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Theory of connections on P
We give the definition of connections on P in terms of elements of Ω 1 (P, g). Covariant derivatives associated to connection 1-forms on P are defined in terms of 1-forms defined on P with values in the space of first order derivatives of sections of a fiber bundle E . Also, the curvature of a connection on P is defined as an element of Ω2 (P, g). Both the passive and the active actions of the gauge group are also detailed.
Connections on principal fiber bundles
The subspace of vertical vector fields is defined as the kernel of T∗π. However, its comple-mentary space in (T P), the so-called horizontal subspace, cannot be uniquely defined: it is related to a choice of a connection on P. From the geometric point of view, a connection on P is the assignment of a subspace HP of the tangent bundle T P such that, for any u ∈ P, we have:
• TuP = VuP ⊕ HuP.
Covariant derivative and curvatures
Let h : (T P) → (HP) be a projector on the horizontal subspace associated to a connection on P. Let α be a differential q-form on P with values in sections of a fiber bundle E . The covariant derivative associated to this connection is a map D acting on q-forms α as (Dα)(X1 , . . . , Xq ) = (dα)(h ◦ X1 , . . . , h ◦ Xq ) (1.3.2) for any X1 , . . . , Xq ∈ (T P). Obviously, the covariant derivative vanishes for vertical vector fields, due to the presence of the projector h. This definition will be used in chapter 4 to define the covariant derivative in the context of transitive Lie algebroids. The curvature R associated to a connection 1-form is an element of Ω2 (P, g) which can be defined by two equivalent distinct ways. The former consists in defining R as the covariant derivative of the connection 1-form i.e. as R(X, Y ) = (dω)(h ◦ X, h ◦ Y ), for any X, Y ∈ (T P). The latter is given in terms of the connection 1-form ω, by the Cartan equation structure, as R(X, Y ) = dω(X, Y ) + [ω(X), ω(Y )] (1.3.3) for any X, Y ∈ (T P). Here the Lie bracket is a graded Lie bracket defined on the space of differential forms Ω•(P, g). 7 By construction, R is horizontal, i.e. R(X, Y ) = 0 if any of the vector fields X or Y are vertical vector fields.
Local expression of a connection 1-form
Connection 1-forms are locally trivialized as elements of Ω1 (Ui, g) by using local cross-sections (si)i∈I on P. Then, the local trivialization of ω over Ui is given by the C∞(Ui)-linear map ωloc, i : (T Ui) → g as ωloc, i(X) = (si∗ω)(X) = ω(si ∗X) (1.3.4) for any X ∈ (T Ui). Over the open set Uij , the gluing transformations of the connection 1-form are ωloc, i = gij−1 ωloc, igij + gij−1 dgij (1.3.5) where d is the Koszul derivative and gij−1 dgij is a g-valued 1-form defined on Uij . We define the local trivialization of the curvature of ω over Ui as 1 Rloc, i = dωloc, i + 2 [ωloc, i, ωloc, i] where the differential operator d is the Koszul derivative acting on Ω•(M). expressions Rloc, i and Rloc, j of the curvature R, associated to ω, are related the homogeneous passive gauge transformation Rloc, i = gij−1 Rloc, j gij (1.3.6) The local over Uij by (1.3.7) These gauge transformations are induced by changes of local trivialization of P. They form the set of the passive gauge transformation. 7 The graded Lie bracket on Ω• (P , g) is defined as [ω1 ⊗ X1 , ω2 ⊗ X2 ] = ω1 ∧ ω2 ⊗ [X1 , X2 ] for any ω1 , ω2 ∈ Ω• (P ) and X1 , X2 ∈ g.
Active gauge transformations on P
The gauge group G associated to a principal bundle P(M, G) is defined as the group of vertical automorphisms f : P → P, along the fiber P, compatible with the action of G in the sense that f (u a) = f (u) a for any a ∈ G and (π ◦ f ) = π. This automorphism can be written in terms of maps g : P → G as f (u) = u g(u), for any u ∈ P. With respect to the right-action of G, the map g : P → G transforms as g(u a) = a−1 g(u)a, for any a ∈ G. This transformation indicates that g defines a section of the associated fiber bundle P ×β G, where β acts by conjugacy on G as βg g′ = g−1 g′g for any g, g′ ∈ G. We denote by G the section of the associated fiber bundle P ×β G. Depending on the context, elements of the gauge group are either G-equivariant maps P → G, or, given a local section of P, a field U → G (see subsection 1.2.3 for the correspondence).
The gauge group G acts on connection 1-forms ω as ω ωg = g−1 ωg + g−1 dg (1.3.8)
where g ∈ G. This transformation corresponds to an active gauge transformation of G. It its straightforward to check that ωg still defines a connection 1-form on P. In this sense, we say that the space of connection 1-forms is compatible with the action of the gauge group. Gauge transformations of ω induce gauge transformations of the curvature R. It transforms as Rg = g−1 Rg for any g ∈ G. In the theory of fiber bundles, active and passive gauge transformations have the same mathematical expression. We will see that it is no longer true in section 6.3: on transitive Lie algebroids, we define an “algebraic” infinitesimal action of the gauge group, distinct from the “geometric” one. This is motivated by the introduction of generalized connections.
Table of contents :
Introduction
Chapter 1 Differential geometry
1.1 Differentiable manifolds
1.2 Principal fiber bundles
1.3 Theory of connections on P
Chapter 2 The theory of Lie algebroids
2.1 Basic notions
2.2 Local trivializations of Lie algebroids
Chapter 3 Differential structures on transitive Lie algebroids
3.1 Representation space
3.2 Differential forms on transitive Lie algebroids
3.3 Local trivializations of differential complexes
Chapter 4 Theory of connections on transitive Lie algebroids
4.1 Ordinary connections
4.2 Generalized connections on transitive Lie algebroids.
Chapter 5 Scalar product for differential forms
5.1 Metric on transitive Lie algebroids
5.2 Integration over A
5.3 Hodge star operator
5.4 Scalar product for forms defined on A.
Chapter 6 Gauge theory based on transitive Lie algebroids
6.1 Geometric action of L on ordinary connections
6.2 Geometric action of L on generalized connections
6.3 Algebraic action of L
Chapter 7 Yang-Mills-Higgs type theories
7.1 Gauge field theories using generalized connections.
7.2 Applications of the gauge field theory model
7.3 Spaces of solutions
Chapter 8 Method of symmetry reduction
8.1 General framework
8.2 Application to SU(N)-gauge theories
8.3 Application for YMH theory based on generalized connections
8.4 Commentaries
Conclusion
Appendices
Chapter A Cech-de Rham bicomplex
A.1 Recalls
A.2 Generalized Cech-de Rham bicomplex
Chapter B Code Mathematica
