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 over a smooth manifold ' is a particular type of connection that is compatible with the action of the group '.
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 ' 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.
Let be a smooth principal G-bundle over a smooth manifold . Then a principal -connection on is a differential 1-form on with values in the Lie algebra of which is -equivariant and reproduces the Lie algebra generators of the fundamental vector fields on .
In other words, it is an element ÃÂ of such that
Sometimes the term principal -connection refers to the pair and itself is called the connection form or connection 1-form of the principal connection.
Most known non-trivial computations of principal '-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let , be a principal '-bundle over .) This means that real-valued 1-forms on the total space are canonically isomorphic to , where is the dual Lie algebra, hence '-connections are in bijection with .
A principal '-connection on determines an Ehresmann connection on in the following way. First note that the fundamental vector fields generating the action on provide a bundle isomorphism (covering the identity of ) from the bundle to , where is the kernel of the tangent mapping which is called the vertical bundle of . It follows that determines uniquely a bundle map which is the identity on . Such a projection is uniquely determined by its kernel, which is a smooth subbundle of (called the horizontal bundle) such that . This is an Ehresmann connection.
Conversely, an Ehresmann connection (or ) on defines a principal -connection if and only if it is -equivariant in the sense that .
A trivializing section of a principal bundle ' is given by a section s of ' over an open subset ' of '. Then the pullback s<sup>*</sup>àof a principal connection is a 1-form on ' with values in . If the section s is replaced by a new section sg, defined by (sg)(x) = s(x)g(x), where g:MâÂÂG is a smooth map, then . The principal connection is uniquely determined by this family of -valued 1-forms, and these 1-forms are also called connection forms or connection 1-forms, particularly in older or more physics-oriented literature.
The group ' acts on the tangent bundle ' 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 dÃÂ:TP/GâÂÂTM. Let ÃÂ:TP/GâÂÂM be the projection onto M. The fibres of the bundle TP/G under the projection àcarry an additive structure.
The bundle TP/G is called the bundle of principal connections . A section àof dÃÂ:TP/GâÂÂTM such that à: TM â 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 àof TP/G.
Finally, let àbe a principal connection in this sense. Let q:TPâÂÂTP/G be the quotient map. The horizontal distribution of the connection is the bundle
If àand ÃÂâ² are principal connections on a principal bundle P, then the difference is a -valued 1-form on P that 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.
For the trivial principal -bundle where , there is a canonical connection<sup>pg 49</sup><blockquote></blockquote>called the Maurer-Cartan connection. It is defined at a point by<blockquote> for </blockquote>which is a composition<blockquote></blockquote>defining the 1-form. Note that<blockquote></blockquote>is the Maurer-Cartan form on the Lie group and .
For a trivial principal -bundle , the identity section given by defines a 1-1 correspondence<blockquote></blockquote>between connections on and -valued 1-forms on <sup>pg 53</sup>. For a -valued 1-form on , there is a unique 1-form on such that
Then given this 1-form, a connection on can be constructed by taking the sum<blockquote></blockquote>giving an actual connection on . This unique 1-form can be constructed by first looking at it restricted to for . Then, is determined by because and we can get by taking<blockquote></blockquote>Similarly, the form<blockquote></blockquote>defines a 1-form giving the properties 1 and 2 listed above.
This statement can be refined<sup>pg 55</sup> even further for non-trivial bundles by considering an open covering of with trivializations and transition functions . Then, there is a 1-1 correspondence between connections on and collections of 1-forms<blockquote></blockquote>which satisfy<blockquote></blockquote>on the intersections for the Maurer-Cartan form on , in matrix form.
For a principal bundle the set of connections in is an affine space<sup>pg 57</sup> for the vector space where is the associated adjoint vector bundle. This implies for any two connections there exists a form such that<blockquote></blockquote>We denote the set of connections as , or just if the context is clear.
We<sup>pg 94</sup> can construct as a principal -bundle where and is the projection map<blockquote></blockquote>Note the Lie algebra of is just the complex plane. The 1-form defined as<blockquote></blockquote>forms a connection, which can be checked by verifying the definition. For any fixed we have<blockquote></blockquote>and since , we have -invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any we have a short exact sequence<blockquote></blockquote>where is defined as<blockquote></blockquote>so it acts as scaling in the fiber (which restricts to the corresponding -action). Taking we get
where the second equality follows because we are considering a vertical tangent vector, and . The notation is somewhat confusing, but if we expand out each term<blockquote></blockquote>it becomes more clear (where ).
For any linear representation W of G there is an associated vector bundle 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 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 is identified with the space of G-equivariant and horizontal W-valued k-forms on P. If ñ is such a k-form, then its exterior derivative dñ, although G-equivariant, is no longer horizontal. However, the combination dñ+ÃÂÃÂñ is. This defines an exterior covariant derivative d<sup>ÃÂ</sup> from -valued k-forms on M to -valued (k+1)-forms on M. In particular, when k=0, we obtain a covariant derivative on .
The curvature form of a principal G-connection àis the -valued 2-form é defined by
It is G-equivariant and horizontal, hence corresponds to a 2-form on M with values in . 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 MaurerâÂÂCartan equations: they are the same equations, but in a different setting and notation.
We say that a connection is flat if its curvature form . There is a useful characterization of principal bundles with flat connections; that is, a principal -bundle has a flat connection<sup>pg 68</sup> if and only if there exists an open covering with trivializations such that all transition functions<blockquote></blockquote>are constant. This is useful because it gives a recipe for constructing flat principal -bundles over smooth manifolds; namely taking an open cover and defining trivializations with constant transition functions.
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 ø, which is an equivariant R<sup>n</sup>-valued 1-form on P, should be taken into account. In particular, the torsion form on P, is an R<sup>n</sup>-valued 2-form àdefined by
ÃÂ 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.
If X is a scheme (or more generally, stack, derived stack, or even prestack), we can associate to it its so-called de Rham stack, denoted X<sub>dR</sub>. This has the property that a principal G bundle over X<sub>dR</sub> is the same thing as a G bundle with *flat* connection over X.