™(„ )

d„ d„ d„ d„

f

which shows that the action S0 is invariant under reparametrizations. 2

EXERCISE 2.3

The action S0 in Eq. (2.5) is also invariant under reparametrizations of the

particle world line. Even though it is not hard to consider ¬nite transfor-

mations, let us consider an in¬nitesimal change of parametrization

„ ’ „ = f („ ) = „ ’ ξ(„ ).

Verify the invariance of S0 under an in¬nitesimal reparametrization.

SOLUTION

The ¬eld X µ transforms as a world-line scalar, X µ („ ) = X µ („ ). Therefore,

22 The bosonic string

the ¬rst-order shift in X µ is

™

δX µ = X µ („ ) ’ X µ („ ) = ξ(„ )X µ .

Notice that the fact that X µ has a space-time vector index is irrelevant

to this argument. The auxiliary ¬eld e(„ ) transforms at the same time

according to

e („ )d„ = e(„ )d„.

In¬nitesimally, this leads to

d

δe = e („ ) ’ e(„ ) =(ξe).

d„

Let us analyze the special case of a ¬‚at space-time metric gµν (X) = ·µν ,

even though the result is true without this restriction. In this case the vector

index on X µ can be raised and lowered inside derivatives. The expression

S0 has the variation

™ ™ ™™

2X µ δ Xµ X µ Xµ

1

δe ’ m2 δe .

’

δ S0 = d„ 2

2 e e

™

Here δ Xµ is given by

d ™™

™ ¨

δ Xµ = δXµ = ξ Xµ + ξ Xµ .

d„

Together with the expression for δe, this yields

™µ ™

™

2X µ ™ ™ ¨ µ ’ X Xµ ξe + ξ e ’ m2 d(ξe) .

1 ™

δ S0 = d„ ξ Xµ + ξ X ™

2

2 e e d„

The last term can be dropped because it is a total derivative. The remaining

terms can be written as

1 d ξ ™µ ™

d„ ·

δ S0 = X Xµ .

2 d„ e

This is a total derivative, so it too can be dropped (for suitable boundary

conditions). Therefore, S0 is invariant under reparametrizations. 2

EXERCISE 2.4

The reparametrization invariance that was checked in the previous exercise

allows one to choose a gauge in which e = 1. As usual, when doing this one

should be careful to retain the e equation of motion (evaluated for e = 1).

What is the form and interpretation of the equations of motion for e and

X µ resulting from S0 ?

2.1 p-brane actions 23

SOLUTION

The equation of motion for e derived from the action principle for S0 is given

by the vanishing of the variational derivative

δ S0 1 ™™

= ’ e’2 X µ Xµ + m2 = 0.

δe 2

Choosing the gauge e(„ ) = 1, we obtain the equation

™™

X µ Xµ + m2 = 0.

™

Since pµ = X µ is the momentum conjugate to X µ , this equation is simply

the mass-shell condition p2 + m2 = 0, so that m is the mass of the particle,

as was shown in Exercise 2.1. The variation with respect to X µ gives the

second equation of motion

d 1

™ ™™

(gµν X ν ) + ‚µ gρ» X ρ X »

’

d„ 2

1

™™ ¨ ™™

= ’(‚ρ gµν )X ρ X ν ’ gµν X ν + ‚µ gρ» X ρ X » = 0.

2

This can be brought to the form

X µ + “µ X ρ X » = 0,

¨ ™™ (2.10)

ρ»

where

1 µν

“µ = g (‚ρ g»ν + ‚» gρν ’ ‚ν gρ» )

ρ» 2

is the Christo¬el connection (or Levi“Civita connection). Equation (2.10)

is the geodesic equation. Note that, for a ¬‚at space-time, “µ vanishes

ρ»

in Cartesian coordinates, and one recovers the familiar equation of motion

for a point particle in ¬‚at space. Note also that the more conventional

™™

normalization (X µ Xµ + 1 = 0) would have been obtained by choosing the

gauge e = 1/m. 2

EXERCISE 2.5

The action of a p-brane is invariant under reparametrizations of the p + 1

world-volume coordinates. Show this explicitly by checking that the action

(2.6) is invariant under a change of variables σ ± ’ σ ± (σ).

SOLUTION

Under this change of variables the induced metric in Eq. (2.8) transforms in

24 The bosonic string

the following way:

‚X µ ‚X ν ‚X µ ’1 δ ‚X ν

gµν = (f ’1 )γ

G±β = (f )β gµν ,

±

‚σ ± ‚σ β ‚σ γ ‚σ δ

where

‚σ ±

±

fβ (σ) = .

‚σ β

De¬ning J to be the Jacobian of the world-volume coordinate transforma-

±

tion, that is, J = det fβ , the determinant appearing in the action becomes

‚X µ ‚X ν ‚X µ ‚X ν

’2

det gµν =J det gµν .

‚σ ± ‚σ β ‚σ γ ‚σ δ

The measure of the integral transforms according to

dp+1 σ = Jdp+1 σ,

so that the Jacobian factors cancel, and the action becomes

‚X µ ‚X ν

dp+1 σ

Sp = ’Tp ’ det gµν .