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 $$J^1\Sigma$$, $$J^1_\Sigma Y$$, 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 $$A_h$$.