String Theory Demystified

98 String Theory Demystifi ed
Generators of Conformal Transformations
To study the generators of a conformal transformation, we consider an infi nitesimal
transformation of the coordinates:
µ µ µ
x′ = x + ε
Now consider an infi nitesimal conformal transformation. That is if g′ µν ( x′
) =
Ω( xg ) µν ( x)
we take Ω( x) = 1 − f ( x)
where f ( x)
is some small departure from the
identity. Then we have g′ µν ( x′ ) = ( 1 − f( x)) gµν ( x) = gµν ( x) − f( x) gµν ( x)
.
µ µ µ
Using x′ = x + ε you can show that
g′ = g −( ∂ ε +∂ ε )
µν µν µ ν ν µ
So, recalling that we are working with a conformal transformation about the fl at
space metric, it must be the case that
∂ ε +∂ ε = f( x) g
µ ν ν µ µν
We can determine the form of f( x)by
multiplying both sides of this equation by
g µν
µν
. In d spacetime dimensions g gµν = d and so on the right we obtain
µν
g f( x) g = d f( x)
. On the left side we have
Hence
µν
µν
µν
µν
g ( ∂ µ εν +∂ ν εµ ) = g ∂ µ εν + g ∂ν
εµ
µ ν
=∂ µ ε +∂ νε
(raise indices with metric)
µ µ
=∂ +∂ (relabel
µ ε µ ε repeated indices
which are dummy indices)
µ
= 2 ∂µ
ε
And we have the relation
f
2
= ∂µ
ε
d
µ
2 ρ 2
∂ µ εν +∂ ν εµ = δµν∂ ρε
= δµν ( ∂⋅ε)
(5.18)
d d