nLab locally variational field theory

Idea

A Lagrangian field theory for fields defined by some field bundle is called locally variational if its equations of motion is determined by a source form on the jet bundle which is locally, i.e. over some neighbourhood of any point of the base space, the Euler-Lagrange variational derivative of a Lagrangian density on this neighbourhood, but not necessarily globally so.

Some common examples of Lagrangian field theories are in fact globally variational (such as the plain scalar field or the uncharged Dirac field), but other common example of Lagrangian field theories are indeed only locally variational (see the Examples below).

Example

Field theories with WZW terms and higher WZW terms are (only) locally variational: the WZW term is the pullback to the jet bundle of a circle n-bundle with connection on the base space (e.g. a line bundle with connection, bundle gerbe with connection, etc.) and the corresponding Euler-Lagrange form is proportional to the horizontal form-projection of the curvature $(n+1)$-form. This is locally exact, but, crucially, not globally so, unless the higher bundle is flat infinity-bundle.

Beware that this is not an exotic situation: already the Lagrangian density for the charged particle, regarded as a field theory on its worldline, coupled to an electromagnetic field with non-trivial magnetic charge (first Chern class) is of this form: the interaction term which gives the Lorentz force is an example of a non-trivial WZW term as above.

References

Exposition is in

• Urs Schreiber, Higher structures in Mathematics and Physics, lecture at Oberwolfach Workshop 1651a, 2016 Dec. 18-23