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

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.

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

- where denotes right multiplication by , and is the adjoint representation on (explicitly, );
- 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 G-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 G-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let , be a principal G-bundle over ) This means that 1-forms on the total space are canonically isomorphic to , where is the dual lie algebra, hence G-connections are in bijection with .

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 , where *VP* = ker(d*π*) is the kernel of the tangent mapping 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 .

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 .
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 . 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 *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** (Kobayashi 1957). 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

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

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.

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

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

for

which is a composition

defining the 1-form. Note that

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

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

- for a vertical vector
- 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.

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 Mauer-Cartan form on , in matrix form.

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.

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 ).

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 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 -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 *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.

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.

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*.

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 _{dR}*. This has the property that a principal