# nLab field bundle

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## Surveys, textbooks and lecture notes

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• Axiomatizations

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• Tools

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• Structural phenomena

• Types of quantum field thories

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• examples

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# Contents

## Idea

In Lagrangian field theory a field history is a section of some fiber bundle $E \overset{fb}{\to} \Sigma$ over spacetime/worldvolume $\Sigma$. This is then called the field bundle.

The de Rham complex of the jet bundle of the field bundle, equipped with its canonical horizontal/vertical bigrading, is the variational bicomplex of the Lagrangian field theory.

For more see at A first idea of quantum field theory the chapter Fields.

###### Remark

The notion of field bundle captures only some aspects of the general notion of spaces of physical fields. For instance the configurations of the B-field do not form sections of a fiber bundle (but of a fiber 2-bundle) and hence for instance the magnetic charge anomaly has no real description by field bundles. See at field (physics) in the section The idea of field bundles and its problems for more.

## Examples

###### Example

For $E \to X$ the trivial bundle $X \times \mathbb{R} \to X$ or $X \times \mathbb{C} \to X$ a smooth section is simply a smooth function on $X$ with values in the real numbers or complex numbers, respectively. In physics this is called a scalar field.

###### Example

More generally, for $V$ a vector space, the trivial field bundle $X \times V \to X$ has as secton $V$-valued functions. These are also called linear sigma-model fields.

Still more generally, if $Y$ is any manifold, then the sections of the tricial field bundle $X \times Y \to X$ are called non-linear sigma-model fields.

###### Example

For $X$ a smooth manifold and $E \to X$ some tensor product of copies of the tangent bundle $T X \to X$ and the cotangent bundle $T^* X \to X$ a section is a tensor field of the corresponding rank.

Many physical fields are related to tensor fields, but few are genuinely tensor fields. See at field (physics) for more on this.

The traditional idea of field bundle is discussed for instance around section 7.3.3 of

• Laurent Claessens, Field theory from a bundle point of view (2011) (pdf) .

Detailed discussion of field bundles in gauge theory with a fixed instanton sector/principal bundle-class is around section 2.5 of

and the issue is highlighted more explicitly in

The issue was then fixed in

precisely by retaining to groupoids/stacks of fields, hence using a higher stacky field bundle.

A discussion of the problems of the traditional notion and its rectification in higher geometry is at