i. quantum field theory and gauge theory ii. conformal field … · conjecture: horowitz,...
TRANSCRIPT
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