In variational calculus
The following discusses the formulation of conserved currents in terms of variational calculus and the variational bicomplex.
Let be a spacetime of dimension , a bundle, its jet bundle and
the corresponding variational bicomplex with being the vertical and the horizontal differential.
The covariant phase space of the Lagrangian is the locus
For any -dimensional submanifold,
is the presymplectic structure on covariant phase space
A conserved current is an element
which is horizontally closed on covariant phase space
For a submanifold of dimension , the charge of the conserved current with respect to is the integral
If are homolous, the associated charge is the same
Every symmetry induces a conserved current.
This is Noether's theorem. See there for more details.
In higher prequantum geometry
The following discusses conserved currently in the context of higher prequantum geometry. This follows (Schreiber 13). Similar observations have been made by Igor Khavkine.
Let be the ambient (∞,1)-topos. For a moduli ∞-stack of fields a local Lagrangian for an -dimensional prequantum field theory is equivalently a prequantum n-bundle given by a map
to the moduli ∞-stack of smooth circle n-bundles with connection. The local connection differential n-form is the local Lagrangian itself as in traditional literature, the rest of the data in is the higher gauge symmetry equivariant structure.
The following is effectively the direct higher geometric analog of the Hamiltonian version of Noether’s theorem.
A transformation of the fields is an equivalence
That the local Lagrangian be preserved by this, up to (gauge) equivalence, means that there is a diagram in of the form
(With equivalently regarded as prequantum n-bundle this is equivalently a higher quantomorphism. These are the transformations studied in (Fiorenza-Rogers-Schreiber 13))
For an infinitesimal operation an locally the Lagrangian -form, this means
hence that the Lagrangian changes under the Lie derivative by an exact term. By Cartan's magic formula this means
In variational calculus
A general discussion as above is around definition 9 of
- G. J. Zuckerman, Action Principles and Global Geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259–284. (pdf)
The relation of conserved currents to moment maps in symplectic geometry is highlighted for instance in
- Huijun Fan, Lecture 8, Moment map and symplectic reduction (pdf)
Higher conserved currents
Higher conserved currents are discussed for instance in
In higher prequantum theory
In the context of higher prequantum geometry conserved currents of the WZW model and in ∞-Wess-Zumino-Witten theory are briefly indicated on the last page of
The same structure is considered in
as higher quantomorphisms and Poisson bracket Lie n-algebras of local currents.