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), let us consider the jet manifolds $J^1\Sigma$ , $J^1_\Sigma Y$ , and $J^1Y$ of the fiber bundles $\Sigma\to X$ , $Y\to \Sigma$ , and $Y\to X$ , respectively. They are provided with the adapted coordinates $( x^\lambda,\sigma^m, \sigma^m_\lambda)$ , $(x^\lambda, \sigma^m, y^i, \widehat y^i_\lambda, y^i_m),$ , and $(x^\lambda, \sigma^m, y^i, \sigma^m_\lambda ,y^i_\lambda).$

There is the canonical map

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

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^i_m\dot\sigma^m)\partial_i

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

\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,

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^i_m\circ h)\partial_\lambda h^m +(A\circ h)^i_\lambda)\partial_i]

on $h^*Y$ . Then the restriction of a vertical covariant differential $\widetilde D$ to $J^1h^*Y\subset J^1Y$ 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