The gauge-¬xing procedure described earlier for the point particle can be

generalized to the case of the string. In this case the auxiliary ¬eld has three

independent components, namely

h00 h01

h±β = , (2.22)

h10 h11

where h10 = h01 . Reparametrization invariance allows us to choose two of

the components of h, so that only one independent component remains. But

this remaining component can be gauged away by using the invariance of the

action under Weyl rescalings. So in the case of the string there is su¬cient

symmetry to gauge ¬x h±β completely. As a result, the auxiliary ¬eld h±β

can be chosen as

’1 0

h±β = ·±β = . (2.23)

01

Actually such a ¬‚at world-sheet metric is only possible if there is no topo-

logical obstruction. This is the case when the world sheet has vanishing

Euler characteristic. Examples include a cylinder and a torus. When a ¬‚at

world-sheet metric is an allowed gauge choice, the string action takes the

simple form

T 2

™

d2 σ(X 2 ’ X ).

S= (2.24)

2

The string actions discussed so far describe propagation in ¬‚at Minkowski

space-time. Keeping this requirement, one could consider the following two

additional terms, both of which are renormalizable (or super-renormalizable)

and compatible with Poincar´ invariance,

e

√ √

S1 = »1 d2 σ ’h S2 = »2 d2 σ ’hR(2) (h).

and (2.25)

S1 is a cosmological constant term on the world sheet. This term is not

allowed by the equations of motion (see Exercise 2.8). The term S2 involves

R(2) (h), the scalar curvature of the two-dimensional world-sheet geometry.

Such a contribution raises interesting issues, which are explored in the next

chapter. For now, let us assume that it can be ignored.

Equations of motion and boundary conditions

Equations of motion

Let us now suppose that the world-sheet topology allows a ¬‚at world-sheet

metric to be chosen. For a freely propagating closed string a natural choice

32 The bosonic string

is an in¬nite cylinder. Similarly, the natural choice for an open string is an

in¬nite strip. In both cases, the motion of the string in Minkowski space is

governed by the action in Eq. (2.24). This implies that the X µ equation of

motion is the wave equation

‚2 ‚2

± µ

X µ = 0.

’

‚± ‚ X = 0 or (2.26)

‚σ 2 ‚„ 2

Since the metric on the world sheet has been gauge ¬xed, the vanishing of the

energy“momentum tensor, that is, T±β = 0 originating from the equation

of motion of the world-sheet metric, must now be imposed as an additional

constraint condition. In the gauge h±β = ·±β the components of this tensor

are

1™

™ T00 = T11 = (X 2 + X 2 ).

T01 = T10 = X · X and (2.27)

2

Using T00 = T11 , we see the vanishing of the trace of the energy“momentum

tensor TrT = · ±β T±β = T11 ’ T00 . This is a consequence of Weyl invariance,

as was mentioned before.

Boundary conditions

In order to give a fully de¬ned variational problem, boundary conditions

need to be speci¬ed. A string can be either closed or open. For convenience,

let us choose the coordinate σ to have the range 0 ¤ σ ¤ π. The stationary

points of the action are determined by demanding invariance of the action

under the shifts

X µ ’ X µ + δX µ . (2.28)

In addition to the equations of motion, there is the boundary term

d„ Xµ δX µ |σ=π ’ Xµ δX µ |σ=0 ,

’T (2.29)

which must vanish. There are several di¬erent ways in which this can be

achieved. For an open string these possibilities are illustrated in Fig. 2.5.

• Closed string. In this case the embedding functions are periodic,

X µ (σ, „ ) = X µ (σ + π, „ ). (2.30)

• Open string with Neumann boundary conditions. In this case the com-

ponent of the momentum normal to the boundary of the world sheet

vanishes, that is,

Xµ = 0 at σ = 0, π. (2.31)

2.3 String sigma-model action: the classical theory 33

If this choice is made for all µ, these boundary conditions respect D-

dimensional Poincar´ invariance. Physically, they mean that no momen-

e

tum is ¬‚owing through the ends of the string.

• Open string with Dirichlet boundary conditions. In this case the positions

of the two string ends are ¬xed so that δX µ = 0, and

µ

X µ |σ=0 = X0 X µ |σ=π = Xπ ,

µ

and (2.32)

µ µ

where X0 and Xπ are constants and µ = 1, . . . , D ’ p ’ 1. Neumann

boundary conditions are imposed for the other p + 1 coordinates. Dirich-

let boundary conditions break Poincar´ invariance, and for this reason

e

they were not considered for many years. But, as is discussed in Chap-

ter 6, there are circumstances in which Dirichlet boundary conditions are

µ µ

unavoidable. The modern interpretation is that X0 and Xπ represent the

positions of Dp-branes. A Dp-brane is a special type of p-brane on which a

fundamental string can end. The presence of a Dp-brane breaks Poincar´ e

invariance unless it is space-time ¬lling (p = D ’ 1).

Solution to the equations of motion

To ¬nd the solution to the equations of motion and constraint equations it

is convenient to introduce world-sheet light-cone coordinates, de¬ned as

σ ± = „ ± σ. (2.33)

In these coordinates the derivatives and the two-dimensional Lorentz metric

take the form

1 1

·++ ·+’ 01

(‚„ ± ‚σ ) =’

‚± = and . (2.34)

·’+ ·’’ 10

2 2

µ

µ

X (σ,„)

X (σ,„)

σ=π

σ=0 σ=0

σ=π

Fig. 2.5. Illustration of Dirichlet (left) and Neumann (right) boundary conditions.

The solid and dashed lines represent string positions at two di¬erent times.

34 The bosonic string

In light-cone coordinates the wave equation for X µ is

‚+ ‚’ X µ = 0. (2.35)

The vanishing of the energy“momentum tensor becomes