Gravity and Strings

4.4 The Cartan–Sciama–Kibble theory 139
torsion, we would find that fermions generate torsion and the solution for the spin connection
would be the standard spin connection plus the corresponding contorsion tensor that
would be a function of the fermions. This is exactly what happens in the CSK theory 15
and in supergravity theories (see e.g. [912] and [221], where the so-called rheonomic approach
for constructing supergravity theories which makes use of the first-order formalism
is explained), for which the first-order formalism seems especially well suited since it leads
to much simpler actions. Furthermore, in the first order formalism, there is an independent
connection and a relation of gravity with Yang–Mills theories and a relation of supergravity
with gauge theories based on supergroups can be established (see Section 4.5 and
Chapter 5).
Now we will study a simple example: a Dirac spinor coupled to gravity in the first-order
formalism. We are going to see that the resulting equations of motion are the same as those
we would have obtained from the second order CSK theory.
Example: a Dirac spinor. The action for a Dirac spinor coupled to gravity in the first-order
formalism is the sum of Eq. (4.132) and Eq. (4.98),
(−1)
S[e,ω,ψ] =
d−1
2 · (d − 2)!χ 2
d d a1a2 a3 xRµ1µ2 (ω)e µ3 ···ead µd ɛa1···ad ɛµ1···µd
+ d d
1
xe 2 (i ¯ψ Dψ − i ¯ψ ←
Dψ) − m ¯ψψ , (4.139)
where D stands for the Lorentz covariant derivative.
By varying the Vierbein, spin connection, and spinor independently in the action we find,
after the use of our previous results, up to total derivatives,
δS = 2
χ 2
d d
xe − Ga µ 2 χ
−
2 Tcan a µ
δe a µ + 3eabc µνρ
Dνe c 2 χ
ρ −
2 Sc
νρ δωµ ab
+ χ 2
2 δ
¯ψ i ∇ψ − i
Ɣµν
2
µ
µ
− γ
µν
ν
ψ − mψ
+ χ 2
−i ¯ψ
2
←
D + i
¯ψγ
ν
Ɣµν
2 µ
µ
− − m ¯ψ δψ ,
µν
(4.140)
where we have introduced an affine connection Ɣ such that the total covariant derivative
∇ satisfies the first Vielbein postulate, which means that it is also metric-compatible as we
have explained before. Then,
Ɣµν µ
µ
− = Kµν
µν
µ = Tνµ µ . (4.141)
15 Observe that, in the first-order formalism, the Vielbein equation is the full Einstein tensor, whereas in the
second-order formalism, it is only the symmetric part of the Einstein tensor. The variation of the matter
action will give automatically the canonical energy–momentum tensor, since there will be no contributions
from the spin connection. Thus, the first-order formalism gives us the equation Ga µ = (χ 2 /2)Tcan a µ in
just one shot.