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The following discusses the formulation of conserved currents in terms of variational calculus and the variational bicomplex.
Let $X$ be a spacetime of dimension $n$, $E \to X$ a bundle, $j:\infty E \to X$ its jet bundle and
the corresponding variational bicomplex with $\delta$ being the vertical and $d = d_{dR}$ the horizontal differential.
For $L \in \Omega^{n,0}(j_\infty E)$ a Lagrangian we have that
for $E$ the Euler-Lagrange operator.
The covariant phase space of the Lagrangian is the locus
For $\Sigma \subset X$ any $(n-1)$-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 $\Sigma \hookrightarrow X$ a submanifold of dimension $n-1$, the charge of the conserved current $j$ with respect to $\Sigma$ is the integral
If $\Sigma, \Sigma' \subset X$ are homolous, the associated charge is the same
By Stokes' theorem.
Every symmetry induces a conserved current.
This is Noether's theorem. See there for more details.
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 $\mathbf{H}$ be the ambient (∞,1)-topos. For $\mathbf{Fields} \in \mathbf{H}$ a moduli ∞-stack of fields a local Lagrangian for an $n$-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 $\mathcal{L}$ 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 $\mathcal{L}$ be preserved by this, up to (gauge) equivalence, means that there is a diagram in $\mathbf{H}$ of the form
(With $\mathcal{L}$ equivalently regarded as prequantum n-bundle this is equivalently a higher quantomorphism. These are the transformations studied in (Fiorenza-Rogers-Schreiber 13))
For $\phi$ an infinitesimal operation an $L$ locally the Lagrangian $n$-form, this means
hence that the Lagrangian changes under the Lie derivative by an exact term. By Cartan's magic formula this means
Hence (…)
The Dickey Lie bracket on conserved currents is due to
and is reviewed in
A general discussion as above is around definition 9 of
The relation of conserved currents to moment maps in symplectic geometry is highlighted for instance in
Higher conserved currents are discussed for instance in
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.