McMahon D. String Theory Demystified
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286
String Theory Demystifi ed
3. The Polyakov action is invariant under a Weyl transformation.
4.
hXX
αβ α
µ
β
ν
µν
η
=∂ ∂
5.
Xx p
cm s
µµ µ
τ
=+2
2
C
6.
1
2
0
C
s
α
µ
Chapter 3
1.
δεε ε
αβ α β β α
ρ
αβ ρ
hh=∂ +∂ −∂
2. The derivative can be evaluated as follows:
dp
d
d
dP
d
d
dP
d
PP
µ
σ
µ
τ
σ
µ
τ
µ
τ
τ
σ
τ
σ
σ
σ
==−==−
∫∫
00
11
0()
µµ
τ
σσ
()=
1
3. Consider
δ
δφ
δ
δ
δ
δφ
αβ
αβ
SS
h
h
pp
=
.
4. Use
TT
++ −−
==0
.
5.
∂+∂= = −
∫∫
τ
τ
µν σ
σ
µν µν
τ
µν
σ
µν
στ
JJJdJdJ0, , boosts and rotations
Chapter 4
1. 0,0
2.
i
η
µν
3. 0
4.
1
1
′
−
α
()a
5.
αα δδ
m
i
n
jij
mn
m,
,
⎡
⎣
⎤
⎦
=
+ 0
6.
αα δδ αα
m
i
n
jij
mn m
i
n
j
m,,,
,
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
=
+ 0
0
Chapter 5
1. 0
2. No,
TTz
az
az b
Tz
ba ab
() ()
(
)
=
+
++
≠
+
1
1
Quiz Solutions
287
3.
i()CC
11
−
4.
1
4
00
42
22
xx z z z
ss
µν
µν µν
ηη
+− −
′
CC
ln ln( )
5.
η
µν
C
s
zz
2
2
2
1
()−
′
6.
00xx
µν
7.
00
2
xx
T
zz
µν
µν
η
π
−+(ln ln )
−−
′
⎛
⎝
⎜
⎞
⎠
⎟
+−
′
⎛
⎝
⎜
⎞
⎠
⎟
+−
′
⎛
η
π
µν
4
111
T
z
z
z
z
z
z
ln ln ln
⎝⎝
⎜
⎞
⎠
⎟
++
′
⎛
⎝
⎜
⎞
⎠
⎟
⎧
⎨
⎩
⎫
⎬
⎭
ln 1
z
z
8.
−−
′
−
′
+−
′
−
′
{}
η
π
µν
4 T
zzzz z zzzln( )( ) ln( )( )
9.
−
−
+
−
∂
⎛
⎝
⎜
⎞
⋅
()
⋅
()
C
C
s
s
ik X w
w
ik X w
k
zw
e
zw
e
2
22
2
2
1
()
⎠⎠
⎟
Chapter 6
1. 0
2.
3
212
2
3
23
∂∂+ −
⎛
⎝
⎜
⎞
⎠
⎟
∂
⎛
⎝
⎜
⎞
⎠
⎟
∫
cc
D
cc
x
3. 26
4.
k
2
1
=
′
α
Chapter 7
1. You will need the supplementary boundary condition
dd
στψδψ ψδψ
π
0
2
0
∫∫
−∞
∞
+−− −+ +
∂+∂
[]
=()()
2.
i
T
2
ψρψ
µ
α
ν
.
288
String Theory Demystifi ed
3.
J
T
XXXXi
α
µν µ
α
νν
α
µµ
α
ν
ψρψ
=∂−∂+
2
()
4. 0
5.
δε δ
β
αβ
µβ
µ
α
βλ
µλ
µ
LXXX
B
=∂ ∂ ∂ − ∂ ∂
⎛
⎝
⎜
⎞
⎠
⎟
X
1
2
6. 0
7. 0
8. 0
9. 1
10. Massless vector boson, spin-3/2 fermion
Chapter 8
1.
′
(
)
+=
′
(
)
+
αα
44
25
2
25
2
pN pN
RR LL
Chapter 9
1. Use
εθ ε
θ
θ
ερ θ θ
α
α
Qi
AB
A
B
BBCC A
,,
⎡
⎣
⎤
⎦
=
∂
∂
+
⎡
⎣
⎤
⎦
∂
2. 0
3.
Γ
µ
4. 0
Chapter 10
1. c
2. b
3. a
4. c
5. b
6. d
Quiz Solutions
289
Chapter 11
1. c
2. b
3. b
4. a
5. c
Chapter 12
1. There are eight states, because there are eight transverse directions in space-
time in the light-cone gauge.
2. These are the 16 states associated with the extra bosonic dimensions.
3. This keeps them in the left-moving sector.
4.
αα αα δ δ
m
i
n
j
m
i
n
j
mn
ij
m,,
,
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
=
+
0
αα δ δ
m
I
n
J
mn
IJ
m,
,
⎡
⎣
⎤
⎦
=
+ 0
5. 0
Chapter 13
1.
φ
=±01,
2.
φ
φ
==− =±=>0
3
1
4
3
0
22
,(),, (m
A
m
A
tachyon nott a tachyon)
3.
φ
φ
=±0
0
,
4.
φ
φ
α
==−
′
0
2
2
0
,m
5. The stretching adds an energy
1
2
00
′
α
αα
aa
to the string where
1
22
00
21
2
′
=
−
′
⎛
⎝
⎜
⎞
⎠
⎟
α
αα
πα
aa
aa
xx
comes from the stretching between the branes.
290
String Theory Demystifi ed
Chapter 14
1.
T
mG Q G
r
=
−
+
()
4
22
4
2
2
π
2.
TT
H
≤=
′
1
4
πα
3.
~10
8−
K
4. 210
68
×
years
5.
SQQnJ=−2
15
2
π
Chapter 15
1.
F
z
ag N
s
≈
2
2. It transforms the metric to the anti-de Sitter form
ds R z dt dx
z
dz
2222 2
2
2
1
≈−−
⎡
⎣
⎢
⎤
⎦
⎥
()
.
3. b
4. c
5. b
Chapter 16
1.
−= =
++ +
−
gt t
pp p
D12 1
$
2. No, because
p
D
j
j
D
2
1
1
1
1
1
=
−
∑
=
−
≠
. If
p
D
j
=
−
1
1
for all j, then this is an isotropic
universe. This shows that the Kasner metric cannot describe an isotropic
universe if Kasner conditions are applied.
3. It is necessary to incorporate the fact that we are applying p to all three
expanding dimensions and q to all n contracting dimensions. So the Kasner
conditions are
−+ = + =313 1
22
pnq p nq,
.
Final Exam
Solutions
1.
P
XX XXX
XX
µ
µν
ν
µ
α
ν
βν
πα
=
′
′
−
′′
∂∂
1
2
2
() ( )
det | |
2. E
l
~
2
′
α
3. TXX hh XX h∂⋅∂− ∂⋅∂
{}
+=
µν µν
αβ
αβ µν
1
2
0()Λ
4. Λ= −
(
)
T
p
2
1
5. S
T
dXX=−
′
∫
2
22 2
σ
()
6. PTX
µµ
=
Copyright © 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use.
292
String Theory Demystifi ed
7. The mass of a closed string must include left movers and right movers.
Hence it is given by
M
nn nn
n
2
1
2
=
′
⋅+ ⋅
−−
=
∞
∑
α
αααα
()
.
8.
[, ]( )LL imnL
mn mn
=−
+
9.
L
nn
n
00
2
1
1
2
=+ ⋅
−
=
∞
∑
ααα
10.
()L
0
10−=
ψ
11.
Jxpxpi
n
nn nn
n
µν µ ν ν µ µ ν ν µ
αα αα
=−− −
(
)
−−
=
∞
∑
00 00
1
1
12.
[, ]pJ i p i p
000
µρλ µρλ µλρ
ηη
=− +
13.
[, ]JJ iJ iJ iJ iJ
µν ρλ µρ νλ νλ µρ νρ µλ µλ νρ
ηηηη
=+−−
14.
[, ]LJ
m
µν
= 0
15.
ξ
⋅=k 0
16. Setting the number of space-time dimensions to
D = 26
17.
TT XX TTXX
00 11
22
01 10
1
2
00== +
′
===⋅
′
=(),
18.
∂
∂
=
X
µ
σ
0
at the string endpoints.
19.
p
mx
xx
µ
µ
ν
ν
=
−
20. Momentum is constant, so
p
µ
= 0
.
21.
S m dt v=− −
∫
1
2
22.
I
I
p
mv
v
=
−1
2
23. As space-time vectors
24. 0
25.
−Γ Γ
0
11
26.
Γ
µ
†
Final Exam Solutions
293
27. Supersymmetry is a symmetry that unifi es fermions and bosons.
28. It takes fermions into bosons and vice versa.
29. RNS formalism uses worldsheet supersymmetry, while GS formalism uses
space-time supersymmetry.
30.
δεψ
δψ ρ ε
µµ
µα
α
µ
X
iX
=
=− ∂
31. 0
32.
JXJX
+−
=∂+∂
(
)
=∂−∂
(
)
1
2
1
2
10
ψψ
µ
τσµ
µ
τσµ
33.
∂−∂
(
)
=∂+∂
(
)
=
+−
τσ τσ
JJ0
34.
ψ
πτ ψ πτ
µµ
+−
=(,) (,)
, periodic boundary condition.
35. e
ψπτ ψπτ
µµ
+−
=−( , ) ( , ), antiperiodic.
36. Fermions
37. Bosons
38. The components of a spinor are real.
39. They have real components, and have half the number of overall components.
40. In the R sector, the summation is over integers, in the NS sector it is over
half integers.
41. Fermion
42. Boson
43. Boson
44. A fermionic operator is added, as in
LLL
mm
B
m
F
→+
() ()
.
45.
FF L m
m n mn mn
,
,
{}
=+
++
25
2
0
δ
46. The tachyon
47.
∂+∂
(
)
=∂−∂
(
)
=
τσ
µ
τσ
µ
ψψ
01
0
48.
T
αβ
= 0
49.
2
3
ρ
50. They anticommute, that is,
ab ba+=0
.
51. To ensure that operators do commute at different points
σ
along the string.
294
String Theory Demystifi ed
52.
τσ
=
+
=
−zz zz
i22
53.
()wz
k
−
−
2
54.
φφ
→
(
)
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
z
dz
dz
dz
dz
hh
2
2
1
1
55. 0
56. 1
57. 2
58.
TzTw
D
zw
() ( )
()
∝
−
−
+
26
4
$
59.
D = 26
60.
∂
∂z
61. Right movers
62. 0
63. 1/2
64.
w
nR
=
′
α
65.
1
2
25 25
pp nR
LR
−
(
)
=
66.
NNnK
RL
−=
67.
′
=
′
⎛
⎝
⎞
⎠
+
⎛
⎝
⎞
⎠
++
(
)
−
α
α
m
nR K
R
NN
RL
2
22
24
68. Kaluza-Klein excitations and winding increase the rest-energy of the string.
The mass is increased.
69. It becomes quantized.
70. It costs less energy to wrap around the small extra dimension.
71. The winding states in one theory become the Kaluza-Klein excitations in the
dual theory, and vice versa.
72. T-duality relates a small compactifi ed dimension in one theory to a large
dimension in the dual theory.
Final Exam Solutions
295
73. Types II A and II B, and the two heterotic theories.
74. You can tell direction along the string.
75.
m
2
1=−
′
/
α
76. It contains open and closed strings, all other superstring theories describe
closed strings.
77. There are two theories, one with
EE
88
×
and one with SO(32).
78. The universe expands in n directions but contracts in D-n directions as time
increases.
79. It makes contracting and expanding solutions dual to one another.
80. It defi nes a minimum radius such that
R
min
=
′
α
.
81. There are two 3-branes separated along a fourth spatial dimension.
82.
β
φ
→ 0
83. The number of space-time dimensions is 26.
84. It is equivalent to Einstein’s equation for a scalar fi eld.
85. Extreme black holes in string theory can have magnetic charge as well.
86. In a theory with
′
=
′
RR
α
/ , the state is transformed to
(, )nK
.
87. They are unchanged.
88. The state has the same mass m in the dual theory.
89.
∂=∂ ∂=−∂
++ − −
XX X X
25 25 25 25
90. It vanishes.
91. The winding numbers are fractional.
92. Neumann
93.
p = 02468,,,,
94. Dirichlet
95.
S
g
T
dx ET
s
=
⎛
⎝
⎞
⎠
−
(
)
∫
1
2
1
5
10
2
π
I
96.
ET==
′
1
2
πα
97. e
g
s
φ
=
1