String Theory Demystified

CHAPTER 2 Equations of Motion 31
This metric determines distances on the worldsheet. We say that this metric is
induced because it includes the metric of the background space-time in its defi nition
(we are taking space-time to be fl at, so are using η µν ). That is to say, on the surface
of the worldsheet, there is a new measure of distance, but that measure of distance
is determined by the background space-time through its metric (which in general,
is not η µν ). Proceeding, we now have
Using the notations
X
µ
2
ds = γ dξ dξ
αβ
α β
µ
µ
X
µ X
= X
τ σ
∂
′ =
∂
∂
∂
We can write the components of the induced metric (for the case of fl at spacetime)
as
γ = η
ττ µν
γ = η
στ µν
γ = η
σσ µν
µ ν
∂ ∂
∂τ
∂ τ
=
X X
X
2
µ ν
∂X
∂X
= X
⋅ X′
= γ = η
∂σ
∂ τ
µ ν
∂X
∂X
= X′
∂σ
∂σ
2
τσ µν
µ ν
∂X
∂X
∂τ
∂σ
Using Eq. (2.15), we can write the induced metric as a matrix in ( τ, σ ) space
γ αβ =
⎛ 2
X X ⋅ X′
⎞
⎜
⎝ X ⋅ X′ 2
X′
⎟
⎠
Notice that the determinant of this matrix is given by
γ = det γ = 2 2
αβ ′ −( 2
XX XX ⋅ ′ )
Let’s go back to where we started, seeking an expression for the action
S =−T∫dA (2.15)
(2.16)
(2.17)