Lectures on String Theory

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The matrix γ i cab be understood as carrying the matrix indices a and ȧ: γaȧ i . The chiral and anti-chiral representations of SO(8) are
constructed then with the help of matrices
γ ij
s = 1
γ i (γ j ) t − γ j (γ i ) t ,
2
γ ij
c = 1
(γ i ) t γ j − (γ j ) t γ i ,
2
The operator Γ 9 = b 1 0 · · · b0 8 is the chirality operator (the analog of the γ 5 -matrix
in 4dim.) and it projects out one of the two Weyl components of the Ramond
ground state |ψ〉. We see that G anti-commutes with any mode b −n : {G, b i −n} = 0
and, therefore, the eigenvalues of G in the Ramond sector are ±1, depending on their
chirality, if we define G|a〉 = |a〉 and G|ȧ〉 = −|ȧ〉. Further, a general state in the R
sector is
|Φ〉 a = α i 1
−n1 · · · α i N
−nN b j 1
−m1 · · · b j M
−mM |a〉
or
We therefore find that
|Φ〉ȧ = α i 1
−n1 · · · α i N
−nN b j 1
−m1 · · · b j M
−mM |ȧ〉
G|Φ〉 a = (−1) M (−1) P i δ m i ,0
|Φ〉 a ,
G|Φ〉ȧ = −(−1) M (−1) P i δ m i ,0
|Φ〉ȧ .
The GSO projection consists in leaving the states which have either G = 1 or G = −1.
To construct the spectrum of the closed superstring we have to tensor left and
right-moving states (such that the level matching constraint ML 2 = M R 2 is satisfied)
and then impose the GSO projection. Here we have to distinguish four different
sectors
(R, R) , (NS, NS) , (R, NS) , (NS, R)
} {{ } } {{ }
space−time bosons
space−time fermions
The GSO projection is imposed separately for the left- and right-moving modes. In
the NS sector one keeps the states with
G = (−1) F = +1 , Ḡ = (−1) ¯F = +1 .
In the Ramond sector there are essentially two possibilities which lead to supersymmetric
and tachyonic-free spectrum. One of them is to take G = Ḡ = 1. The
massless spectrum is
]
]
Bosons :
[(1) + (28) + (35) v +
[(1) + (28) + (35) s
]
]
Fermions :
[(8) c + (56) c +
[(8) c + (56) c .
In total there are 128 bosonic and 128 fermionic states. The GSO projection imposed
in this way defines the so-called Type IIB superstring and its massless spectrum is