# nLab conserved current

### Context

#### Variational calculus

variational calculus

# Contents

## In variational calculus

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

### The context

Let $X$ be a spacetime of dimension $n$, $E\to X$ a bundle, $j:\infty E\to X$ its jet bundle and

${\Omega }^{•,•}\left({j}_{\infty }E\right),\left(D=\delta +d\right)$\Omega^{\bullet,\bullet}(j_\infty E), (D = \delta + d)

the corresponding variational bicomplex with $\delta$ being the vertical and $d={d}_{\mathrm{dR}}$ the horizontal differential.

###### Proposition

For $L\in {\Omega }^{n,0}\left({j}_{\infty }E\right)$ a Lagrangian we have that

$\delta L=E\left(L\right)+d\Theta$\delta L = E(L) + d \Theta

for $E$ the Euler-Lagrange operator.

The covariant phase space of the Lagrangian is the locus

$\left\{\varphi \in \Gamma \left(E\right)\mid E\left(L\right)\left({j}_{\infty }\varphi \right)=0\right\}\phantom{\rule{thinmathspace}{0ex}}.$\{\phi \in \Gamma(E) | E(L)(j_\infty \phi) = 0\} \,.

For $\Sigma \subset X$ any $\left(n-1\right)$-dimensional submanifold,

$\delta \theta :=\delta {\int }_{\Sigma }\Theta$\delta \theta := \delta \int_\Sigma \Theta

is the presymplectic structure on covariant phase space

### Definition

###### Definition

A conserved current is an element

$j\in {\Omega }^{n-1,0}\left({j}_{\infty }E\right)$j \in \Omega^{n-1, 0}(j_\infty E)

which is horizontally closed on covariant phase space

$dj{\mid }_{E\left(L\right)=0}=0\phantom{\rule{thinmathspace}{0ex}}.$d j|_{E(L) = 0} = 0 \,.
###### Definition

For $\Sigma ↪X$ a submanifold of dimension $n-1$, the charge of the conserved current $j$ with respect to $\Sigma$ is the integral

${Q}_{\Sigma }:={\int }_{\Sigma }j\phantom{\rule{thinmathspace}{0ex}}.$Q_\Sigma := \int_\Sigma j \,.

### Properties

###### Proposition

If $\Sigma ,\Sigma \prime \subset X$ are homolous, the associated charge is the same

${Q}_{\Sigma }={Q}_{\Sigma \prime }\phantom{\rule{thinmathspace}{0ex}}.$Q_{\Sigma} = Q_{\Sigma'} \,.
###### Theorem

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.

### Context

Let $H$ be the ambient (∞,1)-topos. For $\mathrm{Fields}\in H$ 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

$ℒ\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\mathrm{Fields}⟶{B}^{\mathrm{nU}}\left(1{\right)}_{\mathrm{conn}}$\mathcal{L} \;\colon\; \mathbf{Fields} \longrightarrow \mathbf{B}^nU(1)_{conn}

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 $ℒ$ is the higher gauge symmetry equivariant structure.

### Symmetries

A transformation of the fields is an equivalence

$\mathrm{Fields}\underset{\simeq }{\overset{\varphi }{⟶}}\mathrm{Fields}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{Fields} \underoverset{\simeq}{\phi}{\longrightarrow} \mathbf{Fields} \,.

That the local Lagrangian $ℒ$ be preserved by this, up to (gauge) equivalence, means that there is a diagram in $H$ of the form

$\begin{array}{ccccc}\mathrm{Fields}& & \underset{\simeq }{\overset{\varphi }{⟶}}& & \mathrm{Fields}\\ & {}_{ℒ}↘& {⇙}_{\alpha }^{\simeq }& {↙}_{ℒ}\\ & & {B}^{n}U\left(1{\right)}_{\mathrm{conn}}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathbf{Fields} &&\underoverset{\simeq}{\phi}{\longrightarrow}&& \mathbf{Fields} \\ & {}_{\mathllap{\mathcal{L}}}\searrow &\swArrow^\simeq_\alpha& \swarrow_{\mathrlap{\mathcal{L}}} \\ && \mathbf{B}^n U(1)_{conn} } \,.

(With $ℒ$ equivalently regarded as prequantum n-bundle this is equivalently a higher quantomorphism. These are the transformations studied in (Fiorenza-Rogers-Schreiber 13))

For $\varphi$ an infinitesimal operation an $L$ locally the Lagrangian $n$-form, this means

${ℒ}_{\delta \varphi }L=d\alpha$\mathcal{L}_{\delta \phi} L = \mathbf{d} \alpha

hence that the Lagrangian changes under the Lie derivative by an exact term. By Cartan's magic formula this means

$d\left(\alpha -{\iota }_{\delta \varphi }ℒ\right)={\iota }_{\delta \varphi }\omega \phantom{\rule{thinmathspace}{0ex}}.$\mathbf{d}\left( \alpha - \iota_{\delta\phi} \mathcal{L} \right) = \iota_{\delta \phi} \omega \,.

Hence (…)

## References

### 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.

Revised on September 13, 2013 18:57:23 by Urs Schreiber (82.169.114.243)