Therefore, the action is invariant under reparametrizations of the world-

volume coordinates. 2

2.2 The string action

This section specializes the discussion to the case of a string (or one-brane)

propagating in D-dimensional ¬‚at 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 = „ , 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 ¬nite

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, „ ), as shown in Fig. 2.4. The action describing a string propagating

in a ¬‚at 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

™ ™

(X · X )2 ’ X 2 X 2 ,

SNG = ’T dσd„ (2.11)

2.2 The string action 25

where

µ ‚X µ

™ µ = ‚X µ

X and X = , (2.12)

‚„ ‚σ

and the scalar products are de¬ned in the case of a ¬‚at space-time by A·B =

·µν Aµ 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.

0

X

1

X

2

X

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, „ ) describe the embedding of the string world sheet

in space-time.

26 The bosonic string

The string sigma model action

Even though the Nambu“Goto action has a nice physical interpretation as

the area of the string world sheet, its quantization is again awkward due to

the presence of the square root. An action that is equivalent to the Nambu“

Goto action at the classical level, because it gives rise to the same equations

of motion, is the string sigma model action.1

The string sigma-model action is expressed in terms of an auxiliary world-

sheet metric h±β (σ, „ ), which plays a role analogous to the auxiliary ¬eld

e(„ ) introduced for the point particle. We shall use the notation h±β for the

world-sheet metric, whereas gµν denotes a space-time metric. Also,

h±β = (h’1 )±β ,

h = det h±β and (2.13)

as is customary in relativity. In this notation the string sigma-model action

is

√

1

Sσ = ’ T d2 σ ’hh±β ‚± X · ‚β X. (2.14)

2

At the classical level the string sigma-model action is equivalent to the

Nambu“Goto action. However, it is more convenient for quantization.

EXERCISES

EXERCISE 2.6

Derive the equations of motion for the auxiliary metric h±β and the bosonic

¬eld X µ in the string sigma-model action. Show that classically the string

sigma-model action (2.14) is equivalent to the Nambu“Goto action (2.11).

SOLUTION

As for the point-particle case discussed earlier, the auxiliary metric h±β ap-

pearing in the string sigma-model action can be eliminated using its equa-

tions of motion. Indeed, since there is no kinetic term for h±β , its equation

of motion implies the vanishing of the world-sheet energy“momentum tensor

1 This action, traditionally called the Polyakov action, was discovered by Brink, Di Vecchia and

Howe and by Deser and Zumino several years before Polyakov skillfully used it for path-integral

quantization of the string.

2.2 The string action 27

T±β , that is,

2 1 δSσ

√

T±β = ’ = 0.

T ’h δh±β

To evaluate the variation of the action, the following formula is useful:

δh = ’hh±β δh±β ,

which implies that

√ 1√

’hh±β δh±β .

δ ’h = ’ (2.15)

2

After taking the variation of the action, the result for the energy“momentum

tensor takes the form

1

T±β = ‚± X · ‚β X ’ h±β hγδ ‚γ X · ‚δ X = 0.

2

This is the equation of motion for h±β , which can be used to eliminate

h±β from the string sigma-model action. The result is the Nambu“Goto

action. The easiest way to see this is to take the square root of minus the

determinant of both sides of the equation

1

h±β hγδ ‚γ X · ‚δ X.

‚± X · ‚ β X =

2

This gives

1√

’h hγδ ‚γ X · ‚δ X.

’ det(‚± X · ‚β X) =

2

Finally, the equation of motion for X µ , obtained from the Euler“Lagrange

condition, is

√

1

∆X µ = ’ √ ‚± ’hh±β ‚β X µ = 0.

’h

2

EXERCISE 2.7

Calculate the nonrelativistic limit of the Nambu“Goto action

G±β = ‚± X µ ‚β Xµ

SNG = ’T ’ det G±β ,