String Theory and M-Theory

256 The heterotic string
Left-movers
The left-moving fermionic fields λ A can have periodic or antiperiodic boundary
conditions, just like the fermionic coordinates ψ µ in the RNS formalism.
Periodic boundary conditions define the P sector, which is the analog of the
R sector of the superstring. The mode expansion in the P sector is
λ A (τ + σ) =
n∈¢ λA n e −2in(τ+σ) . (7.16)
These modes satisfy the anticommutation relations
A
λm, λ B AB
n = δ δm+n,0. (7.17)
Antiperiodic boundary conditions define the A sector, which is the analog
of the NS sector of the superstring. The mode expansion in the A sector is
λ A (τ + σ) =
r∈¢ +1/2
λ A r e −2ir(τ+σ) . (7.18)
These modes satisfy the anticommutation relations
A
λr , λ B AB
s = δ δr+s,0. (7.19)
The left-moving modes of the heterotic string satisfy Virasoro constraints
Lm|φ〉 = ( L0 − ã)|φ〉 = 0, m > 0. (7.20)
If one goes to light-cone gauge and solves the Virasoro constraints, then only
the eight transverse components ˜α i n are relevant. For the left-movers the A
and P sectors need to be treated separately.
• For the P sector
p2 L0 − ãP |φ〉 =
8 + NL
− ãP |φ〉 = 0, (7.21)
where
NL =
∞
(˜α−n · ˜αn + nλ A −nλ A n ). (7.22)
n=1
• In the A sector we have
p2 L0 − ãA |φ〉 =
8 + NL
− ãA |φ〉 = 0, (7.23)
where
NL =
∞
˜α−n · ˜αn +
n=1
∞
r=1/2
rλ A −rλ A r . (7.24)