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Composite bundles $Y\to \mathrm{\Sigma }\to X$ 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 $Y\to X$, $Y\to \mathrm{\Sigma }$ and $\mathrm{\Sigma }\to X$.

Composite bundle

In differential geometry by a composite bundle is meant the composition

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

of fiber bundles

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

It is provided with bundle coordinates $(x^{\lambda },{\sigma }^m,y^i)$, where $(x^{\lambda },{\sigma }^m)$ are bundle coordinates on a fiber bundle $\mathrm{\Sigma }\to X$, 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 $\mathrm{\Sigma }\to X$, 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 $Y\to \mathrm{\Sigma }\to X$ (1), let us consider the jet manifolds $J^1\mathrm{\Sigma }$, $J_{\mathrm{\Sigma }}^{1}Y$, and $J^{1}Y$ of the fiber bundles $\mathrm{\Sigma }\to X$, $Y\to \mathrm{\Sigma }$, and $Y\to X$, respectively. They are provided with the adapted coordinates $(x^{\lambda },{\sigma }^m,{\sigma }_{\lambda }^m)$, $(x^{\lambda },{\sigma }^m,y^i,{\hat{y}}_{\lambda }^i,y_m^i),$, and $(x^{\lambda },{\sigma }^m,y^i,{\sigma }_{\lambda }^m,y_{\lambda }^i).$

There is the canonical map

$J^1\mathrm{\Sigma }{\times }_{\mathrm{\Sigma }}{J}_{\mathrm{\Sigma }}^{1}Y{\to }_Y{J}^{1}Y,y_{\lambda }^i=y_m^i{\sigma }_{\lambda }^m+{\hat{y}}_{\lambda }^i$.

Composite connection

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

$\gamma =d{x}^{\lambda }\otimes ({\partial }_{\lambda }+{\gamma }_{\lambda }^m{\partial }_m+{\gamma }_{\lambda }^i{\partial }_i),$

$A_{\mathrm{\Sigma }}=d{x}^{\lambda }\otimes ({\partial }_{\lambda }+A_{\lambda }^i{\partial }_i)+d{\sigma }^m\otimes ({\partial }_m+A_m^i{\partial }_i),$

$\mathrm{\Gamma }=d{x}^{\lambda }\otimes ({\partial }_{\lambda }+{\mathrm{\Gamma }}_{\lambda }^m{\partial }_m).$

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

$\gamma =d{x}^{\lambda }\otimes ({\partial }_{\lambda }+{\mathrm{\Gamma }}_{\lambda }^m{\partial }_m+(A_{\lambda }^i+A_m^i{\mathrm{\Gamma }}_{\lambda }^m){\partial }_i)$

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 $A_{\mathrm{\Sigma }}(\mathrm{\Gamma }\tau )$ of horizontal lifts of $\tau$ onto $\mathrm{\Sigma }$ by means of a connection $\mathrm{\Gamma }$ and then onto $Y$ by means of a connection $A_{\mathrm{\Sigma }}$.

Vertical covariant differential

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

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

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

$A_{\mathrm{\Sigma }}:TY\supset VY\ni {\dot{y}}^i{\partial }_i+{\dot{\sigma }}^m{\partial }_m\to ({\dot{y}}^i-A_m^i{\dot{\sigma }}^m){\partial }_i$

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

$\tilde{D}:J^{1}Y\to T^{*}X{\otimes }_Y{V}_{\mathrm{\Sigma }}Y,\tilde{D}=d{x}^{\lambda }\otimes (y_{\lambda }^i-A_{\lambda }^i-A_m^i{\sigma }_{\lambda }^m){\partial }_i,$

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

$A_h=d{x}^{\lambda }\otimes [{\partial }_{\lambda }+((A_m^i\circ h){\partial }_{\lambda }{h}^m+(A\circ h{)}_{\lambda }^i){\partial }_i]$

on $h^{*}Y$. Then the restriction of a vertical covariant differential $\tilde{D}$ 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

See also

• Connection (mathematics)
• Connection (fibred manifold)