String Theory and M-Theory

24 The bosonic string
the following way:
where
Gαβ = ∂Xµ
∂σα ∂X ν
∂σβ gµν = (f −1 ) γ ∂X
α
µ
∂σ γ (f −1 ) δ β
f α β
∂σα
(σ) = .
∂σ β
∂X ν
gµν,
∂σ δ
Defining J to be the Jacobian of the world-volume coordinate transformation,
that is, J = det f α β , the determinant appearing in the action becomes
det
= J −2
det
.
gµν
∂X µ
∂σ α
∂X ν
∂σ β
gµν
∂X µ
∂σ γ
The measure of the integral transforms according to
d p+1 σ = Jd p+1 σ,
∂X ν
∂σ δ
so that the Jacobian factors cancel, and the action becomes
Sp = −Tp d p+1
∂X
σ − det gµν
µ
∂σ γ
∂X ν
∂σ δ
.
Therefore, the action is invariant under reparametrizations of the worldvolume
coordinates. ✷
2.2 The string action
This section specializes the discussion to the case of a string (or one-brane)
propagating in D-dimensional flat Minkowski space-time. The string sweeps
out a two-dimensional surface as it moves through space-time, which is called
the world sheet. The points on the world sheet are parametrized by the two
coordinates σ 0 = τ, which is time-like, and σ 1 = σ, which is space-like. If
the variable σ is periodic, it describes a closed string. If it covers a finite
interval, the string is open. This is illustrated in Fig. 2.3.
The Nambu-Goto action
The space-time embedding of the string world sheet is described by functions
X µ (σ, τ), as shown in Fig. 2.4. The action describing a string propagating
in a flat background geometry can be obtained as a special case of the
more general p-brane action of the previous section. This action, called the
Nambu–Goto action, takes the form
SNG = −T dσdτ ( ˙ X · X ′ ) 2 − ˙ X2X ′2 , (2.11)