Connection (principal bundle)

In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal "G"-connection on a principal G-bundle "P" over a smooth manifold "M" is a particular type of connection which is compatible with the action of the group "G".
A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to "P" via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.
Formal definition.
Let formula_1 be a smooth principal "G"-bundle over a smooth manifold formula_2. Then a principal formula_3-connection on formula_4 is a differential 1-form on formula_4 with values in the Lie algebra formula_6 of formula_3 which is formula_3-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on formula_4.
In other words, it is an element "ω" of formula_10 such that
Sometimes the term "principal G-connection" refers to the pair formula_21 and formula_22 itself is called the connection form or connection 1-form of the principal connection.
Computational remarks.
Most known non-trivial computations of principal G-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let formula_23, be a principal G-bundle over formula_24) This means that 1-forms on the total space are canonically isomorphic to formula_25, where formula_26 is the dual lie algebra, hence G-connections are in bijection with formula_27.
Relation to Ehresmann connections.
A principal G-connection "ω" on "P" determines an Ehresmann connection on "P" in the following way. First note that the fundamental vector fields generating the "G" action on "P" provide a bundle isomorphism (covering the identity of "P") from the bundle "VP" to formula_28, where "VP" = ker(d"π") is the kernel of the tangent mapping formula_29 which is called the vertical bundle of "P". It follows that "ω" determines uniquely a bundle map "v":"TP"→"V" which is the identity on "V". Such a projection "v" is uniquely determined by its kernel, which is a smooth subbundle "H" of "TP" (called the horizontal bundle) such that "TP"="V"⊕"H". This is an Ehresmann connection.
Conversely, an Ehresmann connection "H"⊂"TP" (or "v":"TP"→"V") on "P" defines a principal "G"-connection "ω" if and only if it is "G"-equivariant in the sense that formula_30.
Pull back via trivializing section.
A trivializing section of a principal bundle "P" is given by a section "s" of "P" over an open subset "U" of "M". Then the pullback "s"*"ω" of a principal connection is a 1-form on "U" with values in formula_6.
If the section "s" is replaced by a new section "sg", defined by ("sg")("x") = "s"("x")"g"("x"), where "g":"M"→"G" is a smooth map, then formula_32. The principal connection is uniquely determined by this family of formula_6-valued 1-forms, and these 1-forms are also called connection forms or connection 1-forms, particularly in older or more physics-oriented literature.
Bundle of principal connections.
The group "G" acts on the tangent bundle "TP" by right translation. The quotient space "TP"/"G" is also a manifold, and inherits the structure of a fibre bundle over "TM" which shall be denoted "dπ":"TP"/"G"→"TM". Let ρ:"TP"/"G"→"M" be the projection onto "M". The fibres of the bundle "TP"/"G" under the projection ρ carry an additive structure.
The bundle "TP"/"G" is called the bundle of principal connections . A section Γ of dπ:"TP"/"G"→"TM" such that Γ : "TM" → "TP"/"G" is a linear morphism of vector bundles over "M", can be identified with a principal connection in "P". Conversely, a principal connection as defined above gives rise to such a section Γ of "TP"/"G".
Finally, let Γ be a principal connection in this sense. Let "q":"TP"→"TP"/"G" be the quotient map. The horizontal distribution of the connection is the bundle
Affine property.
If "ω" and "ω"′ are principal connections on a principal bundle "P", then the difference is a formula_6-valued 1-form on "P" which is not only "G"-equivariant, but horizontal in the sense that it vanishes on any section of the vertical bundle "V" of "P". Hence it is basic and so is determined by a 1-form on "M" with values in the adjoint bundle
Conversely, any such one form defines (via pullback) a "G"-equivariant horizontal 1-form on "P", and the space of principal "G"-connections is an affine space for this space of 1-forms.
Induced covariant and exterior derivatives.
For any linear representation "W" of "G" there is an associated vector bundle formula_37 over "M", and a principal connection induces a covariant derivative on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of formula_37 over "M" is isomorphic to the space of "G"-equivariant "W"-valued functions on "P". More generally, the space of "k"-forms with values in formula_37 is identified with the space of "G"-equivariant and horizontal "W"-valued "k"-forms on "P". If "α" is such a "k"-form, then its exterior derivative d"α", although "G"-equivariant, is no longer horizontal. However, the combination d"α"+"ω"Λ"α" is. This defines an exterior covariant derivative d"ω" from formula_37-valued "k"-forms on "M" to formula_37-valued ("k"+1)-forms on "M". In particular, when "k"=0, we obtain a covariant derivative on formula_37.
Curvature form.
The curvature form of a principal "G"-connection "ω" is the formula_6-valued 2-form Ω defined by
It is "G"-equivariant and horizontal, hence corresponds to a 2-form on "M" with values in formula_45. The identification of the curvature with this quantity is sometimes called the "(Cartan's) second structure equation". Historically, the emergence of the structure equations are found in the development of the Cartan connection. When transposed into the context of Lie groups, the structure equations are known as the Maurer–Cartan equations: they are the same equations, but in a different setting and notation.
Connections on frame bundles and torsion.
If the principal bundle "P" is the frame bundle, or (more generally) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since the additional structure of the solder form "θ", which is an equivariant R"n"-valued 1-form on "P", should be taken into account. In particular, the torsion form on "P", is an R"n"-valued 2-form Θ defined by
Θ is "G"-equivariant and horizontal, and so it descends to a tangent-valued 2-form on "M", called the "torsion". This equation is sometimes called the "(Cartan's) first structure equation".