bicomplex). The non-trivial cohomology groups of the differential $d_{\tt h}$
acting on $\Omega^{\bullet,\bullet}({\cal R}^{\infty})$ are referred to as
_characteristic cohomology groups:_ they can be related to the so-called
Koszul-Tate differential (see Appendix B.3) and we will see in Section 7 that
they provide important information on the solutions of the field equations. In
particular, they describe non-trivial conservation laws in field theory, see
equations (7.14)-(7.15) and (7.39) below.

###  6.5 Field theory: Relation with multisymplectic approach

The field theoretical generalization of the considerations in Section 6.3 on
mechanics as formulated on the variational bicomplex is tantamount to
considering the base manifold $M=\mathbb{R}^{n}$ and a multisymplectic
dynamical system on $J^{\infty}E$ with $E=\mathbb{R}^{n}\times\mathbb{R}^{N}$.
More precisely, following references [160, 73], we consider a _Lagrangian
multisymplectic system of PDE’s,_ i.e. we assume that the system of PDE’s
represents the Euler-Lagrangian equation ${\cal E}({\cal L}\,d^{n}x)=0$
following from a

$\displaystyle\mbox{Lagrangian $n$-form }\qquad\boxed{\ {\cal
L}\,d^{n}x=p_{a}^{\mu}(x,q_{\textbf{J}})\,d_{\tt h}q^{a}\wedge
d^{n-1}x_{\mu}-{\cal H}\,d^{n}x\ }\,,$ (6.38)

on $J^{\infty}E$. (Note that this expression corresponds to the Poincaré-
Cartan $n$-form on ordinary multiphase space, see Eqn. (3.11). It is
tantamount to the basic relationship ${\cal L}=p_{a}^{\mu}q^{a}_{\mu}-{\cal
H}$, see Eqn. (3.2).) In this setting, it is natural to introduce a symplectic
$2$-form $d_{\tt v}q^{a}\wedge d_{\tt v}p_{a}^{\mu}$ for each space-time
coordinate $x^{\mu}$ (in analogy to the symplectic $2$-form (6.22) for
mechanical systems). One then gathers all of these forms in the so-called
_multisymplectic $(n+1)$-form_ on $J^{\infty}E$, i.e. the following $d_{\tt
v}$-closed $(n-1,2)$-form:

$\displaystyle\mbox{multisymplectic form on $J^{\infty}E$ :}\qquad\boxed{\
\underline{\omega}\equiv d_{\tt v}q^{a}\wedge d_{\tt v}p_{a}^{\mu}\wedge
d^{n-1}x_{\mu}\ }\,.$ (6.39)

The generalization of the calculation (6.22)-(6.24) to field theory, which now
relies on the De Donder-Weyl equations $q^{a}_{\mu}=\frac{\partial{\cal
H}}{\partial p_{a}^{\mu}},\,\partial_{\mu}p^{\mu}_{a}=-\frac{\partial{\cal
H}}{\partial q^{a}}$, yields

$\underline{\omega}\equiv d_{\tt v}q^{a}\wedge d_{\tt v}p_{a}^{\mu}\wedge
d^{n-1}x_{\mu}=dq^{a}\wedge dp_{a}^{\mu}\wedge d^{n-1}x_{\mu}+d{\cal H}\wedge
d^{n}x=\omega_{{\cal H}}\,,$

i.e. we have the “dynamical” multisymplectic $(n+1)$-form (3.12) on multiphase
space, but the present approach does not rely on an extension of phase space.

As for mechanics (see Eqn. (6.25)), we have the following general result for
the multisymplectic form given by (6.39): _the relation_

$\displaystyle\boxed{\ d_{\tt h}\,\underline{\omega}=0\ }\qquad\mbox{on
solutions of the field equations}\,,$ (6.40)

_is tantamount to the existence of a Lagrangian of the form ( 6.38) which
yields these field equations._ Indeed, by virtue of the Poincaré lemma, the
relation $d_{\tt v}\underline{\omega}=0$ for the form
$\underline{\omega}\in\Omega^{n-1,2}$ given by (6.39) implies that there
(locally) exists a form $\underline{\theta}\in\Omega^{n-1,1}$ such that

$\displaystyle\underline{\omega}=-d_{\tt v}\,\underline{\theta}\,.$ (6.41)

The fact that the differentials $d_{\tt v}$ and $d_{\tt h}$ anticommute now
implies that

$d_{\tt v}d_{\tt h}\,\underline{\theta}=-d_{\tt h}d_{\tt
v}\,\underline{\theta}=d_{\tt h}\,\underline{\omega}=0\qquad\mbox{on
solutions.}$

Since $d_{\tt h}\underline{\theta}$ is $d_{\tt v}$-closed (on solutions), we
again apply the Poincaré lemma to infer the (local) existence of a $n$-form
${\cal L}\,d^{n}x\in\Omega^{n,0}$ such that

$\displaystyle d_{\tt h}\,\underline{\theta}=d_{\tt v}({\cal
L}\,d^{n}x)\qquad\mbox{on solutions.}$ (6.42)

The $n$-form ${\cal L}\,d^{n}x$ actually represents a Lagrangian for the
system of PDE’s under consideration since application of the interior Euler
operator ${\cal I}$ defined by Eqn. (6.15) and enjoying the property (6.20),
yields

${\cal E}({\cal L}\,d^{n}x)\equiv{\cal I}\,d_{\tt v}({\cal L}\,d^{n}x)={\cal
I}(d_{\tt h}\,\underline{\theta})=0\qquad\mbox{on solutions,}$

i.e. the Euler-Lagrange equation. According to (6.39) and (6.41), we presently
have the following explicit expression for the form
$\underline{\theta}\in\Omega^{n-1,1}$:

$\displaystyle\underline{\theta}=p_{a}^{\mu}(x,q_{\textbf{J}})\,d_{\tt
v}q^{a}\wedge d^{n-1}x_{\mu}\,.$ (6.43)

This expression (involving the vertical differential) corresponds to the
kinematical multicanonical $n$-form (3.9). As for mechanics, _the
consideration of the variational bicomplex avoids the extension of multiphase
space by an energy-type variable and it allows for the simple coordinate free
expression ( 6.40) for the Hamiltonian flow._

We note that relation (6.42) coincides with equation (6.33) and that (6.41)
coincides with (6.27), i.e.

$\mbox{On ${\cal R}^{\infty}$ :}\qquad j\ \longleftrightarrow\
\underline{\theta}\,,\qquad J\ \longleftrightarrow\ \underline{\omega}\,.$

In fact, (6.43) and (6.39) represent explicit expressions for the forms $j$
and $J$ encountered in the previous subsection: by virtue of
$p^{\mu}_{a}=\frac{\partial{\cal L}}{\partial q^{a}_{\mu}}$, these expressions
precisely correspond to those encountered in Eqn. (6.28) (for our choice of a
first order Lagrangian).

We refer to the work [73] for a discussion of conservation laws (the so-called
multimomentum map) in this geometric set-up of field theory. It turns out that
this approach to PDE’s not only provides new mathematical insights, but also
allows for the derivation of novel predictions concerning various dynamical
systems of physical interest, e.g. in fluid mechanics, see references [160,
73].

###  6.6 Symmetries and gauge field theories

We only discuss variational symmetries (i.e. symmetries of the action) as well
as local (gauge) symmetries and we refer to the literature [31, 74] for a
comprehensive treatment of symmetries and their consequences. We note that in
the literature the notation of Lagrangians or Lagrangian densities is far from
uniform, e.g. see references [154, 17, 22] for different choices. We assume
that _on a space-time manifold $(M,(g_{\mu\nu}))$, the Lagrangian ${\cal L}$
has the expression $\sqrt{|g|}\,{\cal L}$ which is often denoted by $L$ in the
literature,_ e.g. [154, 158]. For the sake of clarity, we recall (cf. text
after Eqn. (6.4)) that the notation $f[q]$ means that $f$ is a smooth real-
valued function of $x,q,q_{\mu},\dots,q_{\mu_{1}\cdots\mu_{K}}$ for some
$K\in\mathbb{N}$.

##### (Global) variational symmetries and Noether’s first theorem:

Suppose we are given a (first order) Lagrangian $n$-form ${\cal
L}\,d^{n}x\in\Omega^{n,0}$. By definition [72], a _(global) variational
symmetry_ of the Lagrangian ${\cal L}\,d^{n}x$ is a so-called _evolutionary
vector field_ $X^{a}\,\frac{\partial\ }{\partial q^{a}}$ with characteristic
$X^{a}[q]$ and prolongation on $J^{\infty}E$ given by

$\displaystyle X=X^{a}\,\frac{\partial\ }{\partial
q^{a}}+\partial_{\mu}X^{a}\,\frac{\partial\ }{\partial
q^{a}_{\mu}}+\partial_{\mu}\partial_{\nu}X^{a}\,\frac{\partial\ }{\partial
q^{a}_{\mu\nu}}+\cdots\,,$ (6.44)

for which there exists a form $K\in\Omega^{n-1,0}$ (depending in general on
$X$, i.e. on the local functions $X^{a}$) such that

$\displaystyle\boxed{\ i_{X}d_{\tt v}({\cal L}\,d^{n}x)=d_{\tt h}K\ }\,.$
(6.45)

We note that the variation of the field variables under $X$ is given by

$\displaystyle\delta_{X}q^{a}=X(q^{a})=X^{a}\,,$ (6.46)

and that the variation $\delta_{X}$ commutes with the total derivative
$\partial_{\mu}$:

$\displaystyle[\partial_{\mu},\delta_{X}]=0\,.$ (6.47)

From $K\equiv k^{\mu}[q]\,d^{n-1}x_{\mu}$, it follows that $d_{\tt
h}K=(\partial_{\mu}k^{\mu})\,d^{n}x$. Evaluation of the left-hand side of
(6.45) yields $i_{X}d_{\tt v}({\cal L}\,d^{n}x)=(\delta_{X}{\cal L})\,d^{n}x$,
hence relation (6.45) is equivalent to the

$\displaystyle\mbox{quasi-invariance of ${\cal L}$ :}\qquad\boxed{\
\delta_{X}{\cal L}=\partial_{\mu}k^{\mu}\
}\,,\quad\mbox{with}\quad\delta_{X}{\cal L}=X^{a}\,\frac{\partial{\cal
L}}{\partial q^{a}}+\partial_{\mu}X^{a}\,\frac{\partial{\cal L}}{\partial
q^{a}_{\mu}}\,.$ (6.48)

We can rewrite $\delta_{X}{\cal L}$ by using the Euler-Lagrange derivative
$E_{a}({\cal L})$ given by (6.16) as well as the Leibniz rule:

$\delta_{X}{\cal L}=X^{a}\,\frac{\partial{\cal L}}{\partial
q^{a}}+\partial_{\mu}X^{a}\,\frac{\partial{\cal L}}{\partial
q^{a}_{\mu}}=X^{a}\,E_{a}({\cal
L})+\partial_{\mu}\Big{(}X^{a}\,\frac{\partial{\cal L}}{\partial
q^{a}_{\mu}}\Big{)}\,.$

Substitution of this expression into (6.48) leads to the relation

$\displaystyle\boxed{\ 0=E_{a}({\cal L})\,X^{a}+\partial_{\mu}{\cal J}^{\mu}\
}\,,\qquad\mbox{with}\qquad\boxed{\ {\cal J}^{\mu}\equiv\frac{\partial{\cal
L}}{\partial q^{a}_{\mu}}\,X^{a}-k^{\mu}\ }\,.$ (6.49)

This result is nothing but _Noether’s first theorem_ which states that a
symmetry of the action $S\equiv\int_{M}{\cal L}\,d^{n}x$ implies a local
conservation law $\partial_{\mu}{\cal J}^{\mu}=0$ for the solutions of the
equations of motion $E_{a}({\cal L})=0$.

##### Example 1 (Geometric transformations):

By way of example, we consider $M=\mathbb{R}^{n}$ and _space-time
translations_ parametrized by constant vectors
$(\varepsilon^{\mu})\in\mathbb{R}^{n}$, i.e. a rigid geometric transformation.
Then $\delta_{X}q^{a}=X^{a}=\varepsilon^{\mu}q_{\mu}^{a}$ and
$\partial_{\nu}X^{a}=\varepsilon^{\mu}q_{\mu\nu}^{a}$, which yields

$\delta_{X}{\cal L}=\varepsilon^{\mu}\left(q^{a}_{\mu}\,\frac{\partial{\cal
L}}{\partial q^{a}}+q^{a}_{\mu\nu}\,\frac{\partial{\cal L}}{\partial
q^{a}_{\nu}}\right)=\varepsilon^{\mu}\left(\partial_{\mu}{\cal
L}-\frac{\partial{\cal L}}{\partial x^{\mu}}\right)\,.$

If ${\cal L}$ does not explicitly depend on $x$ (i.e. $\frac{\partial{\cal
L}}{\partial x^{\mu}}=0$ for all $\mu$), then $\delta_{X}{\cal
L}=\varepsilon^{\mu}\partial_{\mu}{\cal
L}=\partial_{\mu}(\varepsilon^{\mu}{\cal L})$, i.e. $\delta_{X}{\cal
L}=\partial_{\mu}k^{\mu}$ with $k^{\mu}=\varepsilon^{\mu}{\cal L}$. By virtue
of Noether’s first theorem (6.49) we now have the following local conservation
law:

$\displaystyle\mbox{On-
shell:}\quad\partial_{\mu}T^{\mu\nu}=0\,,\qquad\mbox{with}\qquad\boxed{\
{T^{\mu}}_{\nu}\equiv\frac{\partial{\cal L}}{\partial
q^{a}_{\mu}}\,q^{a}_{\nu}-\delta^{\mu}_{\nu}\,{\cal L}\ }\,.$ (6.50)

Here, $T^{\mu\nu}$ is the familiar expression for the _canonical energy-
momentum tensor_ as written on the jet bundle. If ${\cal L}$ explicitly
depends on $x$, then the relations above imply the

$\displaystyle\mbox{balance
equation:}\qquad\partial_{\mu}T^{\mu\nu}=-\frac{\partial{\cal L}}{\partial
x_{\nu}}\,.$ (6.51)

We note that the conserved current density ${\cal J}^{\mu}$ given by (6.49)
and thereby the canonical energy-momentum tensor (6.50) involve the canonical
momentum vector field $\partial{\cal L}/\partial q^{a}_{\mu}$ appearing in the
De Donder-Weyl formulation of dynamics, cf. equations (3.1)-(3.2).

We also remark that there exist other derivations of Noether’s first theorem.
One of them is based on a variation of the region $U$ of space-time in which
the fields are defined, e.g. see [231] and in particular [232] for a
pedagogical account. If the infinitesimal variations are considered to be
generic rather than symmetry variations of an invariant action functional,
then this approach to Noether’s first theorem amounts to an application of the
_Weiss variational principle_ [233] (see reference [234]): this principle,
which allows for variations of end-points in classical mechanics and for
variations of the space-time domain in field theory, provides an alternative
derivation of field equations. It has recently been applied to vary and
interpret the gravitational action including the GHY term [234].

##### Example 2 (Internal transformations):

For later reference, we consider a simple example for a rigid internal
symmetry, i.e. a symmetry transformation which acts in the space of fields,
but not on space-time coordinates. In _Gell-Mann’s model of free quarks_ in
$\mathbb{R}^{4}$, one has the triplet $\varphi\equiv(\varphi^{a})_{a=1,2,3}$
of quarks $u,d,c$. Each of the quark fields $\varphi^{a}$ represents a Dirac
spinor field for which the adjoint is denoted by
$\overline{\varphi^{a}}\equiv(\varphi^{a})^{\dagger}\gamma^{0}$. In this
respect, we recall that the Dirac matrices $\gamma^{\mu}$ are $4\times 4$
matrices satisfying the Clifford algebra relation associated to the Minkowski
metric, i.e. $\\{\gamma_{\mu},\gamma_{\nu}\\}=2\eta_{\mu\nu}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$
where $\\{\cdot,\cdot\\}$ denotes the anticommutator of matrices. The
Lagrangian density describing the dynamics of _free massless quarks_
reads181818In this example, we use the standard physics notation for fields
rather than the notation $q^{a}_{\mu}\equiv(\partial_{\mu}\varphi^{a})(x)$ of
the variational bicomplex approach. We do not spell out the tensor product
symbols, e.g. in the product $\bar{\varphi}\otimes\gamma^{\mu}$.

$\displaystyle{\cal
L}\equiv\textrm{i}\,\bar{\varphi}\gamma^{\mu}\partial_{\mu}\varphi\,.$ (6.52)

This Lagrangian is invariant under the global gauge transformations
$\varphi\leadsto\varphi^{\prime}=U\varphi$ with $U\in
SU(3)_{\textrm{flavor}}$. In terms of a basis $\\{T_{r}\\}_{r=1,\dots,8}$ of
the Lie algebra $su(3)$ with $T_{r}^{\dagger}=T_{r}$ and

$[T_{r},T_{s}]=\textrm{i}\,f_{rst}T_{t}\qquad\mbox{for}\
\;r,s\in\\{1,\dots,8\\}\,,$

we have $U=\textrm{e}^{-\textrm{i}\,f^{r}T_{r}}$ with real parameters $f^{r}$.
Thus, the infinitesimal symmetry transformations of fields write

$\displaystyle\delta_{X_{r}}\varphi^{a}=X_{r}(\varphi^{a})=-\textrm{i}\,(T_{r}\varphi)^{a}=-\textrm{i}\,(T_{r})^{ab}\varphi^{b}=X_{r}^{a}\qquad\mbox{for}\
\;r\in\\{1,\dots,8\\}\,,\ \ a\in\\{1,2,3\\}\,,$ (6.53)

and the corresponding Noether currents (6.49) have the form

$\displaystyle\boxed{\ {\cal
J}^{\mu}_{r}=\bar{\varphi}\gamma^{\mu}T_{r}\varphi\ }\qquad\mbox{for}\
\;r\in\\{1,\dots,8\\}\,.$ (6.54)

##### Local (gauge) symmetries:

A _local symmetry_ of the Lagrangian ${\cal L}\,d^{n}x$ is a variational
symmetry $X_{f}$ depending on local functions $f^{r}[q]$, i.e. [31, 74]

$\displaystyle\delta_{f}{\cal
L}=\partial_{\mu}k^{\mu}\qquad\mbox{and}\qquad\delta_{f}q^{a}=Q^{a}(f)\equiv
Q^{a}_{r}(f^{r})\equiv
Q^{a}_{r}f^{r}+Q^{a\mu}_{r}\partial_{\mu}f^{r}+\cdots\,.$ (6.55)

The latter relation means that the variation $\delta_{f}q^{a}$ depends
linearly on the functions $f^{r}$ and their derivatives. Following P. Bergmann
[235], the coefficients $Q^{a}_{r},Q^{a\mu}_{r},\dots$ are also referred to as
_descriptors_ (of the symmetry transformation). E.g. for the gauge potentials
$A_{\mu}$ of electromagnetism, we have $\delta_{f}A_{\mu}=\partial_{\mu}f$ and
for the YM potentials $A_{\mu}^{r}$ of non-Abelian gauge field theory, we have
$\delta_{f}A^{r}_{\mu}=\partial_{\mu}f^{r}+\textrm{i}{f^{r}}_{st}A_{\mu}^{s}f^{t}$
where $f\equiv f^{r}T_{r}$ represents a Lie algebra-valued function. In
general relativity, the action of the diffeomorphisms generated by a vector
$\xi\equiv\xi^{\mu}\partial_{\mu}$ on the metric tensor field $(g_{\mu\nu})$
is given by
$\delta_{\xi}g_{\mu\nu}=\big{(}L_{\xi}g\big{)}_{\mu\nu}=\xi^{\rho}\partial_{\rho}g_{\mu\nu}+g_{\mu\rho}\partial_{\nu}\xi^{\rho}+g_{\rho\nu}\partial_{\mu}\xi^{\rho}$.
Thus, for many field theories of physical interest, one has at most first
order derivatives of the parameters $f^{r}$ in expansion (6.55). (As a matter
of fact, the expression (6.55) of $Q^{a}_{r}$ is equivalent to the one that we
considered in our discussion of the Peierls bracket, see equations
(4.21)-(4.22).)

##### Symplectic $2$-form in gauge field theories:

From the discussion after Eqn. (5.28) we recall that a presymplectic $2$-form
$\Omega$ is (weakly) degenerate if it admits a non-trivial kernel, i.e. if
there are non-trivial vector fields $V$ for which $i_{V}\Omega=0$. For
covariant phase space $Z$, we noted that such vector fields do exist in gauge
field theories where they correspond to the gauge orbits in $Z$. Furthermore,
we saw that by factoring out the gauge group ${\cal G}$ from $Z$ following
equations (5.24)-(5.25), i.e.

$Z_{\textrm{phys}}\equiv Z/{\cal
G}\qquad\mbox{and}\qquad\Omega=\pi^{*}\Omega_{\textrm{phys}}\,,\quad\mbox{where}\quad\pi:Z\longrightarrow
Z_{\textrm{phys}}\,,$

one can get rid of the kernel of $\Omega$ and thereby obtain a _symplectic
form_ $\Omega_{\textrm{phys}}$ on the quotient space $Z_{\textrm{phys}}$. This
procedure works for all field theoretic models which have been investigated
[236, 152, 59, 60].

Quite generally, for any Lagrangian field theory formulated in terms of the
variational bicomplex, one expects [72] that the presymplectic $2$-form
$\Omega$ is only degenerate if the Lagrangian $n$-form admits some non-trivial
local symmetries and that the latter span the kernel of $\Omega$. (Then, a
non-degenerate $2$-form $\Omega_{\textrm{phys}}$ can be obtained by factoring
out the group of local symmetries from the bundle ${\cal R}^{\infty}$, i.e.

${\cal R}_{\textrm{phys}}\equiv{\cal R}^{\infty}/{\cal
G}\qquad\mbox{and}\qquad\Omega=\pi^{*}\Omega_{\textrm{phys}}\,,\quad\mbox{where}\quad\pi:{\cal
R}^{\infty}\longrightarrow{\cal R}_{\textrm{phys}}\,,$

$\pi$ denoting the projection map.) However, a general proof of this statement
and a proper mathematical characterization of the quotient space ${\cal
R}^{\infty}/{\cal G}$ appear to be missing – see however [237] for some recent
work concerning these issues.

###  6.7 On space-times with boundary

The present subsection is based on references [66, 238] (see also [122, 182]
for related earlier work).

A mathematical tool to deal with differential forms on a smooth manifold $M$
along with the differential forms on its boundary $\partial M$ has been
introduced by R. Bott and L. Tu and dubbed the _relative de Rham complex_
[239]. Recently an extension of this framework to the variational bicomplex
(which is referred to as the _relative variational bicomplex_) has been put
forward by the authors of reference [66]. (We note that other applications of
the relative de Rham complex have been considered in the context of covariant
phase space [149].) This generalization is devised in such a way that it also
allows us to take into account the _boundary terms_ which are added to action
functionals (e.g. to the Einstein-Hilbert action for the gravitational field)
in order to obtain a well-defined action principle [17, 16]. In the following,
we will outline these mathematical notions and illustrate the general set-up
by considering a scalar field which is subject to different boundary
conditions.

####  6.7.1 Relative de Rham complex

The definition given by Bott and Tu [239] is quite general since it applies to
a generic pair $(M,N)$ of manifolds which are related by a smooth map. In view
of the application that we have in mind, namely a manifold $M$ with a boundary
$\partial M$, we focus on a pair $(M,N)$ of manifolds where the manifold
$N\subset\partial M$ is of codimension $1$ (with respect to $M$) and on the
inclusion map191919Here, we adapt some of the notations of references [66,
238] so as to be consistent with our previous notation.
$\imath:N\hookrightarrow M$. The set of _relative $k$-forms on $(M,N)$_ is
then given by

$\displaystyle\boxed{\
\Omega^{k}(M,N)\equiv\Omega^{k}(M)\oplus\Omega^{k-1}(N)\ }\qquad\mbox{for}\
\;k\in\\{0,\dots,n\equiv\textrm{dim}\,M\\}\,,$ (6.56)

and the _differential_ $\underline{d}:\Omega^{k}(M,N)\to\Omega^{k+1}(M,N)$ is
defined for $(\alpha,\beta)\in\Omega^{k}(M,N)$ by

$\displaystyle\boxed{\
\underline{d}\,(\alpha,\beta)\equiv(d\alpha,\imath^{\ast}\alpha-d\beta)\ }\,,$
(6.57)

(where $d\alpha$ and $d\beta$ involve the exterior derivatives on $M$ and $N$,
respectively). The so-defined operator $\underline{d}$ is nilpotent so that
relative cohomology groups can be defined. In this respect we note that
$\underline{d}\,(\alpha,\beta)=0$ is tantamount to saying that the form
$\alpha$ on $M$ is closed and that the form $\imath^{\ast}\alpha$ on $N$ is
exact.

The authors of reference [66] specified various geometric operations for the
relative de Rham complex like the exterior product of relative forms as well
as the interior product and Lie derivative of relative $k$-forms with respect
to vector fields on $M$. Here, we only note the following points. For an
$n$-dimensional manifold $M$ and $N\subset\partial M$, the _integral of a
relative $n$-form $(\alpha,\beta)\in\Omega^{n}(M,N)$ over the pair $(M,N)$_ is
defined by

$\displaystyle\boxed{\
\int_{(M,N)}(\alpha,\beta)\equiv\int_{M}\alpha-\int_{N}\beta\ }\,.$ (6.58)

The _boundary $\underline{\partial}(M,N)$ of the pair $(M,N)$_ is given by

$\displaystyle\underline{\partial}(M,N)\equiv(\partial M\setminus N,\partial
N)\,,$ (6.59)

and its inclusion map (induced by the inclusion $\imath:\partial
M\hookrightarrow M$ is denoted by
$\underline{\imath}:\underline{\partial}(M,N)\hookrightarrow(M,N)$. _Stokes
theorem_ then generalizes to

$\displaystyle\boxed{\
\int_{(M,N)}\underline{d}\,(\alpha,\beta)=\int_{\underline{\partial}(M,N)}\underline{\imath}^{*}\,(\alpha,\beta)\
}\qquad\mbox{for}\ \;(\alpha,\beta)\in\Omega^{n-1}(M,N)\,.$ (6.60)

####  6.7.2 Scalar fields on a globally hyperbolic manifold with boundary

A globally hyperbolic manifold $(M,g)$ with boundary $\partial M$ (see
reference [146] for the mathematical aspects) is homeomorphic to
$\mathbb{R}\times\Sigma$ where $\Sigma$ denotes a Cauchy hypersurface with
boundary $\partial\Sigma$. Thus, the boundary of $M$ is given by the “lateral
boundary” $\partial_{L}M$, i.e. $\partial
M\cong\mathbb{R}\times\partial\Sigma=\partial_{L}M$. In the following, we
consider the dynamics of a real scalar field $\phi$ on $M$, i.e. $\phi\in
C^{\infty}(M)=\Omega^{0}(M)$. Quite generally, the quantities on $\partial M$
are denoted by a bar; for instance, the field $\bar{\phi}$ on $\partial M$ is
defined by pulling back $\phi$ from $M$ to $\partial M$ by the inclusion map
$\imath:\partial M\hookrightarrow M$, i.e.

$\displaystyle\imath^{*}\,:\,\Omega^{0}(M)\ \longrightarrow$ $\displaystyle\
\;\Omega^{0}(\partial M)$ $\displaystyle\phi\ \longmapsto$ $\displaystyle\
\imath^{*}\phi=\phi\circ\imath\equiv\bar{\phi}\,.$

The volume $n$-form $\sqrt{|g|}\,d^{n}x$ on $M$ is denoted by
$\textrm{vol}_{M}$ and similarly the induced volume form on the boundary
$\partial M$ is denoted by $\textrm{vol}_{\partial M}$.

We assume that the

$\mbox{bulk action }\qquad S_{M}\equiv\int_{M}L\equiv\int_{M}{\cal
L}\,\textrm{vol}_{M}\,,$

is supplemented by a

$\mbox{boundary action }\qquad S_{\partial M}\equiv\int_{\partial
M}\bar{\ell}\,.$

By way of _example,_ we consider

$\displaystyle\boxed{\ {\cal
L}=\frac{1}{2}\,g^{\mu\nu}(\nabla_{\mu}\phi)(\nabla_{\nu}\phi)-V(\phi)\
}\,,\qquad\boxed{\
\bar{\ell}=\frac{1}{2}\,f\,\bar{\phi}^{2}\,\textrm{vol}_{\partial M}\ }\,,$
(6.61)

where $V:\mathbb{R}\to\mathbb{R}$ represents a mass/self-interaction function
and $f:\partial M\to\mathbb{R}$ a smooth function which allows us to encode
different boundary conditions for the field $\phi$. The _total action
functional_ (for a real scalar field $\phi$ coupled to the gravitational field
described by the fixed metric $(g_{\mu\nu})$) thus reads

$\displaystyle\boxed{\ \mathbb{S}[\phi]\equiv\int_{(M,\partial
M)}(L,\bar{\ell})=\int_{M}L-\int_{\partial M}\bar{\ell}=S_{M}-S_{\partial M}\
}\,.$ (6.62)

Let us recall the first variational formula on $M$, i.e. Eqn. (6.26): with the
notation $\delta\equiv d_{\tt v}$ and $d\,j_{M}\equiv d_{\tt h}j$, this
relation writes as follows for the bulk:

$\displaystyle\delta L={\cal E}(L)+d\,j_{M}\,.$ (6.63)

By imposing the condition that $\imath^{*}j_{M}$ is decomposable over the
lateral boundary of $M$, we have another relation for the boundary (recall
Eqn. (6.57) for the definition of the differential on $\partial M$)

$\displaystyle\delta\bar{\ell}-\imath^{*}j_{M}=\bar{b}\wedge\delta\phi-d\,j_{\partial
M}\,.$ (6.64)

For our example (6.61), the variational formula (6.63) takes the following
expression (involving the vector field $V\equiv V^{\mu}\partial_{\mu}$ with
$V^{\mu}\equiv\delta\phi\,\nabla^{\mu}\phi$ as well as the operator
$\Box\equiv g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$),

$\displaystyle\delta
L=-[\,\Box\phi+V^{\prime}(\phi)]\,\delta\phi\,\textrm{vol}_{M}+d\,j_{M}\,,$
(6.65)

with

$j_{M}\equiv
i_{V}\textrm{vol}_{M}=\sqrt{|g|}\;V^{\mu}\,d^{n-1}x_{\mu}=\sqrt{|g|}\;(\nabla^{\mu}\phi)\,\delta\phi\,d^{n-1}x_{\mu}\,.$

The variational relation (6.64) then yields

$\displaystyle\bar{b}=-\big{[}\,\imath^{*}(\nabla_{n}\phi)-f\bar{\phi}\,\big{]}\,\textrm{vol}_{\partial
M}\,,\qquad j_{\partial M}=0\,,$ (6.66)

where $n\equiv(n^{\mu})$ denotes an outward pointing _unit normal vector
field_ on the lateral boundary $\partial M$.

The _covariant phase space on $(M,\partial M)$,_ i.e. the space of solutions
of the field equations resulting from the total action functional (6.62) is
presently determined by the conditions ${\cal E}(L)=0=\bar{b}(\phi)$, i.e. by
virtue of equations (6.65)-(6.66),

$\Box\phi=-V^{\prime}(\phi)\,,\qquad\imath^{*}(\nabla_{n}\phi)=f\bar{\phi}\,.$

For $f\neq 0$, the latter relation represents _Robin (i.e. mixed) boundary
conditions_ for $\phi$ on $\partial M$. For the choice $f=0$, this relation
defines _Neumann boundary conditions._

The _Dirichlet boundary condition_ $\bar{\phi}=0$ can be implemented by
starting from $\bar{\ell}=0$ and $\bar{\phi}=0$, i.e. fields $\phi$ on $M$
which vanish on $\partial M$. The equations of motion of $\phi$ and the
symplectic potential $j_{M}$ on the bulk are then the same as above and there
are no boundary terms (i.e. the quantities $\bar{\ell},j_{\partial M}$ and
$\bar{b}$ vanish identically).

For a discussion of the symplectic form and of symmetries as well as the
treatment of other field theoretic models (Yang-Mills, Chern-Simons,
gravity,…) within the presented framework we refer to the work [66, 238, 195]
(see also [19] and references therein). Yet, we note that this approach and
the involved interpretation of boundary variations (as equations of motion)
differ from the view-point adopted in various other works, see footnote 6 of
reference [174] for the case of general relativity. Another approach to
boundaries along with the investigation of _corners_ and related corner terms
will be addressed in Subsection 7.8 below.

##  7 Noether’s theorems, conserved currents, forms and charges in gauge field
theories

Gauge field (-type) theories like Maxwell’s theory of electrodynamics, non-
Abelian Yang-Mills theories or Einstein’s general relativity are characterized
by local symmetries (invariance of the action under gauge transformations or
diffeomorphisms). Natural questions which arise in this context are the ones
whether and how these local symmetries may acquire an observable status, i.e.
the issue of the definition and derivation of conserved local and integral
quantities associated to the local invariances. This subject is obviously
related to Noether’s theorems and over the last decades it has been an impetus
for formulating these theorems in a more general manner and to look for
several generalizations thereof.

The following presentation relies on the work of the Brussels’ school (M.
Henneaux, G. Barnich and their collaborators, see references [23, 229, 240,
230, 154, 74] as well as references therein for earlier contributions,
especially [31]). In Subsection 7.1.1, we consider the basic example of free
Maxwell theory to conclude that _the Noether current associated to gauge
invariance vanishes on-shell up to a superpotential term._ In Subsection
7.1.2, we point out that _the associated charge represents a surface (flux)
integral over the superpotential,_ the existence and properties of this
integral depending on the properties of fields and symmetry parameters of the
underlying theory. The general definition and construction of these so-called
lower degree conservation laws in gauge field theories are then outlined in
the subsequent sections where we again make contact with the (pre-)symplectic
current density encountered in the approaches of covariant phase space and of
the variational bicomplex. More precisely, for non-linear gauge field type
theories like YM-theories or general relativity, generic field configurations
do not have any exact symmetries in general: in this case, the decomposition
of the gauge field $(A_{\mu})$ or metric field $(g_{\mu\nu})$ with respect to
a given background $(\bar{A}_{\mu})$ or $(\bar{g}_{\mu\nu})$ leads to the
study of the _linearized theory (around these background fields)_ or to the
exploration of asymptotic fields and symmetries. In all of these cases, the
definition of charges in terms of surface integrals as well as their
properties are related to the symmetries of the linearized theory [154, 157,
241]. Different approaches to these topics and the relationships between them
are briefly surveyed in Subsection 7.6.

In this context, it is worthwhile recalling that the notion of _asymptotic
fields_ is an important one in classical and quantum field theory, e.g. for
the description of scattering phenomena and their interpretation in terms of
elementary particles. The underlying idea is that the non-linear interaction
of fields in Minkowski space-time $\mathbb{R}^{n}$ is localized in a bounded
region $V\subset\mathbb{R}^{n-1}$ of space and that the fields outside the
region $V$ can be approximated arbitrarily well (though in a prescribed
manner) by almost free fields (background field theory), e.g. see [242] and
references therein. For this instance where the interacting field theory
admits a reasonable linearization (with respect to background fields), one can
associate well-defined conserved charges to the solutions of the interacting
theory, these solutions differing arbitrarily little from the solutions of the
background theory outside the interaction region: each of these so-called
_asymptotic charges_ is given by an integral over a surface that lies outside
the interaction region $V$ and their presence thereby depends on the existence
of a well-defined linearization (admitting symmetries) of the full theory. The
first asymptotic conservation laws which have been constructed in this spirit
are the ADM charges in general relativity in asymptotically flat space-time
[175], the analogous Abbott-Deser charges in asymptotically anti-de Sitter
space-times [243] as well as the color charges in YM-theory [244].

Concerning the _symmetries,_ we note that the local symmetries (local gauge
symmetry or diffeomorphism invariance in gravity) concern the so-called bulk
of space-time whereas the associated charges have to do with the boundary: in
gravity, the latter charges become the conserved charges associated to
Poincaré invariance (energy-momentum and angular momentum) or to the (larger)
BMS symmetry [245]. We note that a similar distinction can be made [197] for a
particle for which the world line is reparametrization invariant (gauge
symmetry) and the ends (i.e. the boundary) correspond to vertices in the
interacting theory: at the latter, the energy-momentum conservation law
(related to Poincaré symmetry) and the associated physical conservation laws
have to be taken into account. The ongoing interests in field theory include
the physical significance of symmetries and conservation laws as well as the
relationship with infrared problems in quantum field theory and with memory
effects, e.g. see the monograph [225] as well as the recent works [246, 247,
248] for detailed discussions of symmetries.

In this section, we will use Dirac’s notation $F\approx 0$ (_“ $F$ vanishes
weakly”_ or _“ $F$ vanishes on-shell”_) for local functions $F$ which vanish
on the so-called _stationary surface_ ${\cal R}^{\infty}$ (where
$0=\frac{\partial S}{\partial q^{a}}=\partial_{\mu}\frac{\partial S}{\partial
q^{a}}=\cdots$). Moreover, we will mostly use the traditional notation of
field theory rather than the corresponding notation introduced for the
variational bicomplex in Subsection 6.1: the dictionary between both notations
has been outlined in Subsection 6.1 and Subsection 6.6.

###  7.1 Motivation (Pedestrian approach)

####  7.1.1 Gauge symmetry and associated Noether current

Consider a pure gauge theory like free Maxwell theory in $n$-dimensional
Minkowski space-time $M=\mathbb{R}^{n}$. The corresponding action functional
$S[A]=-\frac{1}{4}\int_{M}d^{n}x\,F^{\mu\nu}F_{\mu\nu}$ yields the Euler-
Lagrange derivative $\frac{\delta S}{\delta
A_{\mu}}=\partial_{\nu}F^{\nu\mu}$. The Lagrangian and thereby the action are
invariant under the local gauge transformation
$\delta_{f}A_{\mu}=\partial_{\mu}f$ where $x\mapsto f(x)$ is a smooth real-
valued function.

Before determining the conserved current associated to the local gauge
invariance of the action, we make the following observations (based on
reference [154]). The product

$\displaystyle\frac{\delta S}{\delta
A_{\mu}}\,\delta_{f}A_{\mu}=(\partial_{\nu}F^{\nu\mu})\,\partial_{\mu}f\,,$
(7.1)

which occurs in Noether’s first theorem can be rewritten by applying the
Leibniz rule to the derivative $\partial_{\mu}$:

$\displaystyle\frac{\delta S}{\delta A_{\mu}}\,\delta_{f}A_{\mu}$
$\displaystyle\,=(\partial_{\nu}F^{\nu\mu})\,\partial_{\mu}f=\partial_{\mu}\Big{[}(\underbrace{\partial_{\nu}F^{\nu\mu}}_{=\,\frac{\delta
S}{\delta
A_{\mu}}})\,f\Big{]}-(\underbrace{\partial_{\mu}\partial_{\nu}F^{\nu\mu}}_{=\,0})\,f$
$\displaystyle\mbox{i.e.}\qquad\frac{\delta S}{\delta
A_{\mu}}\,\delta_{f}A_{\mu}$
$\displaystyle\,=\partial_{\mu}S_{f}^{\mu}\qquad\mbox{with}\
\;S_{f}^{\mu}\equiv S_{f}^{\mu\nu}\\!\big{(}\frac{\delta S}{\delta
A_{\nu}}\big{)}\equiv\frac{\delta S}{\delta
A_{\mu}}\,f=(\partial_{\nu}F^{\nu\mu})\,f\,.$ (7.2)

Here, the identity $\partial_{\mu}\partial_{\nu}F^{\nu\mu}=0$ follows from the
antisymmetry of the Faraday tensor and actually represents _Noether’s
identity_ $\partial_{\mu}\big{(}\frac{\delta S}{\delta A_{\mu}}\big{)}=0$ (see
Eqn. (4.23)) following from the local gauge invariance of the action
functional by virtue of Noether’s second theorem. The current $S_{f}^{\mu}$
represents a linear combination of Euler-Lagrange derivatives and thus
vanishes on-shell.

Alternatively, we can rewrite the product (7.1) by applying the Leibniz rule
to the derivative $\partial_{\nu}$:

$\displaystyle\frac{\delta S}{\delta A_{\mu}}\,\delta_{f}A_{\mu}$
$\displaystyle\,=(\partial_{\nu}F^{\nu\mu})\,\partial_{\mu}f=\partial_{\nu}\Big{[}F^{\nu\mu}\partial_{\mu}f\Big{]}-(\underbrace{F^{\nu\mu}\partial_{\nu}\partial_{\mu}f}_{=\,0})$
$\displaystyle\mbox{i.e.}\qquad\frac{\delta S}{\delta
A_{\mu}}\,\delta_{f}A_{\mu}$
$\displaystyle\,=\partial_{\mu}\;j_{f}^{\mu}\qquad\mbox{with}\
\;j_{f}^{\mu}\equiv F^{\mu\nu}\partial_{\nu}f\,.$ (7.3)

By subtracting the current densities appearing in equations (7.2),(7.3), we
obtain

$j_{f}^{\mu}-S_{f}^{\mu}=F^{\mu\nu}\partial_{\nu}f-\underbrace{\partial_{\nu}F^{\nu\mu}}_{=\,-\partial_{\nu}F^{\mu\nu}}f\,=\,\partial_{\nu}k_{f}^{\mu\nu}\qquad\mbox{with}\
\ k_{f}^{\mu\nu}\equiv fF^{\mu\nu}\,.$

Here, the current density $\partial_{\nu}k_{f}^{\mu\nu}$ represents a so-
called _superpotential term,_ the antisymmetric tensor $(k_{f}^{\mu\nu})$
being referred to as a _superpotential:_ due to the antisymmetry of
$k_{f}^{\mu\nu}$, the divergence
$\partial_{\mu}(\partial_{\nu}k_{f}^{\mu\nu})$ vanishes identically, i.e. such
a current is identically (off-shell) conserved.

In summary, we have the relation [154]

$\displaystyle\boxed{\
j_{f}^{\mu}=\underbrace{S_{f}^{\mu}}_{\approx\,0}\;-\underbrace{\partial_{\nu}k_{f}^{\nu\mu}}_{\textrm{superpot.
term}}\ }\,.$ (7.4)

Finally, we apply Noether’s first theorem (see equations (6.48)-(6.49)) to the
gauge invariance of Maxwell’s Lagrangian ${\cal L}$ “as if we were dealing
with a global symmetry transformation”:

$\displaystyle 0=\delta_{f}{\cal L}=\frac{\delta S}{\delta
A_{\mu}}\,\delta_{f}A_{\mu}+\partial_{\mu}{\cal J}^{\mu}_{f}\qquad\mbox{with}\
\;{\cal J}^{\mu}_{f}\equiv\frac{\partial{\cal
L}}{\partial(\partial_{\mu}A_{\nu})}\,\delta_{f}A_{\nu}=-F^{\mu\nu}\partial_{\nu}f=-j_{f}^{\mu}\,.$
(7.5)

By virtue of (7.4), we conclude that _the Noether current $({\cal
J}^{\mu}_{f})$ associated to a local gauge symmetry vanishes on-shell up to a
superpotential term_ (which is conserved identically). From the point of view
of Noether’s first theorem, a current (associated to a variational symmetry)
which is on-shell a superpotential term is considered to be a _trivial
current._ Accordingly, a non-trivial _gauge symmetry_ of an action (as well as
a trivial one, i.e. a linear combination of equations of motion and/or of
their derivatives) is to be viewed as a _trivial (global) symmetry_ from the
point of view of Noether’s first theorem. These notions will be formalized
mathematically in Subsection 7.2 by defining equivalence relations for both
symmetries and currents and by formulating Noether’s first theorem in a
general form.

We note that the line of arguments presented above for Maxwell’s theory
generalizes to a generic gauge theory in a space-time of dimension $n\geq 2$
and to possibly field-dependent gauge parameters $x\mapsto f^{r}(x)$ [154,
249].

####  7.1.2 Definition of the conserved charge associated to gauge symmetry

##### Maxwell theory:

As we just noted, the local gauge invariance of the Lagrangian gives rise to a
current $({\cal J}^{\mu}_{f})$ which represents on-shell a superpotential
term, i.e. on-shell we have ${\cal J}^{\mu}_{f}=\partial_{\nu}k_{f}^{\nu\mu}$
with $k_{f}^{\mu\nu}=-k_{f}^{\nu\mu}$. (For the case of Maxwell’s theory, we
found the explicit expression $k_{f}^{\mu\nu}=fF^{\mu\nu}$.) The
superpotential $k_{f}^{\mu\nu}$ determining the current ${\cal J}^{\mu}_{f}$
depends on the arbitrary gauge parameter $f$ and is thus arbitrary itself.
Yet, the charge contained in $V\subset\mathbb{R}^{3}$ (for a four dimensional
space-time) can be expressed as a surface integral over the superpotential
$(k_{f}^{i0})$,

$\displaystyle\boxed{\ Q\equiv\int_{V}d^{3}x\,{\cal
J}^{0}_{f}=\int_{V}d^{3}x\,\partial_{i}k_{f}^{i0}=\oint_{S\equiv\partial
V}k_{f}^{i0}\,dS^{i}\ }\,,$ (7.6)

and one can attribute a well-defined mathematical and physical meaning to this
integral for certain field theories (isolation of specific superpotentials):
_the existence and properties of the flux integral ( 7.6) depend on the
properties of the integrand (i.e. the assumptions made for fields and symmetry
parameters) in the vicinity of the surface $S=\partial V$._ E.g. for Maxwell’s
theory in $\mathbb{R}^{4}$, we have

$\displaystyle Q\equiv\int_{V}d^{3}x\,{\cal
J}^{0}_{f}=\int_{V}d^{3}x\,\partial_{i}(fF^{i0})=\int_{V}d^{3}x\,\partial_{i}(fF_{0i})=\oint_{S\equiv\partial
V}fF_{0i}\,dS^{i}=\oint_{S\equiv\partial
V}f\vec{E}\cdot\overrightarrow{dS}\,,$ (7.7)

where $\vec{E}$ denotes the electric field strength. In this example, trivial
gauge transformations, i.e. $0\approx\delta_{f}A_{\mu}=\partial_{\mu}f$
correspond to a _constant parameter_ $f\equiv\varepsilon$ and the integral
(7.7) then yields a charge
$q\equiv\frac{1}{\varepsilon}\,Q=\oint_{S\equiv\partial
V}\vec{E}\cdot\overrightarrow{dS}$: this flux integral coincides with the
usual expression for the electric charge in electrodynamics.

##### Non-Abelian gauge field theories (and general relativity):

In non-Abelian gauge theories like YM (Yang-Mills) theory, the infinitesimal
gauge transformation $\delta_{f}A_{\mu}=D_{\mu}f$ is field dependent (for non-
trivial gauge parameters, i.e. for $f\neq 0$). Similarly, in general
relativity, the transformation law of the metric tensor field $(g_{\mu\nu})$
under diffeomorphisms generated by a vector field
$\xi\equiv\xi^{\mu}\partial_{\mu}$, i.e.
$\delta_{\xi}g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$ depends
on the metric field. Generic gauge field configurations $(A_{\mu})$ or metrics
$(g_{\mu\nu})$ do not admit any symmetries and thereby the relations
$\delta_{f}A_{\mu}=0$ and $\delta_{\xi}g_{\mu\nu}=0$ do not admit non-trivial
solutions. The situation is different if a background field ($\bar{A}_{\mu}$
or $\bar{g}_{\mu\nu}$) is given which admits symmetries. In this case, the
theory may be _linearized_ around the background field configuration or
_asymptotic symmetries_ may be explored if the fields $A_{\mu}$ (or
$g_{\mu\nu}$) tend to $\bar{A}_{\mu}$ (or $\bar{g}_{\mu\nu}$) in an asymptotic
region, e.g. for $|\vec{x}\,|\to\infty$. In these cases, conserved flux
integrals that are analogous to the ones of Maxwell’s theory may be introduced
as has been pointed out by L. F. Abbott and S. Deser [243, 244] (see also
[154, 157] for the underlying general construction and for references to the
earlier literature which includes in particular the work of ADM [175] on
general relativity): we will address these issues at the end of Subsection 7.4
and in Subsection 7.5.

The gauge parameters $f$ such that $\delta_{f}A_{\mu}$ vanishes on-shell are
referred to as _global reducibility parameters_ or as _(gauge) Killing
vectors_ in reference to general relativity where the corresponding relation
$0\approx\delta_{\xi}g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$
for the metric field $(g_{\mu\nu})$ describes Killing vector fields
$\xi\equiv\xi^{\mu}\partial_{\mu}$. Likewise the gauge parameters $f$ such
that $\delta_{f}A_{\mu}$ vanishes (on-shell) asymptotically are referred to as
_asymptotic reducibility parameters_ [154]. As a matter of fact, by using
general cohomological methods, the authors of references [229, 154] have
established a _generalized version of Noether’s first theorem_ (which we will
spell out in more detail in Subsection 7.4): this result states that in $n\geq
2$ space-time dimensions, there is a 1-1-correspondence between (global)
reducibility parameters and $(n-2)$-forms202020Here and in the following, we
use the standard notation
$k^{[\mu\nu]}\equiv\frac{1}{2}(k^{\mu\nu}-k^{\nu\mu})$ for the
antisymmetrization. $k\equiv k^{[\mu\nu]}(d^{n-2}x)_{\mu\nu}$ which are closed
on-shell (e.g. for $n=4$ we have
$\varepsilon^{\rho\sigma\mu\nu}\partial_{\rho}k_{\mu\nu}\approx 0$), but not
exact (i.e. there is no $(n-3)$-form $l$ such that $k\approx dl$). This result
admits an analog for the case of asymptotic reducibility parameters and
asymptotically conserved $(n-2)$-forms. The surface charges are then given in
terms of the superpotential $k^{[\mu\nu]}$ (cohomological approach of Barnich
and Brandt [154]). For a given gauge field-type theory like YM-theory or
general relativity, the superpotential can be (and has also been) constructed
by other methods, in particular from the (pre-)symplectic potential that we
discussed in the approaches of covariant phase space and of the variational
bicomplex, see Subsection 7.6 below.

###  7.2 Noether’s first theorem

The two fundamental theorems that E. Noether established in her celebrated
article of 1918 [250] have been generalized during the last decades, e.g. see
the monograph [251] for an historical account of these accomplishments. In the
following, we will outline the generalizations of Noether’s first theorem
[252, 154, 74].

Consider a continuous infinitesimal symmetry transformation of the field, i.e.
(cf. (6.55))

$\delta_{f}q^{a}\equiv Q^{a}_{r}(f^{r})\equiv
Q^{a}_{r}f^{r}+Q^{a\mu}_{r}\partial_{\mu}f^{r}+\cdots\,,$

where the functions $x\mapsto f^{r}(x)$ parametrize a local gauge symmetry.
For global symmetry transformations (see equations (6.44),(6.46)), one
introduces the following _equivalence relation_

$\displaystyle X^{a}\sim X^{a}\;+\underbrace{Q^{a}_{r}(f^{r})}_{\textrm{gauge
transf.}}+\;\underbrace{M^{[ba]}\,\frac{\delta S}{\delta
q^{b}}-\partial_{\mu}\Big{(}M^{[b(\nu)a(\mu)]}\partial_{\nu}\,\frac{\delta
S}{\delta q^{b}}\Big{)}\pm\cdots}_{\textrm{e.o.m. symmetry transf.}}\,.$ (7.8)

Here, the coefficients $M^{[ba]}$ are local functions of the fields: the
corresponding terms in the previous relation vanish on-shell. The equivalence
class of $X^{a}$ is denoted by $[X^{a}]$.

For the (on-shell) conserved current densities, one introduces the following
_equivalence relation_ (which amounts to identify conserved currents differing
by trivial currents212121For $n=1$, i.e. for classical mechanics, the current
$(j^{\mu})$ has a single component $j^{0}$ to be interpreted as a charge: in
this case, the equivalence relation also involves an additional real constant
reflecting the fact that charges differing by such a constant are to be
identified in mechanics.)

$\displaystyle j^{\mu}\sim
j^{\mu}\;+\underbrace{\partial_{\nu}k^{[\nu\mu]}}_{\textrm{superpot.
term}}+\;\underbrace{t^{\mu}}_{\approx\,0}\,.$ (7.9)

Here, the antisymmetric tensor field $k^{[\nu\mu]}$ is a local function of the
fields that is twice differentiable whence the identity
$\partial_{\mu}(\partial_{\nu}k^{[\nu\mu]})=0$. For two equivalent currents
$(j_{1}^{\mu})$ and $(j_{2}^{\mu})$, we thus have
$\partial_{\mu}j_{1}^{\mu}\approx\partial_{\mu}j_{2}^{\mu}$.

In this respect we recall that, in the language of differential forms (see
Appendix A), a current density $(j^{\mu})$ corresponds to an $(n-1)$-form
$j\equiv j^{\mu}\,d^{n-1}x_{\mu}$ and an antisymmetric tensor field
$(k^{[\mu\nu]})$ corresponds to an $(n-2)$-form $k\equiv
k^{[\mu\nu]}\,(d^{n-2}x)_{\mu\nu}$. For a $d_{\tt h}$-closed $(n-1)$-form $j$
one has the equivalence

$d_{\tt h}j=0\ \ \Longleftrightarrow\ \ \partial_{\mu}j^{\mu}=0\,,$

and for an $d_{\tt h}$-exact $(n-1)$-form $j$ one has

$j=d_{\tt h}k\ \ \Longleftrightarrow\ \
j^{\mu}=-\partial_{\nu}k^{[\nu\mu]}\,.$

With $t\equiv t^{\mu}\,d^{n-1}x_{\mu}$, the equivalence relation (7.9) then
reads

$\displaystyle j\sim j\;+\underbrace{d_{\tt h}k}_{\textrm{superpot.
term}}+\;\underbrace{t}_{\approx\,0}\,.$ (7.10)

The equivalence class of $j$ will be denoted by $[j]$. We will now use these
notions to spell out generalized formulations of Noether’s first theorem.

####  7.2.1 Complete form of Noether’s first theorem

##### Complete form of Noether’s first theorem:

Consider a local field theory which is defined on a space-time manifold of
dimension $n\geq 2$ and whose dynamics is described by an action functional
$S[\varphi]$. Suppose this action admits continuous global symmetries and
possibly, in addition, local symmetries. Then there is a _one-to-one
correspondence between non-trivial Noether (global variational) symmetries and
non-trivial Noether (on-shell conserved) currents:_

$\displaystyle\boxed{\ [X^{a}]\ \longleftrightarrow\ [j]\ }\,.$ (7.11)

The correspondence $[X^{a}]\mapsto[j]$ is realized by (6.49), i.e.

$\displaystyle\boxed{\ X^{a}\ \longmapsto\ j^{\mu}\equiv\frac{\partial{\cal
L}}{\partial q^{a}_{\mu}}\,X^{a}-k_{X}^{\mu}\
}\qquad\mbox{for}\quad\delta_{X}{\cal L}=\partial_{\mu}k^{\mu}_{X}\,.$ (7.12)

Here, the first term of $j^{\mu}$, i.e.

$\displaystyle V^{\mu}_{a}(X^{a},{\cal L})\equiv\frac{\partial{\cal
L}}{\partial q^{a}_{\mu}}\,X^{a}=\frac{\partial{\cal L}}{\partial
q^{a}_{\mu}}\,\delta_{X}q^{a}\,,$ (7.13)

coincides with the symplectic potential (5.2), viz. the components of the
variational $1$-form (6.28).

We note that _a local gauge symmetry represents the trivial class
$[X^{a}]\equiv[0]$ of global symmetries and corresponds to the trivial class
$[j]=[0]$ of Noether currents,_ see equations (7.4)-(7.5).

##### Outline of proof:

The proof [31, 229] of the given version of Noether’s first theorem consists
of determining the so-called _characteristic cohomology group (associated with
the stationary surface) in form degree_ $n-1$, i.e. the $(n-1)$-th cohomology
group of the exterior space-time differential $d_{\tt h}$ pulled back to the
space of solutions of the field equations: this group is denoted by

$\displaystyle H^{n-1}\Big{(}d_{\tt h},\Omega^{\bullet}(S)\Big{)}\,$ (7.14)

where $\Omega^{\bullet}(S)=\Omega^{\bullet}({\cal R}_{\infty})$ represents the
space of differential forms pull-backed to the stationary surface $S$, (i.e.
solutions of the equations of motion). This means that, for the $(n-1)$-forms
$j\in[j]$ with $[j]\in H^{n-1}\Big{(}d_{\tt h},\Omega^{\bullet}(S)\Big{)}$,
one does not simply consider $d_{\tt h}j=0$ and $j\sim j+d_{\tt h}k$ (as for
the standard cohomology groups), but one rather considers $d_{\tt h}j\approx
0$ and identifies the $(n-1)$-forms $j$ according to relation (7.10). For the
characteristic cohomology group (7.14), one has the general result

$\displaystyle H^{n-1}\Big{(}d_{\tt h},\Omega^{\bullet}(S)\Big{)}\cong
H^{n-1}(d_{\tt h}|\delta)\,,$ (7.15)

i.e. it is equivalent to the $(n-1)$-th cohomology group of $d_{\tt h}$ modulo
the _Koszul-Tate differential_ $\delta$, this differential (discussed in
Appendix B.3) depending on the action and on its symmetries, see references
[23, 229, 240, 74] for the details.

####  7.2.2 Full form of Noether’s first theorem

The vector space of equivalence classes $[X^{a}]$ of non-trivial global
symmetries as well as the vector space of equivalence classes $[j]$ of non-
trivial conserved currents can be endowed with Lie brackets which are
preserved by the correspondence (7.11). Thus, there is a Lie algebra
homomorphism between inequivalent variational symmetries and inequivalent
conserved currents. To present this so-called _full form of Noether’s first
theorem_ [240] (also referred to as _Noether representation theorem_ [249]),
we first introduce the different brackets.

Consider the commutator of two evolutionary vector fields $X_{1},X_{2}$ as
defined by (6.44),(6.46): from the fundamental property
$[\partial_{\mu},\delta_{X}]=0$, it follows that

$\displaystyle\big{[}\delta_{X_{1}},\delta_{X_{2}}\big{]}=\delta_{[X_{1},X_{2}]_{\stackrel{{\scriptstyle\
}}{{\textrm{L}}}}}\,,\qquad\mbox{with}\quad\boxed{\
[X_{1},X_{2}]^{a}_{\stackrel{{\scriptstyle\
}}{{\textrm{L}}}}\equiv\delta_{X_{1}}X_{2}^{a}-\delta_{X_{2}}X_{1}^{a}\ }\,.$
(7.16)

Thus, the vector space of evolutionary vector fields endowed with this bracket
represents an infinite dimensional Lie algebra. The commutator (7.16) is also
qualified as the _Lie bracket of characteristics_ [157].

Now suppose the evolutionary vector fields $X_{1},X_{2}$ represent global
symmetries for a field theory described by a Lagrangian ${\cal L}$, i.e.
$\delta_{X_{1}}{\cal L}=\partial_{\mu}k^{\mu}_{X_{1}}$ and
$\delta_{X_{2}}{\cal L}=\partial_{\mu}k^{\mu}_{X_{2}}$. Then, we have

$\delta_{[X_{1},X_{2}]_{\stackrel{{\scriptstyle\ }}{{\textrm{L}}}}}{\cal
L}=\big{[}\delta_{X_{1}},\delta_{X_{2}}\big{]}{\cal
L}=\delta_{X_{1}}(\partial_{\mu}k^{\mu}_{X_{2}})-(1\leftrightarrow
2)=\partial_{\mu}k^{\mu}_{12}\qquad\mbox{where}\
\;k^{\mu}_{12}\equiv\delta_{X_{1}}k^{\mu}_{X_{2}}-\delta_{X_{2}}k^{\mu}_{X_{1}}\,,$

and where we again used the relation $[\partial_{\mu},\delta_{X}]=0$.
Accordingly, the vector space of global symmetries of a given field theory
represents an (infinite dimensional) Lie subalgebra of the Lie algebra of
evolutionary vector fields. It can be shown [154] that the bracket of global
symmetries induces a well-defined bracket on the vector space of equivalence
classes of these symmetries:

$\displaystyle\big{[}\,[X_{1}],[X_{2}]\,\big{]}^{a}_{\stackrel{{\scriptstyle\
}}{{\textrm{L}}}}\equiv\big{[}\,[X_{1},X_{2}]^{a}_{\stackrel{{\scriptstyle\
}}{{\textrm{L}}}}\,\big{]}\,.$ (7.17)

For the Noether currents $j_{X_{1}},j_{X_{2}}$ associated to global symmetries
$X_{1},X_{2}$ by virtue of (7.12), one can introduce the so-called Dickey
bracket [253] defined by

$\displaystyle\boxed{\ \\{j_{X_{1}},j_{X_{2}}\\}_{\stackrel{{\scriptstyle\
}}{{\textrm{D}}}}\equiv\delta_{X_{1}}j_{X_{2}}=-\delta_{X_{2}}j_{X_{1}}=\frac{1}{2}\big{(}\delta_{X_{1}}j_{X_{2}}-\delta_{X_{2}}j_{X_{1}}\big{)}\
}\,.$ (7.18)

By applying the variation $\delta_{X_{1}}$ to the local conservation law
(6.49) written in terms of differential forms, i.e. $d_{\tt
h}j_{X_{2}}=-E_{a}({\cal L})\,X^{a}_{2}\,d^{n}x$, and by taking into account
the relation $[\partial_{\mu},\delta_{X}]=0$ as well as
$\delta_{X_{1}}\frac{\delta S}{\delta\phi^{i}}\approx 0$, one finds that [74,
249]

$\displaystyle\\{j_{X_{1}},j_{X_{2}}\\}_{\stackrel{{\scriptstyle\
}}{{\textrm{D}}}}=j_{[X_{1},X_{2}]_{\stackrel{{\scriptstyle\
}}{{\textrm{L}}}}}+\mbox{trivial current}\,.$ (7.19)

The trivial current can be eliminated by going over to equivalence classes of
currents: thereby we obtain a Lie bracket on the cohomology groups
$H^{n-1}(d_{\tt h},\Omega^{\bullet}(S))$, i.e. on the vector space of non-
trivial conserved currents:

$\displaystyle\\{[j_{X_{1}}],[j_{X_{2}}]\\}^{\mu}_{\stackrel{{\scriptstyle\
}}{{\textrm{D}}}}=\Big{[}\frac{1}{2}\big{(}\delta_{X_{1}}j_{X_{2}}-\delta_{X_{2}}j_{X_{1}}\big{)}\Big{]}=[\,\omega^{\mu}(X_{1},X_{2})\,]\,.$
(7.20)

Here, $\omega^{\mu}(X_{1},X_{2})$ denotes the contraction of the local
symplectic form (see equations (5.3) and (7.13)), i.e. of $\omega^{\mu}=d_{\tt
v}\big{(}V^{\mu}_{a}(d_{\tt v}q^{a},{\cal L})\big{)}$, with the evolutionary
vector fields $X_{1},X_{2}$ (see references [154, 74]).

From (7.18)-(7.20) we infer that

$\displaystyle\boxed{\
\\{[j_{X_{1}}],[j_{X_{2}}]\\}^{\mu}_{\stackrel{{\scriptstyle\
}}{{\textrm{D}}}}=[\,j^{\mu}_{[X_{1},X_{2}]_{\stackrel{{\scriptstyle\
}}{{\textrm{L}}}}}\,]\ }\,.$ (7.21)

Together with the correspondence (7.11), this result represents [240] the

##### Full form of Noether’s first theorem:

We have a _Lie algebra homomorphism between the Lie algebra of non-trivial
global symmetries (endowed with the Lie bracket ( 7.17)) and the Lie algebra
of non-trivial Noether currents (endowed with the Dickey bracket (7.20))._

We refer to the work [240] for the correspondence (isomorphism in the case of
non-degenerate Lagrangian field theories) between the Dickey bracket of
conserved currents and the standard Hamiltonian Poisson bracket of conserved
charges as well as the Batalin-Vilkovisky anti-bracket for the local BRST
cohomology classes. The first of these correspondences is illustrated by the
following example.

##### Example:

Let us consider the free massless quark model in $\mathbb{R}^{4}$, see
equations (6.52)-(6.54). For the evolutionary vector fields $X_{r},X_{s}$
(with $r,s\in\\{1,\dots,8\\}$) determined by (6.53), it then follows from
(7.16) that

$\displaystyle[X_{r},X_{s}]_{\stackrel{{\scriptstyle\
}}{{\textrm{L}}}}=-f_{rst}X_{t}\,,$ (7.22)

where $f_{rst}$ are the structure constants of the Lie algebra $su(3)$. For
the corresponding conserved currents (6.54), the Dickey bracket (7.18) is
given by

$\displaystyle\\{j_{X_{r}},j_{X_{s}}\\}_{\stackrel{{\scriptstyle\
}}{{\textrm{D}}}}=-f_{rst}j_{X_{t}}\,.$ (7.23)

Obviously, relations (7.22),(7.23) reflect the Lie algebra homomorphism which
is the subject of the full form of Noether’s first theorem.

The Hamiltonian Poisson brackets of the classical quark fields
$\varphi^{a}\equiv(\varphi^{a}_{\alpha})_{\alpha=1,\dots,4}$ have the form

$\\{\varphi^{a}_{\alpha}(t,\vec{x}\,),\varphi_{\beta}^{\dagger
b}(t,\vec{y}\,)\\}=-\textrm{i}\,\delta_{\alpha\beta}\,\delta^{ab}\,\delta(\vec{x}-\vec{y}\,)\qquad\mbox{for}\
\;a,b\in\\{1,2,3\\}\,,\ \ \alpha,\beta\in\\{1,\dots,4\\}\,.$

This implies that the Noether charges associated to the conserved currents
(6.54), i.e. expressions

$Q_{r}\equiv\int_{\mathbb{R}^{3}}d^{3}x\,j^{0}_{r}=\int_{\mathbb{R}^{3}}d^{3}x\;\varphi^{\dagger}T_{r}\varphi\,,$

satisfy the relation

$\displaystyle\\{Q_{r},Q_{s}\\}=f_{rst}\,Q_{t}\,.$ (7.24)

The latter equation reflects the homomorphism between the algebra of
inequivalent conserved charges (endowed with the Poisson bracket) and the
algebra of inequivalent conserved currents (equipped with the Dickey bracket),
_two conserved charges being identified if they coincide on-shell._

###  7.3 Noether identities and Noether’s second theorem

The following considerations provide the general underpinning for the results
that we obtained in Subsection 7.1.1 for the particular case of gauge
invariance of Maxwell’s theory. Our presentation is based on references [74,
249].

####  7.3.1 Noether identities

Suppose we have a local symmetry of a Lagrangian ${\cal
L}(\varphi,\partial_{\mu}\varphi)$ as described by Eqn. (6.55), i.e. we have
$\delta_{f}{\cal L}=\partial_{\mu}k^{\mu}_{f}$ for the variation of ${\cal L}$
induced by the local symmetry transformation $\delta_{f}\varphi^{a}=Q^{a}(f)$.
By using $\delta_{f}(\partial_{\mu}\varphi)=\partial_{\mu}(\delta_{f}\varphi)$
and the Leibniz rule for $\partial_{\mu}$, we find that

$\displaystyle\delta_{f}{\cal L}=\frac{\partial{\cal
L}}{\partial\varphi}\,\delta_{f}\varphi+\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}\,\delta_{f}(\partial_{\mu}\varphi)=\Big{[}\underbrace{\frac{\partial{\cal
L}}{\partial\varphi}-\partial_{\mu}\big{(}\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}\big{)}}_{=\;\frac{\delta
S}{\delta\varphi}}\Big{]}\delta_{f}\varphi+\partial_{\mu}\Big{[}\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}\,\delta_{f}\varphi\Big{]}\,,$ (7.25)

i.e. the “first variational formula” (5.1):

$\displaystyle 0=\delta_{f}\varphi\;\frac{\delta
S}{\delta\varphi}+\partial_{\mu}{\cal J}_{f}^{\mu}\,,\qquad\mbox{with}\
\;{\cal J}_{f}^{\mu}=\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}\,\delta_{f}\varphi-k^{\mu}_{f}\,.$ (7.26)

(For Maxwell’s theory, this is our result (7.5) with $k^{\mu}_{f}\equiv 0$.)

If we consider the general form of $\delta_{f}\varphi$ as given by (6.55),
i.e. $\delta_{f}\varphi^{a}=Q^{a}(f)\equiv Q^{a}_{r}(f^{r})\equiv
Q^{a}_{r}f^{r}+Q^{a\mu}_{r}\partial_{\mu}f^{r}$ (assuming for simplicity that
there are no higher derivatives of $f^{r}$ in this transformation law), then
we obtain another expression for $\delta_{f}\varphi\,\frac{\delta
S}{\delta\varphi}$ (using again the Leibniz rule for $\partial_{\mu}$):

$\displaystyle\delta_{f}\varphi\,\frac{\delta
S}{\delta\varphi}=\Big{[}Q^{a}_{r}f^{r}+Q^{a\mu}_{r}\partial_{\mu}f^{r}\Big{]}\,\frac{\delta
S}{\delta\varphi^{a}}=f^{r}\Big{[}\underbrace{Q^{a}_{r}\frac{\delta
S}{\delta\varphi^{a}}-\partial_{\mu}\big{(}Q^{a\mu}_{r}\,\frac{\delta
S}{\delta\varphi^{a}}\big{)}}_{=\;(Q^{a}_{r})^{\dagger}\big{(}\frac{\delta
S}{\delta\varphi^{a}}\big{)}}\Big{]}+\partial_{\mu}\Big{[}\underbrace{\frac{\delta
S}{\delta\varphi^{a}}\,Q^{a\mu}_{r}f^{r}}_{\equiv\;S^{\mu}_{f}}\Big{]}\,.$
(7.27)

Here, $(Q^{a}_{r})^{\dagger}$ denotes the adjoint operator (with respect to
the $L^{2}$-inner product) of the differential operator $Q^{a}_{r}$ acting on
smooth functions222222By definition, we have
$\langle\psi^{a},Q^{a}_{r}(f^{r})\rangle=\int
d^{n}x\,\psi^{a}\,Q^{a}_{r}(f^{r})=\langle(Q^{a}_{r})^{\dagger}\psi^{a},f^{r}\rangle$..
(For Maxwell’s theory, equation (7.27) coincides with the first line of Eqn.
(7.2).)

Combination of equations (7.26) and (7.27) now leads to the equality

$\displaystyle f^{r}(Q^{a}_{r})^{\dagger}\Big{(}\frac{\delta
S}{\delta\varphi^{a}}\Big{)}=-\partial_{\mu}\big{(}{\cal
J}^{\mu}_{f}+S^{\mu}_{f}\big{)}\,.$ (7.28)

If we apply the Euler-Lagrange derivative with respect to the arbitrary
functions $f^{r}$ (i.e. the derivative
$\frac{\delta\,\boldsymbol{\cdot}}{\delta
f^{r}}\equiv\frac{\partial\,\boldsymbol{\cdot}}{\partial
f^{r}}-\partial_{\mu}\big{(}\frac{\partial\,\boldsymbol{\cdot}}{\partial(\partial_{\mu}f^{r})}\big{)}$)
to this relation, then the right-hand side yields a vanishing result since the
equations of motion are trivial for a total derivative.

Summary: For an action functional $S[\varphi]$ which is invariant under local
symmetry transformations $\delta_{f}\varphi^{a}=Q^{a}_{r}(f^{r})$, we have the
so-called

$\displaystyle\mbox{Noether identities :}\qquad\boxed{\
0=(Q_{r})^{\dagger}\Big{(}\frac{\delta
S}{\delta\varphi}\Big{)}\equiv(Q^{a}_{r})^{\dagger}\Big{(}\frac{\delta
S}{\delta\varphi^{a}}\Big{)}\ }\quad\mbox{for all $r$}\,.$ (7.29)

If $r$ takes $m$ values, then these relations represent $m$ identities
relating the functional derivatives $\delta S/\delta\varphi^{1},\dots,\delta
S/\delta\varphi^{N}$: they show that _the equations of motion are not all
independent in a theory admitting local symmetries._ _These identities hold
off-shell and are trivially satisfied for the solutions of the equations of
motion._ They reflect the fact that if $\varphi$ is a solution of the
equations of motion, then the symmetry transformed solution $\varphi^{\prime}$
is another solution involving $m$ arbitrary functions.

####  7.3.2 Noether’s second theorem

By substituting the Noether identities (7.29) into (7.27) we obtain the
following general result.

##### Noether’s second theorem:

For an action functional $S[\varphi]$ which is invariant under local symmetry
transformations $\delta_{f}\varphi^{a}=Q^{a}_{r}(f^{r})\equiv
Q^{a}_{r}f^{r}+Q^{a\mu}_{r}\partial_{\mu}f^{r}$, we have

$\displaystyle\boxed{\ \delta_{f}\varphi^{a}\,\frac{\delta
S}{\delta\varphi^{a}}=\partial_{\mu}S^{\mu}_{f}\ }\qquad\mbox{with}\
\;S^{\mu}_{f}\equiv S^{\mu a}_{r}\Big{(}\frac{\delta
S}{\delta\varphi^{a}},f^{r}\Big{)}\equiv\frac{\delta
S}{\delta\varphi^{a}}\,Q^{a\mu}_{r}f^{r}\approx 0\,.$ (7.30)

For the solutions of the equations of motion, we thus have a representative
$(S^{\mu}_{f})$ of the conserved Noether current which vanishes weakly. (For
Maxwell’s theory, equation (7.30) coincides with (7.2).)

It is instructive (and useful for the sequel) to spell out the explicit
expressions that one obtains for the basic examples. In each case the
descriptors $Q^{a}_{r}$ and $Q^{a\mu}_{r}$ of local symmetry transformations
can simply be read of from the transformation laws and the current
$(S^{\mu}_{f})$ is conveniently determined by evaluating
$\delta_{f}\varphi^{a}\,\frac{\delta S}{\delta\varphi^{a}}$.

##### Example of Maxwell theory:

As we already mentioned, this is the particular case discussed in Subsection
7.1.1, see equations (7.1)-(7.2):

$\boxed{\ \begin{array}[]{rcl}\mbox{Symmetry transformation of fields
$A_{\mu}$ :}&&\delta_{f}A_{\mu}=\partial_{\mu}f\\\ \mbox{Noether identities
:}&&\partial_{\mu}\partial_{\nu}F^{\mu\nu}=0\\\ \mbox{Current $(S^{\mu}_{f})$
:}&&S_{f}^{\mu}=(\partial_{\nu}F^{\nu\mu})f\,.\end{array}\ }$

##### Example of YM-theory:

With the notation $f\equiv f^{r}T_{r}$ for the Lie algebra valued symmetry
parameters, we have:

$\boxed{\ \begin{array}[]{rcl}\mbox{Symmetry transformation of fields
$A^{r}_{\mu}$
:}&&\delta_{f}A^{r}_{\mu}=(D_{\mu}f)^{r}\equiv\partial_{\mu}f^{r}+\textrm{i}g\,[A_{\mu},f]^{r}\\\
\mbox{Noether identities :}&&D_{\mu}D_{\nu}F^{\mu\nu}=0\\\ \mbox{Current
$(S^{\mu}_{f})$
:}&&S_{f}^{\mu}=\textrm{Tr}\,\big{[}(D_{\nu}F^{\nu\mu})f\big{]}\,.\end{array}\
}$

##### Example of general relativity with a cosmological constant:

In the presence of a cosmological constant $\Lambda$, the action (4.29) for
the metric field $\mathfrak{g}\equiv(g_{\mu\nu})$ becomes

$\displaystyle
S_{\textrm{grav}}[\mathfrak{g}]\equiv\frac{1}{2\kappa}\int_{M}d^{n}x\,\sqrt{|g|}\,(R-2\Lambda)\qquad\mbox{with}\
\;\kappa\equiv 8\pi G\,.$ (7.31)

Variation with respect to the metric field yields the cosmological Einstein
tensor $G_{c}^{\mu\nu}$:

$\displaystyle\frac{\delta S_{\textrm{grav}}}{\delta
g_{\mu\nu}}=-\frac{\sqrt{|g|}}{2\kappa}\,G_{c}^{\mu\nu}\,,\qquad\mbox{with}\
\;G_{c}^{\mu\nu}\equiv R^{\mu\nu}-\frac{1}{2}\,g^{\mu\nu}R+\Lambda
g^{\mu\nu}\,.$ (7.32)

The action $S_{\textrm{grav}}[\mathfrak{g}]$ is invariant under
diffeomorphisms of the space-time manifold $M$ which are given at the
infinitesimal level by
$\delta_{\xi}g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$ where
$\xi\equiv\xi^{\mu}\partial_{\mu}$ is the vector field generating the
diffeomorphisms. By using the symmetry of the tensor $(G_{c}^{\mu\nu})$, the
Leibniz rule and the metricity condition $\nabla_{\lambda}g_{\mu\nu}=0$, we
have

$\displaystyle\delta_{\xi}g_{\mu\nu}\,\frac{\delta S_{\textrm{grav}}}{\delta
g_{\mu\nu}}=-\frac{\sqrt{|g|}}{2\kappa}\,2\,(\nabla_{\mu}\xi_{\nu})\,G_{c}^{\mu\nu}=\partial_{\mu}\Big{(}\underbrace{-\frac{\sqrt{|g|}}{\kappa}\,\xi_{\nu}G_{c}^{\mu\nu}}_{\equiv\;S^{\mu}_{f}}\Big{)}+\frac{\sqrt{|g|}}{\kappa}\;\xi_{\nu}\underbrace{\nabla_{\mu}G_{c}^{\mu\nu}}_{=\;\nabla_{\mu}G^{\mu\nu}}\,.$
(7.33)

To summarize [157]:

$\boxed{\ \begin{array}[]{rcl}\mbox{Symmetry transformation of fields
$g_{\mu\nu}$
:}&&\delta_{\xi}g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}\\\
\mbox{Noether identities :}&&\nabla_{\mu}G^{\mu\nu}=0\\\ \mbox{Current
$(S^{\mu}_{\xi})$
:}&&S_{\xi}^{\mu}=-\frac{\sqrt{|g|}}{\kappa}\,\xi_{\nu}\,G_{c}^{\nu\mu}\,.\end{array}\
}$

The case of Einstein-Maxwell theory is discussed in reference [254]. In view
of the particular form of the Noether identities in the previous examples,
these relations are occasionally referred to as _generalized Bianchi
identities._

##### Relationship with Dirac’s approach to constrained dynamical systems :

For concreteness, we focus on free Maxwell theory in four space-time
dimensions. Integration of the conserved Noether current density
$S_{f}^{\mu}=(\partial_{\nu}F^{\nu\mu})f$ over a three-dimensional space-like
hypersurface $\Sigma$ to be chosen as the hypersurface $t=\textrm{constant}$
(see Eqn. (5.10)) yields the expression

$\displaystyle
G_{f}\equiv-\int_{\Sigma}d\Sigma_{\mu}\,S^{\mu}_{f}=-\int_{\mathbb{R}^{3}}d^{3}x\;S^{0}_{f}=-\int_{\mathbb{R}^{3}}d^{3}x\;(\partial_{\nu}F^{\nu
0})f\,.$ (7.34)

Partial integration and substitution of $F^{\nu 0}=\pi^{\nu}$ (canonical
momentum associated to the gauge field $A_{\nu}$) leads to the result

$\displaystyle\boxed{\
G_{f}\equiv-\int_{\Sigma}d\Sigma_{\mu}\,S^{\mu}_{f}=\int_{\mathbb{R}^{3}}d^{3}x\;\pi^{\nu}(\partial_{\nu}f)\
}\,.$ (7.35)

This quantity represents the _generator of Lagrangian gauge transformations in
the Hamiltonian formulation of Maxwell theory_ (within Dirac’s approach to
constrained Hamiltonian systems), e.g. see [43] and references therein:

$\delta_{f}A_{\mu}\equiv\\{A_{\mu},G_{f}\\}=\partial_{\mu}f\,,\qquad\delta_{f}\pi^{\mu}\equiv\\{\pi^{\mu},G_{f}\\}=0\,.$

##### Superpotentials associated to a local symmetry :

Substitution of the Noether identities (7.29) into relation (7.28) yields the
local conservation law $\partial_{\mu}\big{(}j^{\mu}_{f}-S^{\mu}_{f}\big{)}=0$
(with $j^{\mu}_{f}\equiv-{\cal J}^{\mu}_{f}$, cf. Eqn. (7.5)) or $d_{\tt
h}(j_{f}-S_{f})=0$ for the $(n-1)$-forms $j_{f}\equiv
j^{\mu}_{f}d^{n-1}x_{\mu}$ and $S_{f}\equiv S^{\mu}_{f}d^{n-1}x_{\mu}$. The
algebraic Poincaré lemma (6.14) thus implies the existence of a $(n-2)$-form
$k_{f}\equiv k_{f}^{[\mu\nu]}(d^{n-2}x)_{\mu\nu}$ such that

$\displaystyle\boxed{\ j_{f}=S_{f}+d_{\tt h}k_{f}\ }\,.$ (7.36)

(In the particular case of Maxwell’s theory, this is our equation (7.4).) This
relation states that _the conserved current $(j^{\mu}_{f})$ associated to a
local symmetry vanishes weakly up to a superpotential term_ and is thus
trivial in the sense of Noether’s first theorem (as we already noted above in
that context).

The corresponding (on-shell) conserved charge is obtained by integration over
a $(n-1)$-dimensional space-like hypersurface $\Sigma$ with boundary
$\partial\Sigma$ and by applying Stokes theorem:

$\displaystyle
Q_{f}\equiv\int_{\Sigma}j_{f}=\int_{\Sigma}\underbrace{S_{f}}_{\approx\,0}+\int_{\Sigma}d_{\tt
h}k_{f}\approx\oint_{\partial\Sigma}k_{f}\,.$ (7.37)

Sometimes one also says that $\Sigma$ has _codimension_ $1$ and that
$\partial\Sigma$ has codimension $2$.

_In summary,_ in a pure gauge theory where one does not have any non-trivial
global symmetries, one obtains on-shell a surface charge by integration of the
current $(j^{\mu}_{f})$ over a Cauchy surface $\Sigma$:

$\displaystyle\boxed{\
Q_{f}\equiv\int_{\Sigma}j_{f}^{\mu}\,d^{n-1}x_{\mu}\approx\oint_{\partial\Sigma}k_{f}^{[\mu\nu]}\,(d^{n-2}x)_{\mu\nu}\
}\,.$ (7.38)

The existence and properties of this flux integral (which may be viewed as a
_“lower degree conservation law”_) is determined by the properties of the
tensor field $(k_{f}^{[\mu\nu]})$ (which is a local function of the fields and
symmetry parameters) in the vicinity of the hypersurface $\partial\Sigma$.
_This flux integral provides an appropriate notion of conserved charge in
field theories with local symmetries like YM-theories or general relativity._
Yet, this charge is arbitrary due to the fact that the superpotential
$k_{f}^{[\mu\nu]}$ depending on the arbitrary functions $f$ is arbitrary.
Thus, one has to make more precise the relationship between the symmetry
parameters $f$ and the $(n-2)$-forms $k_{f}$ (next subsection) and in
particular to _isolate specific parameters $f$ so as to define the charges
(7.38): an appropriate choice is given by the Killing vector fields associated
to as fixed background gauge field,_ see next subsections.

##### Converse of Noether’s second theorem and derivation of symmetry
transformations:

Before proceeding further, we note that Eqn. (7.27), i.e.

$\delta_{f}\varphi^{a}\,\frac{\delta
S}{\delta\varphi^{a}}=f^{r}\,(Q^{a}_{r})^{\dagger}\big{(}\frac{\delta
S}{\delta\varphi^{a}}\big{)}+\partial_{\mu}S^{\mu}_{f}\,,$

can be read and applied the other way around. More precisely, suppose we can
find some differential operators $(Q^{a}_{r})^{\dagger}$ (with
$Q^{a}_{r}(f^{r})\equiv Q^{a}_{r}f^{r}+Q^{a\mu}_{r}\partial_{\mu}f^{r}$) which
annihilate the equation of motion functions $\frac{\delta
S}{\delta\varphi^{a}}$, i.e. the Noether identities (7.29) are satisfied. The
contraction of $(Q^{a}_{r})^{\dagger}\big{(}\frac{\delta
S}{\delta\varphi^{a}}\big{)}$ with arbitrary functions $x\mapsto f^{r}(x)$ can
then be rewritten under the form $\delta_{f}\varphi^{a}\,\frac{\delta
S}{\delta\varphi^{a}}-\partial_{\mu}S^{\mu}_{f}$: from this expression (which
vanishes by virtue of the Noether identities) one can read of the local
symmetry transformations $\delta_{f}\varphi^{a}$ leaving the action functional
$S$ invariant as well as the associated (weakly vanishing) Noether current
$(S^{\mu}_{f})$. This line of reasoning has been used by the authors of
reference [255] to determine some novel local symmetries of the first order
action functional describing gravity as well as of some related action
functionals (Holst action and non-minimal coupling of matter fields to
gravity) that we outlined at the end of Subsection 5.1.2 above.

Here, we simply illustrate the line of arguments with the example of Maxwell’s
theory described by the equation of motion function
$\partial_{\nu}F^{\nu\mu}$. Application of $\partial_{\mu}$ yields the Noether
identity $\partial_{\mu}\partial_{\nu}F^{\nu\mu}=0$ by virtue of the
antisymmetry of $F^{\nu\mu}$. By applying the Leibniz rule to
$\partial_{\mu}$, the product of $\partial_{\mu}\partial_{\nu}F^{\mu\nu}$ with
an arbitrary function $x\mapsto f(x)$ can be rewritten as follows:

$f\,\partial_{\mu}\partial_{\nu}F^{\mu\nu}=-f\,\partial_{\mu}\partial_{\nu}F^{\nu\mu}=(\partial_{\mu}f)\underbrace{\partial_{\nu}F^{\nu\mu}}_{=\;\frac{\delta
S}{\delta
A_{\mu}}}-\partial_{\mu}(f\underbrace{\partial_{\nu}F^{\nu\mu}}_{=\;\frac{\delta
S}{\delta A_{\mu}}})\equiv\delta_{f}A_{\mu}\,\frac{\delta S}{\delta
A_{\mu}}-\partial_{\mu}S^{\mu}_{f}\,.$

Thus one recovers the local gauge transformation of $A_{\mu}$ from the obvious
Noether identity for the dynamical system under consideration.

###  7.4 Conserved forms of lower degree and corresponding Noether theorem

####  7.4.1 Generalized form of Noether’s first theorem

In view of relation (7.37), _the non-trivial conservation laws for forms of
lower form degree are determined by the characteristic cohomology group in
degree $n-2$:_ for the latter, one has a result of the form Eqn. (7.15), i.e.

$\displaystyle H^{n-2}\Big{(}d_{\tt h},\Omega^{\bullet}(S)\Big{)}\cong
H^{n-2}(d_{\tt h}|\delta)\,.$ (7.39)

For $n\geq 3$, the calculation of the cohomology group $H^{n-2}(d_{\tt
h}|\delta)$ leads [74] to the determination of physically distinct (global)
reducibility parameters which we have mentioned at the end of Subsection
7.1.2, i.e. _equivalence classes $[f^{r}]$ of local gauge parameters:_ the
equivalence relation is defined by

$\displaystyle f^{r}\sim
f^{r}\,+\,\underbrace{t^{r}}_{\approx\,0}\,,\qquad\mbox{where}\ \;\boxed{\
Q^{a}_{r}(f^{r})\approx 0\ }\,.$ (7.40)

Here, $Q^{a}_{r}(f^{r})\approx 0$ means that all gauge variations
$Q^{a}_{r}(f^{r})$ of the fields $\varphi^{a}$ vanish weakly. (These gauge
transformations which leave the solutions of the equations of motion invariant
are also qualified as _“ineffective gauge transformations”_ [154].) The upshot
of this line of arguments is known as the

##### Generalized form of Noether’s first theorem:

Consider a local field theory which is defined on a space-time manifold of
dimension $n\geq 3$ and whose dynamics is described by an action functional
$S[\varphi]$. Suppose this action admits continuous global symmetries and
possibly, in addition, local symmetries. Then there is a _one-to-one
correspondence between non-trivial global reducibility parameters_ (i.e. gauge
parameters $f^{r}\approx\\!\\!\\!\\!\\!\\!/\;0$ such that
$Q^{a}_{r}(f^{r})\approx 0$ for all fields $\varphi^{a}$ on $M$) _and
$(n-2)$-forms $k\approx\\!\\!\\!\\!\\!\\!/\;0$ which are $d_{\tt h}$-closed
on-shell ($d_{\tt h}k\approx 0$), but not $d_{\tt h}$-exact:_

$\displaystyle\boxed{\ [f^{r}]\ \longleftrightarrow\ [k]\ }\qquad\mbox{with}\
\ Q^{a}_{r}(f^{r})\approx 0\ \ \mbox{and}\ \ d_{\tt h}k\approx 0\,.$ (7.41)

Here, the correspondence $[f^{r}]\mapsto[k]$ is given by the so-called [158,
249] _Barnich-Brandt procedure_ [154, 254] which yields representatives
$k_{f}$ associated to global reducibility parameters $(f^{r})$ which are
constructed from the Euler-Lagrange derivatives of the Lagrangian describing
the dynamics of the theory. To formulate the latter correspondence, we first
introduce a convenient mathematical device (see [254] for references to the
related mathematical and physical literature).

####  7.4.2 Contracting homotopy operators

As in the case of the ordinary Poincaré lemma for the differential
$d=dx^{\mu}\,\frac{\partial\ }{\partial x^{\mu}}$ acting on
$\Omega^{\bullet}(\mathbb{R}^{n})$, a form $\beta$ satisfying $d_{\tt
h}\beta=\alpha$ can be obtained from the $d_{\tt h}$-closed $(n-1)$-form
$\alpha$ by applying the _(contracting) homotopy operator_ [31, 226] which
generalizes the one of the de Rham complex
$(\Omega^{\bullet}(\mathbb{R}^{n}),d)$, see Eqn. (A.19) of Appendix A.5 for
the definition of the latter. Explicit expressions for the homotopy operators
and applications thereof are discussed for instance in appendix A of reference
[157] and in [249]. In the following, we summarize the notions that we will
need in the sequel.

##### General expression:

Consider an infinitesimal field variation $\delta_{Q}\varphi^{a}=Q^{a}$ or (in
terms of the variational bicomplex notation (6.8)) $d_{\tt
v}q^{a}=Q^{a}\in\Omega^{0,1}$. The _contracting homotopy operator of the
horizontal bicomplex (with respect to the characteristic $Q^{a}$)_ has been
introduced by I. M. Anderson [226] and is used for instance in the proof of
the algebraic Poincaré lemma [154, 74, 249]: one defines the map

$\displaystyle I^{k}_{Q}\ :\ \Omega^{k,l}\ $ $\displaystyle\longrightarrow\
\Omega^{k-1,l+1}$ $\displaystyle\alpha\ $ $\displaystyle\longmapsto\
I^{k}_{Q}\alpha\,,$

by

$\displaystyle\boxed{\ I^{k}_{Q}\alpha\equiv\sum_{p\geq
0}\;\frac{p+1}{n-k+p+1}\,\partial_{\mu_{1}}\cdots\partial_{\mu_{p}}\Big{[}Q^{a}\,\frac{\delta\quad}{\delta\varphi^{a}_{\mu_{1}\cdots\mu_{p}\nu}}\Big{(}\frac{\partial\alpha}{\partial(dx^{\nu})}\Big{)}\Big{]}\
}\,,$ (7.42)

i.e.

$I^{k}_{Q}\alpha=\frac{1}{n-k+1}\,Q^{a}\,\frac{\delta\
}{\delta\varphi^{a}_{\nu}}\Big{(}\frac{\partial\alpha}{\partial(dx^{\nu})}\Big{)}+\frac{2}{n-k+2}\,\partial_{\mu}\Big{[}Q^{a}\,\frac{\delta\
\;}{\delta\varphi^{a}_{\mu\nu}}\Big{(}\frac{\partial\alpha}{\partial(dx^{\nu})}\Big{)}\Big{]}+\cdots\,.$

Here,
$\varphi^{a}_{\nu}\equiv\partial_{\nu}\varphi^{a},\,\varphi^{a}_{\mu\nu}\equiv\partial_{\mu}\partial_{\nu}\varphi^{a},...$
Furthermore, the derivatives
$\frac{\delta\quad}{\delta\varphi^{a}_{\mu_{1}\cdots\mu_{p}\nu}}$ represent
_(higher order) Euler-Lagrange operators:_ the action of the latter on
functions $F$ depending at most on second order derivatives of fields is given
by

$\displaystyle\frac{\delta F}{\delta\varphi^{a}}$
$\displaystyle=\frac{\partial
F}{\partial\varphi^{a}}-\partial_{\mu}\Big{(}\frac{\partial
F}{\partial\varphi^{a}_{\mu}}\Big{)}+\partial_{\mu}\partial_{\nu}\Big{(}\frac{\partial
F}{\partial\varphi^{a}_{\mu\nu}}\Big{)}\,,\qquad\frac{\delta
F}{\delta\varphi_{\mu}^{a}}=\frac{\partial
F}{\partial\varphi_{\mu}^{a}}-2\partial_{\nu}\Big{(}\frac{\partial
F}{\partial\varphi^{a}_{\nu\mu}}\Big{)}\,,\qquad\frac{\delta
F}{\delta\varphi_{\mu\nu}^{a}}=\frac{\partial
F}{\partial\varphi_{\mu\nu}^{a}}\,,$ (7.43)

where $\partial/\partial\varphi^{a}_{\mu\nu}$ represents the symmetrized
derivative, i.e.
$\frac{\partial\varphi^{b}_{\alpha\beta}}{\partial\varphi^{a}_{\mu\nu}}=\frac{1}{2}\,\delta^{b}_{a}\,(\delta_{\alpha}^{\mu}\delta_{\beta}^{\nu}+\delta_{\alpha}^{\nu}\delta_{\beta}^{\mu})$.

##### Application 1 (Derivation of (pre-)symplectic potential):

As a simple application [157], we consider the action of
$I^{n}_{\delta\varphi}$ on a first order Lagrangian $n$-form ${\cal
L}\,d^{n}x\in\Omega^{n,0}$: from
$\frac{\partial(d^{n}x)}{\partial(dx^{\nu})}=i_{\partial_{\nu}}(d^{n}x)=d^{n-1}x_{\nu}$
and (7.43) we conclude that

$\displaystyle\boxed{\ I^{n}_{\delta\varphi}({\cal
L}\,d^{n}x)=\delta\varphi^{a}\,\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi^{a})}\;d^{n-1}x_{\mu}\equiv
j^{\mu}[\varphi;\delta\varphi]\;d^{n-1}x_{\mu}\equiv j[\varphi;\delta\varphi]\
\in\Omega^{n-1,1}\ }\,.$ (7.44)

Here, we recognize the _(pre-)symplectic potential_ , see equation (6.28).

##### Application 2 (Corner term):

An explicit expression for the corner term
$\vartheta_{\Lambda}\in\Omega^{n-2,1}$ appearing in the (pre-) symplectic
potential (6.30) is given by

$\displaystyle\boxed{\ \vartheta_{\Lambda}=-I^{n-1}_{\delta\varphi}\Lambda\
}\qquad\mbox{with}\quad\Lambda\equiv\Lambda^{\mu}[q]\,d^{n-1}x_{\mu}\in\Omega^{n-1,0}\,.$
(7.45)

Indeed, from this expression and (7.44), i.e. $j_{{\cal
L}d^{n}x}=I^{n}_{\delta\varphi}({\cal L}\,d^{n}x)$, as well as the properties
of the homotopy operators, it follows that

$j_{d_{\tt h}\Lambda}-d_{\tt h}\vartheta_{\Lambda}=I^{n}_{\delta\varphi}d_{\tt
h}\Lambda+d_{\tt h}I^{n-1}_{\delta\varphi}\Lambda=\delta\Lambda\,.$

More precisely, the relative signs appearing in relation (6.30) are obtained
by choosing signs according to $j_{{\cal
L}d^{n}x}\equiv-I^{n}_{\delta\varphi}({\cal L}\,d^{n}x)$ and

$\displaystyle\vartheta_{\Lambda}\equiv
I^{n-1}_{\delta\varphi}\Lambda=\sum_{p\geq
0}\;\frac{p+1}{p+2}\,\partial_{\mu_{1}}\cdots\partial_{\mu_{p}}\Big{[}\delta\varphi^{a}\,\frac{\delta\quad}{\delta\varphi^{a}_{\mu_{1}\cdots\mu_{p}\nu}}\Big{(}\frac{\partial\Lambda}{\partial(dx^{\nu})}\Big{)}\Big{]}\,.$
(7.46)

With $\frac{\partial\Lambda}{\partial(dx^{\nu})}=i_{\partial_{\nu}}\Lambda$,
this result coincides with the one given in appendix A of the first of
references [180].

##### Application 3 (Derivation of superpotential):

As a second example [154, 157], we consider the action of
$I^{n-1}_{\delta\varphi}$ on a $d_{\tt h}$-closed current form
$J_{f}\in\Omega^{n-1,0}$: the operator (7.42) is defined so as to ensure that
$d_{\tt h}J_{f}=0$ implies that

$\displaystyle J_{f}=d_{\tt h}k_{f}$ (7.47)

with

$\displaystyle\mbox{superpotential}\qquad\boxed{\
k_{f}[\varphi;\delta\varphi]=I^{n-1}_{\delta\varphi}J_{f}\ \in\Omega^{n-2,1}\
}\,,$ (7.48)

this form being defined up to an $d_{\tt h}$-exact $(n-2)$-form. More
specifically, _for_ $J^{\mu}_{f}=S^{\mu}_{f}=\frac{\delta
S}{\delta\varphi^{a}}\,Q^{a\mu}_{r}f^{r}$ (see Eqn. (7.30)) _and a first order
Lagrangian, the definition ( 7.42) yields a simple expression for the
superpotential (7.48):_

$\displaystyle\boxed{\
k_{f}[\varphi;\delta\varphi]=\frac{1}{2}\,\delta\varphi^{a}\,\frac{\delta\
}{\delta\varphi_{\mu}^{a}}\Big{(}\frac{\partial
S_{f}}{\partial(dx^{\mu})}\Big{)}\ }\,.$ (7.49)

The expressions (7.48)-(7.49) depend on the fields $\delta\varphi^{a}$ of
linearized field theory. _An expression depending on the fields of the full
interacting theory can be obtained[254, 249] by integrating the form (7.48)
along a path $\gamma$ in the space ${\cal R}_{\infty}$_ (of solutions of the
field equations) which goes from a given field configuration $\bar{\varphi}$
satisfying $J_{f}[\bar{\varphi}]=0$ to a generic solution $\varphi$:

$\displaystyle\boxed{\
K_{f}[\varphi]\equiv\int_{\gamma}k_{f}[\varphi;\delta\varphi]\ }\,.$ (7.50)

By using $d_{\tt h}J_{f}=0$ and the properties of the operators $I^{k}_{Q}$,
we then have

$d_{\tt h}K_{f}[\varphi]=\int_{\gamma}d_{\tt
h}k_{f}=\int_{\gamma}\underbrace{\big{(}d_{\tt
h}I^{n-1}_{\delta\varphi}+I^{n}_{\delta\varphi}d_{\tt
h}\big{)}J_{f}}_{=\,\delta
J_{f}}=J_{f}[\varphi]-\underbrace{J_{f}[\bar{\varphi}]}_{=\,0}=J_{f}[\varphi]\,,$

in accordance with Eqn. (7.47). Due to the fact that the $(n-2)$-form (7.50)
generically depends on the path which interpolates between the solutions
$\bar{\varphi}$ and $\varphi$ of the field equations [254], the $(n-2)$-form
(7.48) is considered to be more fundamental [249].

Since $k_{f}[\varphi;\delta\varphi]\in\Omega^{n-2,1}$ is a $1$-form in field
space, its integral over a closed $(n-2)$-dimensional surface $\partial\Sigma$
(typically a sphere),
$\delta\\!\\!\\!/Q_{f}[\delta\varphi]\equiv\oint_{\partial\Sigma}k_{f}[\varphi;\delta\varphi]$
yields a _surface charge $1$-form:_ the properties and algebra of these
surface charges are investigated in detail in reference [157]. Here, we only
note that they are _$d_{\tt h}$ -closed on-shell_ and that they do not depend
on the homology class of the closed surface $\partial\Sigma$.

##### Homotopy contraction with respect to gauge parameters:

In the context of gauge field type theories, one also needs the contracting
homotopy operator of the horizontal bicomplex with respect to gauge parameters
$f\equiv(f^{r})$ (e.g. with respect to a diffeomorphism generating vector
field $\xi\equiv\xi^{\mu}\partial_{\mu}$): it is [157, 249] a map

$\displaystyle I^{k}_{f}\ :\ \Omega^{k,l}\ $ $\displaystyle\longrightarrow\
\Omega^{k-1,l}\,,$

whose expression has the same structure as (7.42):

$\displaystyle\boxed{\ I^{k}_{f}\alpha\equiv\sum_{p\geq
0}\;\frac{p+1}{n-k+p+1}\,\partial_{\mu_{1}}\cdots\partial_{\mu_{p}}\Big{[}f^{r}\,\frac{\delta\quad}{\delta
f^{r}_{\mu_{1}\cdots\mu_{p}\nu}}\Big{(}\frac{\partial\alpha}{\partial(dx^{\nu})}\Big{)}\Big{]}\
}\,.$ (7.51)

It enjoys the characteristic property

$I^{k+1}_{f}d_{\tt h}+d_{\tt h}I^{k}_{f}=1\,.$

For later reference, we consider its application to the (pre-)symplectic
potential (7.44): for the case where the gauge parameters are given by a
vector field $\xi\equiv\xi^{\mu}\partial_{\mu}$ (acting on the fields
$\varphi^{a}$ by virtue of the Lie derivative, $\delta_{\xi}\varphi^{a}\equiv
L_{\xi}\varphi^{a}$), we obtain the

$\displaystyle\mbox{Noether-Wald surface charge form :}\qquad
Q_{\xi}[\varphi]\equiv-(I^{n-1}_{\xi}j)[\varphi;L_{\xi}\varphi]\
\in\Omega^{n-2,0}\,.$ (7.52)

We will come back to this expression in Subsection 7.6 (Eqn. (7.75) below).

####  7.4.3 Barnich-Brandt procedure

##### Cohomological determination of superpotentials and of surface charges:

By contracting the _reducibility identity_
$\delta_{f}\varphi^{a}=Q^{a}_{r}(f^{r})\approx 0$ (characterizing the global
reducibility parameters $f^{r}$) with $\frac{\delta S}{\delta\varphi^{a}}$, we
get (see Eqn. (7.30))

$\displaystyle 0\approx\delta_{f}\varphi^{a}\,\frac{\delta
S}{\delta\varphi^{a}}=\partial_{\mu}S^{\mu}_{f}\,,\qquad\mbox{with}\
\;S_{f}^{\mu}\equiv S^{\mu a}_{r}\Big{(}\frac{\delta
S}{\delta\varphi^{a}},f^{r}\Big{)}\,.$ (7.53)

By spelling out the equation of motion term on the right-hand side of relation
$Q^{a}_{r}(f^{r})\approx 0$, one finds that equation (7.53) can be written as
a _divergence identity_ [154]:

$\displaystyle\partial_{\mu}J^{\mu}_{f}=0\qquad\mbox{with}\
\;J_{f}^{\mu}\equiv S^{\mu}_{f}+M^{\mu ba}\Big{(}\frac{\delta
S}{\delta\varphi^{b}},\frac{\delta S}{\delta\varphi^{a}}\Big{)}\,.$ (7.54)

Since $\partial_{\mu}J^{\mu}_{f}=0$ is equivalent to the relation $d_{\tt
h}J_{f}=0$ for the $(n-1)$-form $J_{f}\equiv J^{\mu}_{f}\,d^{n-1}x_{\mu}$ and
since the cohomology of the differential $d_{\tt h}$ is trivial in degree
$n-1$ for $n\geq 2$ (due to the algebraic Poincaré lemma (6.14)), there exists
a $(n-2)$-form $k_{f}\equiv k^{[\mu\nu]}_{f}(d^{n-2}x)_{\mu\nu}$ such that
$J_{f}=d_{\tt h}k_{f}$, i.e. such that

$\displaystyle\partial_{\nu}k^{[\nu\mu]}_{f}=J^{\mu}_{f}\approx 0\,.$ (7.55)

Now suppose that the non-trivial gauge transformations
$\delta_{\bar{f}}\varphi^{a}=Q_{r}^{a}(\bar{f}^{r})=Q_{r}^{a}\bar{f}^{r}+Q_{r}^{a\mu}\partial_{\mu}\bar{f}^{r}+\dots$
contain only field _in_ dependent operators. If the functional derivatives
$\delta S/\delta\varphi^{a}$ are linear and homogenous in the fields, then the
reducibility identity $Q_{r}^{a}(\bar{f}^{r})=0$ holds _off_ -shell (for which
case we put a bar on $f^{r}$ following [154]) and the expression (7.54) for
$J^{\mu}_{\bar{f}}$ reduces to $J^{\mu}_{\bar{f}}=S^{\mu}_{\bar{f}}$. If,
furthermore, $S^{\mu}_{\bar{f}}$ contains a most second order derivatives of
the fields, then the expression (7.48) for the superpotential $k_{\bar{f}}$
given by the homotopy contraction of $J^{\mu}_{\bar{f}}$ reduces to a _simple
formula which provides an explicit realization of the correspondence ( 7.41):_

$\displaystyle\boxed{\ \bar{f}\ \longmapsto\
k^{[\nu\mu]}_{\bar{f}}=\frac{1}{2}\,\varphi^{a}\,\frac{\partial
S^{\mu}_{\bar{f}}}{\partial\varphi^{a}_{\nu}}+\left[\frac{2}{3}\,\varphi^{a}_{\lambda}-\frac{1}{3}\,\varphi^{a}\partial_{\lambda}\right]\frac{\partial
S^{\mu}_{\bar{f}}}{\partial\varphi^{a}_{\lambda\nu}}\,-\,(\mu\longleftrightarrow\nu)\
}\,.$ (7.56)

This expression is determined by the Lagrangian and by the on-shell vanishing
Noether current $S^{\mu}_{\bar{f}}$. Accordingly, the associated surface
charges do not depend on total divergences that might be added to the
Lagrangian or to the Noether current [157].

##### Example of Maxwell’s theory:

All assumptions made before equation (7.56) are satisfied for Maxwell’s theory
in $\mathbb{R}^{n}$ (which represents a linear gauge theory). The general
solution of $0=\delta_{\bar{f}}A_{\mu}\equiv\partial_{\mu}\bar{f}$ is given by
$\bar{f}=\varepsilon=\textrm{const.}$ For
$J^{\mu}_{\bar{f}}=S^{\mu}_{\bar{f}}=(\partial_{\nu}F^{\nu\mu})\,\bar{f}=(\partial_{\nu}F^{\nu\mu})\,\varepsilon$
(see Eqn. (7.2)) we presently have
$J^{\mu}_{\bar{f}}=\partial_{\nu}k^{[\nu\mu]}_{\bar{f}}$ with a superpotential
$k^{[\nu\mu]}_{\bar{f}}=\varepsilon F^{\nu\mu}$ (in agreement with the general
expression (7.56)). For the associated conserved surface charge in
$\mathbb{R}^{4}$ we thus obtain the result (7.6)-(7.7) which represents (upon
division by $\varepsilon$ or derivation with respect to $\varepsilon$) the
usual expression for the electric charge in electrodynamics.

##### Examples of YM-theory and of gravity:

For Yang-Mills theory with a non-Abelian structure group and for general
relativity (which represent non-linear gauge theories), the identity
$Q_{r}^{a}(\bar{f}^{r})=0$ defining _exact Killing vectors_ (or _exact
reducibility parameters_) $(\bar{f}^{r})$ reads

$0=D_{\mu}\bar{f}=\partial_{\mu}\bar{f}+\textrm{i}g\,[A_{\mu},\bar{f}]\qquad\mbox{with}\
\bar{f}\equiv\bar{f}^{r}T_{r}\,,$

and

$0=(L_{\bar{\xi}}g)_{\mu\nu}={\nabla}_{\mu}\bar{\xi}_{\nu}+{\nabla}_{\nu}\bar{\xi}_{\mu}\qquad\mbox{with}\
\bar{\xi}\equiv\bar{\xi}^{\mu}\partial_{\mu}\,,$

respectively. Since these equations have to hold for arbitrary gauge fields
$(A_{\mu})$ and metrics $(g_{\mu\nu})$, the only solutions are the trivial
ones $\bar{f}^{r}=0$ and $\bar{\xi}^{\mu}=0$ [74]. These results reflect the
fact that generic gauge field configurations in YM-theories or generic
geometries in curved space-time do not admit any symmetries. This obstacle for
the definition of surface charges can be overcome in the case where a
_background field configuration_ having symmetries is given and in the case
where the theory is linearized around this configuration (next paragraph) or
where the linearized theory describes the full theory asymptotically in the
vicinity of some boundary (next subsection): in the latter case, one deals
with the corresponding asymptotic symmetries in a general way.

##### Examples of linearized YM-theory and gravity:

For YM-theory on $\mathbb{R}^{n}$ with $n\geq 3$, suppose we are given a
background field configuration $(\bar{A}_{\mu})$ with which the gauge fields
$A_{\mu}$ coincide for $|\vec{x}\,|\to\infty$, i.e. the deviation
$a_{\mu}\equiv A_{\mu}-\bar{A}_{\mu}$ tends to zero for
$|\vec{x}\,|\to\infty$. Upon _linearization of the theory around this
background field,_ we then have [74] the gauge variation
$\delta_{f}a_{\mu}=\bar{D}_{\mu}f\equiv\partial_{\mu}f+\textrm{i}g\,[\bar{A}_{\mu},f]$
which is independent of $a_{\mu}$. For a flat background, i.e.
$\bar{F}_{\mu\nu}=0$, the solution of $\bar{D}_{\mu}f=0$ and the associated
surface charges (“color charges”) will be derived below, see equation (7.70).

Similarly, in general relativity, the _linearization of the metric field
around a given geometric background_ $(\bar{g}_{\mu\nu})$, i.e.
$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$ (where $h_{\mu\nu}$ vanishes
asymptotically) leads to the local symmetry transformation
$\delta_{\xi}h_{\mu\nu}=(L_{\xi}\bar{g})_{\mu\nu}$ which does not depend on
$h_{\mu\nu}$. For a flat background $\bar{g}_{\mu\nu}=\eta_{\mu\nu}$, the
Killing vector fields $\xi=\xi^{\mu}\partial_{\mu}$ (i.e. solutions of
$0=(L_{\xi}\bar{g})_{\mu\nu}=\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}$)
are those generating Poincaré transformations, i.e.
$\xi^{\mu}(x)=a^{\mu}+\varepsilon^{\mu\nu}x_{\nu}$ with constant real
parameters $a^{\mu}$ and $\varepsilon^{\mu\nu}=-\varepsilon^{\nu\mu}$: the
associated surface charges yield the famous ADM charges of linearized gravity
[176, 154, 74].

###  7.5 Asymptotic symmetries and asymptotically conserved forms

##### Generalities:

Suppose some background fields $\bar{\varphi}^{a}$ are given and that the
deviation $\phi^{a}\equiv\varphi^{a}-\bar{\varphi}^{a}$ of fields
$\varphi^{a}$ from these background fields is small (but not necessarily zero)
in a prescribed manner in the _asymptotic region:_ For the latter region, one
can consider spatial infinity or the boundary of a finite domain like the
horizon of a black hole, e.g. see [154] and the review [249] for various
examples. (For spatial infinity and a radial variable $r$, one typically
assumes an asymptotic behavior of the form $\phi^{a}\to{\cal O}(1/r^{m_{a}})$
for some number $m_{a}$ that may depend on the field under consideration.
Analogous assumptions are made for the asymptotic symmetry parameters. In the
following we will not spell out these technical details for which we refer to
the cited references.) Upon making appropriate general assumptions on the
boundary conditions that hold in the asymptotic region, _the linearized theory
describes the full theory asymptotically in the vicinity of the boundary._

Within this general setting, G. Barnich and F. Brandt [154] have extended the
correspondence (7.41) (which is the subject of the generalized form of
Noether’s first theorem) to the case of asymptotic symmetries and
asymptotically conserved $(n-2)$-forms (see also references [155, 157]). The
_asymptotic symmetry parameters_ $\tilde{f}$ are defined to be field
independent gauge parameters $\tilde{f}^{r}$ for which
$\delta_{\tilde{f}}\phi^{a}$ vanishes asymptotically in a prescribed manner.
In this case, the superpotential $\tilde{k}^{[\nu\mu]}_{\tilde{f}}$ associated
to the parameters $\tilde{f}$ is still given by relation (7.56), but with the
fields $\varphi^{a}$ replaced by their deviation $\phi^{a}$ from the
background and with $S^{\mu}_{\tilde{f}}[\varphi]$ replaced by its linearized
expression $s^{\mu}_{\tilde{f}}[\phi;\bar{\varphi}]$ (see example below), i.e.
we have the _correspondence_ [154]

$\displaystyle\boxed{\ \tilde{f}\ \longmapsto\
\tilde{k}^{[\nu\mu]}_{\tilde{f}}=\frac{1}{2}\,\phi^{a}\,\frac{\partial
s^{\mu}_{\tilde{f}}}{\partial\phi^{a}_{\nu}}+\left[\frac{2}{3}\,\phi^{a}_{\lambda}-\frac{1}{3}\,\phi^{a}\partial_{\lambda}\right]\frac{\partial
s^{\mu}_{\tilde{f}}}{\partial\phi^{a}_{\lambda\nu}}\,-\,(\mu\longleftrightarrow\nu)\
}\,.$ (7.57)

##### Summary:

The asymptotic symmetry parameters $\tilde{f}$ represent Killing vector fields
associated to a fixed background gauge field and the $(n-2)$-form
$\tilde{k}_{\tilde{f}}$ is associated to the free theory obtained by
linearization of the full interacting theory around the given background. By
construction, $\tilde{k}^{[\nu\mu]}_{\tilde{f}}$ _depends linearly on the
deviations $\delta\varphi^{a}\equiv\varphi^{a}-\bar{\varphi}^{a}=\phi^{a}$
(which are small in the asymptotic region) and on their derivatives up to a
finite order_ (as well as on a finite number of derivatives of the background
fields $\bar{\varphi}^{a}$). As pointed out in reference [254], the
integration of $\tilde{k}_{\tilde{f}}$ along a path $\gamma$ in the space of
solutions of field equations (see Eqn. (7.50)) allows us to get, under
suitable assumptions, a _$(n-2)$ -form $K_{\tilde{f}}[\varphi]$ depending on
the fields $\varphi$ of the full interacting theory._

##### Example of non-Abelian YM-theories:

For YM-theory in $\mathbb{R}^{n}$ with $n\geq 3$, we can determine [244, 154]
_conserved local and integral quantities associated to symmetries of a
background gauge field which satisfies the vacuum YM equations._ The conserved
quantities are constructed from the gauge Killing vectors of the background
gauge field and the conserved charges can be expressed as flux integrals
having the same form as those encountered in electrodynamics.

To construct all of these quantities, we start by considering the gauge field
configuration $(A^{\mu})$ produced by a source $(j^{\mu})$ which is bounded in
space, i.e. _$(j^{\mu})$ vanishes outside of a finite volume._ The YM equation
reads $D_{\mu}F^{\mu\nu}=j^{\nu}$ and implies $D_{\nu}j^{\nu}=0$. Now the
gauge field $A_{\mu}$ is decomposed as

$\displaystyle A_{\mu}=\bar{A}_{\mu}+a_{\mu}\qquad\textrm{with}\
\;a_{\mu}(t,\vec{x}\,)\stackrel{{\scriptstyle|\vec{x}|\to\infty}}{{\longrightarrow}}0\,.$
(7.58)

Here, $\bar{A}_{\mu}$ is viewed as a _background field corresponding to the
source $j=0$,_ i.e.

$\displaystyle
0=\bar{D}_{\mu}\bar{F}^{\mu\nu}\equiv\partial_{\mu}\bar{F}^{\mu\nu}+\textrm{i}g\,[\bar{A}_{\mu},\bar{F}^{\mu\nu}]\qquad\textrm{with}\
\;\bar{F}_{\mu\nu}\equiv\partial_{\mu}\bar{A}_{\nu}-\partial_{\nu}\bar{A}_{\mu}+\textrm{i}g\,[\bar{A}_{\mu},\bar{A}_{\nu}]\,.$
(7.59)

The _deviation_ $a_{\mu}=A_{\mu}-\bar{A}_{\mu}$ is small for
$|\vec{x}|\to\infty$, but it is not assumed to be small otherwise. For a
detailed treatment of the decaying properties of fields and parameters, we
refer to the work [154]. Here, we only note that (7.58) means that the fields
$a_{\mu}$ vanish on the spatial $(n-2)$-sphere at infinity,
$\partial\mathbb{R}^{n-1}=\lim_{R\to\infty}S_{R}$: instead of the whole space
$\mathbb{R}^{n-1}$ one can consider more generally an $(n-1)$-dimensional
domain $\Sigma\subset\mathbb{R}^{n-1}$ and assume in equation (7.58) that the
fields $a_{\mu}$ vanish for $\vec{x}\in\partial\Sigma$.

A short calculation shows that we have the expansion

$\displaystyle{F}_{\mu\nu}=\bar{F}_{\mu\nu}+f_{\mu\nu}+\textrm{i}g\,[a_{\mu},a_{\nu}]\qquad\textrm{with}\
\;f_{\mu\nu}\equiv\bar{D}_{\mu}a_{\nu}-\bar{D}_{\nu}a_{\mu}\,.$ (7.60)

The computation that we made for free Maxwell theory in Eqn. (7.2) can be
generalized to the present setting of YM-theory:

$\displaystyle\textrm{Tr}\,\Big{[}(\delta_{f}A_{\mu})\,\frac{\delta S}{\delta
A_{\mu}}[A]\Big{]}=\textrm{Tr}\,\big{[}(D_{\mu}f)D_{\nu}F^{\nu\mu}\big{]}=\partial_{\mu}\underbrace{\textrm{Tr}\,\big{[}f\,D_{\nu}F^{\nu\mu}\big{]}}_{\equiv\;S_{f}^{\mu}[A]}-\textrm{Tr}\,\big{[}f\,(\underbrace{D_{\mu}D_{\nu}F^{\nu\mu}}_{=\,0})\big{]}\,.$
(7.61)

From $\bar{D}_{\nu}\bar{F}^{\nu\mu}=0$ it follows that
$D_{\nu}F^{\nu\mu}=\bar{D}_{\nu}f^{\nu\mu}+\textrm{i}g\,[a_{\nu},\bar{F}^{\nu\mu}]$
plus terms which are quadratic or of higher order in $a_{\nu}$. Thus, the
linearized expression of
$S_{\tilde{f}}^{\mu}[A]=\textrm{Tr}\,\big{[}{\tilde{f}}\,D_{\nu}F^{\nu\mu}\big{]}$
is given by

$\displaystyle
s^{\mu}_{\tilde{f}}[a;\bar{A}]=\textrm{Tr}\,\Big{[}\tilde{f}\,\big{(}\bar{D}_{\nu}f^{\nu\mu}+\textrm{i}g\,[a_{\nu},\bar{F}^{\nu\mu}]\big{)}\Big{]}\,.$
(7.62)

Now assume that there are _asymptotic symmetries of the gauge field_ , i.e.
that there exist infinitesimal gauge transformations parametrized by one or
several independent $\mathtt{\mathbf{g}}$-valued fields
$x\mapsto\tilde{f}(x)\equiv\tilde{f}^{a}(x)T_{a}\in\mathtt{\mathbf{g}}$ with

$\displaystyle\boxed{\
0=\delta_{\tilde{f}}\bar{A}_{\mu}\equiv\bar{D}_{\mu}\tilde{f}\
}\qquad\mbox{for}\ \;\mu\in\\{0,1,\dots,n-1\\}\,.$ (7.63)

In other words, there exist $\mathtt{\mathbf{g}}$-valued fields $\tilde{f}$
which are _covariantly constant with respect to the background field
$\bar{A}_{\mu}$_. The matrices $\tilde{f}$ (i.e. gauge parameters) are
referred to as _gauge Killing vectors_ [244, 256] or as _asymptotic
reducibility parameters_ [154]. (In YM-theories and in general relativity on
space-times of dimension $n\geq 3$, they are field-independent and satisfy an
off-shell condition [154, 74].)

We note that $\bar{D}_{\mu}\tilde{f}=0$ and the general identity
$[\bar{D}_{\mu},\bar{D}_{\nu}]\tilde{f}=\textrm{i}g[\bar{F}_{\mu\nu},\tilde{f}]$
imply the integrability conditions $[\bar{F}_{\mu\nu},\tilde{f}]=0$ for the
system of differential equations $\bar{D}_{\mu}\tilde{f}=0$. From relation
$[\bar{F}_{\mu\nu},\tilde{f}]=0$ and the cyclicity of the trace it follows
that the second term in expression (7.62) vanishes. The relation
$\bar{D}_{\mu}\tilde{f}=0$ and the fact that the trace is a gauge singlet then
imply that the current (7.62) represents a superpotential term:

$\displaystyle
s^{\mu}_{\tilde{f}}=\partial_{\nu}\tilde{k}^{[\nu\mu]}_{\tilde{f}}\qquad\textrm{with}\quad\boxed{\
\tilde{k}^{[\nu\mu]}_{\tilde{f}}=\textrm{Tr}\,(\tilde{f}f^{\nu\mu})\ }\,.$
(7.64)

(This result for $\tilde{k}^{[\nu\mu]}_{\tilde{f}}$ can also [154] be obtained
from the general relation (7.57).)

As a matter of fact, the property $\bar{D}_{\mu}\tilde{f}=0$ and the relation
$f^{\nu\mu}=\bar{D}^{\nu}a^{\mu}-\bar{D}^{\mu}a^{\nu}$ can be used to rewrite
the superpotential in terms of $a^{\mu}$:

$\displaystyle\boxed{\ \tilde{k}^{[\nu\mu]}_{\tilde{f}}=\partial^{\nu}{\cal
A}^{\mu}-\partial^{\mu}{\cal A}^{\nu}\equiv{\cal F}^{\nu\mu}\
}\qquad\textrm{with}\ \;\boxed{\ {\cal
A}^{\mu}\equiv\textrm{Tr}\,(\tilde{f}a^{\mu})\ }\,.$ (7.65)

This expression for ${\cal F}^{\nu\mu}$ in terms of ${\cal A}^{\mu}$ has the
same form as the field strength tensor in electrodynamics (i.e. Abelian gauge
theory) where $({\cal A}^{\mu})$ represents the electromagnetic potentials. In
particular, we have ${\cal F}_{0i}=\textrm{Tr}\,(\tilde{f}f_{0i})\equiv{\cal
E}_{x^{i}}$ with $\vec{\cal E}=-\overrightarrow{\textrm{grad}}\,{\cal
A}^{0}-\partial_{t}\vec{\cal A}$.

To the conserved current $(s^{\mu}_{\tilde{f}})$ we can associate an “electric
charge”, e.g. for $n=4$, we have the _flux integral_ over the $2$-sphere at
infinity, $\partial\mathbb{R}^{3}$,

$\displaystyle
Q_{\tilde{f}}\equiv\int_{\mathbb{R}^{3}}d^{3}x\,s^{0}_{\tilde{f}}=\int_{\mathbb{R}^{3}}d^{3}x\,\partial_{i}\textrm{Tr}\,(\tilde{f}f^{i0})=\oint_{\partial\mathbb{R}^{3}}dS^{i}\,\textrm{Tr}\,(\tilde{f}f^{i0})\,,$
(7.66)

or

$\displaystyle\boxed{\ Q_{\tilde{f}}=\oint_{\partial\mathbb{R}^{3}}\vec{\cal
E}\cdot\overrightarrow{dS}\ }\qquad\textrm{with}\ \;\boxed{\ \vec{\cal
E}=-\overrightarrow{\textrm{grad}}\,{\cal A}^{0}-\partial_{t}\vec{\cal A}\
}\,.$ (7.67)

Here, the final result (7.67) for the conserved charge has the same form as
the one in electrodynamics, see equation (7.7), but we may presently have
several charges $Q_{\tilde{f}}$ namely one for each asymptotic symmetry, i.e.
gauge Killing vector $\tilde{f}$ of the background field $\bar{A}$.

##### Particular case of an asymptotically flat connection:

In relation with the structure group $G$ of the theory we use the notation
$n_{G}$ for the dimension of the Lie group $G$ and the symbol ${\cal G}$ for
the (infinite-dimensional) group of gauge transformations $x\mapsto U(x)\in
G$. Furthermore, $\\{T_{a}\\}_{a=1,\dots,n_{G}}$ denotes a basis of the Lie
algebra $\mathtt{\mathbf{g}}$ of $G$ and $f_{abc}$ the corresponding structure
constants, i.e. $[T_{a},T_{b}]=\textrm{i}f_{abc}T_{c}$.

A particularly important example of background fields in YM-theory is the one
where the background field strength vanishes (case of an _asymptotically flat
connection_),

$\displaystyle\bar{F}_{\mu\nu}=0\,,\qquad\mbox{i.e.}\ \;\exists\,U\in{\cal
G}\,|\,\bar{A}_{\mu}=-\frac{\textrm{i}}{g}\,U\partial_{\mu}U^{-1}\,.$ (7.68)

The “gauge Killing equation” $\bar{D}_{\mu}\tilde{f}=0$ can then be solved
[244, 154] by multiplying it from the left by $U^{-1}$ and from the right by
$U$ and by using (7.68):

$0=U^{-1}(\bar{D}_{\mu}\tilde{f})U=\partial_{\mu}(U^{-1}\tilde{f}U)\,.$

Thus, $U^{-1}\tilde{f}U=c^{a}T_{a}$ with some constants $c^{a}$ and thereby we
obtain $n_{G}$ gauge Killing vectors (one for each group generator $T_{a}$):

$\displaystyle\forall x\in\mathbb{R}^{n}\;:\qquad\tilde{f}_{(a)}(x)\equiv
U(x)T_{a}U^{-1}(x)\in\mathtt{\mathbf{g}}\,.$ (7.69)

In the present case, the gauge Killing vectors define a representation of the
underlying Lie algebra,

$[\tilde{f}_{(a)},\tilde{f}_{(b)}]=\textrm{i}f_{abc}\,\tilde{f}_{(c)}\,,$

and we have $n_{G}$ charges
$Q_{\tilde{f}_{(a)}}\equiv\oint_{\partial\mathbb{R}^{3}}\vec{\cal
E}_{(a)}\cdot\overrightarrow{dS}$ with $\vec{\cal
E}_{(a)}=-\overrightarrow{\textrm{grad}}\,{\cal
A}_{(a)}^{0}-\partial_{t}\vec{\cal A}_{(a)}$ where ${\cal
A}_{(a)}^{\mu}\equiv\textrm{Tr}\,(\tilde{f}_{(a)}a^{\mu})$.

Using $\bar{F}_{\mu\nu}=0$ and
$a_{\mu}(t,\vec{x}\,)\stackrel{{\scriptstyle|\vec{x}|\to|\infty}}{{\longrightarrow}}0$,
we conclude from (7.60) that $F_{\mu\nu}=f_{\mu\nu}$ for $|\vec{x}|\to\infty$.
By substituting the latter result and expression (7.69) into (7.66) we
conclude that the $n_{G}$ _conserved charges (“color charges”)_ are given, for
$n=4$, by

$\displaystyle\boxed{\
Q_{\tilde{f}_{(a)}}=\oint_{\partial\mathbb{R}^{3}}dS^{i}\,\textrm{Tr}\,\big{[}U(x)T_{a}U^{-1}(x)F^{i0}(x)\big{]}\
}\qquad\textrm{for}\ \;a\in\\{1,\dots,n_{G}\\}\,,$ (7.70)

with $U$ such that (7.68) holds. The Poisson bracket for the charges (7.70)
again reflects the underlying Lie algebra,
$\\{Q_{\tilde{f}_{(a)}},Q_{\tilde{f}_{(b)}}\\}=f_{abc}\,Q_{\tilde{f}_{(c)}}$
and these charges generate gauge transformations [154].

The charges $Q_{\tilde{f}_{(a)}}$ vanish if the components of the field
strength tensor decrease faster than $1/r^{2}$ at spatial infinity (for
$n=4$): this is the case of _instanton_ configurations due to the fact that
they have a finite action [154]. The electric charge of the so-called _Julia-
Zee dyon_ is discussed in reference [244]. We note that one can also define
_magnetic charges_ which are non-zero for non-Abelian _monopole-type
configurations_ [244], see also reference [257].

##### Example of general relativity (with or without a cosmological constant):

The definition and construction of conserved charges in general relativity and
the exploration of its asymptotic structure has a long history going back to
Einstein’s work, e.g. see the recent monograph [22]. The interest in this
subject has been revived in the sixties with the work of BMS [245] and in more
recent years with the discovery of its connection with the so-called _soft
theorems_ (related to the infrared structure of quantized gauge theories) and
_memory effects_ related in particular to gravitational waves and the black
hole information paradox, e.g. see [225, 158, 249, 258] for reviews of
different aspects. Here, we only mention that G. Barnich and F. Brandt [154]
also applied their cohomological procedure (described above for YM-theories)
to asymptotically flat and non-flat space-time manifolds while relating the
resulting expressions to earlier results (see also [158] and [249] for reviews
of this approach). The case of asymptotically flat space-times is of
particular interest in that it allows us to make sense of the important notion
of an isolated system in curved space [259, 260].

The starting point of the approach of Barnich and Brandt is to decompose the
metric $(g_{\mu\nu})$ into a background metric $(\bar{g}_{\mu\nu})$ and a
deviation $(h_{\mu\nu})$ from the latter which is asymptotically small:

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}\,.$

The conserved charges are then constructed from the _Killing vector fields of
the background metric_ $(\bar{g}_{\mu\nu})$ (these vector fields parametrizing
the asymptotic symmetries). More precisely, application of (7.57) to the
Noether current
$S^{\mu}_{\xi}\equiv-\frac{\sqrt{|g|}}{\kappa}\,\xi_{\nu}G^{\nu\mu}_{c}$
determined in Eqn. (7.33) yields [154, 249] the following superpotential
associated to Killing vector fields
$\tilde{\xi}\equiv\tilde{\xi}^{\mu}\partial_{\mu}$ of the background metric
$(\bar{g}_{\mu\nu})$:

$\displaystyle\tilde{k}_{\tilde{\xi}}^{[\mu\nu]}[g;h]$
$\displaystyle=\frac{\sqrt{|g|}}{\kappa}\,\Big{[}\tilde{\xi}^{\nu}\,\nabla^{\mu}h+\tilde{\xi}^{\mu}\,\nabla_{\sigma}h^{\sigma\nu}+\tilde{\xi}_{\sigma}\,\nabla^{\nu}h^{\sigma\mu}$
(7.71)
$\displaystyle\qquad\qquad+\frac{1}{2}\big{(}h\nabla^{\nu}\tilde{\xi}^{\mu}+h^{\mu\sigma}\nabla_{\sigma}\tilde{\xi}^{\nu}+h^{\nu\sigma}\nabla^{\mu}\tilde{\xi}_{\sigma}\big{)}\,-\,(\mu\leftrightarrow\nu)\Big{]}\,.$

Here, $h\equiv{h^{\mu}}_{\mu}$ and the indices are lowered by $(g_{\mu\nu})$
and raised with $(g^{\mu\nu})$. By integrating the $(n-2)$-form
$\tilde{k}_{\tilde{\xi}}[g;h=\delta g]$ along a path in solution space (see
Eqn. (7.50)) one finds [254] the _Komar integrand_ (encountered in equation
(5.64) of the covariant phase space approach) plus an extra term which is
linear in $\tilde{\xi}^{\mu}$.

Concerning the choice of the background metric, let us mention an important
class of examples involving a cosmological constant. A spherically symmetric
matter distribution (of mass $M$ in Newtonian gravity) produces on its outside
a gravitational field which is described by the so-called _Schwarzschild-de
Sitter metric_ [261]: in terms of spherical coordinates (i.e.
$(r,\theta,\varphi)$ for $n=4$), we have

$\displaystyle\boxed{\
ds^{2}=\Big{(}1-\frac{2MG}{r}-\frac{\Lambda}{3}\,r^{2}\Big{)}\,dt^{2}-\Big{(}1-\frac{2MG}{r}-\frac{\Lambda}{3}\,r^{2}\Big{)}^{-1}dr^{2}-r^{2}d\Omega\
}\,,$ (7.72)

with $d\Omega\equiv d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}$ for $n=4$). This
metric represents a solution of the vacuum cosmological Einstein equations
$G_{\textrm{c}}^{\mu\nu}=0$ (or equivalently
$R_{\mu\nu}=\frac{2}{n-2}\,\Lambda g_{\mu\nu}$). For $\Lambda=0$, the metric
(7.72) represents the _Schwarzschild solution_ describing an asymptotically
flat space-time with spherical symmetry. For $M\to 0$ or for large values of
$r$, the metric (7.72) reduces to the _de Sitter metric_

$\displaystyle
ds^{2}=\Big{(}1-\frac{\Lambda}{3}\,r^{2}\Big{)}\,dt^{2}-\Big{(}1-\frac{\Lambda}{3}\,r^{2}\Big{)}^{-1}dr^{2}-r^{2}d\Omega\,.$
(7.73)

For $\Lambda>0$, one speaks about the _de Sitter space_ and for $\Lambda<0$
about the _anti-de Sitter (AdS) space_. These spaces generalize Minkowski
space (as a basic solution of the vacuum Einstein equations) to the case where
$\Lambda\neq 0$: in the latter instance, the space-time is no longer
asymptotically flat. (Anti-) de Sitter space-time has received a lot of
attention during the last decades, in particular in cosmology involving a
cosmological constant [262] and in relationship with string theory (notably
the AdS/CFT correspondence [263]).

###  7.6 Different constructions of lower degree conservation laws

In the following, we survey different methods and their interrelationships.
For a chronological presentation and further references to the original
literature, we refer to [222] and to the preamble of the memoir [241].

####  7.6.1 Cohomological methods 1 : Barnich-Brandt and Wald-Iyer

##### Barnich-Brandt procedure:

This cohomological approach that we described in the previous section is quite
general and applies to field theories with internal and/or geometric
symmetries like YM-theories or general relativity. The resulting conserved
charges are referred to as the _Barnich-Brandt charges_ [158]: their explicit
expression has been determined for instance for electrodynamics, YM-theories
and general relativity [154].

##### Wald-Iyer procedure:

This cohomological method [63, 64, 65] which is based on the _covariant phase
space approach_ does not apply to arbitrary gauge theories, but only to
diffeomorphism invariant theories like general relativity. Thus, the gauge
parameters $f^{r}$ are given by the components $\xi^{\mu}$ of a vector field
$\xi\equiv\xi^{\mu}\partial_{\mu}$ which is assumed to act on the fields
$\varphi^{a}$ by virtue of the Lie derivative, i.e.
$\delta_{\xi}\varphi^{a}=L_{\xi}\varphi^{a}$, e.g.
$\delta_{\xi}\varphi^{a}=\xi^{\mu}\partial_{\mu}\varphi^{a}$ for scalar fields
$\varphi^{a}$. Our presentation is based on [249], see also references [157,
158].

The starting point is the first variational formula (5.1) written in terms of
differential forms (see (6.26) with (6.16)): for an arbitrary variation of the
Lagrangian $n$-form ${\cal L}\,d^{n}x$ we have232323We recall from (6.8) that
the vertical differential $d_{\tt v}$ represents an infinitesimal field
variation. For convenience, we changed the global (conventional) sign of $j$.

$\delta({\cal L}\,d^{n}x)=\delta\varphi^{a}\,\frac{\delta
S}{\delta\varphi^{a}}-d_{\tt h}j\,,$

where the (pre-)symplectic potential form
$j[\varphi;\delta\varphi]\in\Omega^{n-1,1}$ is given by (7.44), i.e.
application of the contracting homotopy operator $I^{n}_{\delta\varphi}$ to
${\cal L}\,d^{n}x$. The (pre-)symplectic current $J^{\mu}=\delta j^{\mu}$ (see
equations (5.3)-(5.8) and (6.27)-(6.28)) is then given by the

$\displaystyle\mbox{(pre-)symplectic current form :}\qquad\boxed{\
J[\varphi;\delta\varphi;\delta\varphi]\equiv\delta
j[\varphi;\delta\varphi;\delta\varphi]\,\in\Omega^{n-1,2}\ }\,.$ (7.74)

_For asymptotic symmetries generated by a vector field $\xi$,_ one uses the
contracting homotopy (7.51) to define (see Eqn. (7.52)) the

$\displaystyle\mbox{Noether-Wald surface charge form :}\qquad\boxed{\
Q_{\xi}[\varphi]\equiv-(I^{n-1}_{\xi}j)[\varphi;L_{\xi}\varphi]\
\in\Omega^{n-2,0}\ }\,.$ (7.75)

Now consider _general relativity_ as described by the Einstein-Hilbert action
functional 242424For simplicity, we will write $S[g]$ and $j[g;h]$ for the
functional dependence on the fields $g_{\mu\nu}$ and $h_{\mu\nu}$, the
distinction with $g\equiv\textrm{det}\,(g_{\mu\nu})$ and
$h\equiv\textrm{det}\,(h_{\mu\nu})$ being clear from the context.
$S[g]=\frac{1}{2\kappa}\int_{M}d^{n}x\,\sqrt{|g}\,R$ with $\kappa\equiv 8\pi
G$ (see Eqn. (7.31)). For this theory we already determined the symplectic
potential $j$ in our discussion of covariant phase space, see equations
(5.41),(5.61): from

$\displaystyle
j[g;h]=\frac{\sqrt{|g|}}{2\kappa}\,\big{(}\nabla_{\nu}h^{\nu\mu}-\nabla^{\mu}h\big{)}\,d^{n-1}x_{\mu}\,,$
(7.76)

one finds [249] that the Noether-Wald surface charge form (7.75)
writes252525We will come back to this quantity in equation (7.123) below. As
noted by R. Wald [63], it has been introduced earlier by W. Simon [264] who
referred to it as the “gravitational field strength”.

$\displaystyle\boxed{\
Q_{\xi}[g]=\frac{\sqrt{|g|}}{\kappa}\,(\nabla^{\mu}\xi^{\nu})\,(d^{n-2}x)_{\mu\nu}=\frac{1}{2}\,\frac{\sqrt{|g|}}{\kappa}\,(\nabla^{\mu}\xi^{\nu}-\nabla^{\nu}\xi^{\mu})\,(d^{n-2}x)_{\mu\nu}\
}\,.$ (7.77)

This expression obviously agrees with the _Komar integrand_ which also results
from the line of arguments followed in the covariant phase space approach, see
equations (5.63)-(5.64).

The so-called _Iyer-Wald $(n-2)$-superpotential form_ associated to asymptotic
symmetries in a diffeomorphism invariant field theory is now defined by the
following expression [154, 249]:

$\displaystyle\mbox{Iyer-Wald $(n-2)$-superpotential form :}\qquad\boxed{\
k^{IW}_{\xi}\equiv-\delta Q_{\xi}+i_{\xi}j\,\in\Omega^{n-2,1}\ }\,,$ (7.78)

or, more explicitly,

$k^{IW}_{\xi}[\varphi;\delta\varphi]\equiv-\delta\,Q_{\xi}[\varphi]+i_{\xi}j[\varphi;\delta\varphi]\,.$

Here, $i_{\xi}j$ is the interior product of the form $j\in\Omega^{n-1,1}$ with
respect to the vector field $\xi$ and $\delta$ denotes the variation of forms
induced by the field variations $\delta\varphi^{a}$. The $(n-2)$-form (7.78)
is defined up to a $d_{\tt h}$-exact term262626Another ambiguity results from
the freedom to add a term $d_{\tt h}\alpha$ to the (pre-)symplectic potential
as discussed after Eqn. (6.29). and it involves an extra term $Q_{\delta\xi}$
if the asymptotic Killing vector fields $\xi$ depend on the fields. _For
general relativity_ (possibly including a cosmological constant), the
expressions (7.76)-(7.78) lead to [249]

$\displaystyle
k^{IW}_{\xi}[g;h]=\frac{\sqrt{|g|}}{\kappa}\,\Big{[}\xi^{\mu}\nabla_{\sigma}h^{\sigma\nu}-\xi^{\mu}\nabla^{\nu}h+\xi_{\sigma}\nabla^{\nu}h^{\mu\sigma}+\frac{1}{2}\,h\nabla^{\nu}\xi^{\mu}-h^{\nu\sigma}\nabla_{\sigma}\xi^{\mu}\Big{]}\,(d^{n-2}x)_{\mu\nu}\,.$
(7.79)

##### Comparison of expressions:

The relationship between the superpotentials $k^{BB}_{\xi}$ of Barnich-Brandt
and $k^{IW}_{\xi}$ of Iyer-Wald has been addressed in references [154, 157,
249]. In this respect one introduces the form [157, 249]

$E[\varphi;\delta\varphi;\delta\varphi]\equiv-\frac{1}{2}\,I^{n-1}_{\delta\varphi}j=-\frac{1}{2}\,I^{n-1}_{\delta\varphi}I^{n}_{\delta\varphi}({\cal
L}\,d^{n}x)\,\in\Omega^{n-2,2}\,.$

Up to an $d_{\tt h}$-exact $(n-2)$-form, one then has the relation

$\boxed{\ k^{IW}_{\xi}[\varphi;\delta\varphi]\approx
k^{BB}_{\xi}[\varphi;\delta\varphi]+E[\varphi;\delta\varphi;L_{\xi}\varphi]\
}\,.$

E.g. _for general relativity_ where $k^{BB}_{\xi}$ and $k^{IW}_{\xi}$ are
given by (7.71) and (7.79), respectively, one has

$E[g;\delta g;\delta g]=\frac{\sqrt{|g|}}{4\kappa}\,{\delta
g^{\mu}}_{\sigma}\wedge\delta g^{\sigma\nu}\,(d^{n-2}x)_{\mu\nu}\,,$

hence

$E[g;\delta
g;L_{\xi}g]=-\frac{\sqrt{|g|}}{2\kappa}\,\big{[}\nabla^{\mu}\xi_{\sigma}+\nabla_{\sigma}\xi^{\mu}\big{]}\,\delta
g^{\sigma\nu}\,(d^{n-2}x)_{\mu\nu}\,.$

We note that the latter expression (introduced in reference [154]) vanishes
for Killing vector fields $\xi^{\mu}\partial_{\mu}$ as well as for some
particular gauge choices for the metric field $(g_{\mu\nu})$, e.g. see
reference [249]. The relationships with the covariant phase expressions of B.
Julia and S. Silva [205] is discussed in appendix A of [157]. The case of
spatially bounded regions was addressed in particular by the authors of
reference [265], see also [266, 267]. For the _algebra of charges_ (in
particular in the presence of gravitational radiation) we refer to [268] (see
also [242]).

####  7.6.2 Cohomological methods 2 : antifield formalism

An alternative algebraic approach to the determination of lower-dimensional
conservation laws in gauge field-type theories was recently discussed in
reference [242]. It relies on the descent equations appearing in the extension
of the variational bicomplex [240, 269] which is brought about the
introduction of antifields into gauge field theories in view of their
quantization along the lines of the Batalin-Vilkovisky (BV) formalism.

##### Summary of antifield formalism:

We recall that the BV approach [270, 23, 271, 272, 273, 74, 274] amounts to a
symplectic-type reformulation and generalization of the usual BRST
quantization procedure [275, 276] for which the interplay between Lagrangian
symmetries and equations of motion plays a central role. In the following, we
outline some of its basic points for the case of an irreducible gauge theory
[23] while taking pure YM-theory in four-dimensional Minkowski space-time as
an explicit example. Thus, we consider a compact matrix Lie group $G$ and an
anti-Hermitian basis $(T_{r})$ of the associated Lie algebra $\mathfrak{g}$
satisfying

$[T_{r},T_{s}]=f_{rst}T_{t}\,,\qquad\textrm{Tr}\,(T_{r}T_{s})=\delta_{rs}\,.$

The basic fields $(\varphi^{a})$ are presently given by the components of the
connection $1$-form $A\equiv A_{\mu}dx^{\mu}$ with $A_{\mu}(x)\equiv
A^{r}_{\mu}(x)T_{r}$. The associated curvature $2$-form reads $F\equiv
dA+\frac{g}{2}\,[A,A]\equiv\frac{1}{2}\,F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$
with $F_{\mu\nu}(x)\equiv F_{\mu\nu}^{r}(x)T_{r}$ and $g$ denoting the
coupling constant of YM-theory. The _action functional_ (cf. Eqn. (6.19))

$S_{\textrm{inv}}[A]\equiv\frac{1}{2}\,\int_{M}\textrm{Tr}\,(F\wedge\star
F)=-\frac{1}{4}\,\int_{M}d^{4}x\,\textrm{Tr}\,(F^{\mu\nu}F_{\mu\nu})\,,$

is invariant under local gauge transformations which are given at the
infinitesimal level by $\delta A=Df\equiv df+g[A,f]$ where $x\mapsto
f(x)\equiv f^{r}(x)T_{r}$ denotes a smooth Lie algebra-valued function.

In the BRST approach to the quantization of gauge field theories, the symmetry
parameter $f$ is turned into a so-called _Faddeev-Popov ghost field_ $c$ of
ghost-number one. For any Lie algebra-valued $p$-form of ghost number
$g\in\mathbb{Z}$, i.e. $\alpha^{p}_{g}\in\Omega^{p}_{g}(M,\mathfrak{g})$, the
_total degree_ is defined by

$\widetilde{\alpha^{p}_{g}}\equiv p+g\,.$

This $\mathbb{Z}$-grading induces a _$\mathbb{Z}_{2}$ -grading_ or _Grassmann
parity_

$\textrm{grading}\,(\alpha^{p}_{g})\equiv(-1)^{\widetilde{\alpha^{p}_{g}}}\,.$

The commutator $[\alpha,\beta]$ of two Lie algebra-valued forms $\alpha,\beta$
is now supposed to be graded, i.e.

$[\alpha,\beta]\equiv\alpha\wedge\beta-(-1)^{\tilde{\alpha}\tilde{\beta}}\beta\wedge\alpha\,.$

Finally, the _BRST transformations_ of $A$ and $c$ are defined by

$\displaystyle sA=-Dc\,,\qquad sc=-\frac{1}{2}\,g\,[c,c]\,,$ (7.80)

hence $sA_{\mu}=D_{\mu}c$ by virtue of
$dc=dx^{\mu}\partial_{\mu}c=-(\partial_{\mu}c)\,dx^{\mu}$. The so-defined
_BRST operator_ $s$ is assumed to anticommute with the exterior derivative $d$
(i.e. their graded commutator $[s,d]=sd+ds$ vanishes) and it acts on the
algebra of fields as a graded derivation which increases the ghost-number by
one. Its nilpotency, i.e. $s^{2}=0$, reflects the closure of the gauge
algebra. Since the action of the $s$-operator on $A$ amounts to an
infinitesimal gauge transformation (with symmetry parameter replaced by ghost
field), it leaves the action $S_{\textrm{inv}}[A]$ invariant. In summary, the
first transformation law in (7.80) describes the local symmetries of the
action $S_{\textrm{inv}}[A]$ and the second one reflects the non-trivial (non-
Abelian) structure of the gauge algebra.

The gauge fixing procedure (which is considered for obtaining a well-defined
propagator for the gauge field) [276] leads to the addition of the _Faddeev-
Popov ghost term_

$\int_{M}d^{4}x\,\textrm{Tr}\,(\bar{c}\,Dc)=-\int_{M}d^{4}x\,\textrm{Tr}\,(\bar{c}\,sA)\,,$

to the $s$-invariant action $S_{\textrm{inv}}[A]$. The BV formalism in
$n$-dimensional space-time now consist of associating a so-called _antifield_
$\Phi^{*}$ to each field $\Phi\in\\{A,c\\}$ (i.e. to the basic field
$\varphi=A$ as well as to the ghost field $c$) such that
$\textrm{Tr}\,(\Phi^{*}s\Phi)$ represents a $n$-form of ghost-number zero.
Thus, if $\Phi$ is a $p$-form with ghost-number $g$, i.e.
$\textrm{gh}\,(\Phi)=g$, then $\Phi^{*}$ is a $(n-p)$-form with
$\textrm{gh}\,(\Phi^{*})=-(g+1)$. In particular, for YM-theory in four
dimensions we have

$\displaystyle A\in\Omega^{1}_{0}(M,\mathfrak{g})\,,\qquad\quad\
c\in\Omega^{0}_{1}(M,\mathfrak{g})\,,$ $\displaystyle
A^{*}\in\Omega^{3}_{-1}(M,\mathfrak{g})\,,\qquad C^{*}\equiv
c^{*}dx^{0}\wedge\cdots\wedge dx^{3}\in\Omega^{4}_{-2}(M,\mathfrak{g})\,.$

Accordingly, $A^{*}$ corresponds (by Hodge duality) to a $\mathfrak{g}$-valued
vector field $(A_{\mu}^{*})$ of ghost-number $-1$ and $C^{*}$ to the
$\mathfrak{g}$-valued function $c^{*}$ of ghost-number $-2$. For gauge field-
type theories with _closed_ algebras involving structure _constants_ (like YM-
theories) or field-independent structure operators (like general relativity),
the so-called

minimal action $\displaystyle\qquad\boxed{\ S[\Phi,\Phi^{*}]\equiv
S_{\textrm{inv}}[A]+\sum_{\Phi}\int_{M}\textrm{Tr}\,(\Phi^{*}s\Phi)\equiv\int_{M}d^{4}x\,{\cal
L}\ }\,,$ (7.81)

i.e. for YM-theories

$\displaystyle{\cal
L}=\textrm{Tr}\,\Big{[}-\frac{1}{4}\,F^{\mu\nu}F_{\mu\nu}+A_{\mu}^{*}D^{\mu}c+c^{*}(-\frac{g}{2}\,[c,c])\Big{]}\,,$
(7.82)

is a solution of the _(non-linear) Slavnov-Taylor identity_ or

$\displaystyle\mbox{BV master equation}\qquad
0=\frac{1}{2}\,[\\![S,S]\\!]=\sum_{\Phi}\int_{M}\frac{\delta
S}{\delta\Phi^{*}}\,\frac{\delta S}{\delta\Phi}\,.$ (7.83)

Here, the _BV bracket_ (or _antibracket_) of two functionals $X,Y$ of the
variables $\Phi,\Phi^{*}$ (viewed as differential forms on $M$) is the graded
bracket defined as follows in $n$ dimensions272727All of our functional
derivatives are left derivatives [272], but we note that the BV bracket is
traditionally expressed in terms of both left and right derivatives which
implies the presence of other sign factors.:

$\displaystyle\mbox{BV bracket}\qquad\boxed{\
[\\![X,Y]\\!]=\sum_{\Phi}\int_{M}\left[(-1)^{\tilde{X}\widetilde{\Phi^{*}}}\frac{\delta
X}{\delta\Phi^{*}}\,\frac{\delta
Y}{\delta\Phi}+(-1)^{\tilde{X}\tilde{\Phi}+n(\tilde{\Phi}+1)}\frac{\delta
X}{\delta\Phi}\,\frac{\delta Y}{\delta\Phi^{*}}\right]\ }\,.$ (7.84)

We remark that the grading of the functional derivative $\frac{\delta
X}{\delta\Phi}$ is given by $(-1)^{\tilde{X}+\tilde{\Phi}+n}$ whence the last
equality in Eqn. (7.83). From (7.84) we deduce in particular that
$(\Phi,\Phi^{*})$ can be viewed as a conjugate pair with respect to the BV
bracket:

$\displaystyle[\\![\Phi^{*}(x),\Phi(y)]\\!]=(-1)^{\widetilde{\Phi^{*}}}\,\delta(x-y)\,.$
(7.85)

The BV bracket $(X,Y)\mapsto[\\![X,Y]\\!]$ on the graded vector space of
functionals $(\Phi,\Phi^{*})\mapsto F[\Phi,\Phi^{*}]$ (graded by the ghost-
number) represents a _graded Poisson structure of degree one,_ i.e. we have
the homogeneity or

$\mbox{grading
property}\qquad\textrm{gh}\,[\\![X,Y]\\!]=\textrm{gh}\,(X)+\textrm{gh}\,(Y)+1\,,$

as well as the properties of $\mathbb{R}$-bilinearity, graded antisymmetry,
graded Jacobi identity and graded derivation (Leibniz) rule.

From (7.84) we can also infer that the _“linearized Slavnov-Taylor operator”
${\cal S}_{S}$ associated to the minimal action $S$,_ i.e. the infinitesimal
canonical transformation induced by $S$ and the BV bracket,

$\displaystyle{\cal S}_{S}\equiv[\\![S,\boldsymbol{\cdot}\,]\\!]\,,$ (7.86)

acts on fields and antifields (in four dimensions) by the

$\displaystyle\mbox{BRST transformations}\qquad\boxed{\ {\cal
S}_{S}\Phi=\frac{\delta S}{\delta\Phi^{*}}=s\Phi\,,\qquad{\cal
S}_{S}\Phi^{*}=\frac{\delta S}{\delta\Phi}\equiv s\Phi^{*}\ }\,.$ (7.87)

Here, the last equality of the first relation follows from the expression
(7.81) of the minimal action $S$ whereas the last equality in the second
relation amounts to the extension of the BRST operator $s$ from fields (as
defined by Eqn. (7.80)) to antifields. _The operator ${\cal S}_{S}$ raises the
ghost-number by one unit, it is nilpotent and it leaves the minimal action $S$
invariant._ (The nilpotency of ${\cal S}_{S}$ readily follows from the graded
Jacobi identity satisfied by the BV bracket and from the BV master equation
(7.83).) If all antifields are set to zero (along with their $s$-variations),
then $S[\Phi,\Phi^{*}]$ reduces to $S_{\textrm{inv}}[A]$ and we recover the
initial BRST transformations (7.80) together with the equation of motion of
the basic field $A$.

The ghosts and antifields are incorporated into the variational bicomplex
formulation by extending the double complex
$(\Omega^{\bullet,\bullet}(J^{\infty}E),d_{\tt h},d_{\tt v})$ to the

$\displaystyle\boxed{\ \mbox{variational
tricomplex}\qquad\left(\Omega^{\bullet,\bullet}_{\bullet},d_{\tt h},d_{\tt
v},\delta_{Q}\right)\ }\,.$ (7.88)

Here, $\Omega^{\bullet,\bullet}_{\bullet}$ is a short-hand notation for
$\Omega^{\bullet,\bullet}_{\bullet}(J^{\infty}E)$ where the field space $E$ is
now assumed to be $\mathbb{Z}$-graded by the ghost-number. The operator
$\delta_{Q}$ represents a further differential (anticommuting with both
$d_{\tt h}$ and $d_{\tt v}$) to be identified with the BRST operator [240,
269, 242] which increases the ghost-number by one unit.

The BV bracket is a (non-degenerate) graded Poisson bracket and thus
corresponds to a _graded symplectic form_

$\displaystyle\Omega\equiv\int_{M}\omega\,,$ (7.89)

where the local form $\omega\in\Omega^{n,2}_{-1}$ has the expression (cf. Eqn.
(7.85))

$\displaystyle\boxed{\
\omega=\sum_{\Phi}\textrm{Tr}\,[\delta\Phi\wedge\delta\Phi^{*}]\ }\,.$ (7.90)

For four-dimensional YM-theory, we obtain (with $C\equiv c$)

$\displaystyle\omega=\textrm{Tr}\,[\delta A\wedge\delta A^{*}+\delta
C\wedge\delta C^{*}]\,,$ (7.91)

i.e.

$\displaystyle\omega=\hat{\omega}\,d^{4}x\qquad\mbox{with}\quad\hat{\omega}=\textrm{Tr}\,[\delta
A^{\mu}\wedge\delta A_{\mu}^{*}+\delta c\wedge\delta
c^{*}]\,,\quad\textrm{gh}\,(\hat{\omega})=-1\,.$ (7.92)

We note [248] that, from a geometric point of view, the fields $\Phi$ and
antifields $\Phi^{*}$ can be viewed, respectively, as base and fibre
coordinates of the _odd cotangent bundle_ $T^{*}[-1]{\cal E}$: here, ${\cal
E}$ denotes the graded (by the ghost-number) vector space of fields
$x\mapsto\Phi(x)\in E$ on space-time and $-1$ represents the shift
$g\leadsto-(g+1)$ of ghost-number which occurs upon passage from fields to
antifields. With this geometric interpretation, the two-form (7.89) defined by
Eqn. (7.90) represents the _canonical shifted symplectic structure on
$T^{*}[-1]{\cal E}$._

##### Application of antifield formalism:

We will now make contact with the local symplectic form
$J=J^{\mu}\,d^{3}x_{\mu}$ encountered in the covariant phase space approach.
Application of the first variational formula (5.1) to the minimal Lagrangian
(7.82) yields

$\displaystyle\delta{\cal L}\approx\partial_{\mu}j^{\mu}\qquad\mbox{with}\quad
j^{\mu}=\textrm{Tr}\,[-F^{\mu\nu}\delta A_{\nu}+A^{*\mu}\delta c]\,,$ (7.93)

hence

$\displaystyle\boxed{\ J^{\mu}\equiv-\delta j^{\mu}=\textrm{Tr}\,[\delta
A_{\nu}\wedge\delta F^{\nu\mu}+\delta c\wedge\delta A^{*\mu}]\ }\,.$ (7.94)

The fact that this expression extends the one obtained for YM-theory without
antifields (see (5.16)) is due to the “boundary condition” ${\cal
L}(\Phi,\Phi^{*}=0)={\cal L}_{\textrm{inv}}(A)$. Equivalently [269, 242], the
_symplectic current density_ $(J^{\mu})$ can be extracted from the graded
symplectic structure (7.92) corresponding to the BV bracket by application of
the BRST differential $\delta_{Q}$: with (7.87) we have

$\displaystyle\delta_{Q}A_{\mu}$
$\displaystyle=D_{\mu}c\,,\qquad\qquad\qquad\delta_{Q}c=-\frac{g}{2}\,[c,c]\,,$
(7.95) $\displaystyle\delta_{Q}A^{*\mu}$ $\displaystyle=\frac{\delta S}{\delta
A_{\mu}}=D_{\nu}F^{\nu\mu}-g\,[A^{*\mu},c]\,,\qquad\delta_{Q}c^{*}=\frac{\delta
S}{\delta c}=D^{\mu}A_{\mu}^{*}+g\,[c^{*},c]\,,$

and a short calculation leads to the result

$\displaystyle\delta_{Q}\hat{\omega}$
$\displaystyle=-\partial_{\mu}J^{\mu}\,,\qquad\ \mbox{with}\quad
J^{\mu}\equiv\textrm{Tr}\,[\delta A_{\nu}\wedge\delta F^{\nu\mu}+\delta
c\wedge\delta A^{*\mu}]\,,\quad\ \textrm{gh}\,(J^{\mu})=0\,,$