String Theory Demystified

58 String Theory Demystifi ed
The jacobian for a change in coordinates σ → σ ′ is defi ned by
⎛ ∂ ′
J = det
⎝
⎜ ∂
σ
σ
α
µ
The jacobian shows up in two places that turn out to cancel themselves to leave the
form of the Polyakov action invariant. It shows up when calculating the determinant
of the metric as
⎞
⎠
⎟
2
det( h′ ) = J det( h )
αβ αβ
You may recall from calculus that it also shows up in the integration measure:
2 2
d σ′ = Jd σ
These cancel out in the terms that appear in the Polyakov action [Eq. (3.2)]. That is,
2 2
d σ′ − det h′ = d σ −det
h
Putting all of these results together, we see that a change of worldsheet coordinates
(a reparameterization) leaves the Polyakov action invariant. Therefore a reparameterization
is a symmetry of the action. Since a reparameterization depends on the
worldsheet coordinates ( σ, τ ) , these are local symmetries.
WEYL TRANSFORMATIONS
A Weyl transformation or Weyl rescaling is a conformal transformation of the
worldsheet metric (see Chap. 5) of the form:
φσ τ
h → e h
( , ) (3.12)
µν
αβ α
µν − φ( σ , τ ) µν
Since h hβγ
= δγ
it follows from Eq. (3.12) that h → e h . Now we recall
two facts about determinants, where we let A, B be n× n matrices:
µν
det( AB) = det Adet B
n
det( αA) = det( αI A) =
α det A
n