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} .

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} .

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.

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}} .

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 }} .

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}} .

• 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.

• 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

• Connection (mathematics)
• Connection (fibred manifold)