Covariant presymplectic current

Let $M$ be an oriented smooth four-manifold and let $\varphi$ be a field configuration in a specified smooth field bundle. A local Lagrangian four-form $\mathcal{L}(\varphi )$ is a smooth four-form whose coordinate coefficients depend smoothly on the field components and finitely many spacetime derivatives at each point. A chosen first-variation decomposition consists of an Euler functional $E_{\varphi }[\eta ]$ with values in four-forms and a three-form $\theta (\varphi ;\eta )$. In local bundle charts, both are smooth finite-order local differential expressions in the field configuration $\varphi$ and its variation $\eta$: their coefficients depend smoothly on the spacetime coordinates, finitely many spacetime derivatives of $\varphi$, and any fixed smooth background fields, and their dependence on $\eta$ is linear in finitely many of its spacetime derivatives. Every smooth one-parameter field family then satisfies

$\delta \mathcal{L}=E_{\varphi }[\delta \varphi ]+d_M\theta (\varphi ;\delta \varphi )\mathrm{.}$

The choice is part of the data; it is not asserted to be unique. The stated smoothness makes its parameter derivatives defined along every smooth finite-dimensional field family. In a parameter family, ${\delta }_i{E}_{\varphi }$ means the derivative of the coefficients of $E_{\varphi }$ with its variation input held fixed. Thus the total derivative of $E_{\varphi (p)}[\eta (p)]$ is $({\delta }_i{E}_{\varphi })[\eta ]+E_{\varphi }[{\delta }_i\eta ]$. For a smooth finite-dimensional field family with parameter exterior derivative $d_P$, the one-form $\mathrm{\Theta }$ on parameter space with values in spacetime three-forms is ${\mathrm{\Theta }}_p(v)=\theta ({\varphi }_p;{\delta }_v{\varphi }_p)$. Define the covariant presymplectic current by $\omega :=-d_P\mathrm{\Theta }$. For commuting parameter coordinates $s_1,s_2$ this is

$\omega ({\delta }_1\varphi ,{\delta }_2\varphi )={\delta }_2\theta (\varphi ;{\delta }_1\varphi )-{\delta }_1\theta (\varphi ;{\delta }_2\varphi )\mathrm{.}$

For a finite oriented smooth singular three-chain $\mathrm{\Sigma }$, define the integrated two-form by ${\mathrm{\Omega }}_{\mathrm{\Sigma }}({\delta }_1\varphi ,{\delta }_2\varphi )={\int }_{\mathrm{\Sigma }}\omega$.

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §§2.2–2.3

Covariant field families and variations

A smooth differential $k$-form

The exterior derivative by the invariant vector-field formula

Smooth singular chain and cochain complexes

Integral of a form over a smooth singular simplex

The definition uses finite-dimensional smooth families to specify all variations and their derivatives. It does not assume a global smooth infinite-dimensional configuration or solution manifold. No asymptotic falloff, boundary condition, or uniqueness of $\theta$ is included.

The sign conventions of the cited sources require care. Iyer--Wald's equation (44) uses the ordering in this definition. Compère--Fiorucci's equation (1.25) is written as a field-space form; after the contraction convention of (1.26), their classical first variation has the same $+d\theta$ potential convention as this definition. Their component formula in equation (1.28) is ${\delta }_1\theta ({\delta }_2\varphi )-{\delta }_2\theta ({\delta }_1\varphi )$, the negative of the current defined here. Harlow--Wu also use the classical $+d\mathrm{\Theta }$ convention and define $\omega =\delta (\mathrm{\Theta }-dC)$ in equation (2.23) (with $C=0$ in their scalar example, equations (3.8) and (3.13)). When the potential is matched as $\theta =\mathrm{\Theta }-dC$, standard ordered evaluation of the field-space exterior derivative gives ${\omega }_{\mathrm{H}\mathrm{W}}({\delta }_1,{\delta }_2)={\delta }_1\theta ({\delta }_2\varphi )-{\delta }_2\theta ({\delta }_1\varphi )$, again the negative of the convention here. Translate these signs when using their formulas.