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 ${\displaystyle \pi :P\to M}$ be a smooth principal G-bundle over a smooth manifold ${\displaystyle M}$. Then a principal ${\displaystyle G}$-connection on ${\displaystyle P}$ is a differential 1-form on ${\displaystyle P}$ with values in the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of ${\displaystyle G}$ which is ${\displaystyle G}$-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on ${\displaystyle P}$.

In other words, it is an element ω of ${\displaystyle \Omega ^{1}(P,{\mathfrak {g}})\cong C^{\infty }(P,T^{*}P\otimes {\mathfrak {g}})}$ such that

1. ${\displaystyle {\hbox{Ad}}_{g}(R_{g}^{*}\omega )=\omega }$ where ${\displaystyle R_{g}}$ denotes right multiplication by ${\displaystyle g}$, and ${\displaystyle \operatorname {Ad} _{g}}$ is the adjoint representation on ${\displaystyle {\mathfrak {g}}}$ (explicitly, ${\displaystyle \operatorname {Ad} _{g}X={\frac {d}{dt}}g\exp(tX)g^{-1}{\bigl |}_{t=0}}$);
2. if ${\displaystyle \xi \in {\mathfrak {g}}}$ and ${\displaystyle X_{\xi }}$ is the vector field on P associated to ξ by differentiating the G action on P, then ${\displaystyle \omega (X_{\xi })=\xi }$ (identically on ${\displaystyle P}$).

Sometimes the term principal G-connection refers to the pair ${\displaystyle (P,\omega )}$ and ${\displaystyle \omega }$ 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 ${\displaystyle G\to H\to H/G}$, be a principal G-bundle over ${\displaystyle H/G}$) This means that 1-forms on the total space are canonically isomorphic to ${\displaystyle C^{\infty }(H,{\mathfrak {g}}^{*})}$, where ${\displaystyle {\mathfrak {g}}^{*}}$ is the dual lie algebra, hence G-connections are in bijection with ${\displaystyle C^{\infty }(H,{\mathfrak {g}}^{*}\otimes {\mathfrak {g}})^{G}}$.

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 ${\displaystyle P\times {\mathfrak {g}}}$, where VP = ker(dπ) is the kernel of the tangent mapping ${\displaystyle {\mathrm {d} }\pi \colon TP\to TM}$ 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 ${\displaystyle H_{pg}=\mathrm {d} (R_{g})_{p}(H_{p})}$.

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 ${\displaystyle {\mathfrak {g}}}$. 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 ${\displaystyle (sg)^{*}\omega =\operatorname {Ad} (g)^{-1}s^{*}\omega +g^{-1}dg}$. The principal connection is uniquely determined by this family of ${\displaystyle {\mathfrak {g}}}$-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 (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

${\displaystyle H=q^{-1}\Gamma (TM)\subset TP.}$ We see again the link to the horizontal bundle and thus Ehresmann connection.

Affine property

If ω and ω' are principal connections on a principal bundle P, then the difference ω' - ω is a ${\displaystyle {\mathfrak {g}}}$-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

${\displaystyle {\mathfrak {g}}_{P}:=P\times ^{G}{\mathfrak {g}}.}$

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 ${\displaystyle P\times ^{G}W}$ 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 ${\displaystyle P\times ^{G}W}$ 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 ${\displaystyle P\times ^{G}W}$ 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 d^ω from ${\displaystyle P\times ^{G}W}$-valued k-forms on M to ${\displaystyle P\times ^{G}W}$-valued (k+1)-forms on M. In particular, when k=0, we obtain a covariant derivative on ${\displaystyle P\times ^{G}W}$.

Curvature form

The curvature form of a principal G-connection ω is the ${\displaystyle {\mathfrak {g}}}$-valued 2-form Ω defined by

${\displaystyle \Omega =d\omega +{\tfrac {1}{2}}[\omega \wedge \omega ].}$

It is G-equivariant and horizontal, hence corresponds to a 2-form on M with values in ${\displaystyle {\mathfrak {g}}_{P}}$. The identification of the curvature with this quantity is sometimes called the (Cartan's) second structure equation.[1] 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 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

${\displaystyle \Theta =\mathrm {d} \theta +\omega \wedge \theta .}$

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.

References

1. 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