String Theory and M-Theory

122 Strings with world-sheet supersymmetry
to find the currents associated with the supersymmetry transformations in
Eqs (4.14)–(4.16). It is sufficient to consider the ε− transformations, since
the ε+ ones work in an identical way. Therefore, we consider
Using these rules,
δ−X µ = iε−ψ µ
+ ,
δ−ψ µ
+ = −2∂+X µ ε− and δ−ψ µ
− = 0.
δ− (2∂+X · ∂−X + iψ− · ∂+ψ− + iψ+ · ∂−ψ+) = −4iε−∂−(ψ+ · ∂+X)
up to a total derivative. Thus, choosing the normalization appropriately,
this shows that J+ = ψ+ · ∂+X. Similarly, the expression J− = ψ− · ∂−X is
obtained by considering an ε+ transformation. ✷
4.4 Boundary conditions and mode expansions
The possible boundary conditions and mode expansions for the bosonic fields
X µ are exactly the same as for the case of the bosonic string theory, so that
discussion is not repeated here.
Suppressing the Lorentz index µ, the action for the fermionic fields ψ µ in
light-cone world-sheet coordinates is
Sf ∼ d 2 σ (ψ−∂+ψ− + ψ+∂−ψ+) . (4.47)
By considering variations of the fields ψ± one finds that the action is stationary
if the equations of motion (4.10) are satisfied. The boundary terms
in the variation of the action,
δS ∼ dτ (ψ+δψ+ − ψ−δψ−) |σ=π − (ψ+δψ+ − ψ−δψ−) |σ=0, (4.48)
must also vanish. There are several ways to achieve this, which are discussed
in the next two subsections.
Open strings
In the case of open strings the two terms in (4.48), corresponding to the two
ends of the string, must vanish separately. This requirement is satisfied if
at each end of the string
ψ µ
+ = ±ψµ
− . (4.49)