Lectures on String Theory

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Now we note that in addition to the conserved charges generated by the stress tensor:
∫
Q ξ ± = dσ ξ ± (σ ± )T ±± (σ, τ)
we will have the conserved charges generated by the supercurrent
∫
G ɛ ± = dσ ɛ ± (σ ± )G ± (σ, τ) .
6.5 Boundary conditions
Varying the action to derive the equations of motion we will get the following boundary
term
∫ (
)
dσ∂ σ ψ + δψ + − ψ − δψ − .
For the case of closed string, to make this term vanishing one has to impose the
following condition
)
)
(ψ + δψ + − ψ − δψ − (σ) −
(ψ + δψ + − ψ − δψ − (σ + 2π) = 0 .
Since the fermions ψ + and ψ − are independent this equation implies that
The following terminology is standard
ψ + (σ) = ±ψ + (σ + 2π) ,
ψ − (σ) = ±ψ − (σ + 2π) .
• Periodic boundary conditions in σ are called Ramond boundary conditions and
they are denoted by the letter “R”.
• Anti-periodic boundary conditions in σ are called Nevew-Schwarz boundary
conditions and they are denoted by the letter “NS”.
Universally, all fermionic quantities on the world-sheet have the following boundary
conditions
ψ(σ + 2π) = e 2πiθ ψ(σ) ,
where θ = 0 in the R-sector and θ = 1/2 in the NS sector.
Boundary conditions for ψ + and ψ − can be chosen independently, which gives in
total four possibilities
(R, R) , (NS, NS) , (R, NS) , (NS, R)
The boundary conditions for the two components of the supersymmetry parameter
should be chosen in such a way as to make the variation δX µ = i¯ɛψ µ periodic.