Via the variational bicomplex
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 homologous, 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 currents in the context of higher prequantum geometry, closely related to Azcarraga-Izquierdo 95, section 8.1. This follows (classicalinhigher, section 3.3., going back to Schreiber 13). Similar observations have been made by Igor Khavkine.
this section needs much polishing. For the moment better see classicalinhigher, section 3.3
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 that the Lie derivative of has a potential,
hence that the Lagrangian changes under the Lie derivative by an exact term, hence by a divergence on the worldvolume (since the degree of the Lagrangian form is the dimension of the worldvolume). See also (Azcarraga-Izquierdo 95 (8.1.13)).
This is the situation of the Noether theorem for the general case of “weak” symmetries (see at Noether theorem – schematic idea – weak symmetries).
By Cartan's magic formula the above means
and hence the combination (a Hamiltonian form for with respect to ) is conserved on trajectories in the kernel of the n-plectic form (which are indeed the classical trajectories of , see (Azcarraga-Izquierdo 95 (8.1.14))).
This is the first stage in the Poisson bracket Lie n-algebra, the current algebra (see there at As a homotopy Lie algebra).
Of Green-Schwarz super -brane sigma models
The WZW term of the Green-Schwarz super p-brane sigma models is invariant under supersymmetry only up to a divergence, hence here the general Noether theorem for “weak” symmetries applies and yields a current algebra which is an polyvector extension of the supersymmetry algebra. See at Green-Schwarz action functional – Conserved currents for more.
Dickey-Lie bracket on currents
The Dickey Lie bracket on conserved currents is due to
- Leonid Dickey, Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, Vol. 12 (World Scientific 1991).
and is reviewed in
The statement that the algebra of currents is a central Lie algebra extension of the algebra of symmetries by de Rham cohomology of the jet bundle appears as theorem 11.2 in (Part II of)
Alexandre Vinogradov, The -spectral sequence, Lagrangian formalism, and conservation laws. I. the linear theory, Journal of Mathematical Analysis and Applications 100, 1-40 (1984) (doi90071-4))
Alexandre Vinogradov, The -spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, Journal of Mathematical Analysis and Applications 100, Issue 1, 30 April 1984, Pages 41-129 (publisher)
and is stated as exercise 2.28 on p. 203 of
A lift of the Dickey Lie bracket on cohomologically trivial spaces to an equivalent L-infinity equivalent L-infinity bracket is constructed, under some assumptions, in
Glenn Barnich, Ronald Fulp, Tom Lada, Jim Stasheff, The sh Lie structure of Poisson brackets in field theory, Communications in Mathematical Physics 191, 585-601 (1998) (arXiv:hep-th/9702176)
Martin Markl, Steve Shnider, Differential Operator Endomorphisms of an Euler-Lagrange Complex, Contemporary Mathematics, Volume 231, 1999 (arXiv:9808105)
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
Conserved currents for Lagrangians written as WZW terms are discussed in
Building on that, 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.