3.3.2.1 The Connection One-Form

Definition 3.3.6 Let P (M , G) be a principle fibre-bundle with structural Lie group G, whose Lie algebra we denote by G. A connection one-form on this bundle is a Lie algebra valued section of the cotangent bundle A G Γ (T P , P ) which satisfies the following defining properties:

(i)X G : A X# = X.

(ii)g G : LgA = Adjg−1 A ≡ g−1Ag.

Given the connection one-form A the splitting between vertical and horizontal subspaces is performed through the following

Definition 3.3.7 At any u P , the horizontal subspace HuP of the tangent space to the bundle is provided by the kernel of A:

The actual meaning of Definitions 3.3.6, 3.3.7 and their coherence with Definition 3.3.5 becomes clear if we consider a local trivialization of the bundle.

Just as before, let xμ be the coordinates on the base-manifold and αi the group parameters. With respect to this local patch, the connection one-form A has the following appearance:

where, a priori, both the components A2μ(x, α) and Ai (x, α) could depend on both xμ and αi . The second equality in (3.3.79) and (3.3.78) is due to the fact that A and A are Lie algebra valued, hence they can be expanded along a basis of generators of G. Actually the dependence on the group parameters or better on the group element g(α) is completely fixed by the defining axioms. Let us consider

(r)

Therefore AIi =ΣiA(α) and the vertical component of the one-form connection is nothing else but the right-invariant one-form:

Consider now the implications of defining property (ii). In the chosen local trivialization the left action of any element γ of the structural group G is Lγ : (x, g(α)) → (x, γ · g(α)) and its pull-back acts on the right-invariant one-form σ(R) by adjoint transformations:

Property (ii) of Definition 3.3.6 requires that the same should be true for the complete connection one-form A. This fixes the g dependence of A2, namely we must have:

is a G Lie algebra valued one-form on the base manifold M . Consequently we can summarize the content of Definition 3.3.6 by stating that in any local trivialization u = (x, g) the connection one-form has the following structure:

Gauge Transformations From (3.3.86) we easily work out the action on the oneform connection of the transition function from one local trivialization to another one. Let us recall (2.4.8) and (2.4.9). In the case of a principal bundle the transition function tαβ (x) is just a map from the intersection of two open neighborhoods of the base space into the structural group:

Correspondingly, combining this information with (3.3.86) we conclude that:

Aα → Aβ

Aα = g · Aα · g−1 + dg · g−1

(3.3.89)

Aβ = g · Aβ · g−1 + dg · g−1

Aβ = tαβ (x) · Aα · t−αβ1(x) + dtαβ (x) · t−αβ1(x)

The last equation of (3.3.89) is the most significant one for the bridge from Mathematics to Physics and provides the common stem of Yang’s picture in Fig. 3.1. The one-form A defined over the base-manifold M , which in particular can be space-time, is what physicists name a gauge potential, the prototype of which is the gauge-potential of electrodynamics; the transformation from one local trivialization to another one, encoded in the last of (3.3.89), is what physicists name a gauge transformation. The basic idea about gauge-transformations is that of local symmetry. Some physical system or rather some physical law is invariant with respect to the transformations of a group G but not only globally, rather also locally, namely the transformation can be chosen differently from one point of space-time to the next one. The basic mathematical structure realizing this physical idea is that of fibre-bundle and the one-form connection on a fibre-bundle is the appropriate mathematical structure which encompasses all mediators of all physical interactions.

Horizontal Vector Fields and Covariant Derivatives Let us come back to Definition 3.3.7 of horizontal vector fields. In a local trivialization a generic vector field on P is of the form:

where we have used the short-writing ∂μ = ∂/∂xμ and ∂i = ∂/∂αi . The horizontality condition in (3.3.76) becomes:

Multiplying on the left by g−1 and on the right by g, (3.3.91) can be easily solved for Xi obtaining in this way the general form of a horizontal vector field on a principal fibre-bundle P (M , G) endowed with a connection one-form A ↔ A . We get:

To obtain the above result we made use of the following identities:

which follow from the definitions of left-invariant one-forms and vector fields discussed in previous subsections.

Equation (3.3.92) is equally relevant for the bridge between Mathematics and Physics as (3.3.89). Indeed from (3.3.92) we realize that the mathematical notion of horizontal vector fields coincides with the physical notion of covariant derivative. Expressions like that in the bracket of (3.3.92), firstly appeared in the early decades of the XXth century when Classical Electrodynamics was rewritten in the language of Special Relativity and the quadri-potential Aμ was introduced. Indeed with the development first of Dirac equation and then of Quantum Electrodynamics, the covariant derivative of the electron wave-function made its entrance in the language of theoretical physics:

The operator μ might be assimilated with the action of a horizontal vector field (3.3.92) if we just had a one dimensional structural group U(1) with a single generator and if we were acting on functions over P (M , U(1)) that are eigenstates of the unique left-invariant vector field T(L) with eigenvalue i. This is indeed the case

of Electrodynamics. The group U(1) is composed by all unimodular complex num-

−→

bers exp[iα] and the left-invariant vector field is simply T(L) = ∂ /∂α. The wavefunction ψ(x) is actually a section of the principle bundle P (M4, U(1)) since it is a

complex spinor whose overall phase factor exp[iα(x)] takes values in the U(1)-fibre over x M4. Since T(L) exp[iα] = i exp[iα], the covariant derivative (3.3.94) is of

the form (3.3.92).

This shows that the one-form connection coincides, at least in the case of electrodynamics, with the physical notion of gauge-potential which, upon quantization, describes the photon-field, namely the mediator of electromagnetic interactions. The 1954 invention of non-Abelian gauge-theories by Yang and Mills started from the generalization of the covariant derivative (3.3.94) to the case of the su(2) Lie algebra. Introducing three gauge potentials AμA, corresponding to the three standard generators JA of su(2):

From the su(2) × u(1) standard model of electro-weak interactions of Glashow, Salam and Weinberg (see Fig. 3.14) we know nowadays that, in appropriate combinations with Aμ, the three gauge fields AμA, describe the W ± and Z0 particles discovered at CERN in 1983 by the UA1 experiment of Carlo Rubbia. They have spin one and mediate the weak interactions.

The mentioned examples show that the Lie algebra-valued connection one-form defined by Ehresmann on principle bundles is clearly related to the gauge-fields of particle physics since it enters the construction of covariant derivatives via the notion of horizontal vector fields. On the other hand the same one-form must be related to gravitation as well, since also there one deals with covariant derivatives (3.2.8) sustained by the Levi Civita connection and the Christoffel symbols (3.2.7). The key to understand the unifying point of view offered by the notion of Ehresmann connection on a principle bundle is obtained by recalling what we just emphasized in Chap. 2 when we introduced the very notion of fibre-bundles. Following Definition 2.4.5 let us recall that to each principle bundle P (M , G) one can associate as many associated vector bundles as there are linear representations of the structural

Fig. 3.14 Sheldon Glashow, Steven Weinberg and Abdus Salam shared the 1979 Nobel Prize in Physics for their 1968 formulation of a gauge theory of the electro-weak interactions based on the choice of a gauge group G = SU(2) × U(1). In the language of Mathematics this is the structural group of a principle-bundle, whose one-form connection contains four fields. These fields correspond to as many elementary particles: the W + , W − , the photon γ and another neutral massive particle Z0. While the existence of the W ± particles could be indirectly inferred from the known weak decays, the Z0 particle had no experimental motivation at the time when the SU(2) × U(1) was formulated. Yet in 1973 the neutral current interactions were discovered at CERN. Their effect was revealed by otherwise unexplicable electron tracks photographed at the Gargamelle Bubble Chamber. The direct production of the W ± and Z0 particles had to wait the construction of the Super Proton Synchrotron and the UA1 experiment of 1983 which revealed them. In 1984 Carlo Rubbia and the accelerator engineer Simon van der Meer, who invented the adiabatic cooling system at the basis of the experimental success, obtained the Nobel Prize in Physics for the experimental verification of the gauge theory description of electro-weak interactions

group G, namely infinitely many. It suffices to use as transition functions the corresponding linear representations of the transition functions of the principle bundle as displayed in (2.4.20). In all such associated bundles the fibres are vector spaces of dimension r (the rank of the bundle) and the transition functions are r × r matrices. Every linear representation of a Lie group G of dimension r induces a linear representation of its Lie algebra G, where the left- (right-)invariant vector fields T(L/R)A are mapped into r × r matrices:

It is therefore tempting to assume that given a one-form connection A ↔ A on a principle-bundle one can define a one-form connection on every associated vector bundle by taking its r × r matrix representation D(A) ↔ D(A ). In which sense this matrix-valued one-form defines a connection on the considered vector bundle? To answer such a question we obviously need first to define connections on generic vector bundles, which is what we do in the next section.