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

External links

See also

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