Introduction to the Ads/CFT Correspondence Horatiu Nastase
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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 . 0 •Classical scale invariance: xµ ! xµ = λxµ. •QM scale invariance ! usually conformal invariance. β fct. =0. •Conformal transformation of flat space: x0(x) such that 2 0 0µ −2 µ ds = dxµdx = [Ω(x)] dxµdx : •Is an invariance of flat space under transformations. •Conformal invariance is not general coord. invariance ! in 2d, $ 9 of generalization that is diff. and Weyl invariant Z Z 2 2 d z(@µϕ) = dzdz@ϕ¯ @ϕ¯ inv: Z Z d2z m2ϕ2 = dzdz¯ m2ϕ2 4 •So: conformal transf. = generalization of scale transf. of flat .space that change distance between points by local factor. •Infinitesimal: 0 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 z0 = f(z), 0 0 0 0 − ds2 = dz dz¯ = f (z)f¯(¯z)dzdz¯ ≡ [Ω(z; z¯)] 2dzdz¯ •d > 2: most general solution is 2 vµ(x) = aµ + !µνxν + λxµ + bµx − 2xµb · x σν(x) = λ − 2b · x 5 •Pµ $ aµ;Jµν $ !µν: ISO(d − 1; 1) (Poincar´e). Also D $ λ dilatation; Kµ $ bµ special conformal . •Form generators of SO(2; d) 0 1 ¯ ¯ Jµν Jµ,d+1 Jµ,d+2 B ¯ C JMN = @−Jν;d+1 0 D A ¯ −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? 0 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 •. ! 0 Covariant GR tensor: under zi zi(z), ~z = (z1; z2), 0j 0j 0 @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) ! ! 0 h 0 ~h (h;~h) 0 dz dz¯ ϕ (z; z¯) ≡ T ¯ ¯ = T z:::zz:::z z:::zz:::¯ z¯ dz dz¯ •Operator product expansion (OPE) ! in any QFT, X k Oi(xi)Oj(xj) = C ij(xi − xj)Ok(xj) k •In CFT, conformal invariance gives for n-point correlators ( ) Cδ X Ck x + x hO (x )O (x ) = ij ) hO (x )O (x ):::i = ij hO i j :::i i i j j 2∆ i i j j ∆ +∆ −∆ k jxi − xjj i jxi − xjj 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 . X X Lm L~m Tzz(z) = ; T~ (¯z) = m+2 z¯z¯ ¯m+2 m2Z z m2Z z Then we have the Virasoro algebra c 3 [Lm;Ln] = (m − n)L + (m − m)δ − m+n 12 m; n y and idem for L~m. Here Lm = L−m and c=central charge. •fL0;L1;L−1g form a closed subalgebra: Sl(2; C). •Representations: given by "highest weight state" jhi. In CFT, h 9 operator-state correspondence: jhi = limz!0 ϕ (z)j0i: primary field. L−n: gives "descendants": L0jhi = hjhi; Lnjhi = 0;L0(L−njhi) = (h + n)(L−njhi) 8 d > 2 conformal fields and correlators •Eigenfcts. of D with eigenv.−i∆ (like L0 and "energy" of state) 0 ∆ . ϕ(x) ! ϕ (x) = λ ϕ(λx): y Then Kµ ∼ Ln, Pµ ∼ L−n: a and a : create representations [D; Pµ] = −iPµ ) D(Pµjϕ) = −i(∆ + 1)(Pµϕ) [D; Kµ] = +iKµ ) D(Kµjϕ) = −i(∆ − 1)(Kµϕ) •Inversion generates conf. transf. Defined by orthog. matrix 0µ @x 0 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 hO (x)O (y)O (z)i = ijk i j k ∆ +∆ −∆ ∆ +∆ −∆ ∆ +∆ −∆ jx − yj i j k jy − zj j k i jz − xj k i j amd currents ! analyze properties under inversion ab δ Iµν(x − y) hJa(x)Jb(y)i = C µ ν jx − yj2(d−1) 9 •QCD at high energies ' conformal. •Condensed matter systems: Euclidean CFT appears near criti- cal point. Correlation length ! 1. ( )−α ( )−γ ( )−ν T − Tc T − Tc m T − Tc C / ; χm / ; ξ / ! 1 Tc Tc Tc •Near Tc : conformal field theory: correlation functions defined by X lat lat lat −1 lat lat −βH(fsg) hϕ1 (r1)ϕ2 (r2)...ϕn (rn)i = Z ϕ1 (r1)...ϕn (rn)ie fsg and take scaling limit a ! 0 with ξ fixed, such that 0 1 Yn @ 1 A lat lat lat hϕ1(r1)...ϕn(rn)i = lim hϕ1 (r1)ϕ2 (r2)...ϕn (rn)i a!0 ∆i i=1 a •So: critical point: relativistic. 10 2. Conformal field theories: non-relativistic •Condensed matter ! 9 also nonrelativistic scaling near "Lifshitz . points": t ! λzt; ~x ! λ~x, z= dynamical critical exponent. e.g. Lifshitz field theory with z = 2, Z d 2 2 2 2 L = d xdt[(@tϕ) − k (r~ 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. •. 9 "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, 9 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 •9 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 0 −1 Fµν = U (x)FµνU(x) •Couple to fermions and scalars. In Euclidean space, Z E E 4 ¯ ∗ µ S = SA + d x[ (D= + m) + (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) Z R −S [ϕ]+i ddxJ(x)ϕ(x) ZE[J] = Dϕe E . by δ δ (E) (E) Gn (x1; :::; xn) = ::: Z [J] ZδJ(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), Z R −S + ddxO(x)J(x) ZO[J] = Dϕe E giving correlation functions δ δ hO O i (x1)::: (xn) = ::: ZO[J] ZδJ(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 0 Classically, @µj = 0. Quantum mechanically, if Dϕ = Dϕ , Z µ a −SE[ϕ] µ a h@ jµi = 0 = Dϕe @ jµ(x) = 0 •If Dϕ =6 Dϕ0 ! 9 anomaly. •Chiral anomaly: (x) ! eiαγ5 (x); ¯(x) ! eiαγ5 ¯(x), 5 ¯ jµ = γµγ5 ) e 1 h@µj5i = (2d) ϵµνF ext µ 4π 2 µν e2 1 = (4d) ϵµνρσF extF ext 16π2 2 µν ρσ • i a a i j ) Global nonabelian: δ = ϵ (T ) j a i a 1 + γ5 j j = ¯ γµ(T ) : µ ij 2 15 4. Nonperturbative issues . •Strong coupling: difficult. In particular, correlators (n-point functions). Also for currents n Z R δ − µ h a1µa( ) anµn( )i = 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): fAµ; ;X , I;J = 1; :::; 4. SU(4) = SO(6) global symmetry. Is exactly conformal (β = 0). From KK dimensional reduction of N = 1 SYM, Z [ ] 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. Z Z " # . 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 .