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\begin{document}

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\section*{conserved current}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{variational_calculus}{}\paragraph*{{Variational calculus}}\label{variational_calculus}

[[!include variational calculus - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{via_the_variational_bicomplex}{Via the variational bicomplex}\dotfill \pageref*{via_the_variational_bicomplex} \linebreak
\noindent\hyperlink{the_context}{The context}\dotfill \pageref*{the_context} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{InHigherPrequantumGeometry}{In higher prequantum geometry}\dotfill \pageref*{InHigherPrequantumGeometry} \linebreak
\noindent\hyperlink{context_2}{Context}\dotfill \pageref*{context_2} \linebreak
\noindent\hyperlink{symmetries}{Symmetries}\dotfill \pageref*{symmetries} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{energymomentum_tensor}{Energy-momentum tensor}\dotfill \pageref*{energymomentum_tensor} \linebreak
\noindent\hyperlink{dirac_current}{Dirac current}\dotfill \pageref*{dirac_current} \linebreak
\noindent\hyperlink{of_greenschwarz_super_brane_sigma_models}{Of Green-Schwarz super $p$-brane sigma models}\dotfill \pageref*{of_greenschwarz_super_brane_sigma_models} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesDickeyBracket}{Dickey-Lie bracket on currents}\dotfill \pageref*{ReferencesDickeyBracket} \linebreak
\noindent\hyperlink{in_variational_calculus}{In variational calculus}\dotfill \pageref*{in_variational_calculus} \linebreak
\noindent\hyperlink{higher_conserved_currents}{Higher conserved currents}\dotfill \pageref*{higher_conserved_currents} \linebreak
\noindent\hyperlink{in_higher_prequantum_theory}{In higher prequantum theory}\dotfill \pageref*{in_higher_prequantum_theory} \linebreak


\hypertarget{via_the_variational_bicomplex}{}\subsection*{{Via the variational bicomplex}}\label{via_the_variational_bicomplex}

The following discusses the formulation of conserved currents in terms of [[variational calculus]] and the [[variational bicomplex]].

\hypertarget{the_context}{}\subsubsection*{{The context}}\label{the_context}

Let $X$ be a [[spacetime]] of [[dimension]] $n$, $E \to X$ a [[bundle]], $j_\infty E \to X$ its [[jet bundle]] and

\begin{displaymath}
\Omega^{\bullet,\bullet}(j_\infty E), (D = \delta + d)
\end{displaymath}
the corresponding [[variational bicomplex]] with $\delta$ being the vertical and $d = d_{dR}$ the horizontal [[differential]].

\begin{prop}
\label{VariationOfTheLagrangian}\hypertarget{VariationOfTheLagrangian}{}
For $L \in \Omega^{n,0}(j_\infty E)$ a [[Lagrangian]] we have that

\begin{displaymath}
\delta L = E(L) + d \Theta
\end{displaymath}
for $E$ the [[Euler-Lagrange equations|Euler-Lagrange operator]].

\end{prop}
The [[covariant phase space]] of the Lagrangian is the locus

\begin{displaymath}
\{\phi \in \Gamma(E) | E(L)(j_\infty \phi) = 0\}  
  \,.
\end{displaymath}
For $\Sigma \subset X$ any $(n-1)$-dimensional submanifold,

\begin{displaymath}
\delta \theta := \delta \int_\Sigma \Theta
\end{displaymath}
is the [[presymplectic structure]] on covariant phase space

\hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{conserved current} is an element

\begin{displaymath}
j \in \Omega^{n-1, 0}(j_\infty E)
\end{displaymath}
which is horizontally closed on [[covariant phase space]]

\begin{displaymath}
d j|_{E(L) = 0} = 0
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
For $\Sigma \hookrightarrow X$ a submanifold of dimension $n-1$, the \textbf{[[charge]]} of the conserved current $j$ with respect to $\Sigma$ is the [[integral]]

\begin{displaymath}
Q_\Sigma := \int_\Sigma j
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
If $\Sigma, \Sigma' \subset X$ are homologous, the associated charge is the same

\begin{displaymath}
Q_{\Sigma} = Q_{\Sigma'}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By [[Stokes' theorem]].

\end{proof}
\begin{theorem}
\label{}\hypertarget{}{}
Every [[symmetry of the Lagrangian]] induces a conserved current.

\end{theorem}
This is [[Noether's theorem]]. See there for more details.

\hypertarget{InHigherPrequantumGeometry}{}\subsection*{{In higher prequantum geometry}}\label{InHigherPrequantumGeometry}

The following discusses conserved currents in the context of [[higher prequantum geometry]], closely related to \hyperlink{AzcarragaIzquierdo95}{Azcarraga-Izquierdo 95, section 8.1}. This follows (\hyperlink{classicalinhigher}{classicalinhigher, section 3.3.}, going back to \hyperlink{Schreiber13}{Schreiber 13}). Similar observations have been made by [[Igor Khavkine]].

\begin{quote}%
this section needs much polishing. For the moment better see \hyperlink{classicalinhigher}{classicalinhigher, section 3.3}


\end{quote}
\hypertarget{context_2}{}\subsubsection*{{Context}}\label{context_2}

Let $\mathbf{H}$ be the ambient [[(∞,1)-topos]]. For $\mathbf{Fields} \in \mathbf{H}$ a [[moduli ∞-stack]] of [[field (physics)|fields]] a [[local Lagrangian]] for an $n$-dimensional [[prequantum field theory]] is equivalently a [[prequantum n-bundle]] given by a map

\begin{displaymath}
\mathbf{L} \;\colon\; \mathbf{Fields} \longrightarrow \mathbf{B}^n U(1)_{conn}
\end{displaymath}
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 theory|higher]] [[gauge symmetry]] [[equivariance|equivariant]] structure.

The following is effectively the direct higher geometric analog of the \href{Noether's+theorem#HamiltonianNoetherTheorem}{Hamiltonian version of Noether's theorem}.

\hypertarget{symmetries}{}\subsubsection*{{Symmetries}}\label{symmetries}

A transformation of the [[field (physics)|fields]] is an [[equivalence]]

\begin{displaymath}
\mathbf{Fields} 
    \underoverset{\simeq}{\phi}{\longrightarrow}
  \mathbf{Fields}
  \,.
\end{displaymath}
That the [[local Lagrangian]] $\mathcal{L}$ be preserved by this, up to ([[gauge symmetry|gauge]]) equivalence, means that there is a [[diagram]] in $\mathbf{H}$ of the form

\begin{displaymath}
\itexarray{
    \mathbf{Fields} &&\underoverset{\simeq}{\phi}{\longrightarrow}&&
    \mathbf{Fields}
    \\
    & {}_{\mathllap{\mathbf{L}}}\searrow &\swArrow^\simeq_\alpha& \swarrow_{\mathrlap{\mathbf{L}}}
    \\
    && \mathbf{B}^n U(1)_{conn}
  }
  \,.
\end{displaymath}
(With $\mathbf{L}$ equivalently regarded as [[prequantum n-bundle]] this is equivalently a \href{quantomorphism%20group#InHigherGeometry}{higher quantomorphism}. These are the transformations studied in (\hyperlink{FiorenzaRogersSchreiber13}{Fiorenza-Rogers-Schreiber 13}))

For $\phi$ an [[infinitesimal object|infinitesimal]] operation an $L$ locally the Lagrangian $n$-form, this means that the [[Lie derivative]] $\mathcal{L}_{\delta \phi}$ of $L$ has a potential,

\begin{displaymath}
\mathcal{L}_{\delta \phi} L = \mathbf{d} \alpha
\end{displaymath}
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). This defines an \emph{[[infinitesimal symmetry of the Lagrangian]]}. See also (\hyperlink{AzcarragaIzquierdo95}{Azcarraga-Izquierdo 95 (8.1.13)}).

This is the situation of the [[Noether theorem]] for the general case of ``weak'' symmetries (see at \href{Noether+theorem#WeakSymmetrySchematicIdea}{Noether theorem -- schematic idea -- weak symmetries}).

By [[Cartan's magic formula]] the above means

\begin{displaymath}
\mathbf{d}\left(
    \alpha - \iota_{\delta\phi} \mathbf{L} 
  \right)
   = 
  \iota_{\delta \phi} \omega  
  \,.
\end{displaymath}
and hence the combination $j \coloneqq \alpha - \iota_{\delta\phi} \mathbf{L}$ (a \emph{[[Hamiltonian form]]} for $\delta \phi$ with respect to $\omega$) is conserved on trajectories in the kernel of the [[n-plectic form]] $\omega$ (which are indeed the classical trajectories of $\mathbf{L}$, see (\hyperlink{AzcarragaIzquierdo95}{Azcarraga-Izquierdo 95 (8.1.14)})).

This is the first stage in the [[Poisson bracket Lie n-algebra]], the \emph{[[current algebra]]} (see there at \emph{\href{current%20algebra#AsAHomotopyLieAlgebra}{As a homotopy Lie algebra}}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{energymomentum_tensor}{}\subsubsection*{{Energy-momentum tensor}}\label{energymomentum_tensor}

\begin{itemize}%
\item [[energy-momentum tensor]] -- see at \emph{[[geometry of physics -- A first idea of quantum field theory]]} \href{geometry+of+physics+--+A+first+idea+of+quantum+field+theory#ScalarFieldEnergyMomentum}{this example}

\end{itemize}
\hypertarget{dirac_current}{}\subsubsection*{{Dirac current}}\label{dirac_current}

\begin{itemize}%
\item [[Dirac current]] -- see at [[geometry of physics -- A first idea of quantum field theory]] \href{geometry+of+physics+--+A+first+idea+of+quantum+field+theory#DiracCurrent}{this example}

\end{itemize}
\hypertarget{of_greenschwarz_super_brane_sigma_models}{}\subsubsection*{{Of Green-Schwarz super $p$-brane sigma models}}\label{of_greenschwarz_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 \href{super+Poincare+Lie+algebra#PolyvectorExtensions}{polyvector extension} of the [[supersymmetry]] algebra. See at \emph{\href{Green-Schwarz+action+functional#ConservedCurrents}{Green-Schwarz action functional -- Conserved currents}} for more.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[conservation law]]


\item [[current algebra]]


\item [[chiral anomaly]]


\item [[classical observable]]


\item [[Noether theorem]]


\item [[charge]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{ReferencesDickeyBracket}{}\subsubsection*{{Dickey-Lie bracket on currents}}\label{ReferencesDickeyBracket}

The [[Dickey Lie bracket]] on conserved currents is due to

\begin{itemize}%
\item [[Leonid Dickey]], \emph{Soliton equations and Hamiltonian systems}, Advanced Series in Mathematical Physics, Vol. 12 (World Scientific 1991).

\end{itemize}
and is reviewed in

\begin{itemize}%
\item [[Glenn Barnich]], [[Marc Henneaux]], section 3 of \emph{Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket}, Journal of Mathematical Physics 37, 5273-5296 (1996) (\href{http://arxiv.org/abs/hep-th/9601124}{arXiv:hep-th/9601124}, \href{http://dx.doi.org/10.1063/1.531726}{DOI 10.1063/1.531726})

\end{itemize}
The statement that the Dickey bracket Lie 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)

\begin{itemize}%
\item [[Alexandre Vinogradov]], \emph{The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. I. the linear theory}, Journal of Mathematical Analysis and Applications \textbf{100}, 1-40 (1984) (\href{http://dx.doi.org/10.1016/0022-247x(84}{doi}90071-4))


\item [[Alexandre Vinogradov]], \emph{The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory}, Journal of Mathematical Analysis and Applications \textbf{100}, Issue 1, 30 April 1984, Pages 41-129 (\href{http://www.sciencedirect.com/science/article/pii/0022247X84900726}{publisher})



\end{itemize}
and is stated as exercise 2.28 on p. 203 of

\begin{itemize}%
\item [[Alexandre Vinogradov]], [[Joseph Krasil'shchik]] (eds.), \emph{Symmetries and conservation laws for differential equations of mathematical physics}, vol. 182 of Translations of Mathematical Monographs, AMS (1999)

\end{itemize}
A lift of the Dickey Lie bracket on cohomologically trivial spaces to an equivalent \emph{[[model structure for L-infinity algebras|L-infinity equivalent]]} [[L-infinity algebra|L-infinity bracket]] is constructed, under some assumptions, in

\begin{itemize}%
\item [[Glenn Barnich]], [[Ronald Fulp]], [[Tom Lada]], [[Jim Stasheff]], \emph{The sh Lie structure of Poisson brackets in field theory}, Communications in Mathematical Physics 191, 585-601 (1998) (\href{http://arxiv.org/abs/hep-th/9702176}{arXiv:hep-th/9702176})


\item [[Martin Markl]], [[Steve Shnider]], \emph{Differential Operator Endomorphisms of an Euler-Lagrange Complex}, Contemporary Mathematics, Volume 231, 1999 (\href{http://arxiv.org/abs/math/9808105}{arXiv:9808105})



\end{itemize}
The cohomologically non-trivial lift is discussed in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Lie n-algebras of BPS charges]]}, J. High Energ. Phys. (2017) 2017: 87 (\href{http://arxiv.org/abs/1507.08692}{arXiv:1507.08692})

\end{itemize}
\hypertarget{in_variational_calculus}{}\subsubsection*{{In variational calculus}}\label{in_variational_calculus}

A general discussion as above is around definition 9 of

\begin{itemize}%
\item [[Gregg Zuckerman|G. J. Zuckerman]], \emph{Action Principles and Global Geometry} , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259--284. ([[ZuckermanVariation.pdf:file]])

\end{itemize}
The relation of conserved currents to [[moment maps]] in [[symplectic geometry]] is highlighted for instance in

\begin{itemize}%
\item Huijun Fan, \emph{Lecture 8, Moment map and symplectic reduction} (\href{http://www.math.pku.edu.cn/teachers/fanhj/courses/symp-8.pdf}{pdf})

\end{itemize}
\hypertarget{higher_conserved_currents}{}\subsubsection*{{Higher conserved currents}}\label{higher_conserved_currents}

Higher conserved currents are discussed for instance in

\begin{itemize}%
\item [[Glenn Barnich]], [[Friedemann Brandt]], \emph{Covariant theory of asymptotic symmetries, conservation laws and central charges}, Nucl.Phys.B633:3-82, 2002 (\href{http://arxiv.org/abs/hep-th/0111246}{arXiv:hep-th/0111246})

\end{itemize}
\hypertarget{in_higher_prequantum_theory}{}\subsubsection*{{In higher prequantum theory}}\label{in_higher_prequantum_theory}

Conserved currents for Lagrangians written as [[WZW terms]] are discussed in

\begin{itemize}%
\item [[José de Azcárraga]], Jos\'e{} M. Izquierdo, section 8.1 of \emph{[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]} , Cambridge monographs of mathematical physics, (1995)

\end{itemize}
Building on that, in the context of [[higher prequantum geometry]] conserved currents of the [[WZW model]] and in [[schreiber:∞-Wess-Zumino-Witten theory]] are briefly indicated on the last page of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{Higher geometric prequantum theory and The Brane Bouquet}, notes for a [[schreiber:The brane bouquet|talk]] at \href{http://hep.itp.tuwien.ac.at/~miw/bzell2013/}{Bayrischzell 2013} (\href{http://ncatlab.org/schreiber/files/hpqWZWintro.pdf}{pdf notes})


\item [[Urs Schreiber]], \emph{[[schreiber:Classical field theory via Cohesive homotopy types]]}



\end{itemize}
The same structure is considered in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory]]}, 2013

\end{itemize}
as higher [[quantomorphisms]] and [[Poisson bracket Lie n-algebras]] of local currents.

category: physics [[!redirects conserved current]] [[!redirects conserved currents]]

[[!redirects local current]] [[!redirects local currents]]

[[!redirects Noether current]] [[!redirects Noether currents]]

[[!redirects Noether current algebra]] [[!redirects Noether current algebras]]



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