Gravity and Strings

16 Differential geometry
Local GL(d, R) covariance implies the inhomogeneous transformation law for the connection:
ω ′ ab c
= c dωef d −1 f
b − −1c d∂e d
b ( −1 ) e a. (1.78)
The curvature of this connection can be defined through the Ricci identities in the standard
fashion (observe that there are no torsion terms here):
Dµ, Dν φ = 0,
a Dµ, Dν ξ = Rµνb aξ b ,
Dµ, Dν εa =−εb Rµνa b (1.79)
,
and then the curvature is given by 10
Rµνa b = 2∂[µ ων]a b − 2ω[µ|a c ω|ν]c b . (1.81)
At this point we have introduced a new connection ω that is independent of the metric.
In the previous section we managed to relate the connection Ɣ to the metric via the metric
postulate. Here we are going to generalize the metric postulate first to relate the two
connections (the first Vielbein postulate) and then to relate them to the metric (the second
Vielbein postulate). Before we enunciate these postulates we introduce the total covariant
derivative, covariant with respect to all the indices of the object it acts on. We denote it by
∇ again, and, for instance, acting on Vielbeins it is
∇µea ν = ∂µea ν + Ɣµρ ν ea ρ − eb ν ωµa b . (1.82)
We can motivate the first Vielbein postulate as follows: we would like to be able to
convert tangent into world indices and vice-versa inside the total covariant derivative, so
e a ν∇µξ ν = Dµξ a and D is just the projection of ∇ onto the Vielbein basis. To have this
property we impose the first Vielbein postulate,
∇µea ν = 0. (1.83)
It is worth stressing that this does not imply the covariant constancy of the metric ∇µgνρ =
0. The above postulate implies the following relation between the connections:
ωµa b = Ɣµa b − ea ν ∂ b µ eν. (1.84)
Furthermore, the curvatures of the two connections are now related by
Rµνρ σ (Ɣ) = e a ρeb σ Rµνa b (ω). (1.85)
The first Vielbein postulate also gives an important relation between the torsion and the
Vielbein: on taking the antisymmetric part of ∇µea ν = 0, we obtain
2D[µe a ν] = 2 ∂[µe a ν] − ω[µ a
ν] =−Tµν a . (1.86)
10 Observe that, with all Latin indices, Rabc d = ea µ eb ν Rµνc d and, therefore,
Rabc d = 2∂[a ωb]c d − 2ω[a|c e ω|b]e d + 2ab e ωec d . (1.80)