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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and especially
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
and
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, a connection is a device that defines a notion of
parallel transport In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on ...
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 In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
''M'' is a particular type of connection that is compatible with the
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
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 In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...
associated to ''P'' via the
associated bundle Associated may refer to: *Associated, former name of Avon, Contra Costa County, California *Associated Hebrew Schools of Toronto, a school in Canada *Associated Newspapers, former name of DMG Media, a British publishing company See also *Associatio ...
construction. In particular, on any associated vector bundle the principal connection induces a
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
, an operator that can differentiate
sections Section, Sectioning, or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
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 In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...
of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
.


Formal definition

Let \pi : P \to M be a smooth principal ''G''-bundle over a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra \mathfrak g of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P. In other words, it is an element ''ω'' of \Omega^1(P,\mathfrak g)\cong C^\infty(P, T^*P\otimes\mathfrak g) such that # \hbox_g(R_g^*\omega)=\omega where R_g denotes right multiplication by g, and \operatorname_g is the adjoint representation on \mathfrak g (explicitly, \operatorname_gX = \fracg\exp(tX)g^\bigl, _); # if \xi\in \mathfrak g and X_\xi is the vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P'', then \omega(X_\xi)=\xi (identically on P). Sometimes the term ''principal G-connection'' refers to the pair (P,\omega) and \omega itself is called the
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Carta ...
or connection 1-form of the principal connection.


Computational remarks

Most known non-trivial computations of principal ''G''-connections are done with
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
s because of the triviality of the (co)tangent bundle. (For example, let G \to H \to H/G, be a principal ''G''-bundle over H/G.) This means that 1-forms on the total space are canonically isomorphic to C^\infty(H,\mathfrak^*), where \mathfrak^* is the dual lie algebra, hence ''G''-connections are in bijection with C^\infty(H,\mathfrak^*\otimes \mathfrak)^G.


Relation to Ehresmann connections

A principal ''G''-connection \omega 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 V to P\times\mathfrak g, where V=\ker(d\pi) is the kernel of the tangent mapping \pi\colon TP\to TM which is called the
vertical bundle In mathematics, the vertical bundle and the horizontal bundle are Vector bundle, vector bundles associated to a Fiber bundle#Differentiable fiber bundles, smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the verti ...
of P. It follows that \omega determines uniquely a bundle map v:TP\rightarrow 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\oplus H. This is an Ehresmann connection. Conversely, an Ehresmann connection H\subset TP (or v:TP\rightarrow V) on P defines a principal G-connection \omega if and only if it is G-equivariant in the sense that H_=\mathrm d(R_g)_p(H_).


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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
''s''*''ω'' of a principal connection is a 1-form on ''U'' with values in \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 (sg)^* \omega = \operatorname(g)^s^* \omega + g^ dg. The principal connection is uniquely determined by this family of \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 A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
''TP'' by right translation. The quotient space ''TP''/''G'' is also a manifold, and inherits the structure of a
fibre bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
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 . A
section Section, Sectioning, or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
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 :H = q^\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 is a \mathfrak g-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 :\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 In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
for this space of 1-forms.


Examples


Maurer-Cartan connection

For the trivial principal G-bundle \pi:E \to X where E = G\times X, there is a canonical connectionpg 49
\omega_ \in \Omega^1(E,\mathfrak)
called the Maurer-Cartan connection. It is defined at a point (g,x) \in G\times X by
(\omega_)_ = (L_\circ \pi_1)_* for x \in X, g \in G
which is a composition
T_E \xrightarrow T_gG \xrightarrow T_eG = \mathfrak
defining the 1-form. Note that
\omega_0 = (L_)_*: T_gG \to T_eG = \mathfrak
is the Maurer-Cartan form on the Lie group G and \omega_ = \pi_1^*\omega_0.


Trivial bundle

For a trivial principal G-bundle \pi:E \to X, the identity section i: X \to G\times X given by i(x) = (e,x) defines a 1-1 correspondence
i^*:\Omega^1(E,\mathfrak) \to \Omega^1(X,\mathfrak)
between connections on E and \mathfrak-valued 1-forms on Xpg 53. For a \mathfrak-valued 1-form A on X, there is a unique 1-form \tilde on E such that # \tilde(X) = 0 for X \in T_xE a vertical vector # R_g^*\tilde = \text(g^) \circ \tilde for any g \in G Then given this 1-form, a connection on E can be constructed by taking the sum
\omega_ + \tilde
giving an actual connection on E. This unique 1-form can be constructed by first looking at it restricted to (e,x) for x \in X. Then, \tilde_ is determined by A because T_E = ker(\pi_*)\oplus i_*T_xX and we can get \tilde_by taking
\tilde_ = R^*_g\tilde_ = \text(g^)\circ \tilde_
Similarly, the form
\tilde_ = \text(g^) \circ A_x \circ \pi_*: T_E \to \mathfrak
defines a 1-form giving the properties 1 and 2 listed above.


Extending this to non-trivial bundles

This statement can be refinedpg 55 even further for non-trivial bundles E \to X by considering an open covering \mathcal = \_ of X with trivializations \_ and transition functions \_. Then, there is a 1-1 correspondence between connections on E and collections of 1-forms
\_
which satisfy
A_b = Ad(g_^)\circ A_a + g_^*\omega_0
on the intersections U_ for \omega_0 the Maurer-Cartan form on G, \omega_0 = g^dg in matrix form.


Global reformulation of space of connections

For a principal G bundle \pi: E \to M the set of connections in E is an affine spacepg 57 for the vector space \Omega^1(M,E_\mathfrak) where E_\mathfrak is the associated adjoint vector bundle. This implies for any two connections \omega_0, \omega_1 there exists a form A \in \Omega^1(M, E_\mathfrak) such that
\omega_0 = \omega_1 + A
We denote the set of connections as \mathcal(E), or just \mathcal if the context is clear.


Connection on the complex Hopf-bundle

Wepg 94 can construct \mathbb^n as a principal \mathbb^*-bundle \gamma:H_\mathbb \to \mathbb^n where H_\mathbb = \mathbb^-\ and \gamma is the projection map
\gamma(z_0,\ldots,z_n) = _0,\ldots,z_n/math>
Note the Lie algebra of \mathbb^* = GL(1,\mathbb) is just the complex plane. The 1-form \omega \in \Omega^1(H_\mathbb,\mathbb) defined as
\begin \omega &= \frac \\ &= \sum_^n\fracdz_i \end
forms a connection, which can be checked by verifying the definition. For any fixed \lambda \in \mathbb^* we have
\begin R_\lambda^*\omega &= \frac \\ &= \frac \end
and since , \lambda, ^2 = \overline, we have \mathbb^*-invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any z \in H_\mathbb we have a short exact sequence
0 \to \mathbb \xrightarrow T_zH_\mathbb \xrightarrow T_\mathbb^n \to 0
where v_z is defined as
v_z(\lambda) = z\cdot \lambda
so it acts as scaling in the fiber (which restricts to the corresponding \mathbb^*-action). Taking \omega_z\circ v_z(\lambda) we get \begin \omega_z\circ v_z(\lambda) &= \frac(z\lambda) \\ &= \frac \\ &= \lambda \end where the second equality follows because we are considering z\lambda a vertical tangent vector, and dz(z\lambda) = z\lambda. The notation is somewhat confusing, but if we expand out each term
\begin 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 &= \overline + \cdots + \overline \end
it becomes more clear (where a_i \in \mathbb).


Induced covariant and exterior derivatives

For any
linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
''W'' of ''G'' there is an associated vector bundle P\times^G W over ''M'', and a principal connection induces a
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of 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 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 On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
d''α'', although ''G''-equivariant, is no longer horizontal. However, the combination d''α''+''ω''Λ''α'' is. This defines an
exterior covariant derivative In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of ...
d''ω'' from P\times^G W-valued ''k''-forms on ''M'' to P\times^G W-valued (''k''+1)-forms on ''M''. In particular, when ''k''=0, we obtain a covariant derivative on P\times^G W.


Curvature form

The curvature form of a principal ''G''-connection ''ω'' is the \mathfrak g-valued 2-form Ω defined by :\Omega=d\omega +\tfrac12 omega\wedge\omega It is ''G''-equivariant and horizontal, hence corresponds to a 2-form on ''M'' with values in \mathfrak g_P. The identification of the curvature with this quantity is sometimes called the ''(Cartan's) second structure equation''. Historically, the emergence of the structure equations are found in the development of the
Cartan connection In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...
. When transposed into the context of
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s, 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

We say that a connection \omega is flat if its curvature form \Omega = 0. There is a useful characterization of principal bundles with flat connections; that is, a principal G-bundle \pi: E \to X has a flat connectionpg 68 if and only if there exists an open covering \_ with trivializations \left\_ such that all transition functions
g_: U_a\cap U_b \to G
are constant. This is useful because it gives a recipe for constructing flat principal G-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 In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...
, or (more generally) if it has a
solder form In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuiti ...
, then the connection is an example of an
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
, 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 : \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''.


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


References

* * * {{Manifolds Connection (mathematics) Differential geometry Fiber bundles Maps of manifolds Smooth functions