Connection (composite bundle)

Composite bundles ${\displaystyle Y\to \Sigma \to X}$ play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory, non-autonomous mechanics where ${\displaystyle X=\mathbb {R} }$ is the time axis, e.g., mechanics with time-dependent parameters, and so on. There are the important relations between connections on fiber bundles ${\displaystyle Y\to X}$, ${\displaystyle Y\to \Sigma }$ and ${\displaystyle \Sigma \to X}$.

Composite bundle

In differential geometry by a composite bundle is meant the composition

${\displaystyle \pi :Y\to \Sigma \to X\qquad \qquad (1)}$

of fiber bundles

${\displaystyle \pi _{Y\Sigma }:Y\to \Sigma ,\qquad \pi _{\Sigma X}:\Sigma \to X.}$

It is provided with bundle coordinates ${\displaystyle (x^{\lambda },\sigma ^{m},y^{i})}$, where ${\displaystyle (x^{\lambda },\sigma ^{m})}$ are bundle coordinates on a fiber bundle ${\displaystyle \Sigma \to X}$, i.e., transition functions of coordinates ${\displaystyle \sigma ^{m}}$ are independent of coordinates ${\displaystyle y^{i}}$.

The following fact provides the above mentioned physical applications of composite bundles. Given the composite bundle (1), let ${\displaystyle h}$ be a global section of a fiber bundle ${\displaystyle \Sigma \to X}$, if any. Then the pullback bundle ${\displaystyle Y^{h}=h^{*}Y}$ over ${\displaystyle X}$ is a subbundle of a fiber bundle ${\displaystyle Y\to X}$.

Composite principal bundle

For instance, let ${\displaystyle P\to X}$ be a principal bundle with a structure Lie group ${\displaystyle G}$ which is reducible to its closed subgroup ${\displaystyle H}$. There is a composite bundle ${\displaystyle P\to P/H\to X}$ where ${\displaystyle P\to P/H}$ is a principal bundle with a structure group ${\displaystyle H}$ and ${\displaystyle P/H\to X}$ is a fiber bundle associated with ${\displaystyle P\to X}$. Given a global section ${\displaystyle h}$ of ${\displaystyle P/H\to X}$, the pullback bundle ${\displaystyle h^{*}P}$ is a reduced principal subbundle of ${\displaystyle P}$ with a structure group ${\displaystyle H}$. In gauge theory, sections of ${\displaystyle P/H\to X}$ are treated as classical Higgs fields.

Jet manifolds of a composite bundle

Given the composite bundle ${\displaystyle Y\to \Sigma \to X}$ (1), consider the jet manifolds ${\displaystyle J^{1}\Sigma }$, ${\displaystyle J_{\Sigma }^{1}Y}$, and ${\displaystyle J^{1}Y}$ of the fiber bundles ${\displaystyle \Sigma \to X}$, ${\displaystyle Y\to \Sigma }$, and ${\displaystyle Y\to X}$, respectively. They are provided with the adapted coordinates ${\displaystyle (x^{\lambda },\sigma ^{m},\sigma _{\lambda }^{m})}$, ${\displaystyle (x^{\lambda },\sigma ^{m},y^{i},{\widehat {y}}_{\lambda }^{i},y_{m}^{i}),}$, and ${\displaystyle (x^{\lambda },\sigma ^{m},y^{i},\sigma _{\lambda }^{m},y_{\lambda }^{i}).}$

There is the canonical map

${\displaystyle J^{1}\Sigma \times _{\Sigma }J_{\Sigma }^{1}Y\to _{Y}J^{1}Y,\qquad y_{\lambda }^{i}=y_{m}^{i}\sigma _{\lambda }^{m}+{\widehat {y}}_{\lambda }^{i}}$.

Composite connection

This canonical map defines the relations between connections on fiber bundles ${\displaystyle Y\to X}$, ${\displaystyle Y\to \Sigma }$ and ${\displaystyle \Sigma \to X}$. These connections are given by the corresponding tangent-valued connection forms

${\displaystyle \gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\gamma _{\lambda }^{m}\partial _{m}+\gamma _{\lambda }^{i}\partial _{i}),}$

${\displaystyle A_{\Sigma }=dx^{\lambda }\otimes (\partial _{\lambda }+A_{\lambda }^{i}\partial _{i})+d\sigma ^{m}\otimes (\partial _{m}+A_{m}^{i}\partial _{i}),}$

${\displaystyle \Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{m}\partial _{m}).}$

A connection ${\displaystyle A_{\Sigma }}$ on a fiber bundle ${\displaystyle Y\to \Sigma }$ and a connection ${\displaystyle \Gamma }$ on a fiber bundle ${\displaystyle \Sigma \to X}$ define a connection

${\displaystyle \gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{m}\partial _{m}+(A_{\lambda }^{i}+A_{m}^{i}\Gamma _{\lambda }^{m})\partial _{i})}$

on a composite bundle ${\displaystyle Y\to X}$. It is called the composite connection. This is a unique connection such that the horizontal lift ${\displaystyle \gamma \tau }$ onto ${\displaystyle Y}$ of a vector field ${\displaystyle \tau }$ on ${\displaystyle X}$ by means of the composite connection ${\displaystyle \gamma }$ coincides with the composition ${\displaystyle A_{\Sigma }(\Gamma \tau )}$ of horizontal lifts of ${\displaystyle \tau }$ onto ${\displaystyle \Sigma }$ by means of a connection ${\displaystyle \Gamma }$ and then onto ${\displaystyle Y}$ by means of a connection ${\displaystyle A_{\Sigma }}$.

Vertical covariant differential

Given the composite bundle ${\displaystyle Y}$ (1), there is the following exact sequence of vector bundles over ${\displaystyle Y}$:

${\displaystyle 0\to V_{\Sigma }Y\to VY\to Y\times _{\Sigma }V\Sigma \to 0,\qquad \qquad (2)}$

where ${\displaystyle V_{\Sigma }Y}$ and ${\displaystyle V_{\Sigma }^{*}Y}$ are the vertical tangent bundle and the vertical cotangent bundle of ${\displaystyle Y\to \Sigma }$. Every connection ${\displaystyle A_{\Sigma }}$ on a fiber bundle ${\displaystyle Y\to \Sigma }$ yields the splitting

${\displaystyle A_{\Sigma }:TY\supset VY\ni {\dot {y}}^{i}\partial _{i}+{\dot {\sigma }}^{m}\partial _{m}\to ({\dot {y}}^{i}-A_{m}^{i}{\dot {\sigma }}^{m})\partial _{i}}$

of the exact sequence (2). Using this splitting, one can construct a first order differential operator

${\displaystyle {\widetilde {D}}:J^{1}Y\to T^{*}X\otimes _{Y}V_{\Sigma }Y,\qquad {\widetilde {D}}=dx^{\lambda }\otimes (y_{\lambda }^{i}-A_{\lambda }^{i}-A_{m}^{i}\sigma _{\lambda }^{m})\partial _{i},}$

on a composite bundle ${\displaystyle Y\to X}$. It is called the vertical covariant differential. It possesses the following important property.

Let ${\displaystyle h}$ be a section of a fiber bundle ${\displaystyle \Sigma \to X}$, and let ${\displaystyle h^{*}Y\subset Y}$ be the pullback bundle over ${\displaystyle X}$. Every connection ${\displaystyle A_{\Sigma }}$ induces the pullback connection

${\displaystyle A_{h}=dx^{\lambda }\otimes [\partial _{\lambda }+((A_{m}^{i}\circ h)\partial _{\lambda }h^{m}+(A\circ h)_{\lambda }^{i})\partial _{i}]}$

on ${\displaystyle h^{*}Y}$. Then the restriction of a vertical covariant differential ${\displaystyle {\widetilde {D}}}$ to ${\displaystyle J^{1}h^{*}Y\subset J^{1}Y}$ coincides with the familiar covariant differential ${\displaystyle D^{A_{h}}}$ on ${\displaystyle h^{*}Y}$ relative to the pullback connection ${\displaystyle A_{h}}$.

References

• Saunders, D., The geometry of jet bundles. Cambridge University Press, 1989. ISBN 0-521-36948-7.
• Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory. World Scientific, 2000. ISBN 981-02-2013-8.

External links

• Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013. ISBN 978-3-659-37815-7; arXiv:0908.1886

See also

• Connection (mathematics)
• Connection (fibred manifold)