The AdS/CFT Correspondence

PI It from Qubit Summer School: Mukund Rangamani

1 Lecture 1

Q1. Large N expansion: Consider the following Lagrangian for a zero dimensional field

theory (matrix model):

L =1

g2(Tr (ΦΦ) + Tr (ΦΦΦ) + Tr (ΦΦΦΦ)) (1.1)

where Φ is an adjoint valued field of the group U(N).

(i) Write down the Feynman diagrams for this theory.

(ii) Inspection of the Feynman diagrams should tell you that it is useful to regard the matrix

indices of the adjoint field Φ explicitly. Writing Φ = φab, express the Feynman diagrams

in a double line notation, where the two lines correspond to the upper and lower matrix

index of Φ.

(iii) Draw the general Feynman diagram which contributes to the free energy of the matrix

model and show that the free energy of the theory has a perturbation expansion as a

double sum:

F =∞∑g=0

N2−2 g fg(λ) , fg(λ) =∞∑i=0

αg,i λi . (1.2)

where g is the genus of the two dimensional surface on which the Feynman diagram can

be drawn. You will find it useful to recall the Gauss-Bonnet theorem, which gives the

Euler character χ of a graph (and hence its genus) in terms of the vertices V, edges E

and faces F:

χ = 2− 2 g = V − E + F . (1.3)

(iv) Use the result (1.2) to show that in the ’t Hooft scaling limit:

N →∞ with λ = fixed, (1.4)

the free energy is given in terms of just the planar diagrams 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 ∼ 1N .

(vi) Bonus: How would you deal with multi-trace terms, viz., hTr (ΦΦ) Tr (Φ Φ)?

1

a

a

b

a c c b a

ba

c

b

c

d

d

Fig. 1: The standard Feynman diagram for the interaction vertices of the Lagrangian (1.1).

Fig. 2: The Feynman diagrams for the matrix model (1.1) written in the double line notation.

Soln # 1.

The Feynman diagrams are given in Fig. 1 and the double line notation, making clear

the matrix structure is in Fig. 2.

Inspection of the diagrams leads to a the stated counting or at least a way to organize the

perturbation expansion; see Fig. 3 for some typical low order diagrams. Consider the large

N theory where we take the following scaling limit of the parameters

λ = g2YM

N fixed , with N →∞ (1.5)

In this limit, it is more useful to write the Lagrangian (1.1) as

Lmatrix ∼N

λTr(Φ2 + Φ3 + Φ4 + · · ·

)=N

λL (1.6)

where we have ensured that there are no stray powers of N in L. The N and λ dependence

of the relevant parts are then:

Propagators ∼ λ

N

Interaction vertices ∼ N

λ

Loops ∼ N

(1.7)

2

∝ N2 ∝ λN2 ∝ λ3N2

∝ λN0

Fig. 3: Feynman diagrams for the free energy of the matrix model (1.1) written in the double line

notation. The top line is the planar diagrams and the bottom depicts a non-planar diagram

One can prove the following using the double-line notation: Feynman graphs provide a

discretization (triangulation) of a two dimensional surface, see Fig. 3. The dictionary between

the triangulation and the Feynman diagram language can be summarized as

• Feynman vertices ←→ vertices of the triangulation (V)

• propagators ←→ edges of the triangulation (E)

• loops ←→ faces of the triangulation (F)

It then follows that a given Feynman diagram contributes to the vacuum-vacuum amplitude

as

F ∼(N

λ

)V ( λN

)ENF ∼ λE−V NV−E+F (1.8)

Using the well known fact that for a two-dimensional surface the Euler character χ, which is

a topological invariant (´R) is given by

χ = V − E + F = 2− 2 handles− boundary (1.9)

means that we can classify the powers of N via the topology of the Feynman diagram. In

fact, for closed surfaces without a boundary we have

χ = 2− 2 g , g = genus (1.10)

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which implies that the expression for the free energy can be written as a double-sum

F =

∞∑g=0

N2−2g∞∑i=0

αi,g λi =

∞∑g=0

N2−2g fg(λ) (1.11)

Feynman diagrams split into topological classes with the leading contribution coming form

the genus g = 0 i.e., diagrams with the sphere topology. These are the planar diagrams, see

the top line of Fig. 3.

By inspection we see that ~eff = 1N . For the double-traces convince yourself that h ∼ 1

N

owing to the extra factor in the second trace.

Q2. AdSd+1 can be embedded in Rd,2 as an hyperboliod

X2d −X2

−1 −X20 = −`2AdS , (1.12)

with Xd = X1, X2, · · ·Xd are the standard Cartesian coordinates on Rd. The metric on

AdSd+1 is induced from the standard flat metric on Rd,2 with signature (d, 2).

(i) Use this embedding to show that the metric on AdSd+1 can be written either in terms

of the global coordinates:

ds2 = −f(r) dt2 +dr2

f(r)+ r2 dΩ2

d−1 , f(r) =

(1 +

r2

`2AdS

). (1.13)

or in terms of the Poincare coordinates

ds2 =`2AdS

z2

(−dt2 + dz2 + dy2

d−1

). (1.14)

(ii) Exploit the embedding to find a coordinate transformation between the Poincare coor-

dinate (t, z, yi) and the global coordinates (t, r,Ωi), with Ωi representing the angles of

the Sd−1.

(iii) Discuss the region of the global AdS spacetime covered by the Poincare coordinates and

the symmetries manifest in the two coordinate systems.

(iv) Show that the boundary of global AdSd+1 is R × Sd−1 and that of the Poincare patch

is R1,d−1.

(v) Bonus: Describe the Penrose diagram of AdSd+1 by embedding the spacetime into the

Einstein Static Universe (the Lorentzian cylinder)

ds2 = −dt2 + dΩ2d . (1.15)

starting from the global coordinates (1.13).

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(vi) Bonus 2: Find coordinate charts which foliate the AdSd+1 spacetime in slices preserving

(a) SO(d, 1)×R: This give a foliation where the spatial leaves are Euclidean hyperbolic

space and the boundary is a copy of the hyperbolic cylinder Hd−1 × R.

(b) SO(d− 1, 2): This foliates AdSd+1 in terms of AdSd slices.

(c) SO(d, 1): This foliates the spacetime in de Sitterd slices.

Soln # 2.

Global coordinates: To find coordinates on AdSd+1 itself we just need to solve for the

coordinates XI which can be easily done for instance by:

X−1 = R cosh ρ cos τ , X0 = R cosh ρ sin τ , Xi = R sinh ρΩi (1.16)

where Ωi are unit vectors on Sd−1 called direction cosines and they satisfy Ωi Ωi = 1. Plugging

in this into the metric it is easy to check that the canonical metric on AdSd+1 is given by

ds2 = `2AdS

(− cosh2 ρ dτ2 + dρ2 + sinh2 ρ dΩ2

d−1

)= −

(r2

`2AdS

+ 1

)dτ2 +

dr2

r2

`2AdS+ 1

+ r2 dΩ2d−1 (1.17)

where we have made a trivial coordinate change to rewrite the metric in the second line.

These coordinates are referred to as global coordinates on AdSd+1 since they provide a global

coordinate chart. In fact, one can use them to infer the causal structure of the spacetime

and draw a ‘Penrose diagram’. The boundary of the spacetime is at ρ → ∞ or equivalently

r →∞.

Boundary of AdS: To understand the boundary of the AdSd+1 spacetime, define tan θ =

sinh ρ so that metric (1.17) takes the form:

ds2 =`2AdS

cos2 θ

(−dτ2 + dθ2 + sin2 θ dΩ2

d−1

)(1.18)

By this change of coordinates the boundary of the spacetime at ρ =∞ has been brought to

finite coordinate distance θ = π2 .

The metric (1.18) is clearly conformal to the Einstein Static Universe (ESU), R × Sd

whose metric is the part in the parentheses. The part where the conformal factor blows up

corresponds to the boundary of the AdSd+1. This happens at the equator of the Sd of the

ESU, so we can think of AdSd+1 corresponding to half of the ESU. In fact, we can also infer

from this construction what the conformal boundary of AdSd+1 is: it is a timelike hypersurface

which itself is a lower dimensional ESU:

∂ (AdSd+1) = R× Sd−1 (1.19)

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Poincare coordinates: There is another coordinate chart which we have already encoun-

tered and plays a role in the AdS/CFT correspondence. This is the so called Poincare

coordinates which are given in terms of the coordinates of Rd,2 as follows:

X−1 = Ru t

X0 =1

2u

(1 + u2

(`2AdS + x2

d−1 − t2))

Xi = Ruxi , for i = 1, · · · , d− 1

Xd =1

2u

(1− u2

(`2AdS − x2

d−1 + t2))

(1.20)

It is then trivial to check that the flat metric on Rd,2 induces the following metric on AdSd+1:

ds2 = `2AdS

(−u2 dt2 + u2 dx2

d−1 +du2

u2

)(1.21)

which is precisely what we encountered from the near-horizon geometry of AdS in (??). It is

also conventional to define u = 1z so that we have

ds2 =`2AdS

z2

(−dt2 + dx2

d−1 + dz2)

=`2AdS

z2

(dxµ dx

µ + dz2)

(1.22)

Fig. 4: Illustration of Poincare coordinates superposed on conformally compactified AdS. The surfaces

t = 0, 1, 5 (left), z = 1, 5 (middle), and x = 0, 1, 5 (right) are plotted, with colour-coding blue

for 0, green for 1, and red for 5. To guide the eye, surperposed are also the boundary of the I+CS

corresponding to t = ±∞ and the t = 0 boundary slice (black curves).

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Fig. 4 from gives a plot of the constant Poincare coordinates in the AdS spacetime. The

constant Poincare time t surfaces (left plot) are all pinned at the red reference point i0CS(which in global coordinates corresponds to τ = 0, φ = π, and r = ∞), with the t = ±∞surfaces coinciding with the Poincare edge. The constant z surfaces (middle plot) interpolate

from the I+CS at z = 0 to the Poincare edge at z =∞, with the surfaces having round cross-

sections at t = 0 slice, tangent to i0CS . Finally, the constant x slices (right plot) are pinned

at i0CS , and interpolate from a section of φ = 0, π plane at x = 0 to half of the Poincare edge

at x = ±∞.

Q3. Consider a Klein-Gordon scalar field propagating in the global AdSd+1 geometry (1.13)

with Lagrangian

L =

ˆdd+1x

√−g 1

2

(∂Aφ∂

Aφ−m2 φ2)

(1.23)

Solve the equation of motion for the scalar field and show the eigenfrequencies ω associated

with the Killing field(∂∂t

)Aare quantized as in a harmonic oscillator

ωnR = λ± + l + 2n , n = 0, 1, 2, · · · (1.24)

λ± are related to the dimension d of the spacetime and the mass of the field via

λ± =d

2±√d2

4+m2 `2AdS , (1.25)

and l labels the spherical harmonics on Sd−1.

You should first argue that eigenstates of the scalar Laplacian on Sd are given by scalar

spherical harmonics Yl(Ωd) with eigenvalues l (l + d− 1) which is a natural generalization of

standard spherical harmonics on S2 and then proceed to separate variables.

Use the result for the eigenfrequencies to argue that scalar field in AdS are allowed to

have a negative mass squared m2 < 0 provided m2 `2AdS ≥ −d2

4 . This bound on particle

masses is known as the Breitenlohner-Freedman bound and particles violating this bound are

tachyonic in AdSd+1.

Soln # 3.

Consider the scalar wave equation in pure AdSd+1 where we have[1

rd−1

d

dr

(rd−1 (1 + r2)

d

dr

)+

ω2

1 + r2− l (l + d− 2)

r2−∆(∆− d)

]φ(r) = 0 (1.26)

The mode solutions are

φ(r) = C1 rl (1 + r2)

ω2 F (

d+ l + ω −∆

2,l + ∆ + ω

2,d

2+ l,−r2)

+C2 r2−d−l (1 + r2)

ω2 F (

2− l + ω −∆

2,2− d− l + ∆ + ω

2,4− d− 2l

4,−r2)(1.27)

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Boundary conditions at r = 0 force C2 to zero and one has

φsol(r) = rl (1 + r2)ω2 F (

d+ l + ω −∆

2,l + ∆ + ω

2,d

2+ l,−r2)

→ r−∆ Γ(d

2+ 1)

Γ(d2 −∆)

Γ(12(d+ l −∆− ω)) Γ(1

2(d+ l −∆ + ω))

+r−d+∆ Γ(−d2 + ∆)

Γ(12(l + ∆− ω)) Γ(1

2(l + ∆ + ω))(1.28)

Normalizablility now demands that the second piece in the asymptotic expansion vanishes.

One therefore has

ω = 2 s+ ∆ + l (1.29)

leading to

φsol(t, r,Ω) = e−i (2n+l+∆) t rl (1+r2)2n+l+∆

2 F (d+ 2l + 2n

2, l+∆+n,

d

2+ l,−r2)Yl(Ω) (1.30)

Q4. The AdS/CFT correspondence states that the generating function of connected Green’s

functions in the gauge theory WCFT [J ] is given as

WCFT [J ] = − log〈 exp

(ˆd4xJ (x)O(x)

)〉CFT

' extremum ( Igrav [φ])

∣∣∣∣φ(z=ε)=J

(1.31)

where Igrav is the supergravity action evaluated on-shell subject to the specified boundary

condition on the fields φ(z, x). Taking the bulk metric to be given by the Euclidean AdS5

metric

ds2 =`2AdS

z2

(dz2 +

4∑i=1

dxi dxi

)(1.32)

with an explicit cut-off at z = ε, and the bulk field to be a Klein-Gordon scalar of mass m2,

evaluate the on-shell supergravity action

S = Nˆd5x√g

(1

2(∂φ)2 +

1

2m2 φ2

), (1.33)

subject to the aforementioned boundary condition φ(z = ε,x) = J (x) by working in momen-

tum 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

〈O(x)O(y) 〉 = N ε2∆−8 2∆− 4

∆

Γ(∆ + 1)

π2 Γ(∆− 2)

1

|x− y|2∆(1.34)

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with ∆ = 2 +√

4 +m2 `2AdS.

Soln # 4.

Mode decomposing the scalar field as:

φ(z, x) = ei p·x ϕ(pz) , p z ≡ u , p · x = pµxµ (1.35)

we have the wave equation in AdS5[u5 d

du

(1

u3

d

du

)− u2 −m2 `2AdS

]ϕ(u) = 0 (1.36)

which has linearly independent solutions

ϕ(u) = u2 I∆−2(u) , singular at the origin

ϕ(u) = u2K∆−2(u) , regular at the origin (1.37)

with ∆ = 2 +√

4 +m2 `2AdS. Imposing the boundary condition for the scalar that φ(ε, x) =

φ0(x) = ei p·x we find

φ(z, x) =(p z)2K∆−2(p z)

(p ε)2K∆−2(p ε)ei p·x (1.38)

We need to plug in the solution into the action (1.33) and evaluate it. We can simplify this

computation by integrating the action by parts – one term will yield the bulk equation of

motion which vanishes on-shell and we are left with a pure boundary term:

Son-shell =

ˆd4x√g gzz φ(z, x) ∂zφ(z, x)

∣∣∣∣z=ε

(1.39)

where we are using the fact that the boundary is at constant z = ε with our regulator.

The two point function is then given by

〈O(p)O(q) 〉 =∂2W

[φ0 = λ1 e

i p·x + λ2 ei q·x]

∂λ1 ∂λ2

∣∣∣∣λ1=λ2=0

(1.40)

which yields

〈O(p)O(q) 〉 = N ε2∆−8 (2 ∆− 4)Γ(3−∆)

Γ(∆− 1)δ4(p+ q)

(p2

2

)2∆−4

+ analytic (1.41)

where we have refrained from writing the terms that are analytic in (p ε)2. Fourier trans-

forming this expression to position space yields the CFT correlation function in a familiar

form:

〈O(x)O(y) 〉 = N ε2∆−8 2 ∆− 4

∆

Γ(∆ + 1)

π2 Γ(∆− 2)

1

|x− y|2∆(1.42)

The ε2∆−8 is an explicit cut-off dependence which reflects the wavefunction renormalization

of the composite operator O and can be absorbed into it. The result (1.42) is consistent

with the CFT expectations (2 and 3 point functions in CFTs are fixed by conformal Ward

identities).

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