Connection (composite bundle)

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

Composite bundle

In differential geometry by a composite bundle is meant the composition

\pi :Y\to \Sigma \to X\qquad \qquad (1)

of fiber bundles

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

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

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

Composite principal bundle

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

Jet manifolds of a composite bundle

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

There is the canonical map

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 $Y\to X$ , $Y\to \Sigma$ and $\Sigma \to X$ . These connections are given by the corresponding tangent-valued connection forms

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

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

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

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

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

Vertical covariant differential

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

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

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

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

{\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 $Y\to X$ . It is called the vertical covariant differential. It possesses the following important property.

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

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

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

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