There are several different but equivalent formalizations of this idea:
as a parallel transport functor,
as a rule for a covariant derivative,
as a distribution (field) of horizontal subspaces – an Ehresmann connection – and via a connection -form which annihilates the distribution of horizontal subspaces. The connection in that sense induces a smooth version of Hurewicz connection.
The usual textbook convention is to say just connection for the distribution of horizontal subspaces, and the objects of the other three approaches one calls more specifically covariant derivative, connection -form and parallel transport.
In the remainder of this Idea-section we discuss a bit more how to understand connections in terms of parallel transport.
this assignment respects the structure on the fibers (for instance is -equivariant in the case that is a -bundle or that is linear in the case that is a vector bundle);
this assignment is smooth in a suitable sense;
this assignment is functorial in that for all pairs , of composable paths in we have
In other words, a connection on is a functor
The functor is called the parallel transport of the connection. This terminology comes from looking at the orbits of all points in under (i.e. from looking at the category of elements of ): these trace out paths in sitting over paths in and one thinks of the image of a point under as the result of propagating parallel to these curves in .
It may happen that the assignment only depends on the homotopy class of the path relative to its endpoints . In other words: that factors through the functor from the path groupoid to the fundamental groupoid of . In that case the connection is called a flat connection.
of vector bundles. Locally on – meaning: when everything is pulled back to a cover of – this is a -valued 1-form with certain special properties.
In particular, since every -principal bundle canonically trivializes when pulled back to its own total space , a connection in this case gives rise to a 1-form satisfying two conditions. This data is called an Ehresmann connection.
If instead is a vector bundle, then the two conditions satisfies by imply that it defines a linear map
from the space of section of that satisfies the properties of a covariant derivative.
If again the connection is flat, then this is manifestly the datum of a Lie infinity-algebroid representation of the tangent Lie algebroid of on : it defines the action Lie algebroid which is the Lie version of the Lie groupoid classified by the parallel transport functor.
Recall from the discussion at -principal bundle that the -bundle is encoded in a a suitable morphism
It turns out that similarly suitable morphisms
encode in one step the -bundle together with its parallel transport functor.
This is described in great detail in the reference by Schreiber–Waldorf below.
(…am running out of time… )
The groupoid of -principal bundles on is equivalent to the hom-groupoid
Now write for the Lie algebra of . Then consider the functor
There is an evident morphism of (2,1)-sheaves
that forgets the 1-forms on each object .
A connection on a smooth -principal bundle is a lift to
The groupoid of -principal bundles with connection on is
such that on the equation
A lift of this is in addition
such that on the equation
holds, where on the right we have the pullback of the Maurer-Cartan form on (see there).
(existence of connections)
Every -principal bundle admits a connection. In other words, the forgetful functor
By slight abuse of notation we shall write this and similar expressions simply as
Using the that satisfies its cocycle condition, one checks that this satisfies the cocycle condition for the 1-forms:
They play a central role for instance on Riemannian manifolds and pseudo-Riemannian manifolds. From the end of the 19th century and the beginning of the 20th centure originates a language to talk about these in terms of Christoffel symbols.
For more on this see higher category theory and physics.
When the notion of connection on a principal bundle is slightly coarsened, i.e. when more connections are regarded as being ismorphic than usual, one arrives at a structure called a Simons-Sullivan structured bundle. This has the special property that for the unitary group, the corresponding Grothendieck group of such bundles is a model for differential K-theory.
connection on a bundle, universal connection
|higher Atiyah groupoid:||standard higher Atiyah groupoid||higher Courant groupoid||groupoid version of quantomorphism n-group|
|coefficient for cohomology:|
|type of fiber ∞-bundle:||principal ∞-bundle||principal ∞-connection without top-degree connection form||principal ∞-connection|
gauge field: models and components
|physics||differential geometry||differential cohomology|
|gauge field||connection on a bundle||cocycle in differential cohomology|
|instanton/charge sector||principal bundle||cocycle in underlying cohomology|
|gauge potential||local connection differential form||local connection differential form|
|field strength||curvature||underlying cocycle in de Rham cohomology|
|minimal coupling||covariant derivative||twisted cohomology|
|BRST complex||Lie algebroid of moduli stack||Lie algebroid of moduli stack|
|extended Lagrangian||universal Chern-Simons n-bundle||universal characteristic map|
A classical textbook reference is
based on a series of classical observations. I farily comprehensive commented list of related references is here:
Basic facts about connections, such as the existence proof in terms of Cech cocycles, are collected in the brief lecture note