Connection (principal bundle) explained
over a smooth manifold
is a particular type of connection which is compatible with the
action of the group
.
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
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
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
\Omega1(P,akg)\congCinfty(P,T*P ⊗ akg)
such that
where
denotes right multiplication by
, and
is the
adjoint representation on
(explicitly,
\operatorname{Ad}gX=
g\exp(tX)g-1l|t=0
);
- 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
-connection
refers to the pair
and
itself is called the connection form or connection 1-form of the principal connection.Computational remarks
Most known non-trivial computations of principal
-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let
, be a principal
-bundle over
) This means that 1-forms on the total space are canonically isomorphic to
, where
is the dual lie algebra, hence
-connections are in bijection with Cinfty(H,ak{g}* ⊗ ak{g})G
.Relation to Ehresmann connections
A principal
-connection
on
determines an Ehresmann connection on
in the following way. First note that the fundamental vector fields generating the
action on
provide a bundle isomorphism (covering the identity of
) from the bundle
to
, where
is the kernel of the tangent mapping
which is called the vertical bundle of
. It follows that
determines uniquely a bundle map
which is the identity on
. Such a projection
is uniquely determined by its kernel, which is a smooth subbundle
of
(called the horizontal bundle) such that
. This is an Ehresmann connection.Conversely, an Ehresmann connection
(or
) on
defines a principal
-connection
if and only if it is
-equivariant in the sense that
.
Pull back via trivializing section
A trivializing section of a principal bundle
is given by a section s
of
over an open subset
of
. Then the pullback s
*ω
of a principal connection is a 1-form on
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 (sg)*\omega=\operatorname{Ad}(g)-1s*\omega+g-1dg
. 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
acts on the tangent bundle
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
H=q-1\Gamma(TM)\subsetTP.
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 for this space of 1-forms.
Examples
Maurer-Cartan connection
For the trivial principal
-bundle
where
, there is a canonical connection
[1] pg 49\omegaMC\in\Omega1(E,ak{g})
called the Maurer-Cartan connection. It is defined as follows: for a point
define
(\omegaMC)(g,x)=
\circ\pi1)*
for
which is a composition
T(g,x)E\xrightarrow{\pi1*
} T_gG \xrightarrow T_eG = \mathfrak
defining the 1-form. Note that
\omega0=
)*:TgG\toTeG=ak{g}
is the Maurer-Cartan form on the Lie group
and
.
Trivial bundle
For a trivial principal
-bundle
, the identity section
given by
defines a 1-1 correspondence
i*:\Omega1(E,ak{g})\to\Omega1(X,ak{g})
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
T(x,e)E=ker(\pi*) ⊕ i*TxX
and we can get
by taking
\tilde{A}(g,x)=
(e,x)=Ad(g-1)\circ\tilde{A}(e,x)
Similarly, the form
\tilde{A}(x,g)=Ad(g-1)\circAx\circ\pi*:T(x,g)E\toak{g}
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
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
\{Aa\in\Omega1(Ua,ak{g})\}a
which satisfy
on the intersections
for
the
Maurer-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
Wepg 94 can construct
as a principal
-bundle
where
and
is the projection map
\gamma(z0,\ldots,zn)=[z0,\ldots,zn]
Note the Lie algebra of
is just the complex plane. The 1-form
defined as
\begin{align}
\omega&=
2}\\
&=
forms a connection, which can be checked by verifying the definition. For any fixed
we have
&=
| \overline{(zλ) |
td(zλ)}{|zλ| |
2}\\
&=
tdz
}{|λ|2 ⋅ |z|2}
\end{align}
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
0\toC\xrightarrow{vz}TzHZ\xrightarrow{\gamma*}T[z]CPn\to0
where
is defined as
so it acts as scaling in the fiber (which restricts to the corresponding
-action). Taking
we get
\begin{align}
\omegaz\circvz(λ)&=
2}(zλ)\\
&=
2}\\
&=λ
\end{align}
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
\begin{align}
dz&=dz0+ … +dzn\\
z&=a0z0+ … +anzn\\
dz(z)&=a0+ … +an\\
dz(λz)&=λ ⋅ (a0+ … +an)\\
\overline{z}&=\overline{a0}+ … +\overline{an}
\end{align}
it becomes more clear (where
).
Induced covariant and exterior derivatives
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+
ωΛ
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
.
Curvature form
The curvature form of a principal G-connection ω is the
-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
. 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.
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, 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
\Theta=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.
Notes and References
- Web site: Dupont . Johan . August 2003 . Fibre Bundles and Chern-Weil Theory . https://web.archive.org/web/20220331053124/http://www.johno.dk/mathematics/fiberbundlestryk.pdf . 31 March 2022.
- Eguchi. Tohru. Gilkey. Peter B.. Hanson. Andrew J.. 1980. Gravitation, gauge theories and differential geometry. Physics Reports. en. 66. 6. 213–393. 10.1016/0370-1573(80)90130-1. 1980PhR....66..213E.