Lecture 24
8.251 Spring 2007
Lecture 24 - Topics
• Dp-brane
• Parallel Dp’s
Ẋ − ± X −� =
Ẋ I ± X I � =
1 1
(Ẋ I ± X I � )2
2α� 2p+
�
√
2α�
αnI exp(−in(τ ± σ))
(1)
(2)
n∈Z
1
2p p = �
α
+ −
�
∞
1 I I �
α0 α0 +
α−n αnI + a)
2
n=1
Dp-brane
x0 , x1 , . . . , xp : Coordinates on brane
x+ , x− , xi , i = 1, 2, 3, . . . , p ⇒ (p + 1) − 2 values. N N coordinates.
xp+1 , . . . , xd : Normal to brane
xa , a = p + 1, . . . , d ⇒ (d − p) values. DD coordinates.
1
(3)
Lecture 24
8.251 Spring 2007
Thus:
Equation (1) becomes:
1 1
[(ẋi ± xi� )2 + (ẋa ± xa� )2 ]
2α� 2p+
Equation (2) holds. Equation (3):
1
2p p = �
α
+ −
�
�
∞
1 I I �
I
α α +
α−n αn + a
2 0 0 n=1
X a (τ, σ):
a
X a (τ, 0) = X a (τ, π) = X ,
a scalar
�1
√
a
X a (τ, σ) = X + 2α�
αa e−inσ sin nσ
n n
n=0
�
X a� ± X˙ a =
√
2α�
�
αna e−in(τ ±σ)
�
n=0
a
[αm
, αnb ] = mδm+n,0 δ ab
2p+ p− =
∞
1 � i i � i
i
a
a
(α
p
p
+
(α−n α+
n + α−n αn ) − 1)
α�
n=1
M2 =
1
(−1 + N ⊥ )
α�
N ⊥ =�
Nlongitudinal + Ntransverse
∞
i
Nl = �n=1 α−n
αni
∞
a
Nt = n=1 α−n αna
�
�
Ground state: M 2 = − α1� , State �p+ , pi > tachyons, gives rise to ψ(τ, p+ , pi ).
pa = 0.
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Lecture 24
8.251 Spring 2007
Where do these fields live? Reasonable to say, on the brane. Right number of
coordinates p − 1 dim. space. But where did the xa go?
�
�
i � + i
Next state: α−1
p , p , M 2 = 0.
(p + 1) − 2 of these states. What are they? Photon states!
�
�
a � + i
Also, α−1
p , p lives on brane but has xa index. Nothing to do with spacetime.
Inert scalar states. d − p massless scalars. Physical interpretation: Represent
possible excitations casting 0 energy and 0 momentum, displacing the brane.
(See QFT theory of Goldstone).
Parallel Dp ’s
Know everything about strings A and B.
New problem: strings C and D. (note C �= D since orientation matters)
A = [1, 1], B = [2, 2], C = [1, 2], D = [2, 1].
In general, [i, j] with σ = 0 ∈ Di , σ = π ∈ Dj . (Before, always talking about
same D-branes, so there was an implicity [1, 1] always.)
a
X a (τ, σ) = X i +
�1
√
1 a
a
(X 2 − X 1 )σ + 2α�
αa e−inσ sin nσ
π
n n
n=0
�
X a� =
�
√
2α�
αna e−inτ cos(nσ)
n=0
�
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Lecture 24
8.251 Spring 2007
√
1
2α� α0a = (X̄2a − X̄1a )
π
Alternatively, write as:
αa
1
a
a
√0 =
(X 2 − X 1 )
2πα0�
2α�
1 i i i 1 a a
(α p p + α0 α0 + N ⊥ − 1)
α�
2
1
1
2
⊥
M = � (N − 1) + � α0a α0a
α
2α
2p+ p− =
1
⊥
α� (N − 1): Contribution of Quantum Oscillation
1
a a
a
a 2
2α� α0 α0 = (T0 (X2 − X1 )) : Contribution of Tension×Length
Now M 2 quantized
but can
� for any�sector,
�
� choose sectors.
Ground states: �p+ , pi [1, 2] , �p+ , pi [1, 1] (we know how to handle this one)
�
�
i � + i
State α−1
p , p [1, 2]� D-branes separate, so now not massless ⇒ photons
�
a � + i
State α−1
p , p [1, 2] same mass. Always a scalar state.
Suppose have 3 branes:
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Lecture 24
8.251 Spring 2007
String A = [i, j]
String B = [j, k]
Can interact to form one string between Di and Dk : [i, j] × [j, k] = [i, k]. (Then
Dj doesn’t notice the string anymore). Theory of interacting gauge fields.
Dp brane parallel to Dq
Coordinates have split into common D, common N and split N D.
5