having collectively denoted TI = {Pa , Jbc}. Next we can collect the vielbein Ea and the spin-connection ωab into a single object, namely into a principle connection on the Poincaré bundle:

and following the general prescription displayed in (3.7.6) we can construct the curvature two-form of such a Poincaré connection:

Expanding Θ along the generators we discover that the Torsion Ta and the curvature Rab are just the components of the overall Poincaré curvature Θ along the translation and rotation generators respectively.

Thus it appears that the main ingredients apt to describe the gravitational field, once Cartan’s viewpoint is adopted, are encoded into a principal connection, one on the Poincaré bundle with structural group ISO(1, m − 1).2 Such an observation is very tempting since it seems to imply that, at the end of the day, gravitation is explained in terms of a gauge theory just as all the other fundamental interactions of Nature. Up to this point the similarities are several and it is worth summarizing them for comparison. For this reason we open a digression on classical Electrodynamics and classical Yang-Mills theories. We will see that notwithstanding the many striking analogies, gravity involves an extra geometric structure, the soldering, which differentiates the gauge theory of gravitational interactions from all the others and makes it unique and special.

5.3The Structure of Classical Electrodynamics and Yang-Mills Theories

Let us begin by reviewing Classical Electrodynamics in the most frequently utilized approach.

There are some matter fields, typically the spinor field ψα (x), describing the electron/positron, or some other electrically charged fermionic particles3 and one considers the standard action for such fields, the free Dirac action:4

2From now one we choose p = 1 which is the physically most interesting case.

3All leptons and quarks fall into this category.

4For the conventions on spinors and gamma-matrix algebra see Appendix A.

The electrically charged matter fields can also be spin zero fields, namely complex scalars φ(x), in which case the standard free action is the Klein-Gordon one:

The matter action, namely ADirac and /or AKG has a global symmetry under the transformations of a U(1) Lie group defined as follows:

where e, q are the electric charges of the considered fields. Indeed as long as the angle θ is constant and independent from the space-time location, performing the replacements (5.3.3) leaves the actions (5.3.1) and (5.3.2) invariant.

The basic idea of Electrodynamics is to transform this symmetry from a global into a local one, namely to allow for a space-time dependence of the gauge angle θ → θ (x). The appropriate mathematical apparatus for this operation was developed in Chap. 3.

We consider the classical matter fields ψ(x) or φ(x) as sections of a rank one

π

holomorphic vector bundle E = M that has the space-time manifold M as base manifold, U(1) as structural group and C as standard fibre. This vector bundle is associated to a principle bundle P (M , U(1)). Next we introduce a principle connection on this latter bundle, namely a U(1) Lie algebra valued one-form on the total space P , with the structure dictated by (3.3.86):

In the case of U(1) the Lie algebra is made by just one generator that can be identified with the imaginary unity i. According to the discussion of Sect. 3.3.2.1, the existence of the connection allows to define the covariant derivatives:

Global symmetry is promoted to a local one, if ordinary derivatives are replaced by covariant ones in the actions (5.3.1), (5.3.2) which become:

Indeed the transformations (5.3.3) with an x-dependent gauge angle θ (x) are interpreted as the transition functions from a local trivialization of the U(1) bundle

Fig. 5.2 The interaction vertex of electrodynamics

to another one and under such changes of local trivializations the connection oneform changes according the general rule (3.3.89). In the case of electrodynamics this reduces to:

The new Lagrangians (5.3.7), (5.3.8) contain now interaction vertices of the original fields ψ or φ with the gauge field Aμ. For example, in the case of spinor electrodynamics, the interaction vertex is depicted in Fig. 5.2. Hence the gauge field Aμ should be equipped with a kinetic Lagrangian which defines its propagator. The answer is known since the time when Maxwell equations were rewritten in compact relativistic notation. The curvature two-form of the electromagnetic U(1)- connection is:

F = dA = 1 Fμν dxμ dxν

2

(5.3.10)

Fμν = ∂μAν − ∂ν Aμ

and its components Fμν encode the electric and magnetic fields according to the following identifications:

Classical Maxwell equations are reproduced by the following differential statements on the field strength:

where Jν = ψγ μψ is the electric current. There is a fundamental difference between the first equation (5.3.12) and the second (5.3.13). Equation (5.3.12) is just an identity, the Bianchi identity for the curvature of a U(1)-connection:

while (5.3.13) is a true dynamical equation which follows upon variation in δAν of the following action:

One important observation is that Maxwell equations are consistent with the conservation of the electric charge since the left hand side of the second equation satisfies the identity:

which implies the vanishing of the electric current divergence and hence the global conservation of the electric charge. One might wonder whether the actual structure of Maxwell equations is an accident or whether it is uniquely determined by first principles. The correct answer is the second. Indeed we can easily show that, as long as Aμ is interpreted as a connection on a principle U(1)-bundle, namely, as long as the transformation (5.3.9) is required to be a symmetry of the field equations, and as long as the latter are assumed to be Lorentz invariant, linear differential equations of the second order, there is no alternative. Indeed let us see what is the most general form of such equations for the field Aν . Considering all possible index contractions we arrive at the following candidate equation:

where α, m are coefficients to be determined. If we require that this equation should be invariant with respect to the gauge transformation (5.3.9) we obtain:

Exactly the same result is obtained by imposing that the divergence of the right hand side should vanish as a consequence of the field equation. In this way, we realize that gauge invariance and electric current conservation imply each other. On the other hand the choice (5.3.20) of the parameters reproduces (5.3.13). Hence the gauge action (5.3.16), which is quadratic in the curvature tensor Fμν , is uniquely selected by the fundamental physical principle of electric charge conservation encoding the full structure of electrodynamical interactions. Let us analyze the geometrical structure of the action (5.3.16). To this effect we need to introduce a new involutive operation acting on differential forms which is named Hodge duality.

5.3.1 Hodge Duality

Consider a Riemannian manifold (M , g) of dimension m and let us construct the graded algebra of differential forms on M , defined by (2.5.59), which we repeat here for reader’s convenience: