String Theory Demystified

96 String Theory Demystifi ed
“Volume elements” in integrals can be written using coordinate transformations
2
by including the determinant of the metric. Writing dz= dzdzand
using
2
| det gdz | = dτdσ it follows that
Now consider the action
2
dz= 2dd
τ σ (5.13)
1 2 µ
S = d z∂X ∂X
2πα
′ ∫
µ (5.14)
This is, in fact, the Polyakov action [Eq. (5.5)] in a much simpler mathematical
form. To see this, we can use Eq. (5.7) together with Eq. (5.13). Notice that
{ } { ∂ + ∂ }
µ 1
µ 1
∂X ∂ Xµ = ( ∂τ −i∂σ) X ( τ i σ ) Xµ
2
2
1 µ
µ
= ( ∂τX−∂
i σ X )( ∂ τ Xµ +∂ i σ Xµ
)
4
1 µ
µ
µ
µ
= ( ∂τX ∂ τ Xµ + i∂τX ∂σ Xµ
−i∂σX
∂ τ Xµ +∂σX ∂σ
Xµ
)
4
1 µ
µ
= ( ∂τX ∂ τ Xµ +∂σX ∂σ
Xµ
)
4
To move from the third to the fourth line, we used the fact we can raise and lower
indices with the Euclidean metric. That is, X = δ
ν µ
X = X , so
µ µν
µ
µ
µ
−∂ i X ∂ X =−∂ i X ∂ X =−∂ i X ∂ X =−∂ i X ∂ X
σ
τ µ σ µ τ µ σ µ τ
and the middle terms cancel. Therefore
1 2 µ 1
µ
S = d z∂X ∂ X = d d ∂X ∂X
2 ′ ∫ µ
2 ′ ∫2τ
σ
πα πα
1
=
2πα
′
1
=
4πα
′
∫
∫
1 µ
µ
2dττdσ
( ∂τX ∂ τ Xµ +∂σX ∂σ
Xµ
)
4
µ
( τ ττ Xµ +∂σX ∂ σ Xµ )
= SP µ
dτdσ ∂ X ∂
τ
µ
σ µ