is the gauge covariant differential of the 0-form field F I |ab . As stated above, this object is both a section of the adjoint vector bundle associated with the principle bundle P (M , G) and an antisymmetric section of the Lorentz bundle. For this reason its covariant differential involves both the gauge connection A J and the spin connection ωac .

Let us analyze (5.3.44). What is the condition under which it reproduces the standard Yang-Mills equation that we obtain by varying the second order action (5.3.30)? The answer is encoded in the concept of soldering.

5.4 Soldering of the Lorentz Bundle to the Tangent Bundle

Suppose that we enforce the following equation:

what are its consequences?

First of all, with reference to (5.3.44) we note that it reduces to just the first addend. Secondly we note that F I |ab as defined in (5.3.45) is a one-form on the base manifold M . As such it can be expanded along a basis of sections of the cotangent bundle, in particular along the basis provided by the vielbein Ea . Hence we can write:

The tensor cF I |ab is defined by the position (5.4.2) and it is named the intrinsic covariant derivative of the Yang-Mills field strength. Comparison with the definition (5.3.45) implies that:

which justifies its name. Indeed it is the covariant derivative of the tensor F where the Greek holonomic index μ has been anholonomized and converted into a Latin one through multiplication with the inverse vielbein. The meaning of this operation is that cF I |ab is now a section of the adjoint-bundle with respect to the internal gauge group G and a section of the triple power of the Lorentz bundle, while it behaves as a bona-fide scalar function with respect to the tangent bundle T M of the base-manifold.

Inserting (5.4.2) into (5.3.44) we obtain:

Equation D[μEνa] = 0 suffices to completely and uniquely determine the coefficients of the spin connection ωρab in terms of the vielbein derivatives and of its inverse. A simple counting explains why. The total number of unknowns in an m-dimensional manifold is 12 m2(m − 1). The same is the number of linear equations provided by the condition of vanishing torsion. Explicitly the solution can be obtained by means of a trick completely analogous to that used to determine the Levi Civita connection. Introduce the contorsion, defined below:

By summing three copies of the above equations with cyclically permuted indices, using the antisymmetry of the contorsion in its lower indices and that of the spin connection in its upper ones, the solution for ωpab in terms of Kpqa is explicitly obtained. This conclusion is quite momentous. It implies that the Lorentz gauge connection is not an independent dynamical object, rather it is completely determined by the vielbein, just as the Levi Civita connection is completely determined by the metric. The equivalency of the metric and vielbein formalism arises from the already proposed identification of the metric with the quadratic form in the vielbein which was spelled out in (5.2.9). Combining the latter with the covariant constancy of the metric with respect to the Levi Civita connection we obtain:

Implementing once again the standard trick of rewriting three copies of the above equation with permuted indices

and summing them together, upon use of the vanishing torsion equation [ρ Eμa ] = 0 we obtain that also the symmetric combination of derivatives vanishes.

Equation (5.4.16) is of the utmost relevance. It implies that, provided the torsion two-forms Ta vanishes, the tangent bundle and the Lorentz bundle are completely soldered to each other and we can trade the Levi Civita connection for the spinconnection and vice-versa.

In practice this means the following. Given any world tensor:

with r controvariant and s covariant indices, which is therefore a section of the rth power of the cotangent bundle T M and a section of the sth power of the tangent bundle T M , through multiplication with an appropriate number of direct and inverse vielbein we can convert it into a corresponding section of the (r, s)th power of the associated Lorentz vector bundle:

Soldering means that the parallel transport by means of the spin connection in the Lorentz bundle is completely equivalent to the parallel transport on the tangent/cotangent bundle by means of the Levi Civita connection. Introducing the curly capital D as a notation for the Lorentz-covariant differential of any Lorentz-tensor valued p-form, namely:

D T a1...ar ≡ dT a1...ar − ωa1b1 T c1a2...ar ηb1c1 · · · − ωar br T a1a2...cr ηbr cr

(5.4.19)

and reserving the symbol for the Levi Civita covariant differential of any worldtensor valued p-form:

the soldering (5.4.16) guarantees that we can write:

D T a1...ar = Eμa11 . . . Eμarr T μ1...μr

(5.4.21)

T μ1...μr = Eaμ11 . . . Eaμrr D T a1...ar

5.4.1Gravitational Coupling of Spinorial Fields

The results of the previous section show that as far as tensor fields are concerned the use of the Levi Civita connection or of the spin-connection in the vielbein formalism are completely equivalent. This equivalence does not extend to the case of spinorial fields. As we emphasized more than once, spinorial representations are a specific feature of the orthogonal or pseudo-orthogonal algebras, so that spinorial sections of the tangent or cotangent bundle do not exists. On the other hand given the principal Lorentz bundle, we can construct all of its associated spinorial bundles. This

Fig. 5.3 The gravitational interaction vertex of spin 1/2 particles

amounts to saying that via the spin connection ωab we can easily define the covariant derivative of any spinor field and this, thanks to the soldering condition, which expresses ωab in terms of the vielbein, introduces the gravitational coupling of the spinor particles.

We already shew how to rewrite the Yang-Mills Lagrangian in vielbein formalism. Let us now consider the case of the Dirac Lagrangian for a spin one half field. As usual, in preparation for those multi-dimensional theories that constitute the focus of attention in contemporary Theoretical Physics, we deal with the case of spinor fields in m-dimensions. For conventions and properties of spinor fields in diverse dimensions we refer the reader to Appendix A. Here for simplicity we focus on the case of a Dirac spinor which always exists in every dimension. In Minkowski space the free Dirac action is the following one which is the direct generalization to dimension m of the action considered in (5.3.7):

It can be easily recast in a general covariant form and adapted to a curved space-time manifold through the following rewriting:

The reader will notice that the ingredients entering (5.4.23) are just the vielbein Ea and the spin connection ωab , namely the components of the principal connection on the Poincaré bundle. The above action is manifestly invariant against local Lorentz transformations and it is invariant against general diffeomorphisms on the base manifold M , since it is written in terms of differential forms and their exterior products. When the spin connection ωab vanishes and the vielbein reduces to a Krönecker delta function, which is the case of flat Minkowski space, the reader can easily verify that the actions (5.4.23) and (5.4.22) coincide. However, just as the U(1) covariant derivative in the case of electrodynamics produced an interaction vertex of the fermion field with the photon field, in the same way, the Lorentz-covariant derivative used in the action (5.4.23) generates a non-trivial interaction vertex of the same fermionic field with the graviton, which has a universal structure and it is illustrated in Fig. 5.3. Having clarified the advantages and power of the vielbein formalism we just adopt it throughout most of the following exposition, turning to the fundamental question of Einstein equations, namely to the dynamics of the gravitational field.