Editing Connection (principal bundle)

{{Short description|Concept in mathematics}}
{{About|connections on principal bundles|information about other types of connections in mathematics|Connection (mathematics)}}
In [[mathematics]], and especially [[differential geometry]] and [[gauge theory (mathematics)|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 bundle|principal G-bundle]] <math>P</math> over a [[smooth manifold]] ''<math>M</math>'' is a particular type of connection that is compatible with the [[Group action (mathematics)|action]] of the group ''<math>G</math>''.

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 ''<math>P</math>'' via the [[associated bundle]] construction. In particular, on any [[associated vector bundle]] the principal connection induces a [[covariant derivative]], an operator that can differentiate [[section (fiber bundle)|sections]] of that bundle along [[tangent vector|tangent directions]] in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a [[Connection (vector bundle)|linear connection]] on the [[frame bundle]] of a [[smooth manifold]].

==Formal definition==
[[File:Principal bundle connection form projection.png|thumb|300px|A principal bundle connection form <math>\omega</math> may be thought of as a projection operator on the tangent bundle <math>TP</math> of the principal bundle <math>P</math>. The kernel of the connection form is given by the horizontal subspaces for the associated [[Ehresmann connection]].]]
[[File:Ehresmann connection.png|thumb|300px|A connection is equivalently specified by a choice of horizontal subspace <math>H_p\subset T_pP</math> for every tangent space to the principal bundle <math>P</math>.]]
[[File:Equivariance of Principal bundle connection.png|thumb|300px|A principal bundle connection is required to be compatible with the right group action of <math>G</math> on <math>P</math>. This can be visualized as the right multiplication <math>R_g</math> taking the horizontal subspaces into each other. This equivariance of the horizontal subspaces <math>H\subset TP</math> interpreted in terms of the connection form <math>\omega</math> leads to its characteristic equivariance properties.]]
Let <math>\pi : P \to M</math> be a smooth [[principal bundle|principal ''G''-bundle]] over a [[smooth manifold]] <math>M</math>. Then a '''principal''' <math>G</math>'''-connection''' on <math>P</math> is a differential 1-form on <math>P</math> [[Lie algebra valued form|with values in the Lie algebra]] <math>\mathfrak g</math> of <math>G</math> which is <math>G</math>'''-equivariant''' and '''reproduces''' the '''Lie algebra generators''' of the '''fundamental vector fields''' on <math>P</math>.

In other words, it is an element ''ω'' of <math>\Omega^1(P,\mathfrak g)\cong C^\infty(P, T^*P\otimes\mathfrak g)</math> such that
# <math>\hbox{Ad}_g(R_g^*\omega)=\omega</math> where <math>R_g</math> denotes right multiplication by <math>g</math>, and <math>\operatorname{Ad}_g</math> is the [[adjoint representation]] on <math> \mathfrak g</math> (explicitly, <math>\operatorname{Ad}_gX = \frac{d}{dt}g\exp(tX)g^{-1}\bigl|_{t=0}</math>);
# if <math>\xi\in \mathfrak g</math> and <math>X_\xi</math> is [[fundamental vector field|the vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P'']], then <math>\omega(X_\xi)=\xi</math> (identically on <math>P</math>).

Sometimes the term ''principal <math>G</math>-connection'' refers to the pair <math>(P,\omega)</math> and <math>\omega</math> itself is called the '''[[connection form]]''' or '''connection 1-form''' of the principal connection.

=== Computational remarks ===
Most known non-trivial computations of principal ''<math>G</math>''-connections are done with [[homogeneous space]]s because of the triviality of the (co)tangent bundle. (For example, let <math>G \to H \to H/G</math>, be a principal ''<math>G</math>''-bundle over <math> H/G</math>.) This means that 1-forms on the total space are canonically isomorphic to <math>C^\infty(H,\mathfrak{g}^*)</math>, where <math> \mathfrak{g}^*</math> is the dual lie algebra, hence ''<math>G</math>''-connections are in bijection with <math>C^\infty(H,\mathfrak{g}^*\otimes \mathfrak{g})^G</math>.

===Relation to Ehresmann connections===
A principal ''<math>G</math>''-connection <math>\omega</math> on <math>P</math> determines an [[Ehresmann connection]] on <math>P</math> in the following way. First note that the fundamental vector fields generating the <math>G</math> action on <math>P</math> provide a bundle isomorphism (covering the identity of <math>P</math>) from the [[Fiber bundle|bundle]] <math>V</math> to <math>P\times\mathfrak g</math>, where <math>V=\ker(d\pi)</math> is the kernel of the [[Pushforward (differential)|tangent mapping]] <math>{\mathrm d}\pi\colon TP\to TM</math> which is called the [[vertical bundle]] of <math>P</math>. It follows that <math>\omega</math> determines uniquely a bundle map <math>v:TP\rightarrow V</math> which is the identity on <math>V</math>. Such a projection <math>v</math> is uniquely determined by its kernel, which is a smooth subbundle <math>H</math> of <math>TP</math> (called the [[horizontal bundle]]) such that <math>TP=V\oplus H</math>. This is an Ehresmann connection.

Conversely, an Ehresmann connection <math>H\subset TP</math> (or <math>v:TP\rightarrow V</math>) on <math>P</math> defines a principal <math>G</math>-connection <math>\omega</math> if and only if it is <math>G</math>-equivariant in the sense that <math>H_{pg}=\mathrm d(R_g)_p(H_{p})</math>.

===Pull back via trivializing section===
A trivializing section of a principal bundle ''<math>P</math>'' is given by a section ''s'' of ''<math>P</math>'' over an open subset ''<math>U</math>'' of ''<math>M</math>''. Then the [[pullback (differential geometry)|pullback]] ''s''<sup>*</sup>''ω'' of a principal connection is a 1-form on ''<math>U</math>'' with values in <math>\mathfrak g</math>.
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 <math>(sg)^* \omega = \operatorname{Ad}(g)^{-1}s^* \omega + g^{-1} dg</math>. The principal connection is uniquely determined by this family of <math>\mathfrak g</math>-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 ''<math>G</math>'' acts on the [[tangent bundle]] ''<math>TP</math>'' by right translation.  The [[Quotient space (topology)|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''' {{harv|Kobayashi|1957}}.  A [[section (fiber bundle)|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

:<math>H = q^{-1}\Gamma(TM) \subset TP.</math> 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 {{nowrap|''ω''′ − ''ω''}} is a <math>\mathfrak g</math>-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]]
:<math>\mathfrak g_P:=P\times^G\mathfrak g.</math>
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 ==

=== Maurer-Cartan connection ===
For the trivial principal <math>G</math>-bundle <math>\pi:E \to X</math> where <math>E = G\times X</math>, there is a canonical connection<ref name=":0">{{Cite web |last=Dupont |first=Johan |date=August 2003 |title=Fibre Bundles and Chern-Weil Theory |url=http://www.johno.dk/mathematics/fiberbundlestryk.pdf |archive-url=https://web.archive.org/web/20220331053124/http://www.johno.dk/mathematics/fiberbundlestryk.pdf |archive-date=31 March 2022}}</ref><sup>pg 49</sup><blockquote><math>\omega_{MC} \in \Omega^1(E,\mathfrak{g})</math></blockquote>called the Maurer-Cartan connection. It is defined at a point <math>(g,x) \in G\times X</math> by<blockquote><math>(\omega_{MC})_{(g,x)} = (L_{g^{-1}}\circ \pi_1)_*</math> for <math>x \in X, g \in G</math></blockquote>which is a composition<blockquote><math>T_{(g,x)}E \xrightarrow{\pi_{1*}} T_gG \xrightarrow{(L_{g^{-1}})_*} T_eG = \mathfrak{g}</math></blockquote>defining the 1-form. Note that<blockquote><math>\omega_0 = (L_{g^{-1}})_*: T_gG \to T_eG = \mathfrak{g}</math></blockquote>is the [[Maurer-Cartan form]] on the Lie group <math>G</math> and <math>\omega_{MC} = \pi_1^*\omega_0</math>.

=== Trivial bundle ===
For a trivial principal <math>G</math>-bundle <math>\pi:E \to X</math>, the identity section <math>i: X \to G\times X</math> given by <math>i(x) = (e,x)</math> defines a 1-1 correspondence<blockquote><math>i^*:\Omega^1(E,\mathfrak{g}) \to \Omega^1(X,\mathfrak{g})</math></blockquote>between connections on <math>E</math> and <math>\mathfrak{g}</math>-valued 1-forms on <math>X</math><ref name=":0" /><sup>pg 53</sup>. For a <math>\mathfrak{g}</math>-valued 1-form <math>A</math> on <math>X</math>, there is a unique 1-form <math>\tilde{A}</math> on <math>E</math> such that

# <math>\tilde{A}(X) = 0</math> for <math>X \in T_xE</math> a vertical vector
# <math>R_g^*\tilde{A} = \text{Ad}(g^{-1}) \circ \tilde{A}</math> for any <math>g \in G</math>

Then given this 1-form, a connection on <math>E</math> can be constructed by taking the sum<blockquote><math>\omega_{MC} + \tilde{A}</math></blockquote>giving an actual connection on <math>E</math>. This unique 1-form can be constructed by first looking at it restricted to <math>(e,x)</math> for <math>x \in X</math>. Then, <math>\tilde{A}_{(e,x)}</math> is determined by <math>A</math> because <math>T_{(x,e)}E = ker(\pi_*)\oplus i_*T_xX</math> and we can get <math>\tilde{A}_{(g,x)}</math>by taking<blockquote><math>\tilde{A}_{(g,x)} = R^*_g\tilde{A}_{(e,x)} = \text{Ad}(g^{-1})\circ \tilde{A}_{(e,x)}</math></blockquote>Similarly, the form<blockquote><math>\tilde{A}_{(x,g)} = \text{Ad}(g^{-1}) \circ A_x \circ \pi_*: T_{(x,g)}E \to \mathfrak{g} </math></blockquote>defines a 1-form giving the properties 1 and 2 listed above.

==== Extending this to non-trivial bundles ====
This statement can be refined<ref name=":0" /><sup>pg 55</sup> even further for non-trivial bundles <math>E \to X</math> by considering an open covering <math>\mathcal{U} = \{U_a\}_{a \in I}</math> of <math>X</math> with [[Trivialization (mathematics)|trivializations]] <math>\{\phi_a\}_{a \in I}</math> and transition functions <math>\{g_{ab}\}_{a,b\in I}</math>. Then, there is a 1-1 correspondence between connections on <math>E</math> and collections of 1-forms<blockquote><math>\{A_a \in \Omega_1(U_a,\mathfrak{g}) \}_{a \in I}</math></blockquote>which satisfy<blockquote><math>A_b = Ad(g_{ab}^{-1})\circ A_a + g_{ab}^*\omega_0</math></blockquote>on the intersections <math>U_{ab}</math> for <math>\omega_0</math> the [[Maurer–Cartan form|Maurer-Cartan form]] on <math>G</math>, <math>\omega_0 = g^{-1}dg</math> in matrix form.

==== Global reformulation of space of connections ====
For a principal <math>G</math> bundle <math>\pi: E \to M</math> the set of connections in <math>E</math> is an affine space<ref name=":0" /><sup>pg 57</sup> for the vector space <math>\Omega^1(M,E_\mathfrak{g})</math> where <math>E_\mathfrak{g}</math> is the associated adjoint vector bundle. This implies for any two connections <math>\omega_0, \omega_1</math> there exists a form <math>A \in \Omega^1(M, E_\mathfrak{g})</math> such that<blockquote><math>\omega_0 = \omega_1 + A</math></blockquote>We denote the set of connections as <math>\mathcal{A}(E)</math>, or just <math>\mathcal{A}</math> if the context is clear.

=== Connection on the complex Hopf-bundle ===
We<ref name=":0" /><sup>pg 94</sup> can construct <math>\mathbb{CP}^n</math> as a principal <math>\mathbb{C}^*</math>-bundle <math>\gamma:H_\mathbb{C} \to \mathbb{CP}^n</math> where <math>H_\mathbb{C} = \mathbb{C}^{n+1}-\{0\}</math> and <math>\gamma</math> is the projection map<blockquote><math>\gamma(z_0,\ldots,z_n) = [z_0,\ldots,z_n]</math></blockquote>Note the Lie algebra of <math>\mathbb{C}^* = GL(1,\mathbb{C})</math> is just the complex plane. The 1-form <math>\omega \in \Omega^1(H_\mathbb{C},\mathbb{C})</math> defined as<blockquote><math>\begin{align}
\omega &= \frac{\overline{z}^tdz}{|z|^2} \\
&= \sum_{i=0}^n\frac{\overline{z}_i}{|z|^2}dz_i
\end{align}</math></blockquote>forms a connection, which can be checked by verifying the definition. For any fixed <math>\lambda \in \mathbb{C}^*</math> we have<blockquote><math>\begin{align}
R_\lambda^*\omega &= \frac{\overline{(z\lambda)}^td(z\lambda)}{|z\lambda|^2} \\
&= \frac{
  \overline{\lambda}\lambda\overline{z}^tdz
}{|\lambda|^2\cdot |z|^2}

\end{align}</math></blockquote>and since <math>|\lambda|^2 = \overline{\lambda}{\lambda}</math>, we have <math>\mathbb{C}^*</math>-invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any <math>z \in H_\mathbb{C}</math> we have a short exact sequence<blockquote><math>0 \to \mathbb{C} \xrightarrow{v_z} T_zH_\mathbb{C} \xrightarrow{\gamma_*} T_{[z]}\mathbb{CP}^n \to 0</math></blockquote>where <math>v_z</math> is defined as<blockquote><math>v_z(\lambda) = z\cdot \lambda</math></blockquote>so it acts as scaling in the fiber (which restricts to the corresponding <math>\mathbb{C}^*</math>-action). Taking <math>\omega_z\circ v_z(\lambda)</math> we get

<math>\begin{align}
\omega_z\circ v_z(\lambda) &= \frac{\overline{z}dz}{|z|^2}(z\lambda) \\
&= \frac{\overline{z}z\lambda}{|z|^2} \\
&= \lambda

\end{align}</math>

where the second equality follows because we are considering <math>z\lambda</math> a vertical tangent vector, and <math>dz(z\lambda) = z\lambda</math>. The notation is somewhat confusing, but if we expand out each term<blockquote><math>\begin{align}
dz &= dz_0 + \cdots + dz_n \\
z &= a_0z_0 + \cdots +a_nz_n \\
dz(z) &= a_0 + \cdots + a_n \\
dz(\lambda z) &= \lambda\cdot (a_0 + \cdots + a_n) \\
\overline{z} &= \overline{a_0} + \cdots + \overline{a_n}

\end{align}</math></blockquote>it becomes more clear (where <math>a_i \in \mathbb{C}</math>).

==Induced covariant and exterior derivatives==
For any [[linear representation]] ''W'' of ''G'' there is an [[associated vector bundle]] <math> P\times^G W</math> over ''M'', and a principal connection induces a [[Connection (vector bundle)|covariant derivative]] on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of <math> P\times^G W</math> over ''M'' is isomorphic to the space of ''G''-equivariant ''W''-valued functions on ''P''. More generally, the space of ''k''-forms [[vector-valued differential form|with values in]] <math> P\times^G W</math>  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<sup>''ω''</sup> from <math> P\times^G W</math>-valued ''k''-forms on ''M'' to <math> P\times^G W</math>-valued (''k''+1)-forms on ''M''. In particular, when ''k''=0, we obtain a covariant derivative on <math> P\times^G W</math>.

==Curvature form==
The [[curvature form]] of a principal ''G''-connection ''ω'' is the <math>\mathfrak g</math>-valued 2-form Ω defined by
:<math>\Omega=d\omega +\tfrac12 [\omega\wedge\omega].</math>
It is ''G''-equivariant and horizontal, hence corresponds to a 2-form on ''M'' with values in <math>\mathfrak g_P</math>. The identification of the curvature with this quantity is sometimes called the ''(Cartan's) second structure equation''.<ref>{{Cite journal|last1=Eguchi|first1=Tohru|last2=Gilkey|first2=Peter B.|last3=Hanson|first3=Andrew J.|date=1980|title=Gravitation, gauge theories and differential geometry|url=https://www.researchgate.net/publication/234195796|journal=Physics Reports|language=en|volume=66|issue=6|pages=213–393|doi=10.1016/0370-1573(80)90130-1|bibcode=1980PhR....66..213E}}</ref> Historically, the emergence of the structure equations are found in the development of the [[Cartan connection]].  When transposed into the context of [[Lie group]]s, the structure equations are known as the [[Maurer–Cartan equation]]s: they are the same equations, but in a different setting and notation.

=== Flat connections and characterization of bundles with flat connections ===
We say that a connection <math>\omega</math> is '''flat''' if its curvature form <math>\Omega = 0</math>. There is a useful characterization of principal bundles with flat connections; that is, a principal <math>G</math>-bundle <math>\pi: E \to X</math> has a flat connection<ref name=":0" /><sup>pg 68</sup> if and only if there exists an open covering <math>\{U_a\}_{a\in I}</math> with trivializations <math>\left\{ \phi_a \right\}_{a \in I}</math> such that all transition functions<blockquote><math>g_{ab}: U_a\cap U_b \to G</math></blockquote>are constant. This is useful because it gives a recipe for constructing flat principal <math>G</math>-bundles over smooth manifolds; namely taking an open cover and defining trivializations with constant transition functions.

==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 '''R'''<sup>''n''</sup>-valued 1-form on ''P'', should be taken into account. In particular, the [[Torsion (differential geometry)|torsion form]] on ''P'', is an '''R'''<sup>''n''</sup>-valued 2-form Θ defined by
:<math> \Theta=\mathrm d\theta+\omega\wedge\theta. </math>
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==

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''.

==References==

{{reflist|group=note}}
{{reflist}}

* {{citation|last=Kobayashi|first=Shoshichi|title=Theory of Connections|journal=Ann. Mat. Pura Appl.|year=1957|volume=43|pages=119–194|doi=10.1007/BF02411907|s2cid=120972987|doi-access=free}}
* {{citation|last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi|title=[[Foundations of Differential Geometry]]|volume=1| publisher=[[Wiley Interscience]]|year=1996|edition=New|isbn=0-471-15733-3}}
* {{citation|last1=Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|title=Natural operations in differential geometry|year=1993|publisher=Springer-Verlag|access-date=2008-03-25|archive-url=https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf|archive-date=2017-03-30|url-status=dead}}

{{Manifolds}}

[[Category:Connection (mathematics)]]
[[Category:Differential geometry]]
[[Category:Fiber bundles]]
[[Category:Maps of manifolds]]
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