Lectures on String Theory

– 14 –
The Nambu-Goto action
∫ ∫
S NG = −T dA = −T
d 2 σ
√
−det
( )
∂X
µ
∂X ν
∂σ α ∂σ η β µν
(3.1)
This we can write as
[( )]
−det
Ẋµ Ẋ µ Ẋ µ X µ
′
Ẋ µ Ẋ µ X ′µ X µ
′ = (ẊX′ ) 2 − Ẋ2 X ′2 . (3.2)
Thus,
∫
S NG = −T
√
∫
d 2 σ (ẊX′ ) 2 − Ẋ2 X ′2 = −T
d 2 σ √ −Γ
Here Γ = detΓ αβ , where
Γ αβ = ∂Xµ
∂σ α ∂X ν
∂σ β η µν (3.3)
is the metric induced on the string world-sheet.
What is a local characteristic of the string world-sheet? Consider a point on the
world-sheet and the space of all vectors tangent to the surface is at this point. These
vectors sweep two-dim vector space. The physical propagation of the string requires
that in these two-dim vector space there is a basis built over two vectors one of them
is time-like and another is space-like.
Recall the standard definitions from the theory of special relativity. We have the
invariant interval between two infinitezimal events:
−ds 2 = η µν dx µ dx ν = −dx 0 dx 0 +
3∑
dx i dx i . (3.4)
i=1
• If ds 2 > 0 the interval is called time-like. In this case the different events which
happen in the same space-point are always time-separated.
• If ds 2 < 0 the interval is called space-like. In this case events which happen at
the same time are space-separated.
• If ds 2 = 0 the interval is light-like. Vectors v µ obeying the condition v 2 =
η µν v µ v ν = 0 are called light-like or null.
Identify x 0 = t = τ. Then
Ẋ µ Ẋ µ = −1 + ⃗v 2 ≤ 0 ⇐ time − like
X ′µ X ′ µ = (X ′i ) 2 ≥ 0 ⇐ space − like