Composite bundles
play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory, non-autonomous mechanics where
is the time axis, e.g., mechanics with time-dependent parameters, and so on. There are the important relations between connections on fiber bundles
,
and
.
Composite bundle
In differential geometry by a composite bundle is meant the composition
![{\displaystyle \pi :Y\to \Sigma \to X\qquad \qquad (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1dbef6dc8142c9ee3b337f7f0d57d12711eb89)
of fiber bundles
![{\displaystyle \pi _{Y\Sigma }:Y\to \Sigma ,\qquad \pi _{\Sigma X}:\Sigma \to X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e34830046f454ebb0869ead98cc7bb1561ce267)
It is provided with bundle coordinates
, where
are bundle coordinates on a fiber bundle
, i.e., transition functions of coordinates
are independent of coordinates
.
The following fact provides the above-mentioned physical applications of composite bundles. Given the composite bundle (1), let
be a global section
of a fiber bundle
, if any. Then the pullback bundle
over
is a subbundle of a fiber bundle
.
Composite principal bundle
For instance, let
be a principal bundle with a structure Lie group
which is reducible to its closed subgroup
. There is a composite bundle
where
is a principal bundle with a structure group
and
is a fiber bundle associated with
. Given a global section
of
, the pullback bundle
is a reduced principal subbundle of
with a structure group
. In gauge theory, sections of
are treated as classical Higgs fields.
Jet manifolds of a composite bundle
Given the composite bundle
(1), consider the jet manifolds
,
, and
of the fiber bundles
,
, and
, respectively. They are provided with the adapted coordinates
,
, and
There is the canonical map
.
Composite connection
This canonical map defines the relations between connections on fiber bundles
,
and
. These connections are given by the corresponding tangent-valued connection forms
![{\displaystyle \gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\gamma _{\lambda }^{m}\partial _{m}+\gamma _{\lambda }^{i}\partial _{i}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/886ccb451cf8b0594dfc7cabd3ac2b7f8409e5ab)
![{\displaystyle A_{\Sigma }=dx^{\lambda }\otimes (\partial _{\lambda }+A_{\lambda }^{i}\partial _{i})+d\sigma ^{m}\otimes (\partial _{m}+A_{m}^{i}\partial _{i}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43275111771413354c19a2c6c9cd70da8977e9ad)
![{\displaystyle \Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{m}\partial _{m}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6179a1907c5797429f221a15f702bbb3b4b1b08a)
A connection
on a fiber bundle
and a connection
on a fiber bundle
define a connection
![{\displaystyle \gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{m}\partial _{m}+(A_{\lambda }^{i}+A_{m}^{i}\Gamma _{\lambda }^{m})\partial _{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30ebb5f17baa0a531c6f917ffe9319fa37e818f)
on a composite bundle
. It is called the composite connection. This is a unique connection such that the horizontal lift
onto
of a vector field
on
by means of the composite connection
coincides with the composition
of horizontal lifts of
onto
by means of a connection
and then onto
by means of a connection
.
Vertical covariant differential
Given the composite bundle
(1), there is the following exact sequence of vector bundles over
:
![{\displaystyle 0\to V_{\Sigma }Y\to VY\to Y\times _{\Sigma }V\Sigma \to 0,\qquad \qquad (2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5fc02d81da16e3f510162842ab99c4a6b5ffa0)
where
and
are the vertical tangent bundle and the vertical cotangent bundle of
. Every connection
on a fiber bundle
yields the splitting
![{\displaystyle A_{\Sigma }:TY\supset VY\ni {\dot {y}}^{i}\partial _{i}+{\dot {\sigma }}^{m}\partial _{m}\to ({\dot {y}}^{i}-A_{m}^{i}{\dot {\sigma }}^{m})\partial _{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d76d69f8f0c131ad0fddd871125416db97a1acdb)
of the exact sequence (2). Using this splitting, one can construct a first order differential operator
![{\displaystyle {\widetilde {D}}:J^{1}Y\to T^{*}X\otimes _{Y}V_{\Sigma }Y,\qquad {\widetilde {D}}=dx^{\lambda }\otimes (y_{\lambda }^{i}-A_{\lambda }^{i}-A_{m}^{i}\sigma _{\lambda }^{m})\partial _{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67e475d35a2154add9cb9304db412a49d2994d15)
on a composite bundle
. It is called the vertical covariant differential.
It possesses the following important property.
Let
be a section of a fiber bundle
, and let
be the pullback bundle over
. Every connection
induces the pullback connection
![{\displaystyle A_{h}=dx^{\lambda }\otimes [\partial _{\lambda }+((A_{m}^{i}\circ h)\partial _{\lambda }h^{m}+(A\circ h)_{\lambda }^{i})\partial _{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02238115f2310b97bfa76587f1dc931d6f514601)
on
. Then the restriction of a vertical covariant differential
to
coincides with the familiar covariant differential
on
relative to the pullback connection
.
References
External links
See also