String Theory and M-Theory

where
˙X µ = ∂Xµ
∂τ
2.2 The string action 25
and X µ′ = ∂Xµ
, (2.12)
∂σ
and the scalar products are defined in the case of a flat space-time by A·B =
ηµνA µ B ν . The integral appearing in this action describes the area of the
world sheet. As a result, the classical string motion minimizes (or at least
extremizes) the world-sheet area, just as classical particle motion makes the
length of the world line extremal by moving along a geodesic.
X
2
X 0
Fig. 2.3. The world sheet for the free propagation of an open string is a rectangular
surface, while the free propagation of a closed string sweeps out a cylinder.
Fig. 2.4. The functions X µ (σ, τ) describe the embedding of the string world sheet
in space-time.
X
1