Gravity and Strings

4.5 Gravity as a gauge theory 141
in the base manifold, which may but need not be invariant under any translational isometry,
we consider inhomogeneous transformations of Lorentz vectors preserving the Minkowski
metric. The relation between these gauge transformations and GCTs is one of the subtle
points of this formulation of gravity.
To find the generators of the Poincarégroup and their commutation relations, we can use
the representation in position space (as differential operators) or alternatively we can use
the following representation by (d + 1) × (d + 1) matrices of Poincaré transformations
composed of a translation a a and a Lorentz transformation a b:
1 1 0
′ a =
v aa a b
1
vb
. (4.146)
This representation is suggestive because of its (d + 1)-dimensional homogeneous form.
We will later see that there is a reason for its existence.
We give here again the non-vanishing commutators of the generators {Mab, Pa}:
[Mab, Mcd] =−MebƔv(Mcd) e a − MaeƔv(Mcd) e b,
[Pc, Mab] =−PdƔv(Mab) d c.
(4.147)
Here Ɣv(Mab) d c is the matrix corresponding to the generator Mab in the vector representation
of the Lorentz group. The last commutator says that the d generators of translations
Pa can be understood as the components of a Lorentz vector. Observe that Pa acts trivially
on objects with Lorentz indices. It would act non-trivially on objects with a non-trivial
“(d + 1)th” index in the above representation, but by construction they do not exist.
For each generator we would introduce a gauge field: the spin connection ωµ ab for the
Lorentz subalgebra plus d new gauge fields for the translation subalgebra. Our theory has
d Vielbein fields with Lorentz-vector indices and it is natural to try to interpret them as the
gauge fields of translations and the gauge field of the Poincaré group would, tentatively, be,
in some representation Ɣ,
Aµ = 1
2 ωµ ab Ɣ(Mab) + eµ a Ɣ(Pa) . (4.148)
Observe that, since Pa does not act on objects with Lorentz indices, the covariant derivative
contains in practice only the spin connection.
If we can reproduce Einstein’s theory with these elements, we could say that Einstein’s
theory is the pure gauge theory of the Poincaré group. We are going to see whether this
is possible. First we determine the effect of gauge transformations using the standard formalism
of Appendix A. If σ ab and ξ a are the infinitesimal gauge parameters of Lorentz
rotations and translations, then
δωµ ab =−Dµσ ab , δeµ a =−Dµξ a + σ a beµ b . (4.149)
In both cases D stands for the gauge covariant derivative (no Levi-Cività connection is
contained in it because, for the moment, we have no metric but a gauge field eµ a ). It is
useful to compare the last expression with the effect of an infinitesimal GCT generated by