where the symbols denotes saturation of the corresponding indices. Taking a further trace of (5.6.12) we obtain Ta = 0 which, inserted back into the equation, implies the full vanishing of the torsion Tbca = 0. Hence the soldering of the Lorentz bundle with the tangent bundle is no longer an a priori hypothesis, rather a dynamical yield of the classical action principle.

5.6.1.1 Torsionful Connections

Let us now suppose that in dimension m, in addition to the action of pure gravity we have also a contribution from matter, so that the total action is of the form:

where Φ collectively denotes the non-gravitational matter fields. Necessarily the variation with respect to δωab of the matter Lagrangian Lmatter(E, ω, Φ) produces an (m − 1)-form with the following general structure:

The three index tensor Kabf (Φ, D Φ) depends on the matter fields and possibly on their derivatives. In this case, the torsion tensor is not zero, rather it has a very simple expression in terms of Kabf (Φ, D Φ):

In presence of torsion, the spin-connection ωab is modified in the following way:

where 2ωab(E, ∂E), that depends only the vielbein and its derivatives, is the Levi Civita part of the spin connection, while Δωab(ΦD Φ), depending on the matter fields and their derivatives, is the following one:

Substituting (5.6.16) and (5.6.17) back into the action, we obtain new non-linear interactions of the matter fields that are actually a gravitational effect.

The Torsion of Dirac Fields The simplest example is provided by the case of a spin 12 -field. Adding to the gravitational action the spinor action (5.4.23) the tensor

Kabf is immediately calculated:

and for the torsion we find:

Via (5.6.17), this spinor bilinear form of the torsion induces quartic interaction of the Fermi fields that are a consequence of Einstein gravity.

Dilaton Torsion Another notable example of torsion mechanism is provided by the case where we have a dilaton coupling of the Einstein action. Suppose that in addition to the vielbein and the spin connection we have also a scalar field ϕ and that the total action takes the following form:

As defined in (5.6.21), the scalar action Ascalar plays, for the Klein-Gordon action of a scalar field ϕ, the same role that is played by (5.3.35) for the Yang-Mills field, namely its is its transcription in the vielbein formalism without the use of any Hodge dual. Indeed, if we consider the 0-form Φa , which belongs to the vector representation of the Lorentz group, as an independent field and we vary Ascalar with respect to it, we obtain the following algebraic condition:

where, by definition, dϕ = ∂a ϕEa . Substituting this result back into (5.6.21) we obtain:

which is the standard form for the action of a scalar field, with potential W (ϕ), in the background of a metric field gμν .