Lectures on String Theory

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7.1 Light-cone quantization and superstring spectrum
As we have established imposition of the superconformal gauge does not completely
remove the gauge (unphysical) degrees of freedom. The superconformal transformations
which do not destroy the superconformal gauge choice are left. In order to
remove the remaining unphysical degrees of freedom one can try to fix the light-cone
gauge, similar to as was done for bosonic string. We can also fix
X + = α ′ p + τ
in our fermionic theory and this choice will completely remove the reparametrization
invariance. However, the local supersymmetry transformations obeying the equations
∂ + ɛ − = ∂ − ɛ + = 0
are still left over. These transformations can be used in order to completely eliminate
the fermionic field ψ + , where this time ψ ± refer to the target-space light-cone
components
ψ ± = √ 1 (ψ 0 ± ψ d−1 ) .
2
This is equivalent to putting to zero the modes b + r for all r. After this gauge choice
is done we can solve the super-Virasoro constraints and find the longitudinal modes
(remember that T = 1
2πα ′ )
∂ ± X − = 1 (
)
(∂
α ′ p + ± X i ) 2 + iψ±∂ i ± ψ±
i
ψ − ± = 2
α ′ p + ψi ±∂ ± X i .
This shows that only the transversal components X i and ψ i are physical degrees of
freedom. In terms of oscillators the previous equations read
α − m =
1
(
√ : αnα i i
2α′ p + m−n : + ∑ r
b − r = 2 ∑
α
α ′ p
r−qb i i + q .
q
(m
2 − r) : b i rb i m−r : −2aδ m
)
For the case of closed strings these expressions must be supplemented by the analogous
ones for the left-moving modes. Here we also include a normal ordering constant
a which is a = 1 in the NS sector and a = 0 in the R sector.
2
where
The mass operator is
( ∑
α ′ MR 2 = 2
M 2 = M 2 R + M 2 L ,
n>0
α i −nα i n + ∑ r>0
)
rb i −rb i r − a