Connection (composite bundle)
Composite bundles [math]\displaystyle{ Y\to \Sigma \to X }[/math] play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory, non-autonomous mechanics where [math]\displaystyle{ X=\mathbb R }[/math] is the time axis, e.g., mechanics with time-dependent parameters, and so on. There are the important relations between connections on fiber bundles [math]\displaystyle{ Y\to X }[/math], [math]\displaystyle{ Y\to \Sigma }[/math] and [math]\displaystyle{ \Sigma\to X }[/math].
Composite bundle
In differential geometry by a composite bundle is meant the composition
- [math]\displaystyle{ \pi: Y\to \Sigma\to X \qquad\qquad (1) }[/math]
of fiber bundles
- [math]\displaystyle{ \pi_{Y\Sigma}: Y\to\Sigma, \qquad \pi_{\Sigma X}: \Sigma\to X. }[/math]
It is provided with bundle coordinates [math]\displaystyle{ (x^\lambda,\sigma^m,y^i) }[/math], where [math]\displaystyle{ (x^\lambda,\sigma^m) }[/math] are bundle coordinates on a fiber bundle [math]\displaystyle{ \Sigma\to X }[/math], i.e., transition functions of coordinates [math]\displaystyle{ \sigma^m }[/math] are independent of coordinates [math]\displaystyle{ y^i }[/math].
The following fact provides the above-mentioned physical applications of composite bundles. Given the composite bundle (1), let [math]\displaystyle{ h }[/math] be a global section of a fiber bundle [math]\displaystyle{ \Sigma\to X }[/math], if any. Then the pullback bundle [math]\displaystyle{ Y^h=h^*Y }[/math] over [math]\displaystyle{ X }[/math] is a subbundle of a fiber bundle [math]\displaystyle{ Y\to X }[/math].
Composite principal bundle
For instance, let [math]\displaystyle{ P\to X }[/math] be a principal bundle with a structure Lie group [math]\displaystyle{ G }[/math] which is reducible to its closed subgroup [math]\displaystyle{ H }[/math]. There is a composite bundle [math]\displaystyle{ P\to P/H\to X }[/math] where [math]\displaystyle{ P\to P/H }[/math] is a principal bundle with a structure group [math]\displaystyle{ H }[/math] and [math]\displaystyle{ P/H\to X }[/math] is a fiber bundle associated with [math]\displaystyle{ P\to X }[/math]. Given a global section [math]\displaystyle{ h }[/math] of [math]\displaystyle{ P/H\to X }[/math], the pullback bundle [math]\displaystyle{ h^*P }[/math] is a reduced principal subbundle of [math]\displaystyle{ P }[/math] with a structure group [math]\displaystyle{ H }[/math]. In gauge theory, sections of [math]\displaystyle{ P/H\to X }[/math] are treated as classical Higgs fields.
Jet manifolds of a composite bundle
Given the composite bundle [math]\displaystyle{ Y\to \Sigma\to X }[/math] (1), consider the jet manifolds [math]\displaystyle{ J^1\Sigma }[/math], [math]\displaystyle{ J^1_\Sigma Y }[/math], and [math]\displaystyle{ J^1Y }[/math] of the fiber bundles [math]\displaystyle{ \Sigma\to X }[/math], [math]\displaystyle{ Y\to \Sigma }[/math], and [math]\displaystyle{ Y\to X }[/math], respectively. They are provided with the adapted coordinates [math]\displaystyle{ ( x^\lambda,\sigma^m, \sigma^m_\lambda) }[/math], [math]\displaystyle{ (x^\lambda, \sigma^m, y^i, \widehat y^i_\lambda, y^i_m), }[/math], and [math]\displaystyle{ (x^\lambda, \sigma^m, y^i, \sigma^m_\lambda ,y^i_\lambda). }[/math]
There is the canonical map
- [math]\displaystyle{ J^1\Sigma\times_\Sigma J^1_\Sigma Y\to_Y J^1Y, \qquad y^i_\lambda=y^i_m \sigma^m_\lambda +\widehat y^i_\lambda }[/math].
Composite connection
This canonical map defines the relations between connections on fiber bundles [math]\displaystyle{ Y\to X }[/math], [math]\displaystyle{ Y\to\Sigma }[/math] and [math]\displaystyle{ \Sigma\to X }[/math]. These connections are given by the corresponding tangent-valued connection forms
- [math]\displaystyle{ \gamma=dx^\lambda\otimes (\partial_\lambda +\gamma_\lambda^m\partial_m + \gamma_\lambda^i\partial_i), }[/math]
- [math]\displaystyle{ A_\Sigma=dx^\lambda\otimes (\partial_\lambda + A_\lambda^i\partial_i) +d\sigma^m\otimes (\partial_m + A_m^i\partial_i), }[/math]
- [math]\displaystyle{ \Gamma=dx^\lambda\otimes (\partial_\lambda + \Gamma_\lambda^m\partial_m). }[/math]
A connection [math]\displaystyle{ A_\Sigma }[/math] on a fiber bundle [math]\displaystyle{ Y\to\Sigma }[/math] and a connection [math]\displaystyle{ \Gamma }[/math] on a fiber bundle [math]\displaystyle{ \Sigma\to X }[/math] define a connection
- [math]\displaystyle{ \gamma=dx^\lambda\otimes (\partial_\lambda +\Gamma_\lambda^m\partial_m + (A_\lambda^i + A_m^i\Gamma_\lambda^m)\partial_i) }[/math]
on a composite bundle [math]\displaystyle{ Y\to X }[/math]. It is called the composite connection. This is a unique connection such that the horizontal lift [math]\displaystyle{ \gamma\tau }[/math] onto [math]\displaystyle{ Y }[/math] of a vector field [math]\displaystyle{ \tau }[/math] on [math]\displaystyle{ X }[/math] by means of the composite connection [math]\displaystyle{ \gamma }[/math] coincides with the composition [math]\displaystyle{ A_\Sigma(\Gamma\tau) }[/math] of horizontal lifts of [math]\displaystyle{ \tau }[/math] onto [math]\displaystyle{ \Sigma }[/math] by means of a connection [math]\displaystyle{ \Gamma }[/math] and then onto [math]\displaystyle{ Y }[/math] by means of a connection [math]\displaystyle{ A_\Sigma }[/math].
Vertical covariant differential
Given the composite bundle [math]\displaystyle{ Y }[/math] (1), there is the following exact sequence of vector bundles over [math]\displaystyle{ Y }[/math]:
- [math]\displaystyle{ 0\to V_\Sigma Y\to VY\to Y\times_\Sigma V\Sigma\to 0, \qquad\qquad (2) }[/math]
where [math]\displaystyle{ V_\Sigma Y }[/math] and [math]\displaystyle{ V_\Sigma^*Y }[/math] are the vertical tangent bundle and the vertical cotangent bundle of [math]\displaystyle{ Y\to\Sigma }[/math]. Every connection [math]\displaystyle{ A_\Sigma }[/math] on a fiber bundle [math]\displaystyle{ Y\to\Sigma }[/math] yields the splitting
- [math]\displaystyle{ A_\Sigma: TY\supset VY \ni \dot y^i\partial_i + \dot\sigma^m\partial_m \to (\dot y^i -A^i_m\dot\sigma^m)\partial_i }[/math]
of the exact sequence (2). Using this splitting, one can construct a first order differential operator
- [math]\displaystyle{ \widetilde D: J^1Y\to T^*X\otimes_Y V_\Sigma Y, \qquad \widetilde D= dx^\lambda\otimes(y^i_\lambda- A^i_\lambda -A^i_m\sigma^m_\lambda)\partial_i, }[/math]
on a composite bundle [math]\displaystyle{ Y\to X }[/math]. It is called the vertical covariant differential. It possesses the following important property.
Let [math]\displaystyle{ h }[/math] be a section of a fiber bundle [math]\displaystyle{ \Sigma\to X }[/math], and let [math]\displaystyle{ h^*Y\subset Y }[/math] be the pullback bundle over [math]\displaystyle{ X }[/math]. Every connection [math]\displaystyle{ A_\Sigma }[/math] induces the pullback connection
- [math]\displaystyle{ A_h=dx^\lambda\otimes[\partial_\lambda+((A^i_m\circ h)\partial_\lambda h^m +(A\circ h)^i_\lambda)\partial_i] }[/math]
on [math]\displaystyle{ h^*Y }[/math]. Then the restriction of a vertical covariant differential [math]\displaystyle{ \widetilde D }[/math] to [math]\displaystyle{ J^1h^*Y\subset J^1Y }[/math] coincides with the familiar covariant differential [math]\displaystyle{ D^{A_h} }[/math] on [math]\displaystyle{ h^*Y }[/math] relative to the pullback connection [math]\displaystyle{ A_h }[/math].
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
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