Connection (composite bundle)

Composite bundles  Y -> Σ -> X {\displaystyle 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 = R {\displaystyle 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 -> X {\displaystyle Y\to X} ,  Y -> Σ {\displaystyle Y\to \Sigma } and  Σ -> X {\displaystyle \Sigma \to X} .

## Composite bundle

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In differential geometry by a composite bundle is meant the composition

π : Y -> Σ -> X ( 1 ) {\displaystyle \pi :Y\to \Sigma \to X\qquad \qquad (1)}

of fiber bundles

π Y Σ : Y -> Σ , π Σ X : Σ -> X . {\displaystyle \pi _{Y\Sigma }:Y\to \Sigma ,\qquad \pi _{\Sigma X}:\Sigma \to X.}

It is provided with bundle coordinates  ( x λ , σ m , y i ) {\displaystyle (x^{\lambda },\sigma ^{m},y^{i})} , where  ( x λ , σ m ) {\displaystyle (x^{\lambda },\sigma ^{m})} are bundle coordinates on a fiber bundle  Σ -> X {\displaystyle \Sigma \to X} , i.e., transition functions of coordinates  σ m {\displaystyle \sigma ^{m}} are independent of coordinates  y i {\displaystyle y^{i}} .

The following fact provides the above-mentioned physical applications of composite bundles. Given the composite bundle (1), let  h {\displaystyle h} be a global section of a fiber bundle  Σ -> X {\displaystyle \Sigma \to X} , if any. Then the pullback bundle Y h = h ∗ Y {\displaystyle Y^{h}=h^{*}Y} over  X {\displaystyle X} is a subbundle of a fiber bundle  Y -> X {\displaystyle Y\to X} .

## Composite principal bundle

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For instance, let  P -> X {\displaystyle P\to X} be a principal bundle with a structure Lie group  G {\displaystyle G} which is reducible to its closed subgroup  H {\displaystyle H} . There is a composite bundle  P -> P / H -> X {\displaystyle P\to P/H\to X} where  P -> P / H {\displaystyle P\to P/H} is a principal bundle with a structure group  H {\displaystyle H} and  P / H -> X {\displaystyle P/H\to X} is a fiber bundle associated with  P -> X {\displaystyle P\to X} . Given a global section  h {\displaystyle h} of  P / H -> X {\displaystyle P/H\to X} , the pullback bundle  h ∗ P {\displaystyle h^{*}P} is a reduced principal subbundle of  P {\displaystyle P} with a structure group  H {\displaystyle H} . In gauge theory, sections of  P / H -> X {\displaystyle P/H\to X} are treated as classical Higgs fields.

## Jet manifolds of a composite bundle

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Given the composite bundle  Y -> Σ -> X {\displaystyle Y\to \Sigma \to X} (1), consider the jet manifolds J 1 Σ {\displaystyle J^{1}\Sigma } ,  J Σ 1 Y {\displaystyle J_{\Sigma }^{1}Y} , and  J 1 Y {\displaystyle J^{1}Y} of the fiber bundles  Σ -> X {\displaystyle \Sigma \to X} ,  Y -> Σ {\displaystyle Y\to \Sigma } , and  Y -> X {\displaystyle Y\to X} , respectively. They are provided with the adapted coordinates  ( x λ , σ m , σ λ m ) {\displaystyle (x^{\lambda },\sigma ^{m},\sigma _{\lambda }^{m})} ,  ( x λ , σ m , y i , y ^ λ i , y m i ) , {\displaystyle (x^{\lambda },\sigma ^{m},y^{i},{\widehat {y}}_{\lambda }^{i},y_{m}^{i}),} , and  ( x λ , σ m , y i , σ λ m , y λ i ) . {\displaystyle (x^{\lambda },\sigma ^{m},y^{i},\sigma _{\lambda }^{m},y_{\lambda }^{i}).}

There is the canonical map

J 1 Σ × Σ J Σ 1 Y -> Y J 1 Y , y λ i = y m i σ λ m + y ^ λ i {\displaystyle J^{1}\Sigma \times _{\Sigma }J_{\Sigma }^{1}Y\to _{Y}J^{1}Y,\qquad y_{\lambda }^{i}=y_{m}^{i}\sigma _{\lambda }^{m}+{\widehat {y}}_{\lambda }^{i}} .

## Composite connection

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

γ = d x λ ⊗ ( ∂ λ + γ λ m ∂ m + γ λ i ∂ i ) , {\displaystyle \gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\gamma _{\lambda }^{m}\partial _{m}+\gamma _{\lambda }^{i}\partial _{i}),}

A Σ = d x λ ⊗ ( ∂ λ + A λ i ∂ i ) + d σ m ⊗ ( ∂ m + A m i ∂ i ) , {\displaystyle A_{\Sigma }=dx^{\lambda }\otimes (\partial _{\lambda }+A_{\lambda }^{i}\partial _{i})+d\sigma ^{m}\otimes (\partial _{m}+A_{m}^{i}\partial _{i}),}

Γ = d x λ ⊗ ( ∂ λ + Γ λ m ∂ m ) . {\displaystyle \Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{m}\partial _{m}).}

A connection  A Σ {\displaystyle A_{\Sigma }} on a fiber bundle  Y -> Σ {\displaystyle Y\to \Sigma } and a connection  Γ {\displaystyle \Gamma } on a fiber bundle  Σ -> X {\displaystyle \Sigma \to X} define a connection

γ = d x λ ⊗ ( ∂ λ + Γ λ m ∂ m + ( A λ i + A m i Γ λ m ) ∂ i ) {\displaystyle \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 -> X {\displaystyle Y\to X} . It is called the composite connection. This is a unique connection such that the horizontal lift γ τ {\displaystyle \gamma \tau } onto  Y {\displaystyle Y} of a vector field  τ {\displaystyle \tau } on  X {\displaystyle X} by means of the composite connection  γ {\displaystyle \gamma } coincides with the composition  A Σ ( Γ τ ) {\displaystyle A_{\Sigma }(\Gamma \tau )} of horizontal lifts of  τ {\displaystyle \tau } onto  Σ {\displaystyle \Sigma } by means of a connection  Γ {\displaystyle \Gamma } and then onto  Y {\displaystyle Y} by means of a connection  A Σ {\displaystyle A_{\Sigma }} .

## Vertical covariant differential

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Given the composite bundle  Y {\displaystyle Y} (1), there is the following exact sequence of vector bundles over  Y {\displaystyle Y} :

0 -> V Σ Y -> V Y -> Y × Σ V Σ -> 0 , ( 2 ) {\displaystyle 0\to V_{\Sigma }Y\to VY\to Y\times _{\Sigma }V\Sigma \to 0,\qquad \qquad (2)}

where  V Σ Y {\displaystyle V_{\Sigma }Y} and  V Σ ∗ Y {\displaystyle V_{\Sigma }^{*}Y} are the vertical tangent bundle and the vertical cotangent bundle of  Y -> Σ {\displaystyle Y\to \Sigma } . Every connection  A Σ {\displaystyle A_{\Sigma }} on a fiber bundle  Y -> Σ {\displaystyle Y\to \Sigma } yields the splitting

A Σ : T Y ⊃ V Y ∋ y ˙ i ∂ i + σ ˙ m ∂ m -> ( y ˙ i − A m i σ ˙ m ) ∂ i {\displaystyle A_{\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

D ~ : J 1 Y -> T ∗ X ⊗ Y V Σ Y , D ~ = d x λ ⊗ ( y λ i − A λ i − A m i σ λ m ) ∂ i , {\displaystyle {\widetilde {D}}:J^{1}Y\to T^{*}X\otimes _{Y}V_{\Sigma }Y,\qquad {\widetilde {D}}=dx^{\lambda }\otimes (y_{\lambda }^{i}-A_{\lambda }^{i}-A_{m}^{i}\sigma _{\lambda }^{m})\partial _{i},}

on a composite bundle  Y -> X {\displaystyle Y\to X} . It is called the vertical covariant differential. It possesses the following important property.

Let  h {\displaystyle h} be a section of a fiber bundle  Σ -> X {\displaystyle \Sigma \to X} , and let  h ∗ Y ⊂ Y {\displaystyle h^{*}Y\subset Y} be the pullback bundle over  X {\displaystyle X} . Every connection  A Σ {\displaystyle A_{\Sigma }} induces the pullback connection

A h = d x λ ⊗ [ ∂ λ + ( ( A m i ∘ h ) ∂ λ h m + ( A ∘ h ) λ i ) ∂ i ] {\displaystyle A_{h}=dx^{\lambda }\otimes [\partial _{\lambda }+((A_{m}^{i}\circ h)\partial _{\lambda }h^{m}+(A\circ h)_{\lambda }^{i})\partial _{i}]}

on  h ∗ Y {\displaystyle h^{*}Y} . Then the restriction of a vertical covariant differential  D ~ {\displaystyle {\widetilde {D}}} to  J 1 h ∗ Y ⊂ J 1 Y {\displaystyle J^{1}h^{*}Y\subset J^{1}Y} coincides with the familiar covariant differential D A h {\displaystyle D^{A_{h}}} on  h ∗ Y {\displaystyle h^{*}Y} relative to the pullback connection  A h {\displaystyle A_{h}} .

## References

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* 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

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* 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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* Connection (mathematics)
* Connection (fibred manifold)