For the above construction of differential forms, the manifold structure is sufficient, the existence of a metric being completely irrelevant. Yet if a metric structure is present, the algebra A (M ) can be equipped with a linear (anti-)involutive operation, named Hodge duality which maps forms of degree k into forms of degree m − k:

and is explicitly realized as follows. Let ω(k) Ak (M ) be a k-form. In any coordinate patch we have:

The Hodge dual of ω(k), denoted ω(k) is the m − k form with the following components:

Civita symbol whose first k indices have been raised by means of the metric tensor. As a consequence of the above definition we find:

where det(sign) denotes the determinant of the signature of the considered metric which is either +1 or −1.

5.3.2 Geometrical Rewriting of the Gauge Action

Using the Hodge dual operation we can easily rewrite the gauge action (5.3.16) in the following way:

What we learnt from the above discussion is that the structure of Maxwell equations is completely determined by the gauge nature of the electromagnetic field and this fact is just the mathematical transcription of the Physical Conservation Law of Electric Charge. On the other hand the unique structure of Maxwell equations implies an Action Principle (5.3.27) which is quadratic in the curvature F of the

principle connection A and cannot be written without the use of the Hodge dual. The latter contains the metric tensor gμν in disguise. In other words, although the electromagnetic field Aμ coincides with a principle connection on a U(1)-fibre bundle, whose construction involves only the manifold structure of the space-time base manifold M , the classical dynamics of Aμ, i.e. its propagation equations and interaction with other fields, can neither be formulated nor solved without invoking the metric structure of M . In short the electromagnetic U(1)-bundle needs to be constructed on a (pseudo-)Riemannian base manifold.

This is the first signal of the basic difference between gravity and the other fundamental interactions. All interactions are mediated by gauge fields which happen to be principle connections on fibre bundles, yet the connections involved by gravity are special in that they deal with the tangent bundle T M of the spacetime manifold M , which is the base manifold not only for T M , but also for all the other principle-bundles P (M , G) associated with non-gravitational interactions. The structure (5.3.27) implies an obligatory and a priori predetermined interaction of every other gauge field with the gravitational one, namely with the metric. Here is the source of universality for gravity.

5.3.3 Yang-Mills Theory in Vielbein Formalism

In view of our advertised preference for the Cartan viewpoint and of our announced plan of trading the metric for the vielbein, a natural question arises about the formulation of electrodynamics in vielbein formalism. If we are going to discard the metric tensor, how can we rewrite the action principle (5.3.27) without using Hodge duality? The answer we presently provide to such a question illustrates the basic spirit of the Repère Mobile philosophy: all fundamental operations are lifted to the Lorentz fibres, where they take the same appearance as in flat Minkowski space. What happens in the Lorentz bundle is then transmitted to the tangent bundle through the soldering of the two-bundles which is implemented by the vielbein, via the vanishing of the Torsion.

Since there is no difference, in this respect, between the Abelian gauge theory of Electrodynamics and the non-Abelian Yang-Mills theories utilized in the description of electro-weak and strong interactions, we immediately address the most general case. We consider a principle bundle P (M , G) where G is the structural group (= gauge group in physical parlance) and the G -Lie algebra valued one-form A = A I TI is a principle connection (i.e. a gauge field in physical parlance). The corresponding curvature two-form is defined below:

the bracket in the second line of (5.3.28) providing the explicit definition of the two-forms FI . Each of these latter can be expanded along a basis of differentials:5

In terms of the field strength FμνI the Yang-Mills action in m-dimensions takes a form which is the obvious generalization of (5.3.16) and (5.3.27), namely:

In (5.3.30) the symbol γI J denotes the G-invariant Killing metric on the Lie algebra G. Typically, choosing a linear representation of G, the generators TI become matrices that can be normalized in such a way that:6

The above equation explains the second identity in (5.3.30).

Let us then solve the task of recasting the Yang-Mills action (5.3.30) into the framework of the vielbein formalism. To this effect we begin from the volume form, namely from the generally covariant integration measure on an m-dimensional space-time. In metric formalism, if we are supposed to integrate any scalar function f(x) on a (pseudo-)Riemannian manifold, we just write:

f(x)Volg = f(x) − det g dmx (5.3.32)

In view of the identification (5.2.9), the above equation can be rewritten as follows:

5Note that from now on we omit the factor 12 introduced in the electromagnetic analogue of (5.3.29), namely in (5.3.10). There such a factor was used in order for Fμν to have the traditional normalization Fμν = ∂μAν − ∂ν Aμ . Omitting this factor the normalization of Fμν becomes the less traditional one: Fμν = 12 (∂μAν − ∂ν Aμ + A A − terms). Paying this moderate price all formulae become much neater and deprived of annoying prefactors.

6An important comment is obligatory at this point. A fundamental theorem of Lie Algebra Theory, states that the Killing metric of a compact semi-simple Lie algebra is always negative definite. So, as long as, the gauge group G is chosen compact, as it is the case in all standard gauge theories of non-gravitational interactions, the kinetic term of the gauge fields defined by the action (5.3.31) turns out to have correct positive definiteness properties and the corresponding quanta have physical propagators. In the case of non-compact gauge algebras the action (5.3.31) introduces negative norm states in the spectrum. This does not mean that non-compact gauge groups are altogether forbidden. Actually they appear in supergravity theories yet the corresponding kinetic terms have a more sophisticated structure which takes care of unitarity.

where εa1...am denotes the completely antisymmetric Levi Civita tensor with flat indices. Indeed it suffices to observe that:

In the above equation the writing det E denotes the determinant of the square matrix Eμa (x) and the last identity follows from (5.2.9) which implies det g = (det E)2 × (det η) = −(det E)2.

This being clarified the gauge action (5.3.30) of a Yang-Mills field in a m- dimensional, Lorentz signature, space-time can be rewritten in the vielbein approach, utilizing a first order formalism. Consider the following action functional:

where Fab = FabI TI is a 0-form transforming as a section of the second antisym-

π

metric power of the Lorentz vector bundle Vso(1,m−1) → M mentioned in (5.2.13):

gauge bundle P (M , G), to be precise, the vector bundle of the adjoint representation. This object should be regarded as an independent dynamical variable with respect to which we are supposed to vary the action functional (5.3.35). Performing such a variation we find the equation:

The argument goes as follows. Since the curvature FI is a horizontal two-form leaving on the cotangent bundle of the base manifold M it can always be expanded along a basis of local sections of T M , namely along an independent basis of oneforms. The coordinate differentials dxμ provide such a basis but, by construction,