Chapter 2
8.821/8.871 Holographic Duality
Fall 2014
Chapter 2: Deriving AdS/CFT
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2014
Lecture 11
Important equations for this lecture from the previous ones:
1.
The Virasoro constraints in the light-cone gauge, equation (32) and (33) in lecture 10:
2v + ∂τ X −
v + ∂σ X −
=
=
(∂τ X i )2 + (∂σ X i )2 ,
i
∂τ X ∂σ X
i
(1)
,
(2)
2.1.2: LIGHT-CONE QUANTIZATION (cont.)
µ
Since the general classical solution strings (for closed strings XR
and XLµ are independent periodic functions with
µ
µ
period 2π, while for open strings XR = XL ):
µ
X µ (σ, τ ) = xµ + v µ τ + XR
(τ − σ) + XLµ (τ + σ) .
(3)
Then it can be rewritten in terms of Fourier expansion. For closed string:
X µ (σ, τ ) = X µ + v µ τ + i
α
2
n=0
1 µ −in(τ +σ)
α e
+ α̃nµ e−in(τ −σ)
n n
.
(4)
µ
It’s similar for open string, but from XR
= XLµ one arrives at αnµ = α̃nµ :
√
X µ (σ, τ ) = X µ + v µ τ + i 2α
n=0
1 µ −inτ
α e
cos nσ .
n n
(5)
(The physical meaning of αnµ , α̃nµ will be studied in Pset 3, so please do the homework.)
The center off mass motion can be found by averaging the position of the strings at a given timeslice (l = 2π for
closed strings and l = π for open strings):
1
l
ˆ
l
dσX µ (σ, τ ) = xµ + v µ τ .
(6)
0
The constant v µ is identified with the strings’ center of mass velocity.
The classical coefficients αnµ and α̃nµ keep track of the oscillation modes of the strings. While the closed strings
have independent left-moving and right-moving contributions, open string can be described as standing waves so
that left-moving and right-moving are the same.
1
Chapter 2
8.821/8.871 Holographic Duality
Fall 2014
In the light-cone gauge, X + = v + τ and X − can be obtained by writing X − in Fourier expansion, plugging
equations (4) and (5) into equations (1) and (2) then equating the coefficients of different Fourier modes. The 0th
(non-oscillating) mode gives the relations between the strings’ center of mass velocity and the strings’ oscillation
modes. For closed string, from equation (1):
X
i
i
2v + v − = vi2 + α0
α−n
αni + α̃−n
α̃ni
.
(7)
n=0
6=0
For open string, from equation (1):
X
2v + v − = vi2 + 2α0
i
α−n
αni .
(8)
i
α̃−n
α̃ni
(9)
n=0
6=0
For closed string, from equation (2):
X
i
α−n
αni =
n=0
6=0
X
n=0
6=0
This is known as the level matching condition for closed strings. Equation (2) satisfies trivially for open strings.
Poincare global symmetries of the action corresponds to the conserved currents on the worldsheet. For the
moment, let’s look at translation and apply the standard Noether procedure:
Πµa = ∂∂a X µ LP
=
γ ab =η ab
1
∂a X µ .
2πα0
(10)
Also note that ∂ a Πµa = 0, from the equation of motion for X µ .
Πµτ is the momentum density along the string, and the corresponded conserved current is the string momentum in
spacetime:
ˆ l
ˆ l
1
l vµ
pµ =
dσΠµτ =
dσ∂τ X µ =
.
(11)
0
2πα 0
2π α0
0
The mass-squared is related to the spacetime momentum of the strings (the mass shell condition):
M 2 = −pµ pµ = 2p+ p− − pi2 .
(12)
For closed string, from equation (7):
M2 =
1 X i
2 X i
i
i
i
α
α
+
α̃
α̃
α−n αni .
−n n
−n n = 0
0
α
α
(13)
1 X i
α−n αni .
2α0
(14)
n=0
6=0
n=0
6=0
For open string, from equation (8):
M2 =
n=0
6
One can concluded that the mass of a string can be determined from its oscillations.
After understanding the strings at classical level, the next step is to quantization – quantize independent degrees
of freedom X i (σ, τ ) (with canonical momentum density Πi ) in the action:
ˆ
1
1
S=−
d2 σ∂ a X i ∂a X i , Πi =
∂τ X i .
(15)
4πα0
2πα0
Nominate X i to be a quantum operator, with the canonical commutation relation at a given timeslice:
[X i (σ, τ ), X j (σ 0 , τ )] = [Πi (σ, τ ), Πj (σ 0 , τ )] = 0 , [X i (σ, τ ), Πj (σ 0 , τ )] = iδ ij δ(σ − σ 0 ) .
(16)
The results are 0th mode xi , pi and oscillation modes αni , α
˜ ni all become operators:
j
j
[xi , pj ] = iδ ij , [αni , αm
] = [α̃ni , α̃m
] = nδ ij δn+m,0 .
2
(17)
Chapter 2
8.821/8.871 Holographic Duality
Fall 2014
Note that αni , α̃ni can be related to the creation and annihilation operators (similar to canonical quantization of
QFT, creation operators are associated with positive branch while annihilation operators are associated with
negative branch so that the Hamiltonian is bounded from below):
1
1 i
1
1 i
i
√ αni = ani , √ α−n
= (a−n
)† , √ α̃ni = ãin , √ α̃−n
= (ãi−n )† , n > 0 .
n
n
n
n
(18)
Therefore, the oscillator vacuum state (labelled by string spacetime momentum pµ ) satisfies:
αni |0, pµ ) = α̃ni |0, pµ ) = 0 , n > 0 .
(19)
i
i
Excited states can be built from creation operators (α−n
, α̃−n
with n > 0):
j1
i1
α−n
αi2 αi3 ...α̃−m
α̃j2 α̃j3 ...|0, pµ ) .
1 −n2 −n3
1 −m2 −m3
(20)
For closed string, define the oscillation number operator (no summation in i index, and the order of operators is
very important):
1 i
1 i
αni , Ñni = α
˜ α
˜i .
(21)
Nni = α−n
n
n −n n
Hence the level matching condition can be rewritten as:
X
X
nÑni
(22)
nNni =
n=0
6=0
n=0
6=0
i
For open string, only one set of oscillation is needed (let’s pick α−n
).
The quantum version of the mass shell condition for closed strings, from equation (13):
M2 =
D−1
D−1
2 XX
4 XX
i
i
n(N
+
Ñ
)
+
a
=
nNni + ac .
0
n
n
α0 i=2
α0 i=2
6=0
n=0
(23)
6=0
n=0
The constant ac is the zero-point energy for closed string, comes from rearranging the operator to normal ordered
(for αni and α̃ni , negative n to the left and positive n to the right):
ac =
∞
2(D − 2)
D−2
2(D − 2) X
.
n=
ζ(−1) = −
0
0
α
6α0
α
n=1
(24)
For open string, from equation (14):
M2 =
D−1
1 XX
nNni + ao .
α0 i=2
(25)
6=0
n=0
The constant ao is found to be:
ao =
∞
(D − 2) X
(D − 2)
D−2
n=
ζ(−1) = −
.
2α0 n=1
2α0
24α0
The trick used here to find the zero-point energy is the ζ-function regularization:
∞
=
1
ζ(s) =
n−s , ζ(−1) = −
.
12
n=1
(26)
(27)
The sum is only convergant for s > 1, but by using analytical continuation (around the pole at s = 1) one can get a well-define finite
result for s = −1. In path integral approach, the zero-point energy can also be found (on the general worldsheet) after mapping an
1
asymptotic state to a point insertion, and the consistency of the CFT requires − 12
(in that language, it’s the central charge).
Indeed, the divergent in equation (26) is just the artifact of a perturbative description (looking only at an asymptotic part instead of
the worldsheet as a whole), and the analytical continuation trick (which is arise a lot in quantum field theories and statistical field
theories) is used to read-off nonperturbative description from perturbative results.
For example of a fake divergent, consider a Taylor (perturbative) expansion of the following function:
∞
=
1
f (x) =
=
xn .
(28)
1−x
n=0
f (x) is well-define everywhere except for x = 1 (pole). However, the Taylor expansion description gives a summation series with
divergent (ill-defined) whenever |x| > 1, which is not the true property of f (x).
3
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8.821 / 8.871 String Theory and Holographic Duality
Fall 2014
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