Introduction to String Theory and D–Branes - School of Natural ...

3.1.1 Diverse Dimensions
Imagine[9] that we do a change of variables x → x ′ . Such a change, if invertible, is a “conformal transformation”
if the metric is is invariant up to an overall scale Ω(x), which can depend on position:
g ′ µν(x ′ ) = Ω(x)gµν(x) . (119)
The name comes from the fact that angles between vectors are unchanged.
If we consider the infinitessimal change
then since the metric changes as:
we get:
x µ → x ′µ = x µ + ɛ µ (x) , (120)
gµν −→ g ′ µν
∂x
= gαβ
α
∂x ′µ
and so we see that in order for this to be a conformal transformation,
where, by taking the trace of both sides, it is clear that:
∂xβ , (121)
∂x ′ν
g ′ µν = gµν − (∂µɛν + ∂νɛµ) , (122)
∂µɛν + ∂νɛµ = F(x)gµν , (123)
F(x) = 2
D gµν ∂µɛν .
It is enough to consider our metric to be Minkowski space, in Cartesian coordinates, i.e. gµν = ηµν. We can
take one more derivative ∂κ of the expression (123), and then do the permutation of indices κ → µ, µ →
ν, ν → κ twice, generating two more expressions. Adding together any two of those and subtracting the
third gives:
2∂µ∂νɛκ = ∂µFηνκ + ∂νFηκµ − ∂κFηµν , (124)
which yields
2✷ɛκ = (2 − D)∂κF . (125)
We can take another derivative this expression to get 2∂µ✷ɛκ = (2 − D)∂µ∂κF, which should be compared
to the result of acting with ✷ on equation (123) to eliminate ɛ leaving:
ηµν✷F = (2 − D)∂µ∂νF =⇒ (D − 1)✷F = 0 , (126)
where we have obtained the last result by contraction.
For general D we see that the last equations above ask that ∂µ∂νF = 0, and so F is linear in x. This
means that ɛ is quadratic in the coordinates, and of the form:
ɛµ = Aµ + Bµνx ν + Cµνκx ν x κ , (127)
where C is symmetric in its last two indices. The parameter Aµ is obviously a translation. Placing the
B term in equation (127) back into equation (123) yields that Bµν is the sum of an antisymmetric part
ωµν = −ωνµ and a trace part λ:
Bµν = ωµν + ληµν . (128)
This represents a scale transformation by 1 + λ and an infinitessimal rotation. Finally, direct substitution
shows that
Cµνκ = ηµκbν + ηµνbκ − ηνκbµ , (129)
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