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
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the group ''G''.
A principal connection can be viewed as a special case of the notion of an
Ehresmann connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it do ...
, 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 In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with ...
construction. In particular, on any
associated vector bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with ...
the principal connection induces a
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differen ...
, 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
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
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
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 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, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
s 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
.
Relation to Ehresmann connections
A principal G-connection ''ω'' on ''P'' determines an
Ehresmann connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it do ...
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
Vertical is a geometric term of location which may refer to:
* Vertical direction, the direction aligned with the direction of the force of gravity, up or down
* Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
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
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the vertical bundle VE and horizontal bundle HE are subbundles of ...
) 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
.
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
.
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.
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 . 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.
Affine property
If ''ω'' and ''ω''′ are principal connections on a principal bundle ''P'', then the difference 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
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 relate ...
for this space of 1-forms.
Examples
Mauer-Cartan connection
For the trivial principal
-bundle
where
, there is a canonical connection
pg 49called 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
.
Trivial bundle
For a trivial principal
-bundle
, the identity section
given by
defines a 1-1 correspondence
between connections on
and
-valued 1-forms on
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.
Extending this to non-trivial bundles
This statement can be refined
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.
Global reformulation of space of connections
For a principal
bundle
the set of connections in
is an affine space
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.
Connection on the complex Hopf-bundle
We
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
).
Induced covariant and exterior derivatives
For any
linear representation ''W'' of ''G'' there is an
associated vector bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with ...
over ''M'', and a principal connection induces a
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differen ...
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
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.
Definition
Let ''G' ...
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
.
Curvature form
The
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie algebra ...
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''.
Historically, the emergence of the structure equations are found in the development of the
Cartan connection. When transposed into the context of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
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
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
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.
Connections on frame bundles and torsion
If the principal bundle ''P'' is the
frame bundle, 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. Intuitiv ...
, 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
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a cur ...
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''.
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
dR''. This has the property that a principal ''G'' bundle over ''X
dR'' is the same thing as a ''G'' bundle with connection over ''X''.
References
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*
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{{Manifolds
Connection (mathematics)
Differential geometry
Fiber bundles
Maps of manifolds
Smooth functions