The AdS/CFT Correspondence PI It from Qubit Summer School: Mukund Rangamani

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1 Lecture The AS/CFT Corresponence PI It from Qubit Summer School: Mukun Rangamani Q. Large N expansion: Consier the following Lagrangian for a zero imensional fiel theory (matrix moel): L = (Tr (ΦΦ) + Tr (ΦΦΦ) + Tr (ΦΦΦΦ)) (.) g where Φ is an ajoint value fiel of the group U(N). (i) Write own the Feynman iagrams for this theory. (ii) Inspection of the Feynman iagrams shoul tell you that it is useful to regar the matrix inices of the ajoint fiel Φ explicitly. Writing Φ = φ a b, express the Feynman iagrams in a ouble line notation, where the two lines correspon to the upper an lower matrix inex of Φ. (iii) Draw the general Feynman iagram which contributes to the free energy of the matrix moel an show that the free energy of the theory has a perturbation expansion as a ouble sum: F = N g f g (λ), f g (λ) = α g,i λ i. (.) g=0 where g is the genus of the two imensional surface on which the Feynman iagram can be rawn. You will fin it useful to recall the Gauss-Bonnet theorem, which gives the Euler character χ of a graph (an hence its genus) in terms of the vertices V, eges E an faces F: χ = g = V E + F. (.3) (iv) Use the result (.) to show that in the t Hooft scaling limit: i=0 N with λ = fixe, (.4) the free energy is given in terms of just the planar iagrams i.e., g = 0. Compare this expansion with the string perturbation expansion. (v) Using this result argue that the large N limit effective classicalizes the theory, viz., eff N. (vi) Bonus: How woul you eal with multi-trace terms, viz., h Tr (ΦΦ) Tr (Φ Φ)?

2 a b a b b a c c b a c c Fig. : The stanar Feynman iagram for the interaction vertices of the Lagrangian (.). Fig. : The Feynman iagrams for the matrix moel (.) written in the ouble line notation. Soln #. The Feynman iagrams are given in Fig. an the ouble line notation, making clear the matrix structure is in Fig.. Inspection of the iagrams leas to a the state counting or at least a way to organize the perturbation expansion; see Fig. 3 for some typical low orer iagrams. Consier the large N theory where we take the following scaling limit of the parameters λ = g N fixe, with N (.5) YM In this limit, it is more useful to write the Lagrangian (.) as L matrix N λ Tr ( Φ + Φ 3 + Φ 4 + ) = N λ L (.6) where we have ensure that there are no stray powers of N in L. The N an λ epenence of the relevant parts are then: Propagators λ N Interaction vertices N λ (.7) Loops N

3 N λ N λ 3 N λ N 0 Fig. 3: Feynman iagrams for the free energy of the matrix moel (.) written in the ouble line notation. The top line is the planar iagrams an the bottom epicts a non-planar iagram One can prove the following using the ouble-line notation: Feynman graphs provie a iscretization (triangulation) of a two imensional surface, see Fig. 3. The ictionary between the triangulation an the Feynman iagram language can be summarize as Feynman vertices vertices of the triangulation (V) propagators eges of the triangulation (E) loops faces of the triangulation (F) It then follows that a given Feynman iagram contributes to the vacuum-vacuum amplitue as ( ) N V ( ) λ E F N F λ E V N V E+F (.8) λ N Using the well known fact that for a two-imensional surface the Euler character χ, which is a topological invariant ( R) is given by χ = V E + F = hanles bounary (.9) means that we can classify the powers of N via the topology of the Feynman iagram. In fact, for close surfaces without a bounary we have χ = g, g = genus (.0) 3

4 which implies that the expression for the free energy can be written as a ouble-sum F = g=0 N g i=0 α i,g λ i = N g f g (λ) (.) g=0 Feynman iagrams split into topological classes with the leaing contribution coming form the genus g = 0 i.e., iagrams with the sphere topology. These are the planar iagrams, see the top line of Fig. 3. By inspection we see that eff = N. For the ouble-traces convince yourself that h N owing to the extra factor in the secon trace. Q. AS + can be embee in R, as an hyperbolio X X X 0 = l AS, (.) with X = {X, X, X } are the stanar Cartesian coorinates on R. The metric on AS + is inuce from the stanar flat metric on R, with signature (, ). (i) Use this embeing to show that the metric on AS + can be written either in terms of the global coorinates: ) s = f(r) t + r f(r) + r Ω (, f(r) = + r. (.3) or in terms of the Poincaré coorinates l AS s = l AS z ( t + z + y ). (.4) (ii) Exploit the embeing to fin a coorinate transformation between the Poincaré coorinate (t, z, y i ) an the global coorinates (t, r, Ω i ), with Ω i representing the angles of the S. (iii) Discuss the region of the global AS spacetime covere by the Poincaré coorinates an the symmetries manifest in the two coorinate systems. (iv) Show that the bounary of global AS + is R S an that of the Poincaré patch is R,. (v) Bonus: Describe the Penrose iagram of AS + by embeing the spacetime into the Einstein Static Universe (the Lorentzian cyliner) starting from the global coorinates (.3). s = t + Ω. (.5) 4

5 (vi) Bonus : Fin coorinate charts which foliate the AS + spacetime in slices preserving (a) SO(, ) R: This give a foliation where the spatial leaves are Eucliean hyperbolic space an the bounary is a copy of the hyperbolic cyliner H R. (b) SO(, ): This foliates AS + in terms of AS slices. (c) SO(, ): This foliates the spacetime in e Sitter slices. Soln #. Global coorinates: To fin coorinates on AS + itself we just nee to solve for the coorinates X I which can be easily one for instance by: X = R cosh ρ cos τ, X 0 = R cosh ρ sin τ, X i = R sinh ρ Ω i (.6) where Ω i are unit vectors on S calle irection cosines an they satisfy Ω i Ω i =. Plugging in this into the metric it is easy to check that the canonical metric on AS + is given by s = l ( AS cosh ρ τ + ρ + sinh ρ Ω ) ( ) r = l + τ + r r AS + + r Ω (.7) l AS where we have mae a trivial coorinate change to rewrite the metric in the secon line. These coorinates are referre to as global coorinates on AS + since they provie a global coorinate chart. In fact, one can use them to infer the causal structure of the spacetime an raw a Penrose iagram. The bounary of the spacetime is at ρ or equivalently r. Bounary of AS: To unerstan the bounary of the AS + spacetime, efine tan θ = sinh ρ so that metric (.7) takes the form: s = l AS cos θ ( τ + θ + sin θ Ω ) (.8) By this change of coorinates the bounary of the spacetime at ρ = has been brought to finite coorinate istance θ = π. The metric (.8) is clearly conformal to the Einstein Static Universe (ESU), R S whose metric is the part in the parentheses. The part where the conformal factor blows up correspons to the bounary of the AS +. This happens at the equator of the S of the ESU, so we can think of AS + corresponing to half of the ESU. In fact, we can also infer from this construction what the conformal bounary of AS + is: it is a timelike hypersurface which itself is a lower imensional ESU: (AS + ) = R S (.9) 5

6 Poincare coorinates: There is another coorinate chart which we have alreay encountere an plays a role in the AS/CFT corresponence. This is the so calle Poincare coorinates which are given in terms of the coorinates of R, as follows: X = R u t X0 = + u `AS + x t u X i = R u xi, for i =,, X = u `AS x + t u (.0) It is then trivial to check that the flat metric on R, inuces the following metric on AS+ : u (.) s = `AS u t + u x + u which is precisely what we encountere from the near-horizon geometry of AS in (??). It is also conventional to efine u = z so that we have s = `AS `AS + z = xµ xµ + z t + x z z (.) Fig. 4: Illustration of Poincare coorinates superpose on conformally compactifie AS. The surfaces t = 0,, 5 (left), z =, 5 (mile), an x = 0,, 5 (right) are plotte, with colour-coing blue for 0, green for, an re for 5. To guie the eye, surperpose are also the bounary of the I+ CS corresponing to t = ± an the t = 0 bounary slice (black curves). 6

7 Fig. 4 from gives a plot of the constant Poincaré coorinates in the AS spacetime. The constant Poincaré time t surfaces (left plot) are all pinne at the re reference point i 0 CS (which in global coorinates correspons to τ = 0, φ = π, an r = ), with the t = ± surfaces coinciing with the Poincaré ege. The constant z surfaces (mile plot) interpolate from the I + CS at z = 0 to the Poincaré ege at z =, with the surfaces having roun crosssections at t = 0 slice, tangent to i 0 CS. Finally, the constant x slices (right plot) are pinne at i 0 CS, an interpolate from a section of φ = 0, π plane at x = 0 to half of the Poincaré ege at x = ±. Q3. Consier a Klein-Goron scalar fiel propagating in the global AS + geometry (.3) with Lagrangian ˆ L = + x g ( A φ A φ m φ ) (.3) Solve the equation of motion for the scalar fiel an show the eigenfrequencies ω associate with the Killing fiel ( t) A are quantize as in a harmonic oscillator ω n R = λ ± + l + n, n = 0,,, (.4) λ ± are relate to the imension of the spacetime an the mass of the fiel via λ ± = ± 4 + m l AS, (.5) an l labels the spherical harmonics on S. You shoul first argue that eigenstates of the scalar Laplacian on S are given by scalar spherical harmonics Y l (Ω ) with eigenvalues l (l + ) which is a natural generalization of stanar spherical harmonics on S an then procee to separate variables. Use the result for the eigenfrequencies to argue that scalar fiel in AS are allowe to have a negative mass square m < 0 provie m l AS 4. This boun on particle masses is known as the Breitenlohner-Freeman boun an particles violating this boun are tachyonic in AS +. Soln # 3. Consier the scalar wave equation in pure AS + where we have [ r The moe solutions are ( r ( + r ) ) ] + ω l (l + ) r r + r r ( ) φ(r) = 0 (.6) φ(r) = C r l ( + r ) ω F ( + l + ω +C r l ( + r ) ω F ( l + ω, l + + ω, + l, r ), l + + ω, 4 l, r )(.7) 4 7

8 Bounary conitions at r = 0 force C to zero an one has φ sol (r) = r l ( + r ) ω F ( + l + ω, l + + ω, + l, r ) r Γ( + ) Γ( ) Γ( ( + l ω)) Γ( ( + l + ω)) +r + Γ( + ) Γ( (l + ω)) Γ( (l + + ω)) (.8) Normalizablility now emans that the secon piece in the asymptotic expansion vanishes. One therefore has ω = s + + l (.9) leaing to φ sol (t, r, Ω) = e i (n+l+ ) t r l (+r ) n+l+ + l + n F (, l+ +n, +l, r ) Y l (Ω) (.30) Q4. The AS/CFT corresponence states that the generating function of connecte Green s functions in the gauge theory W CFT [J ] is given as (ˆ ) W CFT [J ] = log exp 4 x J (x) O(x) CF T (.3) extremum ( I grav [φ]) φ(z=ɛ)=j where I grav is the supergravity action evaluate on-shell subject to the specifie bounary conition on the fiels φ(z, x). Taking the bulk metric to be given by the Eucliean AS 5 metric ( s = l AS z z + ) 4 x i x i i= (.3) with an explicit cut-off at z = ɛ, an the bulk fiel to be a Klein-Goron scalar of mass m, evaluate the on-shell supergravity action ˆ S = N 5 x ( g ( φ) + ) m φ, (.33) subject to the aforementione bounary conition φ(z = ɛ, x) = J (x) by working in momentum space. From this show that the two-point function of a CFT operator O(x) whose source is J (x) has a two point function 8 4 O(x) O(y) = N ɛ Γ( + ) π Γ( ) x y (.34) 8

9 with = m l AS. Soln # 4. Moe ecomposing the scalar fiel as: we have the wave equation in AS 5 [ u 5 φ(z, x) = e i p x ϕ(pz), p z u, p x = p µ x µ (.35) u ( u 3 which has linearly inepenent solutions ϕ(u) = u I (u), ) ] u m l AS ϕ(u) = 0 (.36) u singular at the origin ϕ(u) = u K (u), regular at the origin (.37) with = m l AS. Imposing the bounary conition for the scalar that φ(ɛ, x) = φ 0 (x) = e i p x we fin φ(z, x) = (p z) K (p z) (p ɛ) K (p ɛ) ei p x (.38) We nee to plug in the solution into the action (.33) an evaluate it. We can simplify this computation by integrating the action by parts one term will yiel the bulk equation of motion which vanishes on-shell an we are left with a pure bounary term: ˆ S on-shell = 4 x g g zz φ(z, x) z φ(z, x) (.39) z=ɛ where we are using the fact that the bounary is at constant z = ɛ with our regulator. The two point function is then given by O(p) O(q) = W [ φ 0 = λ e i p x + λ e i q x] (.40) λ λ which yiels O(p) O(q) = N ɛ 8 ( 4) Γ(3 ) Γ( ) δ4 (p + q) λ =λ =0 ( ) p 4 + analytic (.4) where we have refraine from writing the terms that are analytic in (p ɛ). Fourier transforming this expression to position space yiels the CFT correlation function in a familiar form: O(x) O(y) = N ɛ 8 4 Γ( + ) π Γ( ) x y (.4) The ɛ 8 is an explicit cut-off epenence which reflects the wavefunction renormalization of the composite operator O an can be absorbe into it. The result (.4) is consistent with the CFT expectations ( an 3 point functions in CFTs are fixe by conformal War ientities). 9