Lectures on String Theory

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The values of J and M 2 are quantized and the last inequality implies that the states
lie on the Regge trajectories.
level α ′ mass 2 rep of SO(24) little rep of little
group
group
0 −4 |0〉
}{{}
1
SO(1, 24) 1
1 0 α−1ᾱ i −1|0〉
j
} {{ }
299 s+276 a+1
SO(24) 299 s + 276 a + 1
α i −2ᾱ j −2|0〉
} {{ }
299 s +276 a +1
α i −1α j −1ᾱ k −1ᾱ l −1|0〉
} {{ }
299 s +1+299 s +1
20150 s + 32175
2 +4 SO(25)
α i −2ᾱ j −1ᾱ k 1|0〉
} {{ }
(24)×(299+1)
α i −1α j −1ᾱ k −2|0〉
} {{ }
(299+1)×(24)
52026 + 324 s + 300 a + 1
Tab. 4. The spectrum of closed bosonic string up to level 2.
Now we discuss the spectrum of closed strings. The mass operator for closed
strings is
M 2 = 2 ( ∑ ∞ ∞∑
)
α i
α
−nα i ′ n + ᾱ−nᾱ i n i − 2a
n=1
n=1
In addition one has to impose the level-matching condition
V =
∞∑
α−nα i n i −
n=1
∞∑
ᾱ−nᾱ i n i = 0 ,
which simply means that the excitation (level) number of α-oscillators should be
equal to the excitation number of ᾱ-oscillators.
The ground state is a tachyon which is scalar particle with α ′ M 2 = −4. The
first excited state α−1ᾱ i −1|0〉 j is massless. It can be decomposed into irreducible
n=1