String Theory Demystified

CHAPTER 9 Superstring Theory Continued 175
where,
µ µ µ α µ i α
DY = ψ + θB −iρθ∂ α X + θθρ ∂αψ
2
µ µ µ α µ i
DY = ψ + B θ + iθρρ
∂αX− θθ∂αψ ρ
2
Performing the Grassman integration in the action, we can obtain the component
form, which is
µ
µ α
1 2
α µ µ α
µ
S =− d ∂ X ∂ X −i ∂ −B
B
4πα
′ ∫ σ( ψ ρ ψ )
α µ
α µ µ
The equation of motion for B µ , as mentioned earlier, is B µ = 0, which allows us to
discard the auxiliary fi eld, and we arrive back at the theory described in Chap. 7.
To see how to arrive at this, you can just apply the rules of Grassman integration,
2
considering θ and θ as separate variables and using d θ = dθdθ. We illustrate by
computing a couple of terms. For example,
But ∫ dθθ = 1 and so,
On the other hand,
∫ µ ∫ ∫
2 µ
µ
d θB θψ = dθ dθθB ψ
∫ ∫ µ ∫
µ
µ
dθ dθθB ψ = dθB ψ =0
∫ µ ∫
2 µ
2
µ
µ
d θ( B θ)( θB ) = d θ( θθ)
B B =−2iB
B
Using these types of computations, one can transform the manifestly supersymmetric
action into the coordinate form to recover the theory of the RNS
superstring.
µ
µ
µ
µ
The Green-Schwarz Action
In this section, we use the idea of supersymmetry applied to the space-time coordinates.
α
For worldsheet supersymmetry, we extended the coordinates σ = ( σ, τ)
of the
worldsheet by introducing fermionic super-worldsheet coordinates. Now, we are
going to utilize this same idea but apply it to the actual space-time coordinates,