String Theory Demystified

CHAPTER 9 Superstring Theory Continued 183
Note that there are dotted spinors in the case of type IIA string theory (see Quantum
Field Theory Demystifi ed if you are not familiar with this).
Making the defi nition
P h h
± = ±
αβ 1 αβ αβ
( ε / ) (9.24)
2
we can write down the equations of motion that are derived by adding Eqs. (9.12),
(9.15), and (9.16). These are the equations of motion for the GS superstring, which
are in general quite complicated:
1 γδ
Π ⋅ Π = Π ⋅Π
α β αβ γ δ
2 h h (9.25)
Γ⋅Π αβ 1
P ∂ Θ = 0 (9.26)
α
−
β
αβ 2
Γ⋅Π P ∂ Θ = 0 (9.27)
α
+
β
Remarkably, in the light-cone gauge, the equations of motion turn out to be very
simple. This is because we can simplify the expression:
µ µ µ
Π =∂ X −iΘ Γ ∂ Θ
α
α
A A
α
A µ A
getting the term Θ Γ ∂ αΘ
to drop out in most cases. Using Eq. (9.21), this is
immediate when taking µ =+:
A + A
Θ Γ ∂ Θ = 0
α
There is only one nonvanishing term, when µ =−. For the cases where µ = i , we
can use the following trick. Consider the fact that
+ − 1 0 9 0 9
ΓΓ = ( Γ + Γ)( Γ −Γ)
2
1 0 0 9 0 0 9 9 9
= ( ΓΓ + ΓΓ −ΓΓ −ΓΓ
2
1 00 99 9 0 0 9
2
9 0 0 9
1 1
)
= ( − η + η + ΓΓ −ΓΓ)
= + ( ΓΓ −ΓΓ)
2