ 2 2 g g(x − xi) + ρ a a aij a a a − a where ηµν=’t Hooft symbol, ηij = ϵ , ηi4 = δi , ϵ4i = δi .

•Nonperturbative physics: in Wilson loops, [ ∫ ] µ W [C] = Tr P exp i Aµdx

17 •For C= reactangle T × R, for T → ∞, we have . − ⟨W [C]⟩ ∝ e TVqq¯(R)

•Vqq¯(R)= qq¯ potential. If V ∼ σR ⇒ confinement. In CFT, Vqq¯(R) ∼ α/R.

•Finite temperature: ∫ ( ) −S [ϕ] −βHˆ ZE[β] = Dϕe E = Tr e ϕ(⃗x,tE+β)=ϕ(⃗x,tE) •N = 4 SYM at finite T similar to QCD at finite T . Universality? Finite T QCD: at RHIC, LHC → strongly coupled plasma → like N 2 → ∞ finite T = 4 SYM at gYM .

18 Lecture 2

Strings and Anti-de Sitter space

19 1. AdS space • − −1 Maximally symmetric spaces in signature (1, d 1): Rµν 2gµνR = Λgµν. Minkowski, dS, AdS. Cosmological const. Λ = 0, > 0, < 0. •Sphere (Euclidean signature): embed

. ∑d−1 2 2 2 2 ds = +dX0 + dXi + dXd+1 i=1 ∑d−1 2 2 2 2 R = +X0 + Xi + Xd+1 i=1 •de Sitter (Minkowski signature): embed ∑d−1 2 − 2 2 2 ds = dX0 + dXi + dXd+1 i=1 ∑d−1 2 − 2 2 2 R = X0 + Xi + Xd+1 i=1 •Anti-de Sitter (Minkowski signature): embed ∑d−1 2 − 2 2 − 2 ds = dX0 + dXi dXd+1 i=1 ∑d−1 − 2 − 2 2 − 2 R = X0 + Xi Xd+1 i=1 20

•Poincar´ecoordinates (Poincar´epatch) [ ( ) ] d−2 ∑ du2 ds2 = R2 u2 −dt2 + dx2 + i u2 ( i=1 ) d−2 . R2 ∑ = −dt2 + dx2 + dx2 x2 i 0 0 i=1 −y •Here u = 1/x0. If x0/R = e , ”warped metric”   d∑−2 2 2y − 2 2 2 2 ds = e dt + dxi + R dy i=1 2 •Light ray: ds = 0 at constant xi ⇒ ∫ ∫ ∞ − t = dt = R e ydy < ∞. •→ light takes finite time to reach boundary: can reflect back: This is the only patch of a global space (universal cover)

1,d−1 •Its boundary at x0 = ϵ: R :   − R2 d∑2 2 = − 2 + 2 ds 2 dt dxi ϵ i=1 21

•Global coordinates (whole space): 2 2 − 2 2 2 2 ⃗ 2 . dsd = R ( cosh ρdτ + dρ + sinh ρdΩd−2) similar to sphere 2 2 2 2 2 2 ⃗ 2 ≡ ⃗ 2 dsd = R (cos ρdτ + dρ + sin ρdΩd−2) dΩd •Also by tan θ = sinh ρ ⇒ 2 2 R 2 2 2 2 ds = (−dτ + dθ + sin θdΩ⃗ − ) d cos2 θ d 2 •Boundary of space: θ = π/2 − ϵ ⇒ 2 2 R 2 2 2 ds = (−dτ + sin θdΩ⃗ − ) ϵ2 d 2 •Analytical continuation to Euclidean signature: AdSd → EAdSd. But then, boundary Poincar´evs. global is radial time continua- tion. d∑−2 2 2 2 2 ⃗ 2 2tE 2 ⃗ 2 ds = dtE + dxi = dρ˜+ρ ˜ dΩd−2 = e (dτE + dΩd−2). i=1 22

Penrose diagram .

•Flat space, under ( ) τ + θ u = t  x = tanu ˜ = tan ⇒ 2 2 − 2 2 2 ⃗ 2 ds = dt + dr + r dΩd−2 1 = (− 2 + 2 + sin2 Ω⃗ 2 ) 2 2 dτ dθ θd d−2 4 cos u˜+ cos u˜− ⃗ where |τ θ| ≤ π, θ ≥ 0 ⇒ (τ, θ, Ωd−2) form triangle of revolution.

• 2 AdS space: same, for Poincar´epatch (drop 1/x0).

•Global space: extend to full cylinder. Boundary of global space: cylinder Rt × Sd−2. Related to boundary for conformal patch by conformal transformation by e2tE. 23 2. Holography in AdS space . •In AdS space, the boundary is a finite time away ⇒ natural observables on the boundary → take boundary sources ϕ0(⃗x), and field ∫ 4 ′ ′ ′ ϕ(⃗x,x0) = d ⃗x KB(⃗x,x0; ⃗x )ϕ0(⃗x )

where KB is the bulk-to-boundary propagator, satisfying 2 − 2 ′ d − ′ ( x,x0 m )KB(⃗x,x0; ⃗x ) = δ (⃗x ⃗x ) [ ]∆ ′ Γ(∆) x K (⃗x,x ; ⃗x ) = 0 B,∆ 0 d/2 2 − ′ 2 π Γ (∆ − d/2) x0 + (⃗x ⃗x ) •Massless scalar kinetic on-shell action ∫ ∫ ∫ 1 4 ′ 4 ′ 5 √ ′ ′ µ ′ ′ Sϕ = d xd⃗x d ⃗y d x gϕ0(⃗x )∂µ⃗x,x KB(⃗x,x0; ⃗x )∂ KB(⃗x,x0; ⃗y )ϕ0(⃗y ) 2 0 ⃗x,x0 ∫ ′ ′ C d ϕ0(⃗x )ϕ0(⃗y ) = d dd⃗x′dd⃗y′ 2 |⃗x − ⃗y′|2d

24 •In general, on-shell action for boundary sources defines some

.holographic quantities (on the boundary)

•Bulk-to-bulk propagator: (2 − m2)G(x, y) = −√1 δd+1(x − y) x gy is (ν ≡ m2R2 + d2/4) ∫ d d/2 d k i⃗k·(⃗x−⃗y) < > G(x, y) = (x0y0) e Iν(kx )Kν(kx ) (2π)d 0 0 Constructed out of the two solutions of (2 − m2)Φ = 0,

∝ i⃗k·⃗x d/2 ⃗ ∼ ∆− − Φ e x0 Kν(kx0)ϕ0(k) x0 (non normalizable) ∝ i⃗k·⃗x d/2 ⃗ ∼ ∆+ e x0 Iν(kx0)ϕ0(k) x0 (normalizable) •Lorentzian signature: (Poincar´e)solution

 ∝ i⃗k·⃗x d/2 | | Φ e x0 Jν( k x0) + ∼ ∆+ but now Φ , normalizable mode ( x0 ) is finite in center. 25 3. String theory •String theory: generalize QFT in worldline particle formalism. Action ∫ √ dxµ dXν S = −m dτ − η → . 1 µν ∫ √ dτ dτ ∫ [ ] v2 mv2 → −mc2 dt 1 − ≃ dt −mc2 + c2 2 ( ) • µ ⇒ d dXµ Equation of motion δ/δX dτ m dτ = 0. Free particle. •Couple to background fields by adding ∫ ( ) ∫ µ ρ dX 4 ρ µ ρ dτAµ(X (τ)) q ≡ d xAµ(X (τ))j (X (τ)) dτ √ •First order particle action in terms of einbein e(τ) = −γττ (τ), ∫ ( ) µ ν 1 −1 dX dX 2 SP = dτ e (τ) ηµν − em 2 dτ dτ •Can put m = 0 and choose gauge e(τ) = 1 for reparametriza- tion invariance, but then e(τ) equation of motion is constraint

dXµ dXν ηµν ≡ T = 0. dτ dτ 26

•Nambu-Goto action → generalization of S1: . ∫ √ 1 µ ν a S = − dτdσ − det (∂aX ∂ X gµν(X(ξ ))) NG 2πα′ b •In det, induced metric on worldsheet for (σ, τ).

•Polyakov action → generalization of SP : ∫ √ 1 ab µ ν S = − dτdσ −γγ ∂aX ∂ X ηµν P 4πα′ b •It has invariances under: -spacetime Poincar’e -worldsheet diff. (with X′µ(σ′, τ ′) = Xµ(σ, τ)) ′µ µ ′ 2ω(σ,τ) -Weyl invariance X (σ, τ) = X (σ, τ) and γab(σ, τ) = e γab(σ, τ).

27 •Boundary conditions: -closed strings (periodic in σ ∼ σ + 2π) or -open strings: Neumann ∂σXµ(τ, 0) = ∂σXµ(τ, l) = 0 or .Dirichlet δXµ(τ, σ = 0.l) = 0.

•Fix a gauge, e.g. conformal gauge γab = ηab. Residual invari- ance is conformal invariance.

•Then, equation of motion is 2Xµ(σ, τ) = 0 ⇒ Xµ = Xµ(τ − µ R σ) + XL(τ + σ) (left- and right- moving modes). •Constraints: Tab = 0 give T+−+ = 0 and T−− = 0 → Virasoro constraints (generated by Ln and L˜n).

•Spectrum: closed string √ µ ′ ′ ∑ x α i 2α 1 − − Xµ = + pµ(τ − σ) + αµe in(τ σ) R 2 2 2 n n √ n=0̸ µ ′ ′ ∑ x α i 2α 1 − Xµ = + pµ(τ + σ) + α˜µe in(τ+σ) L 2 2 2 n n=0̸ n

28 •Hamiltonian ∑ 1 µ µ . H = α− α 2 n n n∈Z • µ µ | ⟩ i Spectrum: act with α−n, α˜−n on 0 . But, only α−n physical. •No quantum anomalies (spacetime Lorentz, Weyl, BRST) ⇒ di- mension (number of scalars Xµ) is D = 26 for (bosonic) string. •Supersymmetry → introduce fermions ⇒ D = 10 for super- string. •String theory has ∞ number of modes ↔ worldline particles ↔ µ ∈ N fields. (from α−n, n )

•Background fields: from (massless) modes of (super)string. Closed, bosonic (gµν,Bµν, ϕ) ⇒ ∫ 1 √ − 2 − ab µ ν ρ S = ′ d σ[ γγ ∂aX ∂bX gµν(X ) 4πα √ ′ ab µ ν ρ ′ (2) ρ +α ϵ ∂aX dbX Bµν(X ) − α −γR Φ(X )] •It also contains nonperturbative objects: D-branes. 29 4. AdS as a limit of D-branes . •D-branes = enpoints of strings with D−(p+1) Dirichlet bound- ary conditions δXµ(τ, σ = 0, l) = 0. Wall with p + 1 dimensions.

•A graviton δgµν (or δϕ, etc.) can hit wall and excite modes that live on it → gives action v u ( ) ∫ u µ ν p+1 −ϕt ∂X ∂X ′ ′ S = −Tp d ξe − det (gµν + α Bµν) + 2πα F +S P ∂ξa ∂ξb ab WZ •But: Dp-branes = (same masses and charges as) p-brane so- lutions of supergravity (= low energy of string theory) 2 −1/2 − 2 2 1/2 2 2 2 dsstring = Hp ( dt + d⃗xp) + Hp (dr + r dΩ8−p) p−3 −2ϕ 2 e = Hp 1 − A = − (H 1 − 1) 01...p 2 p

30 •Here the harmonic function is . 4 2CpQp R Hp = 1 + ≡ 1 + r7−p r4 4 ′2 and colorGreenR = 4πgsNα .

• 2 −ϕ/2 2 Note: ”string frame”. ”Einstein frame” dsE = e dss .

•D3-branes (p = 3) for r → 0: H ≃ R4/r4 ⇒

r2 R2 ds2 = (−dt2 + d⃗x2) + dr2 + R2dΩ2 R2 3 r2 5 − 2 2 2 2 dt + d⃗x3 + dx0 2 2 = R 2 + R dΩ5 x0 5 •⇒ AdS5 × S space! Therefore AdS space appears from N (N → ∞) D-branes, in the near-horizon limit r → 0.

31 5. String theory in AdS space

•Supergravity (low energy) limit: . 5 5 •On AdS5 × S → KK reduction on S → gauged supergravity in AdS5 background.

•∃ cosmological constant, SO(6)= symmetry group of S5 is gauged (local), with coupling g ≠ 0.

5 •We have also solitonic objects, e.g. D-instantons in AdS5 × S , − 1 charged under a = a∞ + e ϕ − , with gs 24pi x4x˜2 eϕ = g + 0 0 + ... s 2 2 | − |2 4 N [˜x0 + ⃗x ⃗xa ] •Also Dp-brane that can wrap cycles in geometry → e.g. D5 on 5 5 S for AdS5 × S .

•We also have long (classical) strings that can end on D-branes

32 •If the D-brane is on boundary at infinity, and string ends on .a contour C, the string stretches into AdS by gravity, and is stopped by tension: 2 ′ r ds2 = α (−dt2 + d⃗x2) + ... R2 2 ∼ (1 + 2VNewton)(−dt + ...) •Quantum strings in AdS space → hard to quantize: highly nonlinear worldsheet action.

•e.g. in embedding space for S3 ⊂ S5, ∫ ∫ R2 2π ∑4 − 0 2 i 2 S = ′ dτ dσ[ (∂aX ) + (daX ) ] 0 4πα i=1 ∑ i i where i X X = 1. → is actually nonlinear.

33 5 •Exception: Penrose limit (near null geodesic in AdS5 × S ) . − ds2 = −2dx+dx − µ2(⃗r2 + ⃗y2)(dx+)2 + d⃗y2 + d⃗r2 •Polyakov string action ∫ ∫ 1 l 1√ − − ab − + − S = ′ dσ dτ γγ [ 2∂aX ∂βX 2πα 0 2 − 2 2 + + i i µ Xi ∂aX ∂bX + ∂aX ∂bX ] •In light-cone gauge, X+(σ, τ) = τ, ∫ ∫ [ ] 1 l 1 µ − ab i i 2 S = ′ dτ dσ η ∂aX ∂bX + Xi 2πα 0 2 2

34 Lecture 3

The AdS/CFT map and gauge/gravity duality

35 1. AdS/CFT and state-operator map . 5 •We have seen that AdS5 × S appears in the near-horizon of N D3-branes, and AdS space is likely holographic.

•More precise: two open strings on D3 collide: closed string peels off into bulk as Hawking radiation ⇒ relation between bulk gravity theory and D3-brane field theory in decoupling limit of D-branes.

•So: SU(N) at large N N = 4 SYM = gravity theory at r → 0, ′ for α → 0 (for no string worldsheet corrections) and gs → 0 (for 5 no quantum string corrections): AdS5 × S .

• 2 4 ′2 More precisely: λ = gYM N = R /α large and fixed.

•In fact, duality is believed to hold for all gs and N. 36 • 2 2 4 ′2 Map: couplings 4πgs = gYM , λ = gYM N = R /α . •SO(6) R-symmetry ↔ SO(6) gauge symmetry in AdS5 ↔ isom- etry of S5 on which we KK reduce.

•Conformal symmetry SO(4, 2) ↔ isometry of AdS5.

•Supergravity modes couple to (are sources for) boundary gauge invariant operators. e.g. scalar ϕ with KK expansion (on S5) ∑ ∑ ( ) = In ( ) In ( ) ϕ x, y ϕ(n) x Y(n) y n In then In couples with operator OIn of dimension ϕ(n) (n) √ d d2 ∆ = + + m2R2 2 4

37 •Gravity dual: string theory in the background (U ≡ r/α′) [ ( )] . √ U2 dU2 2 = ′ √ (− 2 + 2) + 4 + Ω2 ds α dt d⃗x3 πgsN 2 d 5 4πgsN U ′2 F5 = 16πgsα N(1 + ∗)ϵ(5) • 2 − 2 ⇒ ∼ ∆ Solve ( m )ϕ = 0 in background we obtain ϕ x0 ϕ0, where ϕ0= boundary source, and √ 2 d d 2 2 ∆ =  + m R 2 4 • ∆− d−∆ ∆+ Then x0 ϕ0 = x0 ϕ0 is non-normalizable mode, and x0 Φ0 is normalizable mode.

38 •In non-normalizable mode, ϕ0 = source for O = composite, gauge. invariant operator → need partition function ZO[ϕ0], ∫ ∫ −S+ O·ϕ ZO[ϕ0] = D[SYM]e 0 •Witten prescription for duality: partition function is same

ZO[ϕ0]CFT = Zϕ[ϕ0]string ′ •Moreover, in α → 0 limit, gs → 0 ⇒ classical, on-shell, super- gravity limit

−Ssugra[ϕ[ϕ0]] Zϕ[ϕ0] = e

•Ssugra is on-shell, for classical ϕ[ϕ0] depending on boundary value ϕ0, i.e., as we saw ∫ 4 ′ ′ ′ ϕ(⃗x,x0) = d ⃗x KB(⃗x,x0; ⃗x )ϕ0(⃗x )

39 Correlators

δn ⟨O O ⟩ (x1)... (xn) = ZO(ϕ0) δϕ (x )...δϕ (xn) 0 1 0 ϕ0=0 . 2 δ Ssugra[ϕ[ϕ0]] ⟨O(x1)O(x2)⟩ = − δϕ (x )δϕ (x ) 0 1 0 2 ϕ0=0 •Given the form of the quadratic part of Ssugra in lecture 2, we obtain C d ⟨O(x )O(x )⟩ = − d 1 2 2d |⃗x1 − ⃗x2| as required by CFT. •In general, ”Witten diagrams” from Ssugra[ϕ[ϕ0]]: tree (classi- cal) Feynman diagrams in x space with endpoints on boundary.

•Lorentzian case: ∃ good normalizable and non-normalizable ⇒ ↔ ∼ d−∆ modes normalizable modes VEVs (states). For ϕ αix0 + ∆ βix0 ,

H = HCFT + αiOi; ⟨βi|Oi|βi⟩ = βi + (αi piece)

40 Nonperturbative states .

5 •e.g. instanton maps to D-instantons in AdS5 × S , by ∫ δS δ 1 √ − 5 − µν = 2 d x gg ∂µϕ∂νϕ δϕ0(⃗x) δϕ0(⃗x)4πκ5 48 z˜4 = − 2 2 4 4πgs [˜z + |⃗x − ⃗xa| ] 1 ⟨ 2 ⟩ = 2 Tr [Fµν(⃗x)] 2gYM matches exactly.

•Also, D5-brane wrapping S5 ↔ baryon vertex operator in SU(N) SYM, for connecting N external quarks.

41 Wilson loops

•Nonperturbative gauge theory → encoded in Wilson loops •⟨W [C]⟩ encodes V (r). . qq¯ •In a susy theory, susy Wilson loop [I √ ] 1 µ I I µ 2 W [C] = Tr P exp (Aµx˙ + θ X (x ) x˙ )dτ N where Xµ(τ) parametrizes the loop, θI unit vector on S5. Then

− − ⟨W [C]⟩ = e Sstring[C] lϕ Here lϕ= renormalization → exract divergence: straight string forming parallelipiped.

•Non-susy loop can also be defined. •Calculation in AdS5 gives √ 2 2 4π 2gYM N V ¯(L) = − qq Γ(1/4)4 L

42

PP wave correspondence . 5 •Penrose limit of AdS5×S → worldsheet string is free (massive).

•In SYM, corresponds to large charge J (for U(1) ⊂ SO(6)R). Z = Φ5 + iΦ6 is charged under it, Φ1, ..., Φ4 aren’t. Vacuum of string: 1 |0, p+⟩ = √ Tr [ZJ] JNJ/2 •String states → insertions of Φ1, ..., Φ4, etc. inside the trace. e.g. ∑J † † + 1 1 3 l 4 J−l eπinl a a |0, p ⟩ = √ Tr [Φ Z ϕ Z ]e J n,4 −n,3 J/2 J l=1 N •∃ Hamiltonian acting on these → Hamiltonian of discretized string on the pp wave: only way to obtain quantum string. 43 2. Gauge/gravity duality .

5 •AdS5 × S best understood example. But ∃ other cases.

•Conformal: -N M2-branes and N M5-branes in 11d M-theory 7 4 (strong coupling string theory) ⇒ AdS4 × S and AdS7 × S . 5 -orbifolds/orientifolds of AdS5 × S → less susy. 4 - ABJM model in 2+1 dimensions: N M2-branes on C /Zk, IR 7 3 limit ↔ AdS4 × S /Z = AdS4 × CP . k k→∞

•Nonconformal, less (or no) susy: ”gravity dual backgrounds” → no AdS factor.

•Extra dimension r ↔ energy scale U. ∃ β function → running with U ↔ r ⇒ nontrivial metric as a function of r.

44 •To holographically simulate QCD-like gauge theory, need:

.-large N gauge theory (e.g. SU(N)): small string corrections -(flat) boundary for field theory , and sections of constant r ↔ U. -compact space Xn ↔ global symmetry of field theory -RG flow in r between low energy (low r) and high energy (large r).

•Map:

-Gauge invariant SYM operators (”glueballs”) ↔ sugra fields in gravity dual -Gauge invariant operators with ”quarks” (”mesons”) ↔ SYM fields on brane in gravity dual -Wavefunctions in field theory, e.g. eik·x ↔ wavefunctions in gravity dual (times wavefunctions on compact space), e.g. ik·x ϕ(x, U, Xm) = e ψ(U, Xm)

45 Phenomenological gauge/gravity duality . •QCD ”bottom-up” models or condensed matter models: no decoupling limit of brane theory.

•Instead, gravity dual with some field theory on it (not string theory) with right properties. (could be truncation of a low energy sugra limit of string theory)

•Could be nonrelativistic, e.g. Lifshitz system (add −iu∂u to dilatation D) ( ) dt2 d⃗x2 du2 ds2 = R2 − + + d+1 u2z u2 u2 or gravity dual of Schr¨odingeralgebra (add −iu∂u to D) ( ) dt2 d⃗x2 du2 2dt dξ ds2 = R2 − + + + d+2 u2z u2 u2 u2

46 3. Finite temperature . •For sQGP and condensedf matter applications, need finite tem- perature.

•Hawking radiation from Wick rotation of black hole ( ) 2 2 2MGN 2 dr 2 2 ds = + 1 − dτ + + r dΩ2 r − 2MGN 1 r 2 2 2 2 has conical singularity (ds ≃ A(dρ + ρ dτ )) near r = 2MGN , unless θ = τ/(4MGN ) has periodicity 2π ⇒ 1 ∂M TBH = ⇒ C = − < 0 8πMGN ∂T So it is not thermodynamically stable system.

•But a black hole in AdS space is. ∃C > 0 branch. 47 •For a black hole in AdS space, ( ) . r2 w M dr2 2 − − n 2 2 2 ds = + 1 − dt + 2 + r dΩn−2 R2 rn 2 r + 1 − wnM R2 rn−2 2 one finds (r is largest solution to r + 1 − wnM = 0) + R2 rn−2 nr + (n − 2)R2 = + T 2 4πR r+ •T (M) has a minimum, followed by a uniformly increasing branch → stable.

•Need to take also R · T → ∞ ⇒ M → ∞.

•More generally, put black hole in gravity dual ⇒ Put dual field theory at finite temperature.

48 4. AdS/CMT and transport

.•Strongly coupled CMT field theories → not easy to describe → phenomenological models. •Need finite temperature ⇒ black holes.

•For transport properties, need spectral functions → retarded Green’s functions GR . OAOB

•Linear response theory:

δ⟨O ⟩(ω, k) = GR (ω, k)δϕ (ω, k) A OAOB B(0) •Im GR (ω, k) is spectral functions for χ, since OAOB ∫ ′ R ′ +∞ dω ImGO O (ω , x ≡ R A B χ lim GO O (ω, x) = ′ ω→0+i0 A B −∞ π ω •To calculate GR holographically, prescription by Son and OAOB Starinets. 49 •For asymptotically AdS gravity dual with black hole ↔ event .horizon H, if

∫ z=z ddk H S = ϕ (−k)F(k, z)ϕ (k) (2π)d (0) (0) z=zB then, GR is (2∆−d) 2∆ − d δϕ norm GR(k) = − 2F(k, z)| = A A, zB R δϕB,norm. •Results: Kubo formulas: R ⃗ R ⃗ iG (ω, k) iGT T (ω, k) σ(ω,⃗k) = JxJx ; η(ω,⃗k) = xy xy ω ω •For gravity duals with black holes, generically one finds η/s = 1/(4π) (s = entropy density).

50