Connection (principal bundle)

Summary

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 which 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.

Formal definition edit

 
A principal bundle connection form   may be thought of as a projection operator on the tangent bundle   of the principal bundle  . The kernel of the connection form is given by the horizontal subspaces for the associated Ehresmann connection.
 
A connection is equivalently specified by a choice of horizontal subspace   for every tangent space to the principal bundle  .
 
A principal bundle connection is required to be compatible with the right group action of   on  . This can be visualized as the right multiplication   taking the horizontal subspaces into each other. This equivariance of the horizontal subspaces   interpreted in terms of the connection form   leads to its characteristic equivariance properties.

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 ω of   such that

  1.   where   denotes right multiplication by  , and   is the adjoint representation on   (explicitly,  );
  2. if   and   is the vector field on P associated to ξ by differentiating the G action on P, then   (identically on  ).

Sometimes the term principal  -connection refers to the pair   and   itself is called the connection form or connection 1-form of the principal connection.

Computational remarks edit

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 1-forms on the total space are canonically isomorphic to  , where   is the dual lie algebra, hence  -connections are in bijection with  .

Relation to Ehresmann connections edit

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  .

Pull back via trivializing section edit

A trivializing section of a principal bundle   is given by a section s of   over an open subset   of  . Then the pullback s*ω of 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:MG 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.

Bundle of principal connections edit

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 :TP/GTM. Let ρ:TP/GM 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 (Kobayashi 1957). A section Γ of dπ:TP/GTM such that Γ : TMTP/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:TPTP/G be the quotient map. The horizontal distribution of the connection is the bundle

  We see again the link to the horizontal bundle and thus Ehresmann connection.

Affine property edit

If ω and ω′ are principal connections on a principal bundle P, then the difference ω′ − ω is a  -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.

Examples edit

Maurer-Cartan connection edit

For the trivial principal  -bundle   where  , there is a canonical connection[1]pg 49

 

called the Maurer-Cartan connection. It is defined as follows: for a point   define

  for  

which is a composition

 

defining the 1-form. Note that

 

is the Maurer-Cartan form on the Lie group   and  .

Trivial bundle edit

For a trivial principal  -bundle  , the identity section   given by   defines a 1-1 correspondence

 

between connections on   and  -valued 1-forms on  [1]pg 53. For a  -valued 1-form   on  , there is a unique 1-form   on   such that

  1.   for   a vertical vector
  2.   for any  

Then given this 1-form, a connection on   can be constructed by taking the sum

 

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

 

Similarly, the form

 

defines a 1-form giving the properties 1 and 2 listed above.

Extending this to non-trivial bundles edit

This statement can be refined[1]pg 55 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

 

which satisfy

 

on the intersections   for   the Maurer-Cartan form on  ,   in matrix form.

Global reformulation of space of connections edit

For a principal   bundle   the set of connections in   is an affine space[1]pg 57 for the vector space   where   is the associated adjoint vector bundle. This implies for any two connections   there exists a form   such that

 

We denote the set of connections as  , or just   if the context is clear.

Connection on the complex Hopf-bundle edit

We[1]pg 94 can construct   as a principal  -bundle   where   and   is the projection map

 

Note the Lie algebra of   is just the complex plane. The 1-form   defined as

 

forms a connection, which can be checked by verifying the definition. For any fixed   we have

 

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

 

where   is defined as

 

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

 

it becomes more clear (where  ).

Induced covariant and exterior derivatives edit

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 α is such a k-form, then its exterior derivative dα, although G-equivariant, is no longer horizontal. However, the combination dα+ωΛα is. This defines an exterior covariant derivative dω from  -valued k-forms on M to  -valued (k+1)-forms on M. In particular, when k=0, we obtain a covariant derivative on  .

Curvature form edit

The curvature form of a principal G-connection ω is the  -valued 2-form Ω 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.[2] 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.

Flat connections and characterization of bundles with flat connections edit

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[1]pg 68 if and only if there exists an open covering   with trivializations   such that all transition functions

 

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.

Connections on frame bundles and torsion edit

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 Rn-valued 1-form on P, should be taken into account. In particular, the torsion form on P, is an Rn-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.

Definition in algebraic geometry edit

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 XdR. This has the property that a principal G bundle over XdR is the same thing as a G bundle with connection over X.

References edit

  1. ^ a b c d e f Dupont, Johan (August 2003). "Fibre Bundles and Chern-Weil Theory" (PDF). Archived from the original (PDF) on 31 March 2022.
  2. ^ Eguchi, Tohru; Gilkey, Peter B.; Hanson, Andrew J. (1980). "Gravitation, gauge theories and differential geometry". Physics Reports. 66 (6): 213–393. Bibcode:1980PhR....66..213E. doi:10.1016/0370-1573(80)90130-1.
  • Kobayashi, Shoshichi (1957), "Theory of Connections", Ann. Mat. Pura Appl., 43: 119–194, doi:10.1007/BF02411907, S2CID 120972987
  • Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 1 (New ed.), Wiley Interscience, ISBN 0-471-15733-3
  • Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operations in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2008-03-25