I. Quantum Field Theory and Gauge Theory

II. Conformal Field Theory

III. Short Introduction to Supersymmetry

IV. General Relativity

V. Some String Theory Introduction

VI. A hand-waving derivation of AdS/CFT

1

A hand-waving derivation of AdS/CFT

• Recall that Dp-branes are subspaces with(p+ 1)− dim worldvolumes, where open strings can end

• light open strings are short, only weakly excited, andare localized on and close to the branes,whose fluctuations they describe

• Dp branes carry their own charges (RR-charges, potentials Cp+1))

satisfying Dirac quantization (like electric charges in the presence ofmagnetic monopoles).

dCp+1 = Fp+2 = ∗F8−p = ∗dC7−p;

p = 3 : F5 =∗F5 self dual,

∫

S5F5 =

∫

S5

∗F5 = N

Some memory refreshing of differential forms:

2

In a D=4 world-volume magnetic 2-flux F (2) is dual toelectric 2-flux ∗F (2) (differ by exchange of electric and magnetic)∫

S2F2 = 0 (no magnetic monopoles enclosed by S2)

∫

S2

∗F2 = Q (Q = charge enclosed by S2)

In IIB string theory in a D=10 world-volume the RR-field F (5) isself-dual

(∗F (5))µνκλρ = ε µνκλρµνκλρ F

(5)

µνκλρ= F

(5)µνκλρ

• On the world-volume of N coincident p-branes, open stringsare free to begin and end on anyone of those;described by N ×N unitary matrices, forming a U(N)-group:a single U(1) = U(N)/SU(N)-factor describes collective motion,the remaining internal symmetry SU(N)− a Yang Mills theory;if all the N branes are split apart, open strings get massive (Higgsmechanism) and U(1)× SU(N) is broken down to U(1)N

3

• emission of RR-bosons and of gravitons

tree-diagram of N Dp branes, emitting a closed string (graviton)or, via an equivalent 1-loop-diagram,an open string (RR gauge boson)

disk-amplitude (with number of handles h=0, and boundaries b=1)∼ tension of brane ∼ α′−D/2Ng−2+2h+b

S ∼ α′−D/2Ng−1S

4

• back-reaction of N coincident Dp-branes on space-time:from Einstein’s equations in D = 10

Rµν −1

2gµν R = 8π GN︸︷︷︸

∼g2S

α′4

∼1gS

N α′−5

︷︸︸︷Tµν ∼ α′−1 gSN︸︷︷︸

≡λ/4π

– small ’t Hooft coupling λ: back-reaction negligible, black branemetamorphs to → Dp brane

then we have very weakly interacting gravitons (closed strings) ne-ar to and far from the braneand interacting gauge-bosons (weakly excited open strings) but on-ly on the brane

– large ’t Hooft coupling λ: now the back-reaction strong:

the N Dp-branes now form a black brane, (take r = 0 on horizon) ,collapsed transverse to and extended along original Dp branes.

5

Black brane metric for special case p=3

3+1 dim. world volume xµ, µ = 0 . . . 3

transverse coordinates yi, horizon at yi = 0, i = 1 . . . 6,

i.e. at r = 0, r2 =6∑

i=1

y2i

Form of metric

ds2 =ηµν

√

H(r)dxµ dxν +

√

H(r)6∑

i=1

dy2i

H(r) is a harmonic function of the transverse y-coordinates

6∑

i=1

dy2i = dr2 + r2dΩ25 in polar coordinates

H(r) = 1 +L4

r4

1 for r → ∞∼ L4

r4for r → 0

L4 = 4πgsNα′2

6

In addition there is the self-dual RR 5-form field F (5)(r)

in the 6-dimensional space transverse to the brane.

Its flux counts the number of 3-branes sourcing it.

∫

S5

F (5)(r) = N

7

Far from the brane, for r → ∞, there is theboundary of AdS5 × S5 at r ∼ L.It is reached by light in a finite time and therefore an excellent place fromwhich to observe what happens in AdS5.Beyond r ∼ L for r ≫ L the metric becomes D = 5 Minkowski.Close to the brane it becomes AdS5 × S5

r ≪ L ds2 =r2

L2ηµν dxµxν + L2 dr2

r2︸ ︷︷ ︸AdS5

+L2 dΩ25︸ ︷︷ ︸

S5

where the black brane itself hides behind the horizon, which is at r=0.(The boundary of the AdS, as purified in the present limit, would now beat r = ∞.)

Due to infinite redshift at the horizon, even arbitrarily high-energy andshort-wavelength excitations (which really means all of type IIB theory)become visible looking from the boundary on a sufficiently close neigh-bourhood of the horizon.

8

large ’t Hooft coupling λ small λ

(large back-reaction, big black-brane formation) (small back-reaction,N D3-branes)

Low-energy excitations

near brane :All excitations of string get open strings, N = 4

red-shifted to low energy for r → 0; SU(N) Super-Yang-Millsquantum-gravity (complete II B superstring-theory) gauge theory on D=4on AdS5 × S5 background. r → ∞ boundary of AdS5

far from brane: except for gravity only closed stringsultralow energy excitations cannotescape from brane to r → ∞(and dont fit into AdS5-throat of radius L)therefore only closed strings can remainfar from the brane.

Maldacena’s idea: subtract the closed strings from both sides and

9

equate what is left

⇒ AdS/CFT correspondence

For λ large: The very strongly interacting open strings problem on theboundary would thus become replacable by the moderately curved GRTdescription in the bulk of AdS.

For λ small: The highly curved (large inverse curvature radia) and ratherill-defined (quantum) gravity problem in the bulk becomes replacable bythe weakly interacting open strings problem without any gravity on theboundary.

10

Matching of parameters (following Polchinski):

• gauge theory (parameters g2Y M , N ):

g2YM = 4πgS YM coupling

L5

∫

S5

∗F (5) = L5

∫

S5

F5 = N integer because of Dirac quantization

• AdS: (parameters L (size of AdS and radius of S5),N number of branes, also integer

Rµν︸︷︷︸

∼L−2

= GN︸︷︷︸

∼g2S

α′4

F(5)µαβγδ F (5) αβγδ

ν︸ ︷︷ ︸

∼ N2

L10

⇒ L4

α′2= 4πgS︸ ︷︷ ︸

g2YM

N = λ ′t Hooft coupling

11

L/α′1/2 = (4πgN)1/4 = (g2YMN)1/4 = (λ)1/4

! λ very large for classical description !

and gravity parameters:Define a reduced Planck length LP,D

such that we have in D dimensions

SEinstein = (1/2LD−2P,D )

∫dDxR.

Then usual Planck length and the reduced one in D=4

LP,4 = (8π)(1/2)LP ;

In string theory L8P,10 = (1/2)(2π)7g2α′4

and L/LP,10 = 2−1/4π−5/8N1/4

! must be large for classical description ! (N very large)

12

Conjecture: Horowitz, Polchinski [gr-qc/0602037]

Hidden inside ‘any‘ non-abelian gauge theory is aquantum theory of gravityi.e. a theory with a massless spin-2 field (graviton)(in some respects like a composite of the gauge boson)

• A no-go theorem (Weinberg-Witten) seems to forbid this(QFT forbids massless particles with spin > 1 in non-abeliangauge theories) .But here this is circumvented becausegraviton and QFT live in different spaces

• This meshes well with the Holographic conjecture :The information of quantum gravity in a given spatial domain can bethought to reside in the boundary of that domain.

That’s because a quantum theory of gravity hasmaximum entropy ∼ (D-2)-dimensional area of boundary A

13

(from Bekenstein/Hawking entropy of black holes SBH = A

4GN)

Meaning of the extra dimension:

any local QFT has an additional dimension, in which it is local,the energy-scale z.Coupling constant(s) depend on the energy scale also in a local way viathe Callan/Symanzik equation(s)

z∂g(z)

∂z= β(g(z))

So r can be interpreted as the inverse (dimensions !) of the energy-scalez

r =1

zSince for r → 0 we look at energy-scale → ∞, all finite energyexcitations become low energy in comparison (just another way to lookat the redshift).

14

Dictionary of the correspondence:gauge field versus type II B string theory

• parameters g2YM = 4πgs, g2Y MN = L4

α′2

• correlation functions of observables (i.e. local gauge-invariantoperators O(~x), ~x = spacetime coords of the field-theory)

〈e∫d4~xφ0(~x)O(~x)〉CFT = Zstringφ(~x, r → ∞) = φ0(~x)

• Operators O(~x) correspond to bulk-fields φ(~x, r) in which the stringspropagate, with the same quantum numbers and symmetries, butotherwise remaining rather unspecified.

• The source of O(~x) is the required value of the bulk-field φ(~x, r) atthe boundary r → ∞

15

Strongest form of the conjecture:it holds for all values of gY M and N

A proof is not in sight, and certainly very difficult. But also notdisproved yet, even though a single counterexample would suffice!

Two limiting versions are important for applications:

• The ’t Hooft limit N → ∞ together with λ = Ng2Y M = 4πgsN fixed, i.e.with gs → 0

• in which case theplanar diagrams of the SU(N) theory become dominant, which re-semble the diagrams of a string-theory.

• The Maldacena-limit: λ → ∞ after the ’t Hooft limit, in which case thecurvature radius L of AdS5 becomes very large and classical super-gravitational theory can effectively replace string theory.

16

The correspondence under simplifying assumptions

• consider non-Abelian gauge theory SU(N) with many colours, N lar-ge

• To make the new coordinate r or z = 1/r simple, consider scale-invariant case

xµ → λ xµ symmetry

on top of Lorentz invariance→ together they imply conformal invariance.

• work in strong coupling limit gsN large

• to avoid instability consider susy theory. (Theory with maximal susyin D = 4 : N = 4 SYM. with conformal symmetry β = 0).

If β = 0, the coupling is arbitrary and allows to adiabatically change itfrom weak to strong coupling.

17

N = 4 susy SU(N) CFT in D = 4

is dual totype II B string theory on AdS5 × S5 with N 3-branes

• effective action, relevant part

S ∼ α′−4

∫

d10 x√−G

(e−2φR− FMNPQR FMNPQR

)

Take the purely spatial (’magnetic’) components of F as independentvariables.

• do KK reduction on S5

FMNPQR ∼ Nα′2 from Dirac quantization∫

S5

F (5) = Nα′2

18

and integrate over 5 coordinates ⊥ AdS5 which are the coordinateson the S5

S(5) ∼ α′−4

∫

d5x√

−G5 r5

(

e−2φR5 + e−2φ 1

r2− α′4N2

r10

)

.

• Perform a conformal rescaling with a suitable scaling factor λ to bringit to Einstein/Hilbert form

GEµν = λG(5)µν ,

√

|GE| = λ5/2√

−G5 , RE =1

λR5 ;

want√

|GE|RE =√

−G5 r5e−2φ︸ ︷︷ ︸

!=λ3/2

R5 ⇒ λ =(r10e−4φ

)1/3

Hence, with (later) change of notation GE, RE → G,R

S(5) ∼ α′−4

∫

d5x√

−GE

RE +

e−2φ

r2λ−5/2 r5 − α′4N2

r10λ−5/2 r5

︸ ︷︷ ︸

−V (r,φ)

19

λ−5/2 =(r10e−4φ

)−5/6

V (r, φ) = α′4 N2 e103 φ r−

403 − e+

43φ r−

163

∼ −x−4/3 + α′4N2x−10/3

րfrom flux, dominates at small x

where x = r4 e−φ

V

r

minimizing does not fix r and gs = eφ separately, but gives negativeminimum (AdS !) at

xmin ∼ α′2N Radius of AdS and of S5 L4 = r4min ∼ α′2 N eφ

20

Check of holography:# degrees of freedom in d = 3 field theory of N ×N matrices= (# volume-cells δ3 of lattice used in regularization in box R(3) )×N2

= R3

δ3N2 should be equal to

Area of boundary of AdS5

4G(5)Newton

since

(Area of boundary for z = δ → 0) =

∫

R(3)

d3xL3

z3=

R3L3

δ3

and

G(5)Newton ∼ L3

N2

both agree,

Area of boundary

G(5)Newton

∼ # degrees of freedom

21