Gravity and Strings

134 Action principles for gravity
form we do not need. 11 Then the Vielbein equation takes the form
χ 2
G αβ [Ɣ(g)] =
2 Tmatter αβ + f (T 2 ). (4.116)
Then, by substituting Eq. (4.114) into this, we obtain
G αβ [Ɣ(g)] =
χ 2
2 Tmatter αβ + O(χ 4 ), (4.117)
which coincides with Einstein’s equation to order χ 2 .Infact, taking into account that the
order-χ 4 correction is associated with the density of intrinsic spins, only under the most
extreme macroscopic conditions [523] can the CSK theory give predictions different from
Einstein’s, which is good. At the microscopic level, the CSK theory gives different predictions:
for instance, it predicts contact interactions between fermions. These have two
origins: the term quadratic in the torsion in the CSK gravity action 12 and the covariant
derivatives in the matter action. All of them are of higher order in χ.
Conceptually, the CSK theory offers clear advantages over Einstein’s. It allows the coupling
to fermions and the relation between the canonical and Vielbein energy–momentum
tensors is clarified. As we are going to see, the simplest supergravity theory (N = 1, d = 4)
has the structure of the CSK theory for a Rarita–Schwinger spinor coupled to gravity (and
torsion). Finally, we are going to see that the separation between GCTs (which can be
seen as the local generalization of translations) and local Lorentz transformations suggests
a reinterpretation of gravity as a gauge theory (in the Yang–Mills sense) of the Poincaré
group.
Before we move on to these developments, we want to derive the complete gauge identities
and Noether currents for matter coupled to gravity in the CSK theory and study the
first-order formalism for it.
4.4.3 Gauge identities and Noether currents
Let us consider the action of matter minimally coupled to Vielbein and torsion e a µ and
Tµν a :
Smatter = 1
c
d d x Lmatter(ϕ, ∇ϕ,e) = 1
c
d d x Lmatter(ϕ, ∂ϕ, e,∂e, T ). (4.119)
(According to the minimal coupling prescription, the dependence on torsion is only through
the covariant derivative.) We assume that our matter fields, generically denoted by ϕ, have
only Lorentz indices and that only their first derivatives occur in the action. Furthermore,
the fundamental fields are assumed to be e a µ and Tµν a (not Tµν ρ ).
11 Actually, the second term on the l.h.s. of this equation also contains terms quadratic in the torsion that we
can include in f (T 2 ) by replacing the modified divergence ∗
∇µ by the Levi-Cività covariant derivative {}
∇µ.
12 Using Eq. (1.56), we can split the CSK action into a standard Einstein–Hilbert action and a piece quadratic
in the torsion plus a total derivative that we can ignore:
SCSK[e a µ, Tµν a ] = 1
χ2
d d
xe R(e) + Kµ µλ Kν ν λ + KνµρK µρν
. (4.118)