Connection (composite bundle) explained

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 = 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 ( 1 )

of fiber bundles

\pi _{Y\Sigma} : Y\to\Sigma, \pi _{\Sigma} : \Sigma\to X.

It is provided with bundle coordinates

(x ^{λ,\sigma} ^{m,y} ^{i)} , where (x ^{λ,\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 sectionof 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 ^{λ,\sigma} ^{m,} \sigma ^{m} _{λ )} , (x ^{λ,} \sigma ^{m,} y ^{i,} \widehat y ^{i} _{λ ,} y ^{i} _{m),} , and (x ^{λ,} \sigma ^{m,} y ^{i,} \sigma ^{m} _{λ} ,y ^{i} _{λ ).}

There is the canonical map

J ^{1 \Sigma x} _{\Sigma} J ^{1} _{\Sigma} Y\to _{Y} J ^{1Y,} y ^{i} _{m} \sigma ^{m} _{λ} + \widehat y ^{i} _{λ} .

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 ^{λ ⊗} (\partial _{λ} + \gamma ^{m\partial} _{m} + \gamma ^{i\partial} _{i),} A ^{λ ⊗} _{\Sigma = dx} (\partial _{λ} + A ^{i\partial} _{i)} + d\sigma ^{m ⊗} (\partial _{m} + A ^{i\partial} _{i),} \Gamma = dx ^{λ ⊗} (\partial _{λ} + \Gamma ^{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 ^{λ ⊗} (\partial _{λ} + \Gamma ^{m\partial} _{m} + (A ^{i} _{λ} + A ^{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 x _{\Sigma} V\Sigma\to 0 , ( 2 )

where

V _{\Sigma} Y and V ^{*Y} _{\Sigma} 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 splittingA _{\Sigma:} TY\supset VY \ni • y ^{i\partial} ^{i} + • \sigma ^{m\partial} ^{m} \to ( • y ^{i} - A ^{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 ⊗} _{Y} V _{\Sigma} Y, \widetilde D = dx ^{λ ⊗ (y} ^{i} ^{λ-} A ^{i} _{λ} - A ^{m} _{λ )\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 connectionA ^{i} _{m\circ} h)\partial _{λ} h ^{m +(A\circ} h) ^{i} _{λ )\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. .
• Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory. World Scientific, 2000. .

External links

• Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013. ;

See also

• Connection (mathematics)
• Connection (fibred manifold)

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