String Theory Demystified

CHAPTER 7 RNS Superstrings 141
OPEN STRING BOUNDARY CONDITIONS
When varying the action, the boundary terms must vanish in order to maintain
Lorentz invariance. In the case of open string, the boundary terms σ = 0 and σ = π
must both vanish independently. We can obtain
at σ = 0 if we take
ψδψ− ψδψ=0
+ + − −
+ −
ψ ( 0, τ) = ψ ( 0 , τ)
(7.25)
µ µ
Now in general, ψ =± ψ will make the boundary terms vanish, but typical convention
+ −
is to fi x the boundary condition at σ = 0 using Eq. (7.25). This leaves the choice of
sign at σ = π ambiguous. Depending on the sign we choose, we obtain two different
boundary conditions. Ramond or R boundary conditions are given by the choice
+ −
ψ ( π , τ) = ψ ( π , τ)
(Ramond) (7.26)
µ µ
The other choice we can make is known as Neveau-Schwarz or NS boundary
conditions:
+ −
ψ ( π , τ) =−ψ
( π , τ)
(Neveau-Schwarz) (7.27)
µ µ
We often refer to the boundary conditions chosen as the sector. The choice of
boundary conditions has dramatic consequences. In particular
• The R sector gives rise to string states that are space-time fermions.
• The NS sector gives rise to string states that are space-time bosons.
OPEN STRING MODE EXPANSIONS
We consider the R sector fi rst. The mode expansions are
µ 1 µ −in( τ−σ) ψ−( σ, τ)
= ∑ dne 2 n
µ 1 µ −in(
τ+ σ)
ψ+ ( σ, τ)
= ∑ dne 2
n
(7.28)