# Lie and Leibniz Algebras
of Lower-Degree Conservation Laws

Boris M. Elfimov & Alexey A. Sharapov

(Department of Quantum Field Theory, Tomsk State University,
Lenin ave. 36, Tomsk 634050, Russia)

###### Abstract

A relationship between the asymptotic and lower-degree conservation laws in
(non-) linear gauge theories is considered. We show that the true algebraic
structure underlying asymptotic charges is that of Leibniz rather than Lie.
The Leibniz product is defined through the derived bracket construction for
the natural Poisson brackets and the BRST differential. Only in particular,
though not rare, cases that the Poisson brackets of lower-degree conservation
laws vanish modulo central charges, the corresponding Leibniz algebra
degenerates into a Lie one. The general construction is illustrated by two
standard examples: Yang-Mills theory and Einstein’s gravity.

##  1 Introduction

Although exact symmetries seldom if ever occur in nature, they do play a
significant role in our understanding of physical laws. The famous Noether’s
theorem, for example, establishes a correspondence between global symmetries
and conservation laws in Lagrangian theories. Local or gauge symmetries, in
turn, govern the structure of fundamental interactions in the Standard Model
of particle physics and General Relativity. Unlike global symmetries, gauge
invariance never leads to nontrivial conserved currents, as was found by Emmy
Noether herself. Yet, in some cases, it is possible to associate certain
conserved charges with gauge symmetries as well. These are given by integrals
over lower-dimensional surfaces rather than entire physical space. Therefore
one calls them the lower-degree conservation laws. For a detailed account of
the subject we refer the reader to [1, 2, 3, 4, 5] and references therein.

It is not rare in field theory to consider a situation where the interaction
of fields is presumably localized in a compact region of space, outside of
which the fields behave as almost free. Such a supposition is a departing
point of the scattering problem and an indispensable element of the multi-
particle interpretation of quantum field states. In that case, one can compute
the lower-degree conserved charges by integrating over surfaces lying entirely
outside the interaction region. As the fields under the integral sign are
supposed to be almost free, one may disregard much of the nonlinearities in
the Lagrangian and charges. Not only does this choice of integration surface
facilitate the actual computation of charges, but it also leads to a further
generalization of the very notion of a conservation law. Indeed, if a
nonlinear gauge theory admits a reasonable linearization, it may happen that
the linear theory enjoys lower-degree conservation laws that are not present
in the full nonlinear theory. Then one may attribute the corresponding lower-
degree charges to the nonlinear theory itself or, more precisely, to those its
solutions that differ arbitrary little from solutions of the linear theory
outside a compact region of space (the interaction region). In the physical
literature, one usually refers to the charges assigned in such a way to a
nonlinear gauge theory as asymptotic charges. The ADM energy in general
relativity and color charges in Yang–Mills theory are prime examples of
asymptotic charges defined through the integration over two-sphere at space
infinity. For an extended discussion of the above approach to asymptotic
conservation laws see [3, 6].

In Lagrangian theories, both the conventional and lower-degree conservation
laws come equipped with a Lie algebra structure w.r.t. some natural Poisson
brackets (canonical or covariant) [7, 8, 9]. Modulo central extension, this
algebra is known to be isomorphic to the Lie algebra of infinitesimal
symmetries. If asymptotic charges are now defined through the linearization
procedure above and do not survive in the full nonlinear theory, one runs into
a problem: Linearization usually implies abelization of the gauge symmetry
algebra, so that the corresponding asymptotic charges may form either an
abelian Lie algebra or a central extension thereof, whatever a nonlinear gauge
theory. This conclusion, however, contradicts to some other approaches to
asymptotic conservation laws that do not appeal directly to linearization [10,
11, 12, 13]. In the general case one expects the nonabelian gauge symmetries
to generate nonabelian Lie algebras of surface charges modulo central
extension. We thus face the following dilemma: a nonabelian gauge theory per
se may have no nontrivial conservation laws of lower degree, while the lower-
degree conservation laws resulting from its linearization cannot form a
nonabelian Lie algebra w.r.t. Poisson brackets.

In this paper, we resolve this puzzle by introducing a new product on the
space of lower-degree conservation laws of a linearized gauge theory. The
product is constructed as a derived bracket [14] and involves the original
Poisson brackets together with the classical BRST differential. It is the
dependence of interaction through the BRST differential that restores a
nonabelian algebra structure on conserved charges. Generally the product we
introduce is not skew-symmetric but satisfies the axioms of a Leibniz algebra.
This suggests that a ‘genuine’ algebraic structure underlying asymptotic
symmetries and conservation laws is that of Leibniz rather than Lie111An
instructive discussion of the origin of Leibniz algebras in the context of
gauge symmetries can be found in [15, Sec. 1].. In order to test and
illustrate our construction we re-derive the well-known algebras of surface
charges in Einstein’s gravity and Yang-Mills theory.

Compared to other approaches our method is purely algebraic: we are not
concerned with the fall-off of fields at infinity or suitable boundary
conditions, nor do we make any assumption about the differential structure of
field equations and/or gauge generators. What we actually use is the
separation of field equations into a free part and interaction. The separation
is clearly ambiguous and we regard it as part of the definition of a classical
field theory. Another advantage of our construction is that it applies equally
well to non-Lagrangian equations of motion endowed with a compatible
presymplectic structure. The presence of a presymplectic structure is known to
be much less restrictive for dynamics than the existence of a Lagrangian.

##  2 Variational tricomplex of a gauge system

This section provides a brief glossary on local gauge systems in the formalism
of variational tricomplex. For a more coherent exposition of these concepts we
refer the reader to [4, 16, 17, 7]. Throughout the paper, we systematically
use the notation and terminology of [5].

#### Classical fields.

In modern language classical fields are sections of a locally trivial fiber
bundle $\pi:E\rightarrow M$ over a spacetime manifold $M$. The typical fiber
of $E$ is called the target space of fields. The space of all field
configurations is thus identified with the space of smooth sections
$\Gamma(E)$. In this paper, we restrict ourselves to the case where $E$ is a
$\mathbb{Z}$-graded vector bundle over $M$. As is customary in the physical
literature, we will refer to this $\mathbb{Z}$-grading as the ghost number and
denote the degree of a homogeneous object $A$ by $\mathrm{gh}(A)$. The
Grassmann parity of fields (which governs the signs) is given by the ghost
number modulo two. In physical terms this means that we restrict ourselves to
gauge theories without fermionic degrees of freedom. The extension of our
results to general theories with bosonic and fermionic fields is
straightforward.

#### Variational bicomplex.

According to the principle of spacetime locality, the classical dynamics of
fields are governed by partial differential equations. The jet-bundle
formalism offers then a natural geometric framework for formulating and
studying local field theories. A relevant jet-bundle for our considerations is
the bundle of infinite jets $\pi_{\infty}:J^{\infty}E\rightarrow M$ associated
with the vector bundle $\pi:E\rightarrow M$. Each section $\varphi$ of $E$
induces a section $j^{\infty}\varphi$ of $J^{\infty}E$ by the following rule.
If $E|_{U}=U\times\mathbb{R}^{N}$ is a trivializing chart with local
coordinates $(x^{i},\phi^{a})$, then
$(x^{i},\phi^{a},\phi^{a}_{i},\phi^{a}_{ij},\ldots)$ are local coordinates in
$J^{\infty}E|_{U}$ and the induced section $j^{\infty}\varphi:M\rightarrow
J^{\infty}E$ is defined by

$x\mapsto(x,\varphi^{a}(x),\partial_{i}\varphi^{a}(x),\partial_{i}\partial_{j}\varphi^{a}(x),\ldots)\,.$
(2.1)

The section $j^{\infty}\varphi$ is called the $\infty$-jet prolongation of
$\varphi$.

The total space of $J^{\infty}E$, defined through the inverse limit
$\lim\limits_{\leftarrow}J^{k}E$ of finite-dimensional jet-bundles, inherits
the structure of a $\mathbb{Z}$-graded manifold. Let
$\Lambda(J^{\infty}E)=\lim\limits_{\rightarrow}\Lambda(J^{k}E)$ denote the
algebra of differential forms on $J^{\infty}E$. It is known that the de Rham
complex of $\Lambda(J^{\infty}E)$ splits naturally into a bicomplex for the
vertical differential $\delta$ and horizontal differential $d$, so that

$\delta^{2}=0\,,\qquad d^{2}=0\,,\qquad\delta d+d\delta=0\,,$ (2.2)

with $\delta+d$ being the exterior differential in $\Lambda(J^{\infty}E)$. In
the adapted coordinates above, every form on $J^{\infty}E$ can be written as a
finite sum of homogeneous forms

$f\delta\phi^{a_{1}}_{I_{1}}\wedge\cdots\wedge\delta\phi^{a_{p}}_{I_{p}}\wedge
dx^{i_{1}}\wedge\cdots\wedge dx^{i_{q}}\in\Lambda^{p,q}(J^{\infty}E)\,,$ (2.3)

where $f$ is a smooth function on $J^{\infty}E|_{U}$ and $I=i_{1}i_{2}\cdots
i_{|I|}$ denotes the multi-index of order $|I|$. The numbers $p$ and $q$ are
called the vertical and horizontal degree of a form, respectively. Since the
vertical differential
$\delta:\Lambda^{p,q}(J^{\infty}E)\rightarrow\Lambda^{p+1,q}(J^{\infty}E)$
implements the action of the variational derivative on fields, one refers to
the bicomplex $\Lambda^{\ast,\ast}(J^{\infty}E;\delta,d)$ as variational.

#### Classical BRST differential.

An evolutionary vector field222Recall that a vertical vector field $X$ is
called evolutionary if $i_{X}d+(-1)^{\mathrm{gh}(X)}di_{X}=0$, where $i_{X}$
is the operation of contraction of $X$ with differential forms. The Lie
algebra of evolutionary vector fields will be denoted by
$\mathfrak{X}_{\mathrm{ev}}(J^{\infty}E)$. $Q$ on $J^{\infty}E$ is called
homological if

$[Q,Q]=0\,,\qquad\mathrm{gh}(Q)=1\,.$ (2.4)

In what follows we will use the symbol $\delta_{Q}$ to denote the Lie
derivative along $Q$. Clearly, $\delta_{Q}^{2}=0$. Moreover, the operator
$\delta_{Q}$ anticommutes with the differentials $d$ and $\delta$:

$\delta_{Q}d+d\delta_{Q}=0\,,\qquad\delta_{Q}\delta+\delta\delta_{Q}=0\,.$
(2.5)

The equalities follow from Cartan’s formula for the Lie derivative and the
fact that the vector field $Q$ is evolutionary. This yields the variational
tricomplex $\Lambda^{\ast,\ast,\ast}(J^{\infty}E;\delta,d,\delta_{Q})$, where
the third differential

$\delta_{Q}:\Lambda^{p,q,r}(J^{\infty}E)\rightarrow\Lambda^{p,q,r+1}(J^{\infty}E)$
(2.6)

increases the ghost number by one. Under certain ‘properness’ conditions (see
e.g. [16]) the operator $\delta_{Q}$ is called the classical BRST
differential. The corresponding homological vector field $Q$ carries all the
information about a classical gauge system. In particular, the true field
configurations are determined by the stationary points of $Q$. Under the
standard regularity assumptions the latter form a graded submanifold
$\Sigma^{\infty}\subset J^{\infty}E$ defined through the inverse limit. By
definition, a field $\varphi\in\Gamma(E)$ satisfies the classical equations of
motion iff $j^{\infty}\varphi\in\Sigma^{\infty}$.

In physics, the submanifold $\Sigma^{\infty}\subset J^{\infty}E$ is usually
referred to as the shell. By definition, the shell is invariant under the
action of the homological vector field $Q$. The restriction of the variational
bicomplex $\Lambda^{\ast,\ast}(J^{\infty}E;\delta,d)$ on $\Sigma^{\infty}$
gives the on-shell bicomplex $\Lambda^{\ast,\ast}(\Sigma^{\infty};\delta,d)$.
The latter inherits the additional grading by the ghost number. In general,
the on-shell bicomplex is not $d$-exact even locally and this gives rise to
interesting invariants associated with gauge dynamics. The most notable among
them are the cohomology groups $H^{0,\ast}(\Sigma^{\infty};d)$ in ghost number
zero. These are known as the characteristic cohomology of a gauge system [1,
2, 3, 4, 13, 18]. The study of natural algebraic structures on characteristic
cohomology is the main subject of the present paper.

#### Presymplectic structure.

Another geometric entity present in most gauge theories is called a
presymplectic structure. This is given by a form
$\omega\in\Lambda^{2,m}(J^{\infty}E)$ obeying the condition

$\delta\omega\simeq 0\,.$ (2.7)

Hereinafter the sign $\simeq$ means equality modulo $d$-exact forms. Notice
that the horizontal degree of a presymplectic form may take any value in the
interval $0\leq m\leq\dim M$. We also impose no restriction on the ghost
number of $\omega$. Two presymplectic forms are considered equivalent if they
differ by a $d$-exact form. By abuse of notation, we will not distinguish
between a presymplectic form $\omega$ and its equivalence class in
$\Lambda^{2,m}(J^{\infty}E)/d\Lambda^{2,m-1}(J^{\infty}E)$. As we are dealing
with a vector bundle $E$, the relative ‘$\delta$ modulo $d$’ cohomology
appears to be trivial in positive vertical degree [7, Sec.19.3.9]. Among other
things this means that each presymplectic structure has a $\delta$-exact
representative, i.e., there exists $\theta\in\Lambda^{1,m}(J^{\infty}E)$ such
that $\omega\simeq\delta\theta$. The form $\theta$ is called a presymplectic
potential. In what follows we will always work with $\delta$-exact
representatives $\omega=\delta\theta$ of presymplectic forms, so that
$\delta\omega=0$.

An evolutionary vector field $X$ is called Hamiltonian relative to $\omega$ if
$L_{X}\omega\simeq 0$. Again, the triviality of relative $\delta$-cohomology
implies that

$i_{X}\omega\simeq\delta\alpha$ (2.8)

for some $\alpha\in\Lambda^{0,m}(J^{\infty}E)$. It is natural to refer to the
form $\alpha$ as Hamiltonian or as a Hamiltonian of the vector field $X$.
Notice that Eq. (2.8) defines $\alpha$ up to a $d$-exact form; hence, two
Hamiltonians $\alpha$ and $\alpha^{\prime}$ are considered equivalent if
$\alpha\simeq\alpha^{\prime}$. The space of all Hamiltonian $m$-forms is a
graded Lie algebra with the bracket

$\\{\alpha,\beta\\}=(-1)^{\mathrm{gh}(X_{\alpha})}i_{X_{\alpha}}i_{X_{\beta}}\omega$
(2.9)

of degree $-\mathrm{gh}(\omega)$. Here $X_{\alpha}$ and $X_{\beta}$ are
Hamiltonian vector fields with the Hamiltonians $\alpha$ and $\beta$. The
bracket enjoys the symmetry property

$\\{\alpha,\beta\\}\simeq-(-1)^{(\mathrm{gh}(\alpha)-\mathrm{gh}(\omega))(\mathrm{gh}(\beta)-\mathrm{gh}(\omega))}\\{\beta,\alpha\\}$
(2.10)

and obeys the Jacobi identity

$\\{\gamma,\\{\alpha,\beta\\}\\}\simeq\\{\\{\gamma,\alpha\\},\beta\\}+(-1)^{(\mathrm{gh}(\gamma)-\mathrm{gh}(\omega))(\mathrm{gh}(\alpha)-\mathrm{gh}(\omega))}\\{\alpha,\\{\gamma,\beta\\}\\}\,.$
(2.11)

For more details see [9, Prop. 2.1].

#### Gauge systems and their descendants.

By a gauge system on $J^{\infty}E$ we mean a pair $(Q,\omega)$ composed of a
homological vector field $Q$ and a $Q$-invariant presymplectic form $\omega$
of type $(2,m)$. In other words, the vector field $Q$ is supposed to be
Hamiltonian relative to $\omega$, so that $\delta_{Q}\omega\simeq 0$. The last
relation is equivalent to

$\delta_{Q}\omega=d\omega_{1}$ (2.12)

for some $\omega_{1}\in\Lambda^{2,m-1}(J^{\infty}E)$. We will refer to the
horizontal degree of the form $\omega$ as the degree of a gauge system
$(Q,\omega)$. Applying the differentials $\delta$ and $\delta_{Q}$ to both
sides of (2.12) and using the acyclicity of $d$ in positive vertical degree,
we find that

$\delta\omega_{1}\simeq 0\,,\qquad\delta_{Q}\omega_{1}\simeq 0\,.$

Hence, $\omega_{1}$ is a $Q$-invariant presymplectic form of type $(2,m-1)$.
Again, without loss of generality we may assume $\omega_{1}=\delta\theta_{1}$
for some presymplectic potential $\theta_{1}\in\Lambda^{1,m-1}(J^{\infty}E)$.
We call the pair $(Q,\omega_{1})$ the descendent gauge system. Iterating the
above construction once and again, one can produce a sequence of gauge systems
$(Q,\omega_{k})$ where the $k$-th presymplectic structure
$\omega_{k}\in\Lambda^{2,m-k}(J^{\infty}E)$ is the descendant of
$\omega_{k-1}$. The minimal $k$ for which $\omega_{k}\simeq 0$ is called the
length of a gauge system.

#### Symmetries.

An evolutionary vector field $X$ is called a symmetry of a gauge system
$(Q,\omega)$ if it commutes with $Q$, i.e., $[X,Q]=0$. Therefore the flow
generated by $X$ preserves the shell $\Sigma^{\infty}\subset J^{\infty}E$
mapping solutions to solutions.

A symmetry $X$ is called trivial or gauge symmetry if their exists an
evolutionary vector field $Y$ such that $X=[Q,Y]$. Clearly, the gauge
symmetries form an ideal in the Lie algebra of all symmetries, so that one may
regard the corresponding quotient algebra, denoted by $\mathrm{Sym}(Q)$, as
the Lie algebra of nontrivial symmetries. Alternatively, we can identify the
nontrivial symmetries $\mathrm{Sym}(Q)$ with the cohomology of the
differential graded Lie algebra
$(\mathfrak{X}_{\mathrm{ev}}(J^{\infty}E),\delta_{Q})$.

A symmetry $X$ is called Hamiltonian if $X$ is a Hamiltonian vector field,
that is, $L_{X}\omega\simeq 0$. Finally, we say that a symmetry $X$ is on-
shell Hamiltonian if

$i_{X}\omega|_{\Sigma^{{}^{\infty}}}\simeq\delta\alpha|_{\Sigma^{{}^{\infty}}}$
(2.13)

for some $\alpha$. Read from right to left, this relation defines an on-shell
Hamiltonian form $\alpha$. Writing $\mathrm{Sym}(Q,\omega)$ and
$\mathrm{Sym}_{\Sigma}(Q,\omega)$ for the Lie algebras of Hamiltonian and on-
shell Hamiltonian symmetries, respectively, we get the following sequence of
subalgebras in the Lie algebra of vector fields:

$\mathrm{Sym}(Q,\omega)\subset\mathrm{Sym}_{\Sigma}(Q,\omega)\subset\mathrm{Sym}(Q)\subset\mathfrak{X}_{\mathrm{ev}}(J^{\infty}E)\subset\mathfrak{X}(J^{\infty}E)\,.$

When dealing with on-shell Hamiltonian symmetries and forms it is convenient
to introduce the following equivalence relation on the space of differential
forms $\Lambda(J^{\infty}E)$:

$\alpha\approx\alpha^{\prime}\quad\Leftrightarrow\quad\alpha|_{\Sigma^{{}^{\infty}}}\simeq\alpha^{\prime}|_{\Sigma^{{}^{\infty}}}.$
(2.14)

Then Eq. (2.13) takes the form $i_{X}\omega\approx\delta\alpha$. Although the
last equation implies $L_{X}\omega\approx 0$, the converse is not always true
as the on-shell bicomplex may not be $\delta$-exact even for vector bundles.
Notice that Eq. (2.13) defines $\alpha$ only modulo $d$-exact and on-shell
vanishing forms. Therefore, it makes sense to consider the equivalence classes
of on-shell Hamiltonian forms defined by Rel. (2.14). Due to the regularity
condition, this implies the existence of forms $\beta$ and $\gamma$ such that
$\alpha-\alpha^{\prime}=i_{Q}\beta+d\gamma$. It is significant that the
equivalence classes of on-shell Hamiltonian forms constitute a Lie algebra,
which we denote by $\Lambda^{H}(Q,\omega)$, for the same Lie bracket (2.9).
The last fact is quite obvious as we can identify the on-shell Hamiltonian
forms with the Hamiltonian forms on $\Sigma^{\infty}$ endowed with the induced
presymplectic structure $\omega|_{\Sigma^{{}^{\infty}}}$.

#### Conservation laws.

A form $\alpha\in\Lambda^{0,m}(J^{\infty}E)$ is said to define a conservation
law of degree $m$ if

$d\alpha|_{\Sigma^{{}^{\infty}}}=0\,.$ (2.15)

A conservation law is called trivial if $\alpha\approx 0$. In other words, the
space of nontrivial conservation laws of degree $m$ is identified with the
cohomology group $H^{0,m}(\Sigma^{{}^{\infty}};d)$ of the on-shell bicomplex.
Due to the standard regularity conditions on $Q$, Eq. (2.15) implies the
existence of a form $\chi\in\Lambda^{1,m+1}(J^{\infty}E)$ such that

$d\alpha=i_{Q}\chi\,.$ (2.16)

The form $\chi$ is called the characteristic of a conservation law $\alpha$.

Let $C\subset M$ be an $m$-cycle, $\alpha$ conservation law of degree $m$, and
$\varphi\in\Gamma(E)$ a solution to the field equations, then the functional

$I[\varphi]=\int_{C}(j^{\infty}\varphi)^{\ast}(\alpha)$ (2.17)

depends only on the homotopy class of $C$ in $M$ and is called the conserved
charge.

#### Flat gauge systems.

We say that a homological vector $Q$ field on $J^{\infty}E$ is flat  if
$i_{Q}\alpha|_{M}=0$ for all $\alpha\in\Lambda(J^{\infty}E)$. Geometrically,
this means that the submanifold $M\subset J^{\infty}E$, identified with the
zero section, belongs to the zero locus of $Q$. Similarly, a form
$\alpha\in\Lambda(J^{\infty}E)$ is called flat if $\alpha|_{M}=0$. The flat
forms constitute an ideal in the exterior algebra $\Lambda(J^{\infty}E)$ and a
subcomplex in the variational tricomplex whenever $Q$ is flat. Let us denote
the latter by
$\Lambda_{\mathrm{{}^{flat}}}^{\ast,\ast,\ast}(J^{\infty}E;\delta,d,\delta_{Q})$.
The horizontal, vertical, and relative cohomology of the corresponding flat
bicomplex $\Lambda_{\mathrm{{}^{flat}}}^{\ast,\ast}(J^{\infty}E;\delta,d)$ are
given by the groups

$\begin{array}[]{rl}H_{{}^{\mathrm{flat}}}^{\ast,p}(J^{\infty}E;d)=0\,,\quad
p<n\,,&\qquad
H_{{}^{\mathrm{flat}}}^{\ast,n}(J^{\infty}E;d)\simeq\Lambda^{\ast,n}_{\mathrm{{}^{flat}}}(J^{\infty}E)/d\Lambda^{\ast,n-1}_{\mathrm{{}^{flat}}}(J^{\infty}E)\,,\\\\[8.53581pt]
H_{{}^{\mathrm{flat}}}^{\ast,\ast}(J^{\infty}E;\delta)=0\,,&\qquad
H_{{}^{\mathrm{flat}}}^{\ast,\ast}(J^{\infty}E;\delta/d)=0\,,\end{array}$
(2.18)

$n$ being the dimension of the base manifold $M$. The proof can be found in
[7, Ch.19]. Amongst the elements of
$H_{{}^{\mathrm{flat}}}^{0,n}(J^{\infty}E;d)$ are the equivalence classes of
Lagrangians. We will say that $(Q,\omega)$ is a flat gauge system if $Q$ is
flat. From the physical viewpoint, flat gauge systems correspond to field
theory models without external sources.

##  3 Lie algebra of conservation laws

The presence of the zero section $j^{\infty}(0):M\rightarrow J^{\infty}E$
together with the canonical projection $\pi_{\infty}:J^{\infty}E\rightarrow M$
allows us to split the complex of purely horizontal forms into the direct sum

$\begin{array}[]{c}\Lambda^{0,\ast}(J^{\infty}E;d)=\Lambda^{0,\ast}_{{}^{\mathrm{flat}}}(J^{\infty}E;d)\bigoplus\pi^{\ast}_{\infty}\Lambda^{\ast}(M;d)\,.\end{array}$
(3.1)

For flat gauge systems, this results in the natural spitting of the cohomology
groups

$\begin{array}[]{c}H^{0,\ast}(\Sigma^{\infty};d)=H^{0,\ast}_{{}^{\mathrm{flat}}}(\Sigma^{\infty};d)\bigoplus
H^{\ast}(M;d)\end{array}$ (3.2)

associated with the equivalence classes of conservation laws. The second
summand in (3.2) owes its existence to the topology of $M$ rather than field
dynamics. Therefore, in the sequel, we will ignore it, focusing on the
subspace of flat conservation laws
$H^{0,\ast}_{{}^{\mathrm{flat}}}(\Sigma^{\infty};d)$. By definition, the
elements of the latter subspace vanish identically when evaluated on the zero
field configuration.

Let us denote by $\hat{\alpha}$ the projection of a purely horizontal form
$\alpha$ on the first summand in (3.1). The elements of the second summand in
(3.1), being in the kernel of the differential $\delta$, belong to the center
of the Lie algebra $\Lambda^{H}(Q,\omega)$ of on-shell Hamiltonian forms. This
gives the short exact sequence of Lie algebras

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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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0.0pt\raise
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161.92087pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\hat{\Lambda}{}^{H}(Q,\omega)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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where $i$ is the natural embedding and $p$ is the canonical projection induced
by the assignment $\alpha\mapsto\hat{\alpha}$. The exact sequence does not
split in general, in which case $\Lambda^{H}(Q,\omega)$ is a nontrivial
central extension of $\hat{\Lambda}{}^{H}(Q,\omega)$.

With the definitions above we are ready to formulate a generalization of
Noether’s first theorem to gauge systems.

######  Theorem 3.1 ([5]).

Let $(Q,\omega)$ be a flat gauge system of degree $m$. Then each Hamiltonian
symmetry $X$ gives rise to a uniquely defined sequence
$\alpha_{1},\alpha_{2},\ldots,\alpha_{m}$ of flat conservation laws. Trivial
symmetries generate trivial conservation laws.

###### Proof.

Let $\alpha_{0}$ denote the Hamiltonian of the symmetry $X$, then

$i_{X}\omega\simeq\delta\alpha_{0}\,.$ (3.4)

Applying $\delta_{Q}$ to this equality yields
$\delta\delta_{Q}\alpha_{0}\simeq 0$. Since the form $\delta_{Q}\alpha_{0}$ is
flat, we can write it as

$\delta_{Q}\alpha_{0}=d\alpha_{1}$ (3.5)

for some $\alpha_{1}\in\Lambda^{0,m-1}(J^{\infty}E)$. By definition,
$\alpha_{1}$ is a conservation law with characteristic $\delta\alpha_{0}$. It
follows from Eq. (3.5) that $d\delta_{Q}\alpha_{1}=0$. Since
$\delta_{Q}\alpha_{1}$ is flat, the triviality of the group
$H^{0,m-1}_{{}^{\mathrm{flat}}}(J^{\infty}E;d)$ implies the existence of a
form $\alpha_{2}$ such that $\delta_{Q}\alpha_{1}=d\alpha_{2}$. Hence,
$\alpha_{2}$ is a conservation law with characteristic $\delta\alpha_{1}$.
Iterating this construction once and again, we get the desired sequence of
conservation laws $\alpha_{1},\alpha_{2},\ldots,\alpha_{m}$. These are not
claimed to be all nontrivial; quite the opposite: if the $k$-th conservation
law happens to be trivial, $\alpha_{k}\approx 0$, then so are all its
descendants $\alpha_{k+1},\ldots,\alpha_{m}$.

If now $X=[Q,Y]$ for some Hamiltonian vector field $Y$, then

$\delta\alpha_{0}\simeq
i_{X}\omega\simeq\delta_{Q}i_{Y}\omega\simeq-\delta\delta_{Q}\beta\,,$

where $\beta$ is the Hamiltonian of $Y$. The group
$H^{0,m}_{{}^{\mathrm{flat}}}(J^{\infty}E;\delta/d)$ being trivial,
$\alpha_{0}=-\delta_{Q}\beta+d\gamma$ for some
$\gamma\in\Lambda^{0,m-1}(J^{\infty}E)$. Eq. (3.5) then gives
$d(\delta_{Q}\gamma-\alpha_{1})=0$. Since the group
$H^{0,m-1}_{{}^{\mathrm{flat}}}(J^{\infty}E;d)$ is zero, we conclude that
$\alpha_{1}\approx 0$ and the conservation law $\alpha_{1}$ is trivial
together with all its descendants. ∎

It is worth noting that the assumption of flatness is rather technical; in
many cases, one can replace it with other mild conditions that ‘kill’ the de
Rham cohomology of the base manifold $M$. In particular, the above theorem
holds for any nonflat $Q$ whenever $M$ is contractible.

######  Theorem 3.2 ([5]).

Let $\\{\alpha_{k}\\}$ be a sequence of conservation laws associated with a
Hamiltonian symmetry $X$. Then the form $\alpha_{k}$ is on-shell Hamiltonian
relative to the $k$-th descendent presymplectic structure $\omega_{k}$.

###### Proof.

As above, we let $\alpha_{0}$ denote the Hamiltonian of $X$. Then Eq. (3.4)
implies that

$i_{X}\omega=\delta\alpha_{0}+d\alpha^{\prime}_{0}$ (3.6)

for some $\alpha^{\prime}_{0}\in\Lambda^{1,m-1}(J^{\infty}E)$. Acting by
$\delta_{Q}$ on both sides of the equality, we find

$(-1)^{\mathrm{gh}(X)}i_{X}\delta_{Q}\omega=\delta\delta_{Q}\alpha_{0}+d\delta_{Q}\alpha^{\prime}_{0}\,.$
(3.7)

Combining this with Eqs. (2.12) and (3.5), we get
$di_{X}\omega_{1}=d\delta\alpha_{1}-d\delta_{Q}\alpha^{\prime}_{0}$. The
acyclicity of $d$ in positive vertical degree implies that

$i_{X}\omega_{1}=\delta\alpha_{1}-\delta_{Q}\alpha^{\prime}_{0}+d\alpha^{\prime}_{1}$
(3.8)

for some $\alpha_{1}^{\prime}\in\Lambda^{1,m-2}(J^{\infty}E)$. Therefore,
$i_{X}\omega_{1}\approx\delta\alpha_{1}$ and the form $\alpha_{1}$ is on-shell
Hamiltonian relative to the descendent presymplectic structure $\omega_{1}$.
Applying the operator $\delta_{Q}$ once again, we obtain from (3.8)

$(-1)^{\mathrm{gh}(X)}i_{X}\delta_{Q}\omega_{1}=\delta\delta_{Q}\alpha_{1}+d\delta_{Q}\alpha^{\prime}_{1}\,.$
(3.9)

This equation coincides in form with (3.7). Hence, there is a form
$\alpha^{\prime}_{2}$ such that

$i_{X}\omega_{2}=\delta\alpha_{2}-\delta_{Q}\alpha^{\prime}_{1}+d\alpha^{\prime}_{2}$
(3.10)

and the conservation law $\alpha_{2}$ is on-shell Hamiltonian w.r.t.
$\omega_{2}$. Proceeding in such a way, we obtain the sequence of relations
$i_{X}\omega_{k}=\delta\alpha_{k}-\delta_{Q}\alpha^{\prime}_{k-1}+d\alpha^{\prime}_{k}$,
which imply that all forms $\alpha_{k}$ are on-shell Hamiltonian.

∎

######  Theorem 3.3.

Let $\\{\alpha_{k}\\}$ be the sequence of flat conservation laws associated
with a Hamiltonian symmetry $X$, then the assignment $X\mapsto\alpha_{k}$
defines a Lie algebra homomorphism

$h_{k}:\mathrm{Sym}(Q,\omega)\rightarrow\hat{\Lambda}^{H}(Q,\omega_{k})\,.$
(3.11)

###### Proof.

For every pair of Hamiltonian symmetries $X,Y\in\mathrm{Sym}(Q,\omega)$ there
corresponds a Hamiltonian symmetry $[X,Y]$ satisfying the equation

$i_{[X,Y]}\omega\simeq\delta\\{\alpha_{0},\beta_{0}\\}\,.$

Here $\alpha_{0}$, $\beta_{0}$, and
$\\{\alpha_{0},\beta_{0}\\}=(-1)^{\mathrm{gh}(X)}i_{X}i_{Y}\omega$ are the
Hamiltonians of $X$, $Y$, and $[X,Y]$, respectively. Let $\\{\alpha_{k}\\}$
and $\\{\beta_{k}\\}$ denote the sequences of conservation laws associated
with $X$ and $Y$. By definition,

$\delta_{Q}\alpha_{k}=d\alpha_{k+1}\,,\qquad\delta_{Q}\beta_{k}=d\beta_{k+1}\,,\quad
k=0,1,\ldots,m\,.$

Applying the Lie derivative $L_{X}$ to the second sequence of equations, we
get

$\delta_{Q}L_{X}\beta_{k}=dL_{X}\beta_{k+1}\quad\Rightarrow\quad\delta_{Q}\\{\alpha_{k},\beta_{k}\\}=d\\{\alpha_{k+1},\beta_{k+1}\\}\,,$
(3.12)

where $\\{\alpha_{k},\beta_{k}\\}=(-1)^{\mathrm{gh}(X)}i_{X}i_{Y}\omega_{k}$.
For $k>0$, this yields a uniquely defined sequence
$\\{\alpha_{k},\beta_{k}\\}$ of flat conservation laws associated with the
Hamiltonian symmetry $[X,Y]$. ∎

The homomorphism (3.11) defines a Lie subalgebra
$\mathrm{Im}\,h_{k}\subset\hat{\Lambda}^{H}(Q,\omega_{k})$. By the $k$-th Lie
algebra of conservation laws, denoted by $\mathrm{CL}_{k}(Q,\omega)$, we will
understand the preimage of $\mathrm{Im}\,h_{k}$ in
$\Lambda^{H}(Q,\omega_{k})$, that is,
$\mathrm{CL}_{k}(Q,\omega)=p^{-1}(\mathrm{Im}\,h_{k})$. In view of (3.3), the
Lie algebra $\mathrm{CL}_{k}(Q,\omega)$ is nothing more than a central
extension of $\mathrm{Im}\,h_{k}$. A necessary condition for the existence of
a nontrivial central extension of $\mathrm{Im}\,h_{k}$ is that
$H^{m-k}(M;d)\neq 0$.

One more version of Noether’s first theorem is given by the next statement.

######  Proposition 3.4.

Let $(Q,\omega)$ be a regular gauge system. Suppose the following two
conditions are satisfied:

  * (i)

$d\omega|_{\Sigma^{{}^{\infty}}}=0$,

  * (ii)

$H_{\rm flat}^{0,\ast}(\Sigma^{\infty};\delta)=0$;

then every on-shell Hamiltonian form is a conservation law and Eq. (2.13)
establishes a homomorphism
$h:\mathrm{Sym}_{\Sigma}(Q,\omega)\rightarrow\Lambda^{H}(Q,\omega)$ of the Lie
algebras.

###### Proof.

The second part of the statement is obvious. To prove the first consider an
on-shell Hamiltonian vector field $X$ with Hamiltonian
$\alpha\in\Lambda^{H}(Q,\omega)$. Applying the horizontal differential $d$ to
both sides of Eq. (2.13), we get

$\delta
d\alpha|_{\Sigma^{{}^{\infty}}}=(-1)^{\mathrm{gh}X}i_{X}d\omega|_{\Sigma^{{}^{\infty}}}\,.$
(3.13)

Regularity implies that $d\omega=i_{Q}\beta+\delta_{Q}\gamma$ for some forms
$\beta$ and $\gamma$, whence

$i_{X}d\omega=i_{Q}i_{X}\beta-(-1)^{\mathrm{gh}X}\delta_{Q}i_{X}\gamma\,.$
(3.14)

Here we used the equality $[Q,X]=0$. Therefore,
$\delta(d\alpha)|_{\Sigma^{{}^{\infty}}}=0$. Condition (ii) then means that
$d\alpha|_{\Sigma^{{}^{\infty}}}=0$, i.e., $\alpha$ is a flat conservation
law. ∎

Notice that all descendent presymplectic structures $\omega_{k}$ meet
condition (i), see Eq. (2.12). Condition (ii) is fulfilled e.g. for
homogeneous linear systems of field equations; in that case, the components of
the homological vector field $Q$ are homogeneous linear functions of the
vertical coordinates $\phi^{a}_{ij\cdots}$.

######  Example 3.5.

All the notions above are best illustrated by the example of Chern–Simons
theory. Let us consider a $U(1)$-vector bundle over a three-dimensional
manifold $M$. For simplicity, we assume that the bundle is trivial. Then the
affine space of all $U(1)$-connections is naturally isomorphic to the space of
$1$-forms $\Lambda^{1}(M)$, that is, the sections of the cotangent bundle
$T^{\ast}M$. In the Batalin–Vilkovisky (BV) formalism333aka field-antifield
formalism, see e.g. [4]., the field spectrum of abelian Chern–Simons theory
includes the connection $1$-form $A\in\Lambda^{1}(M)$, ghost field
$C\in\Lambda^{0}(M)$ and their antifields $A^{\ast}\in\Lambda^{2}(M)$ and
$C^{\ast}\in\Lambda^{3}(M)$. By definition,

$\mathrm{gh}(C)=1\,,\qquad\mathrm{gh}(A)=0\,,\qquad\mathrm{gh}(A^{\ast})=-1\,,\qquad\mathrm{gh}(C^{\ast})=-2\,.$
(3.15)

The space of fields and antifields carries the canonical symplectic structure

$\omega=\delta A\wedge\delta A^{\ast}+\delta C\wedge\delta
C^{\ast}\,,\qquad\mathrm{gh}(\omega)=-1\,,$ (3.16)

of top horizontal degree. Here $\delta$ stands for the usual variational
differential on fields, whose properties are identical to those of the
vertical differential above, hence the notation. The action of the classical
BRST differential is given by

$\delta_{Q}C=0\,,\qquad\delta_{Q}A=dC\,,\qquad\delta_{Q}A^{\ast}=dA\,,\qquad\delta_{Q}C^{\ast}=dA^{\ast}\,.$
(3.17)

As is seen, only flat connections belong to the stationary surface of $Q$. The
zero-curvature equation $dA=0$ enjoys the standard gauge symmetry
$\delta_{\varepsilon}A=d\varepsilon$, which manifests itself through the
second equation in (3.17). The Hamiltonian of the homological vector field $Q$
is given by the BV master Lagrangian of Chern–Simons theory:

$i_{Q}\omega\simeq\delta L\,,\qquad L=\frac{1}{2}A\wedge dA+dC\wedge
A^{\ast}\,.$ (3.18)

Thus, Eqs. (3.16, 3.17) define a gauge system of degree three.

The canonical symplectic structure (3.16) generates the full sequence of
descendants:

$\begin{array}[]{lll}\delta_{Q}\omega=d\omega_{1}\,,&\omega_{1}=\frac{1}{2}\delta
A\wedge\delta A+\delta C\wedge\delta
A^{\ast}\,,&\mathrm{gh}(\omega_{1})=0\,,\\\\[8.53581pt]
\delta_{Q}\omega_{1}=d\omega_{2}\,,&\omega_{2}=\delta C\wedge\delta
A\,,&\mathrm{gh}(\omega_{2})=1\,,\\\\[8.53581pt]
\delta_{Q}\omega_{2}=d\omega_{3}\,,&\omega_{3}=\frac{1}{2}\delta C\wedge\delta
C\,,&\mathrm{gh}(\omega_{3})=2\,.\end{array}$ (3.19)

Hence, the length of Chern–Simons theory is four. Notice that $\omega_{1}$
coincides with the canonical presymplectic structure associated with the
Lagrangian (3.18). The classical BRST differential enjoys a nontrivial global
symmetry $Y$ in ghost number $-1$, whose action on fields is given by

$L_{Y}C=1\,,\qquad L_{Y}A=0\,,\qquad L_{Y}A^{\ast}=0\,,\qquad
L_{Y}C^{\ast}=0\,.$ (3.20)

The variational vector field $Y$ is Hamiltonian, $i_{Y}\omega=\delta
C^{\ast}$, and gives rise to the sequence of conservation laws $A^{\ast}$,
$A$, and $C$. The latter fact is readily seen from (3.17). The corresponding
Lie brackets read

$\\{A^{\ast},A^{\ast}\\}_{1}=0\,,\qquad\\{A,A\\}_{2}=0\,,\qquad\\{C,C\\}_{3}=1\,.$
(3.21)

The last equality exemplifies the phenomenon of central extension: the abelian
subalgebra $[Y,Y]=0$ of global symmetries gets a nontrivial central extension
$1\in H^{0}(M;d)$ when evaluated on the corresponding conservation laws. Among
the conservation laws above only the $1$-form $A$ has ghost number zero.
Integrating it over a loop $\gamma\subset M$ yields the conserved charge
$I[A]=\oint_{\gamma}A$, which is nothing but the holonomy invariant of the
flat connection $A$.

##  4 Leibniz algebras of symmetries and conservation laws

Most of the gauge systems encountered in physics depend on some numerical
parameters such as masses of particles, coupling constants, etc. This
motivates us to consider families of gauge systems rather than isolated
systems. For the sake of simplicity we will restrict ourselves to one-
parameter families $(Q_{t},\omega_{t})$. Also, it is convenient to be a bit
sloppy about the class of functions of $t$. The homological vector field
$Q_{t}$, for instance, may be a smooth function of $t$ or a formal power
series

$Q_{t}=Q_{0}+tQ_{1}+t^{2}Q_{2}+\cdots\,.$ (4.1)

Such formal expansions are at the heart of perturbation theory: If one regards
$t$ as a coupling constant, then the leading term $Q_{0}$ defines a ‘free
gauge system’ $(Q_{0},\omega_{0})$, while the higher order terms
$Q_{1},Q_{2},\ldots$ describe a ‘consistent interaction’. In many instances,
the corresponding presymplectic form $\omega_{t}$ does not depend on $t$ at
all.

Even though the gauge system $(Q_{t},\omega_{t})$ varies ‘smoothly’ with $t$,
this may not be the case with the corresponding cohomology groups. For
example, the inclusion of interaction may violate some nontrivial symmetries
of a free gauge system. It is the differences in cohomology for different
$t$’s that give rise to interesting algebraic constructions that we consider
below.

#### Symmetries.

Differentiating the defining equality $[Q_{t},Q_{t}]=0$, we obtain

$[Q_{t},\dot{Q}_{t}]=0\,,\qquad[\dot{Q}_{t},\dot{Q}_{t}]=-[Q_{t},\ddot{Q}_{t}]\,,$
(4.2)

where the overdots stand for the $t$-derivatives. The first equation says that
the evolutionary vector field $\dot{Q}_{t}$ is $Q_{t}$-invariant; hence, it
defines an odd symmetry of the gauge system $(Q_{t},\omega_{t})$. According to
the second equation, the symmetry $\dot{Q}_{t}$ always squares to a trivial
symmetry. Thus, our first observation is that any one-parameter family of
gauge systems possesses a canonical (perhaps trivial) symmetry generated by
$\dot{Q}_{t}$. Furthermore, this symmetry makes $\mathrm{Sym}(Q_{t})$ into a
differential graded Lie algebra with the deferential

$\partial_{t}\tilde{X}=\widetilde{[\dot{Q}_{t},X]}\,,\qquad\partial_{t}^{2}=0\,.$
(4.3)

Here $\tilde{X}$ denotes the equivalence class of the symmetry generated by a
$Q$-invariant vector field $X\in\mathfrak{X}_{\mathrm{ev}}(J^{\infty}E)$.

The main purpose of this section is to equip the space of symmetries
$\mathrm{Sym}(Q_{t})$ with the structure of a graded Leibniz algebra. The
latter is defined as follows.

######  Definition 4.1.

A graded Leibniz algebra is a graded vector space
$V=\bigoplus_{n\in\mathbb{Z}}V_{n}$ together with a bilinear product

$\circ:V_{n}\otimes V_{m}\rightarrow V_{n+m}$ (4.4)

satisfying the Leibniz identity

$a\circ(b\circ c)=(a\circ b)\circ c+(-1)^{|a||b|}b\circ(a\circ c)$ (4.5)

for all $a,b,c\in V$; here $|a|$ denotes the degree of a homogeneous element
$a\in V$.

It is convenient to split the $\circ$-product into its graded symmetric and
skew-symmetric parts444In the following, we will often omit the word
‘graded’.:

$\begin{array}[]{c}a\circ b=[\\![a,b]\\!]+\\{[a,b]\\}\,,\\\\[14.22636pt]
\displaystyle[\\![a,b]\\!]=\frac{1}{2}\big{(}a\circ b-(-1)^{|a||b|}b\circ
a\big{)}\,,\qquad\\{[a,b]\\}=\frac{1}{2}\big{(}a\circ b+(-1)^{|a||b|}b\circ
a\big{)}\,.\end{array}$ (4.6)

Then it follows from the Leibniz identity (4.5) that the subspace $I\subset V$
spanned by all elements of the form $\\{[a,b]\\}$ is an ideal of the Leibniz
algebra $(V,\circ)$. Moreover, $I\circ V=0$. The condition $I=0$ is clearly
equivalent to the skew-symmetry of the $\circ$-product. In that case, Rel.
(4.5) boils down to the Jacobi identity for the Lie bracket
$[\\![a,b]\\!]=a\circ b$. Therefore, a graded Leibniz algebra with skew-
symmetric product is the same as a graded Lie algebra and vice versa. In
general, the skew-symmetric part of the $\circ$-product does not satisfy the
Jacobi identity, but it always induces a Lie bracket in the quotient space
$V/I$.

Central to our construction is a functor

$\mathbf{F}:\mathbf{dgLie}\rightarrow\mathbf{gLeib}$ (4.7)

from the category of differential graded Lie algebras to that of graded
Leibniz algebras. It is defined in terms of the so-called derived bracket
[14]. Given a differential graded Lie algebra
$L=\bigoplus_{n\in\mathbb{Z}}L_{n}$ with bracket $[-,-]$ of degree $n$ and
differential $\partial$ of degree $1$, we write

$a\circ b=(-1)^{|a|-n-1}[\partial a,b]\,.$ (4.8)

It is straightforward to check the $\circ$-product makes the graded vector
space $L[-n-1]$ into a Leibniz algebra555For every $\mathbb{Z}$-graded vector
space $V=\bigoplus V_{n}$ and $k\in\mathbb{Z}$, $V[k]$ is a graded vector
space with $V[k]_{n}=V_{n+k}$.. One can also see that the symmetric part of
the $\circ$-product (4.8) is in the image of the differential:

$\\{[a,b]\\}=\frac{1}{2}(-1)^{|a|-n-1}\partial[a,b]\,.$ (4.9)

As a consequence, the skew-symmetric part of $\circ$ induces a Lie bracket in
the quotient space $L/\partial L$. Furthermore, the differential $\partial$
defines a homomorphism of the Leibniz algebras $(L[-n-1],\circ)$ and
$(L,[-,-])$.

Applying the construction of the derived bracket (4.8) to our situation, we
can turn the differential graded Lie algebra of symmetries
$(\mathrm{Sym}(Q_{t}),\partial_{t})$ into a graded Leibniz algebra w.r.t. the
product

$\tilde{X}\circ_{t}\tilde{Y}=(-1)^{\mathrm{gh}(X)-1}\widetilde{[[\dot{Q}_{t},X],Y]}=\widetilde{[[X,\dot{Q}_{t}],Y]}\,,$
(4.10)

$X$ and $Y$ being $Q_{t}$-invariant evolutionary vector fields. We will denote
this Leibniz algebra by $\mathbf{Sym}(Q_{t})$; as a graded vector space
$\mathbf{Sym}(Q_{t})=\mathrm{Sym}(Q_{t})[-n-1]$. In general, the product
(4.10) is not skew-symmetric and $\mathbf{Sym}(Q_{t})$ is not a Lie algebra.
An important particular situation when it does degenerate to a Lie algebra
structure is the following. Let $\mathcal{L}\subset\mathrm{Sym}(Q_{t})$ be a
commutative subalgebra in the Lie algebra of symmetries such that
$[\partial\mathcal{L},\mathcal{L}]\subset\mathcal{L}$. Then
$\mathbf{F}\mathcal{L}$ is again a Lie algebra. Moreover, the derived Lie
algebra $\mathbf{F}\mathcal{L}$ may well be noncommutative.

######  Remark 4.2.

The functor (4.7) is not the only one that can be attributed to the category
of differential graded Lie algebras. In [19], Getzler constructed a functor
from $\mathbf{dgLie}$ to the category of $L_{\infty}$-algebras. A remarkable
property of Getzler’s functor is that it extends the skew-symmetric part of
the derived bracket (4.8), restricted to the subspace $L_{-1}$, to the full
$L_{\infty}$-structure. The discussion of other interesting functors from the
category of Leibniz algebras to that of $L_{\infty}$-algebras can be found in
the recent paper [20].

######  Remark 4.3.

Differentiating the identity $\delta_{Q_{t}}\omega_{t}=0$ w.r.t. the parameter
$t$, one can see that the symmetry $\dot{Q}_{t}$ is on-shell Hamiltonian. If,
as often happens, $\omega_{t}$ does not depend on $t$, the symmetry
$\dot{Q}_{t}$ is Hamiltonian and the Leibniz product (4.10) restricts
consistently onto the subspace of Hamiltonian symmetries
$\mathrm{Sym}(Q_{t},\omega)$. In that case, one can speak of the Leibniz
algebras of Hamiltonian symmetries and the corresponding conservation laws.

#### From Lie to Leibniz and back.

Let us return to the interpretation of the expansion (4.1) as a formal
deformation of the free gauge system $(Q_{0},\omega_{0})$ by interaction.
Evaluating (4.2) at $t=0$, we find

$[Q_{0},Q_{1}]=0\,,\qquad[Q_{1},Q_{1}]=-2[Q_{0},Q_{2}]\,.$ (4.11)

These equalities say that the first-order interaction $Q_{1}$ defines an odd
symmetry whose square is a trivial symmetry of the free system. Hence,
$\tilde{Q}_{1}\in\mathrm{Sym}(Q_{0})$. Formula (4.10) makes then the space of
free symmetries into the Leibniz algebra with product

$\tilde{X}\circ\tilde{Y}=\widetilde{[[X,Q_{1}],Y]}\,.$ (4.12)

We denote this Leibniz algebra by $\mathbf{Sym}(Q_{0},Q_{1})$ to emphasise
that the product (4.12) on the symmetries of the free system
$(Q_{0},\omega_{0})$ depends on the first-order interaction $Q_{1}$. At the
level of graded vector spaces we have the isomorphism
$\mathbf{Sym}(Q_{0},Q_{1})=\mathrm{Sym}(Q_{0})[-1]$. As is often the case, the
inclusion of interaction breaks some symmetries of the free system, and those
that survive form a subalgebra $\mathcal{L}_{\mathrm{int}}$ in the Lie algebra
$\mathrm{Sym}(Q_{0})$. The quotient
$\mathcal{L}_{\mathrm{br}}=\mathrm{Sym}(Q_{0})/\mathcal{L}_{\mathrm{int}}$,
identified with the space of broken symmetries, carries no natural Lie algebra
structure unless $\mathcal{L}_{\mathrm{int}}$ is an ideal in
$\mathrm{Sym}(Q_{0})$.

From the viewpoint of Leibniz algebras, the unbroken symmetries constitute a
central ideal in $\mathbf{Sym}(Q_{0},Q_{1})$, that is,

$\mathbf{Sym}(Q_{0},Q_{1})\circ\mathcal{L}_{\mathrm{int}}\subset\mathcal{L}_{\mathrm{int}}\,,\qquad\mathcal{L}_{\mathrm{int}}\circ\mathbf{Sym}(Q_{0},Q_{1})=0\,.$
(4.13)

In view of Rel. (4.9), this includes the ideal
${I}\subset\mathcal{L}_{\mathrm{int}}$ spanned by the unbroken symmetries of
the form

$\\{[\tilde{X},\tilde{Y}]\\}=\frac{1}{2}(-1)^{\mathrm{gh}(X)-1}\widetilde{[Q_{1},[X,Y]]}\,,$
(4.14)

$X$ and $Y$ being arbitrary $Q_{0}$-invariant evolutionary vector fields. As a
result, the Leibniz product (4.12) canonically induces a Lie bracket on the
vector space $\mathcal{L}_{\mathrm{br}}[-1]$, the suspended space of broken
symmetries.

Let us summarize the results of this subsection by the following thesis. Given
a consistent first-order interaction $Q_{1}$, we can split (noncanonically)
the symmetries of the corresponding free system $(Q_{0},\omega_{0})$ into two
groups – those that preserve $Q_{1}$ and those that do not – and make them
both into graded Lie algebras. We emphasize that the Lie brackets in
$\mathcal{L}_{\mathrm{int}}$ and $\mathcal{L}_{\mathrm{br}}[-1]$ are
essentially different: the former is just the restriction of that in
$\mathrm{Sym}(Q_{0})$, while the latter, being induced by the Leibniz product
(4.12), involves the interaction $Q_{1}$. For free gauge systems, the Lie
algebra of symmetries $\mathrm{Sym}(Q_{0})$ is normally commutative in nonzero
ghost number [13, Sec. 3.9] and so is its subalgebra
$\mathcal{L}_{\mathrm{int}}$. By contrast, the Lie algebra of broken
symmetries $\mathcal{L}_{\mathrm{br}}[-1]$ may well be nonabelian due to the
interaction $Q_{1}$. Notice that whenever $\mathrm{Sym}(Q_{0})$ is abelian the
corresponding Leibniz algebra $\mathbf{Sym}(Q_{0},Q_{1})$ is a Lie algebra
with $\mathcal{L}_{\mathrm{int}}[-1]$ belonging to its centre.

#### Conservation laws.

The same construction of the derived bracket (4.8) allows one to make the Lie
algebra of on-shell Hamiltonian forms $\Lambda^{H}(Q_{0},\omega_{0})$ into a
Leibniz algebra. We can proceed from the simple observation that each first-
order interaction $Q_{1}$ defines an on-shell Hamiltonian symmetry. Indeed,
evaluating the equation $i_{Q_{t}}\omega_{t}\simeq\delta H_{t}$ at first order
in $t$, we readily find

$i_{Q_{1}}\omega_{0}\simeq\delta H_{1}-i_{Q_{0}}\omega_{1}\approx\delta
H_{1}\,.$ (4.15)

Hence, $Q_{1}$ is an on-shell Hamiltonian symmetry of the free gauge system
$(Q_{0},\omega_{0})$. It follows from the second relation in (4.11) that the
Lie bracket (2.9) of the on-shell Hamiltonian form $H_{1}$ with itself is
equivalent to zero:

$\\{H_{1},H_{1}\\}\simeq-2\\{H_{0},H_{2}\\}=2i_{Q_{0}}i_{Q_{2}}\omega_{0}\approx
0\,.$ (4.16)

Notice that the degree of the bracket is opposite to the ghost number of the
presymplectic structure. Considering now the adjoint action of $H_{1}$, we
turn $\Lambda^{H}(Q_{0},\omega_{0})$ into a differential graded Lie algebra
with the differential $\partial$ defined by the relation

$\partial\alpha=\\{H_{1},\alpha\\}\qquad\forall\alpha\in\Lambda^{H}(Q_{0},\omega_{0})\,.$
(4.17)

Then the functor (4.7) gives us immediately the Leibniz product

$\alpha\circ\beta=(-1)^{\mathrm{gh}(\alpha)+\mathrm{gh}(\omega_{0})-1}\\{\\{H_{1},\alpha\\},\beta\\}=\\{\\{\alpha,H_{1}\\},\beta\\}$
(4.18)

for all $\alpha,\beta\in\Lambda^{H}(Q_{0},\omega_{0})$. With the definition of
an on-shell Hamiltonian form (2.13) we can rewrite this product in several
equivalent ways:

$\alpha\circ\beta\approx(-1)^{\mathrm{gh}(\alpha)+\mathrm{gh}(\omega_{0})}\\{L_{Q_{1}}\alpha,\beta\\}\approx
i_{[Q_{1},X]}i_{Y}\omega\approx L_{[Q_{1},X]}\beta\,.$ (4.19)

Here $X$ and $Y$ are the on-shell Hamiltonian symmetries associated with
$\alpha$ and $\beta$. We will denote this Leibniz algebra by
${\boldsymbol{\Lambda}}^{H}(Q_{0},Q_{1},\omega_{0})$. As usual the passage
from the Lie algebra $\Lambda^{H}(Q_{0},\omega_{0})$ to the Leibniz algebra
${\boldsymbol{\Lambda}}^{H}(Q_{0},Q_{1},\omega_{0})$ implies the shift in
degree of on-shell Hamiltonian forms.

The above considerations apply then to all the descendent gauge systems
$(Q_{0},\omega_{0}^{k})$ associated with the free gauge system
$(Q_{0},\omega_{0})$. Indeed, replacing $\omega_{0}$ and $\omega_{1}$ in
(4.15) with $\omega_{0}^{k}$ and $\omega_{1}^{k}$, respectively, we readily
conclude that the vector field $Q_{1}$ is on-shell Hamiltonian relative to
$(Q_{0},\omega_{0}^{k})$, that is,

$i_{Q_{1}}\omega^{k}_{0}\simeq\delta
H^{k}_{1}-i_{Q_{0}}\omega^{k}_{1}\approx\delta H^{k}_{1}\,,$ (4.20)

$H_{1}^{k}$ being the Hamiltonian. The derived bracket construction gives then
the sequence of Leibniz products

$\alpha\circ_{k}\beta=\\{\\{\alpha,H^{k}_{1}\\}_{k},\beta\\}_{k}\,,\quad
k=1,2,\ldots,m,$ (4.21)

for all $\alpha,\beta\in\Lambda^{H}(Q_{0},\omega_{0}^{k})$. We denote the
corresponding Leibniz algebras by
${\boldsymbol{\Lambda}}^{H}(Q_{0},Q_{1},\omega_{0}^{k})$. If the assumption is
made that $H^{0,\ast}_{\rm flat}(\Sigma^{\infty};\delta)=0$, the elements of
these algebras become conservation laws of the free gauge system, see
Proposition 3.4.

In special, but not rare, instances where the presymplectic structure of the
family $(Q_{t},\omega_{t})$ does not depend on $t$, the vector field $Q_{1}$
is Hamiltonian relative to $\omega_{0}=\omega_{t}$. Being a symmetry, it
generates the sequence of conservation laws
$H^{k}_{1}\in\mathrm{CL}_{k}(Q_{0},\omega_{0})$, $k=1,\ldots,m$. In that case
the product (4.21) restricts onto the subspace
$\mathrm{CL}_{k}(Q_{0},\omega_{0})\subset\Lambda^{H}(Q_{0},\omega^{k}_{0})$
making it into a Leibniz algebra, which we denote by
$\mathbf{CL}_{k}(Q_{0},Q_{1},\omega_{0})$.

##  5 Applications

In this section, we exemplify the general approach developed above by two
fundamental physical models: Yang–Mills theory and Einstein’s gravity without
matter. Both the theories enjoy asymptotic conservation laws with nontrivial
Leibniz algebras. The technical simplicity of these models combined with the
Batalin–Vilkovisky (BV) formalism enables us to exhibit explicitly all the
relevant presymplectic structures, conserved currents and their algebras. As
the third example we consider the gravity field subject to the so-called
unimodularity condition. This model, being highly nonlinear, is intended to
demonstrate that lower-degree conservation laws are not prerogative of free
gauge theories alone.

###  5.1 Yang–Mills fields

Let $M$ be a $4$-dimensional spacetime manifold, $\mathcal{G}$ a compact Lie
algebra, and $\mathrm{Tr}$ an invariant nondegenerate trace on the universal
enveloping algebra $\mathcal{U}(\mathcal{G})$. Denote by
$\Lambda_{q}^{p}(M,\mathcal{G})$ the space of differential $p$-forms on $M$
with values in $\mathcal{G}$ that have ghost number $q\in\mathbb{Z}$. Recall
that in the BV formalism the spectrum of Yang–Mills (YM) theory without matter
consists of the following fields and antifields:

$A\in\Lambda_{0}^{1}(M,\mathcal{G})\,,\quad
C\in\Lambda_{1}^{0}(M,\mathcal{G})\,,\quad
A^{\ast}\in\Lambda_{-1}^{3}(M,\mathcal{G})\,,\quad
C^{\ast}\in\Lambda_{-2}^{4}(M,\mathcal{G})\,.$ (5.1)

The space of fields and antifields carries the canonical symplectic
structure666Hereinafter the wedge product combines the exterior product of
forms with the product in $\mathcal{U}(\mathcal{G})$.

$\omega=\mathrm{Tr}\left(\delta A\wedge\delta A^{\ast}+\delta C\wedge\delta
C^{\ast}\right)\,,\qquad\mathrm{gh}(\omega)=-1\,.$ (5.2)

The $2$-form $\omega$ being nondegenerate, the corresponding homological
vector field $Q$ is uniquely defined by the relation
$i_{Q}\omega=\delta\mathcal{L}$, where the Hamiltonian form $\mathcal{L}$ is
given by the standard BV extension of the YM Lagrangian, namely,

$\mathcal{L}=\mathrm{Tr}\left(\frac{1}{2}F\wedge\tilde{F}+A^{\ast}\wedge
DC+gC^{\ast}CC\right)\,.$ (5.3)

Here $\tilde{F}$ is the Hodge dual of the curvature $2$-form $F=dA+gA\wedge A$
and $D$ stands for the covariant derivative, e.g. $DC=dC+g[A,C]$. Regarding
the YM coupling constant $g$ as deformation parameter, we can write

$\mathcal{L}=\mathcal{L}_{0}+g\mathcal{L}_{1}+g^{2}\mathcal{L}_{2}\,,$ (5.4)

and similar expansion takes place for the homological vector field
$Q=Q_{0}+gQ_{1}+g^{2}Q_{2}$. One can easily check that $Q^{2}=0$ or, what is
the same, $\\{\mathcal{L},\mathcal{L}\\}\simeq 0$, where the braces stand for
the canonical BV bracket associated with the symplectic structure (5.2). The
leading term $\mathcal{L}_{0}$ describes the dynamics of free YM fields, while
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ introduce a consistent interaction.

Applying the BRST differential $\delta_{Q}$ to the symplectic structure (5.2),
one can find the following descendants:

$\begin{array}[]{lll}\delta_{Q}\omega=d\omega_{1}\,,&\omega_{1}=\mathrm{Tr}(\delta
A\wedge\delta\tilde{F}+\delta C\wedge\delta
A^{\ast})\,,&\omega_{1}\in\Lambda^{3}(M)\,,\qquad\mathrm{gh}(\omega_{1})=0\,,\\\\[8.53581pt]
\delta_{Q}\omega_{1}=d\omega_{2}\,,&\omega_{2}=\mathrm{Tr}(\delta
C\wedge\delta\tilde{F})\,,&\omega_{2}\in\Lambda^{2}(M)\,,\qquad\mathrm{gh}(\omega_{2})=1\,.\end{array}$
(5.5)

Since $\delta_{Q}\omega_{2}=0$, the length of YM theory is equal to $3$.
Unlike (5.2), the descendent presymplectic forms $\omega_{1}$ and $\omega_{2}$
depend on the coupling constant $g$. The first descendant is nothing but the
canonical presymplectic structure associated with the Lagrangian (5.3), that
is, $\omega_{1}=\delta\theta_{1}$, where the presymplectic potential
$\theta_{1}$ comes from the variation
$\delta\mathcal{L}=i_{Q}\omega+d\theta_{1}$. The second descendant
$\omega_{2}$ gives rise to the Lie and Leibniz algebra structures on lower-
degree conservation laws (surface charges).

In order to define these algebraic structures we note that $Q_{1}$ is a
Hamiltonian vector field generating a symmetry of the free Lagrangian
$\mathcal{L}_{0}$. By Theorem 3.1, it yields a pair of conservation laws
$\mathbf{J}_{1}$ and $\mathbf{J}_{2}$ defined by

$\begin{array}[]{llll}\delta_{Q_{0}}\mathcal{L}_{1}=d\mathbf{J}_{1}\,,&\quad\mathbf{J}_{1}=\mathrm{Tr}\big{(}\tilde{F}_{0}\wedge[A,C]+A^{\ast}CC\big{)}\,,&\quad\mathbf{J}_{1}\in\Lambda^{3}(M)\,,&\quad\mathrm{gh}(\mathbf{J}_{1})=1\,,\\\\[8.53581pt]
\delta_{Q_{0}}\mathbf{J}_{1}=d\mathbf{J}_{2}\,,&\quad\mathbf{J}_{2}=\mathrm{Tr}\big{(}\tilde{F}_{0}CC\big{)}\,,&\quad\mathbf{J}_{2}\in\Lambda^{2}(M)\,,&\quad\mathrm{gh}(\mathbf{J}_{2})=2\,,\\\\[8.53581pt]
\delta_{Q_{0}}\mathbf{J}_{2}=0\,,&&&\end{array}$ (5.6)

$F_{0}=dA$ being the strength of free YM fields. In the calculations above we
used the following formulas for the action of the free BRST differential on
fields and antifields:

$\delta_{Q_{0}}C^{\ast}=dA^{\ast}\,,\qquad\delta_{Q_{0}}A^{\ast}=d\tilde{F}_{0}\,,\qquad\delta_{Q_{0}}A=dC\,,\qquad\delta_{Q_{0}}C=0\,.$
(5.7)

With the help of $\mathbf{J}$’s we can turn the Lie algebras of on-shell
Hamiltonian forms of degree three and two into a pair of Leibniz algebras by
setting

$\alpha\circ_{k}\beta=\\{\\{\alpha,\mathbf{J}_{k}\\}_{k},\beta\\}_{k}\,.$
(5.8)

Here the braces with the subscript $k=1,2$ stand for the Lie brackets
determined by the presymplectic structures (5.5).

The $2$-form $\mathbf{J}_{2}$ is not the only surface current that one can
attribute to free YM fields. The free Lagrangian $\mathcal{L}_{0}$ obviously
enjoys the shift symmetry $C\rightarrow C+\epsilon\xi$, where $\xi$ is a
vector of $\mathcal{G}$ and $\epsilon$ is a constant parameter of ghost number
one ($d\epsilon=0$). By Noether’s theorem, we get immediately the conserved
current

$J_{1}^{\xi}=\mathrm{Tr}(\xi
A^{\ast})\in\Lambda^{3}(M)\,,\qquad\mathrm{gh}(J_{1}^{\xi})=-1\,,$ (5.9)

and its descendant

$J_{2}^{\xi}=\mathrm{Tr}(\xi\tilde{F}_{0})\in\Lambda^{2}(M)\,,\qquad\mathrm{gh}(J_{2}^{\xi})=0\,.$
(5.10)

Obviously,

$dJ^{\xi}_{1}\approx 0\,,\qquad dJ^{\xi}_{2}\approx
0\,,\qquad\delta_{Q_{0}}J_{1}^{\xi}=dJ_{2}^{\xi}\,,\qquad\delta_{Q_{0}}J^{\xi}_{2}=0\,.$
(5.11)

The conserved current $J^{\xi}_{2}$, being of ghost number zero, admits a
straightforward physical interpretation. If $S$ is a closed space-like surface
in $M$, then the integral

$q^{\xi}=\int_{S}J_{2}^{\xi}$ (5.12)

defines the net (color) charge enclosed by the surface $S$. The surface
currents (5.9) and (5.10) form the abelian Lie algebras

$\\{J^{\xi}_{k},J_{k}^{\xi^{\prime}}\\}_{k}=0\,,\qquad k=1,2,$ (5.13)

as is usually the case for free theories. The first-order interaction
$\mathcal{L}_{1}$ breaks the shift symmetry above, leading thus to nontrivial
Leibniz products of surface currents:

$J_{k}^{\xi}\circ_{k}J_{k}^{\xi^{\prime}}=J_{k}^{[\xi,\xi^{\prime}]}\,,\qquad\forall\xi,\xi^{\prime}\in\mathcal{G}\,,\qquad
k=1,2.$ (5.14)

As is seen, both the products are skew-symmetric and define the Lie algebra
structure isomorphic to $\mathcal{G}$. In such a way we are able to reproduce
the color Lie algebra at the level of surface currents. We emphasize that the
currents $J_{k}^{\xi}$ are conserved on the free equations of motion $Q_{0}=0$
and cannot be promoted to conservation laws of full YM theory. Nevertheless,
to make them into a nonabelian Lie algebra we need the cubic interaction
vertices accommodated in $\mathcal{L}_{1}$. These vertices are essentially
responsible for the ‘nonabelian part’ of the gauge generators.

In order to present a genuine example of Leibniz algebra which is not Lie, we
note that the free Lagrangian $\mathcal{L}_{0}$ is invariant under orthogonal
transformations. Indeed, as the Lie algebra $\mathcal{G}$ is supposed to be
compact, there is a basis $\\{t_{a}\\}_{a=1}^{n}$ in $\mathcal{G}$ such that
$\mathrm{Tr}(t_{a}t_{b})=\delta_{ab}$. Expanding now the fields in terms of
this basis, e.g. $A=A^{a}t_{a}$, we see that the quadratic in fields
Lagrangian $\mathcal{L}_{0}$ is determined by the Euclidean metric
$\delta_{ab}$. As a result, it appears to be invariant under the
$O(n)$-rotations of fields: $A^{a}\rightarrow A^{\prime a}=R^{a}{}_{b}A^{b}$
and similar transformations for the other fields; here $R=(R^{a}{}_{b})$ is an
orthogonal matrix from $O(n)$. Applying Noether’s theorem to infinitesimal
rotations gives the following conserved currents for the free YM fields:

$j^{ab}_{1}=A^{[a}\wedge\tilde{F}_{0}^{b]}+C^{[a}A^{\ast\,\,b]}\,,\qquad
j^{ab}_{1}\in\Lambda^{3}(M)\,,\qquad\mathrm{gh}(j^{ab}_{1})=0\,.$ (5.15)

These have descendants of the first generation:

$\delta_{Q_{0}}j^{ab}_{1}=dj_{2}^{ab}\,,\qquad
j^{ab}_{2}=C^{[a}\tilde{F}_{0}^{b]}\,,\qquad
j^{ab}_{2}\in\Lambda^{2}(M)\,,\qquad\mathrm{gh}(j_{2}^{ab})=1\,.$ (5.16)

As usual the square brackets stand for skew-symmetrization of indices. By
Theorem 3.3, the Lie brackets of the currents reproduce the commutation
relations of the Lie algebra $o(n)$:

$\\{j_{k}^{ab},j_{k}^{cd}\\}_{k}=\delta^{bc}j_{k}^{ad}-\delta^{ac}j_{k}^{bd}+\delta^{ad}j_{k}^{bc}-\delta^{bd}j_{k}^{ac}\,,\qquad
k=1,2.$ (5.17)

The currents $j^{ab}_{k}$ generate the Hamiltonian action of $o(n)$ on the
space of fields and antifields. Evaluating now the symmetric part of the
Leibniz products $j_{k}^{ab}\circ_{k}j_{k}^{cd}$, one easily finds

$\\{[j^{ab}_{k},j^{cd}_{k}]\\}_{k}=-\frac{1}{2}\\{\mathbf{J}_{k},\\{j_{k}^{ab},j_{k}^{cd}\\}_{k}\\}_{k}=\delta^{bc}{J}^{ad}_{k}-\delta^{ac}{J}_{k}^{bd}+\delta^{ad}{J}^{bc}_{k}-\delta^{bd}{J}^{ac}_{k}\,,$
(5.18)

where $J^{ab}_{k}=-\frac{1}{2}\\{\mathbf{J}_{k},j^{ab}_{k}\\}_{k}$ are new
conserved currents of ghost number $k$. Generally the structure constants of
the Lie algebra $\mathcal{G}$, entering the currents $\mathbf{J}_{k}$, are not
$O(n)$-invariant, so that the new currents $J^{ab}_{k}$ are different from
zero.

Writing the currents (5.9) and (5.10) in terms of the basis above,
$J^{\xi}_{k}=\delta_{ab}J^{a}_{k}\xi^{b}$, we obtain

$\begin{array}[]{rcl}j^{ab}_{k}\circ_{k}J^{c}_{k}&=&\displaystyle
f^{ac}{}_{d}\,j^{db}_{k}-f^{bc}{}_{d}\,j^{da}_{k}+\delta^{ac}f^{b}_{de}\,j_{k}^{de}-\delta^{bc}f^{a}_{de}\,j_{k}^{de}\,,\\\\[8.53581pt]
J^{c}_{k}\circ_{k}j^{ab}_{k}&=&f^{bc}{}_{d}\,j^{da}_{k}-f^{ac}{}_{d}\,j^{db}_{k}\,,\qquad
k=1,2.\end{array}$ (5.19)

These Leibniz products also have nonzero symmetric parts.

###  5.2 Einstein gravity

In the vierbein formalism, the dynamics of the gravity field are described by
ten $1$-form fields: the vierbein $e^{a}$ and the spin-connection
$w^{ab}=-w^{ba}$. As usual, we use the Minkowski metric $\eta_{ab}$ for
raising and lowering the Lorentz indices $a,b=0,1,2,3$. In order to have
explicit control over the gauge symmetries corresponding to the
diffeomorphisms of the spacetime manifold $M$ and local Lorentz invariance,
one extends the field content with the ghost fields $c^{a}$ and
$c^{ab}=-c^{ba}$ as well as the corresponding antifields labeled by star.
Table 1 collects the full spectrum of fields and antifields together with
their form degrees and ghost numbers.

| $c^{*}_{a}$ | $c^{*}_{ab}$ | $e^{*}_{a}$ | $w^{*}_{ab}$ | $e^{a}$ | $w^{ab}$ | $c^{a}$ | $c^{ab}$
---|---|---|---|---|---|---|---|---
$\mathrm{deg}$ | 4 | 4 | 3 | 3 | 1 | 1 | 0 | 0
$\mathrm{gh}$ | -2 | -2 | -1 | -1 | 0 | 0 | 1 | 1
Table 1: Fields and antifields in the vierbein formulation of gravity.

The fields and antifields are canonically conjugate to each other w.r.t. the
BV symplectic structure

$\omega=\delta e^{*}_{a}\wedge\delta e^{a}+\delta w^{*}_{ab}\wedge\delta
w^{ab}+\delta c^{*}_{a}\wedge\delta c^{a}+\delta c^{*}_{ab}\wedge\delta
c^{ab}\,.$ (5.20)

The BV master Lagrangian of Einstein’s gravity now reads

$\mathcal{L}=\frac{1}{2}\epsilon_{abcd}e^{a}\wedge e^{b}\wedge
R^{cd}+e^{\ast}_{a}\wedge Dc^{a}+w^{\ast}_{ab}\wedge
Dc^{ab}+e^{\ast}_{a}\wedge
c^{a}{}_{b}e^{b}+c^{\ast}_{a}c^{a}{}_{b}c^{b}+\frac{1}{2}c^{\ast}_{ab}c^{a}{}_{c}c^{cb}.$
(5.21)

Here $D=d+w$ is the Lorentz covarian differential with the curvature $2$-form
$R^{ab}=dw^{ab}+w^{a}{}_{c}\wedge w^{cb}$. The homological vector field $Q$
underlying Einstein’s gravity is just the Hamiltonian vector field generated
by the master Lagrangian (5.21) and the symplectic structure (5.20). As is
well known, see e.g. [21], the characteristic cohomology of pure gravity is
empty. The nontrivial conservation laws arise only upon linearization of the
Lagrangian (5.21) about a suitable geometric background. Consider, for
simplicity, the flat background geometry with a vierbein $h^{a}$ obeying
$dh^{a}=0$. Then we can put $e^{a}=h^{a}+\tilde{e}^{a}$, where the $1$-forms
$\tilde{e}^{a}$ describe fluctuations over the flat background. On
substituting this decomposition into (5.21), we get777Rescaling all the fields
and antifields by $g$, while multiplying $\mathcal{L}$ by $g^{-2}$, we can
bring the Lagrangian (5.22) into the form (5.4) with $g$ playing the role of
coupling constant.

$\mathcal{L}\simeq\mathcal{L}_{0}+\mathcal{L}_{1}+\mathcal{L}_{2}\,,$ (5.22)

where

$\begin{array}[]{rcl}\mathcal{L}_{0}&=&\epsilon_{abcd}dw^{ab}\wedge\tilde{e}^{c}\wedge
h^{d}+\frac{1}{2}\epsilon_{abcd}w^{a}{}_{e}\wedge w^{eb}\wedge h^{c}\wedge
h^{d}+e^{*}_{a}\wedge(dc^{a}+c^{ab}h_{b})+w^{*}_{ab}\wedge
dc^{ab}\,,\\\\[8.53581pt]
\mathcal{L}_{1}&=&\frac{1}{2}\epsilon_{abcd}dw^{ab}\wedge\tilde{e}^{c}\wedge\tilde{e}^{d}+\epsilon_{abcd}w^{a}{}_{e}\wedge
w^{eb}\wedge\tilde{e}^{c}\wedge
h^{d}+e^{*}_{a}\wedge(w^{a}{}_{e}c^{e}+c^{ab}\tilde{e}_{b})+w^{*}_{ab}\wedge
w^{a}{}_{e}c^{eb}\,+\\\\[8.53581pt]
&&+\,c^{\ast}_{a}c^{a}{}_{b}c^{b}+\frac{1}{2}c^{\ast}_{ab}c^{a}{}_{c}c^{cb}\,,\\\\[8.53581pt]
\mathcal{L}_{2}&=&\frac{1}{2}\epsilon_{abcd}w^{a}{}_{e}\wedge
w^{eb}\wedge\tilde{e}^{c}\wedge\tilde{e}^{d}\,.\end{array}$

It is convenient to introduce the following background $1$\- and $2$-forms:

$h_{abc}=\epsilon_{abcd}h^{d}\,,\qquad H_{ab}=\epsilon_{abcd}h^{c}\wedge
h^{d}\,.$ (5.23)

Then the action of the free BRST differential is given by

$\begin{array}[]{ll}\delta_{Q_{0}}e^{a}=dc^{a}+c^{a}{}_{b}h^{b}\,,&\delta_{Q_{0}}e^{*}_{a}=dw^{bc}\wedge
h_{abc}\,,\\\\[8.53581pt]
\delta_{Q_{0}}w^{ab}=dc^{ab}\,,&\delta_{Q_{0}}w^{*}_{ab}=de^{c}\wedge
h_{abc}+H_{ac}\wedge w^{c}{}_{b}-H_{bc}\wedge w^{c}{}_{a}\,,\\\\[8.53581pt]
\delta_{Q_{0}}{c}^{a}=0\,,&\delta_{Q_{0}}{c}^{*}_{a}=de^{*}_{a}\,,\\\\[8.53581pt]
\delta_{Q_{0}}{c}^{ab}=0\,,&\delta_{Q_{0}}{c}^{*}_{ab}=dw^{*}_{ab}-e^{\ast}{}_{a}\wedge
h_{b}+e^{\ast}{}_{b}\wedge h_{a}\,.\end{array}$ (5.24)

The BV symplectic structure (5.20) has the following descendants:

$\small\begin{array}[]{lll}\delta_{Q_{0}}\omega=d\omega_{1}\,,&\omega_{1}=\delta
e^{a}\wedge\delta w^{bc}\wedge h_{abc}+\delta c^{a}\wedge\delta
e^{*}_{a}+\delta c^{ab}\wedge\delta w^{*}_{ab}\,,\\\\[8.53581pt]
\delta_{Q_{0}}\omega_{1}=d\omega_{2}\,,&\omega_{2}=(\delta c^{a}\wedge\delta
w^{bc}+\delta c^{ab}\wedge\delta e^{c})\wedge h_{abc}\,,\\\\[8.53581pt]
\delta_{Q_{0}}\omega_{2}=d\omega_{3}\,,&\omega_{3}=\delta c^{ab}\wedge\delta
c^{c}\wedge h_{abc}\,.\end{array}$ (5.25)

The free Lagrangian $\mathcal{L}_{0}$ is invariant under the shifts

$c^{a}\rightarrow c^{a}+\xi^{a}\,,\qquad c^{ab}\rightarrow c^{ab}+\xi^{ab}\,,$
(5.26)

where the transformation parameters $\xi=(\xi^{a},\xi^{ab})$ obey the
conditions

$\xi^{ab}=-\xi^{ba}\,,\qquad d\xi^{ab}=0\,,\qquad d\xi^{a}=\xi^{ab}h_{b}\,.$
(5.27)

In local coordinates where $h^{a}=dx^{a}$, we can solve these equations as
$\xi^{a}=\zeta^{a}+\zeta^{ab}x_{a}$ and $\xi^{ab}=\zeta^{ab}$, with
$\zeta^{a}$ and $\zeta^{ab}=-\zeta^{ba}$ being arbitrary constant parameters.
The symmetry (5.26) is Hamiltonian relative to the BV symplectic structure
(5.20) and is generated by the Hamiltonian form
$H=\xi^{a}c^{\ast}_{a}+\xi^{ab}c^{\ast}_{ab}$. By Noether’s first theorem, we
obtain the $10$-parameter family of conserved currents. We find

$\delta_{Q_{0}}H=dJ^{\xi}_{1}\,,\qquad
J^{\xi}_{1}=\xi^{a}e^{\ast}_{a}+\xi^{ab}w^{\ast}_{ab}=\zeta_{a}P^{a}_{1}+\zeta_{ab}M^{ab}_{1}\,.$
(5.28)

Applying the BRST differential (5.24) yields the following descendants:

$\begin{array}[]{ll}\delta_{Q_{0}}J^{\xi}_{1}=dJ_{2}^{\xi}\,,&J^{\xi}_{2}=\zeta_{a}P_{2}^{a}+\zeta_{ab}M^{ab}_{2}=(\xi^{ab}e^{c}+\xi^{a}w^{bc})\wedge
h_{abc}\,,\\\\[8.53581pt]
\delta_{Q_{0}}J^{\xi}_{2}=dJ_{3}^{\xi}\,,&J^{\xi}_{3}=\zeta_{a}P_{3}^{a}+\zeta_{ab}M^{ab}_{3}=(\xi^{ab}c^{c}+\xi^{a}c^{bc})\wedge
h_{abc}\,,\end{array}$ (5.29)

and $\delta_{Q_{0}}J_{3}^{\xi}=0$. The conserved currents $J^{\xi}_{2}$, being
$2$-forms of ghost number zero, are used to define the total energy-momentum
$\mathcal{P}$ and the angular momentum $\mathcal{M}$ of an asymptotically flat
universe. These are given by the charges

$\mathcal{P}^{a}=\int_{S}P_{2}^{a}\,,\qquad\mathcal{M}^{ab}=\int_{S}M^{ab}_{2}\,,$
(5.30)

where the integrals are over a closed space-like surface $S\subset M$ at
infinity. In particular, $\mathcal{P}^{0}$ gives the ADM energy [3, 13]. From
the physical viewpoint, it is quite natural to identify the above charges with
the generators of the Poincaré group – the isometry group of flat spacetime –
and to expect them to form the Poincare algebra w.r.t. the Lie brackets.
However, this is not the case. A straightforward calculation yields

$\\{J^{\xi}_{1},J^{\xi^{\prime}}_{1}\\}_{1}=0\,,\qquad\\{J_{2}^{\xi},J_{2}^{\xi^{\prime}}\\}_{2}=0\,,\qquad\\{J_{3}^{\xi},J_{3}^{\xi^{\prime}}\\}_{3}=-(\xi^{a}\xi^{\prime
bc}+\xi^{\prime a}\xi^{bc})h_{abc}\,.$ (5.31)

As is seen, the abelian Lie algebra of the symmetry transformations (5.26)
gets a central extension when evaluated at the level of the $1$-form currents
$J_{3}^{\xi}$. This is the phenomenon of central extension that we discussed
in Sec. 3.

In order to reproduce the commutation relations of the Poincaré algebra the
cubic interaction of gravitons needs to be taken into account. As usual, the
cubic part of the Lagrangian (5.22) leads to a sequence of conserved currents
of the corresponding free theory. Explicitly,

$\small\begin{array}[]{ll}\delta_{Q_{0}}\mathcal{L}_{1}=d\mathbf{J}_{1}\,,&\mathbf{J}_{1}=\big{(}\frac{1}{2}c^{a}w^{b}{}_{d}\wedge
w^{dc}-w^{a}{}_{d}\wedge e^{d}c^{bc}\big{)}\wedge
h_{abc}+e^{*}_{a}c^{ab}c_{b}+\frac{1}{2}w^{*}_{ab}c^{a}{}_{c}c^{cb}\,,\\\\[8.53581pt]
\delta_{Q_{0}}\mathbf{J}_{1}=d\mathbf{J}_{2}\,,&\mathbf{J}_{2}=\big{(}c^{a}w^{b}{}_{d}c^{dc}+\frac{1}{2}e^{a}c^{b}{}_{d}c^{c}\big{)}\wedge
h_{abc}\,,\\\\[8.53581pt]
\delta_{Q_{0}}\mathbf{J}_{2}=d\mathbf{J}_{3}\,,&\mathbf{J}_{3}=\frac{1}{2}c^{a}c^{b}{}_{d}c^{dc}h_{abc}\,.\end{array}$
(5.32)

By formulas (4.18, 4.21), these currents define Leibniz algebra structures on
the spaces of on-shell Hamiltonian forms of degree $3$, $2$, and $1$. Since
the currents $J^{\xi}_{k}$ commute to each other modulo central elements, the
symmetric part of the corresponding Leibniz products

$J^{\xi}_{k}\circ_{k}J^{\xi^{\prime}}_{k}=\\{\\{J^{\xi}_{k},\mathbf{J}_{k}\\}_{k},J^{\xi^{\prime}}_{k}\\}_{k}\,,\qquad
k=1,2,3,$ (5.33)

vanishes identically, while the skew-symmetric one reproduces the structure
relations of the Poincaré algebra:

$\begin{array}[]{l}[\\![M_{k}^{ab}\,,M_{k}^{a^{\prime}b^{\prime}}]\\!]_{k}=\eta^{aa^{\prime}}M_{k}^{bb^{\prime}}-\eta^{ba^{\prime}}M_{k}^{ab^{\prime}}+\eta^{bb^{\prime}}M_{k}^{aa^{\prime}}-\eta^{ab^{\prime}}M_{k}^{ba^{\prime}}\,,\\\\[8.53581pt]
[\\![P_{k}^{a}\,,M_{k}^{bc}]\\!]_{k}=\eta^{ac}P_{k}^{b}-\eta^{ab}P_{k}^{c}\,,\\\\[8.53581pt]
[\\![P_{k}^{a},P_{k}^{b}]\\!]_{k}=0\,.\end{array}$ (5.34)

It should be recognized that the construction of this Lie algebra involves
essentially the cubic part $\mathcal{L}_{1}$ of the Lagrangian, and not just
the conserved currents of linearized gravity.

###  5.3 Unimodular gravity

Unimodular gravity provides an example of a nonlinear gauge theory with lower-
degree conservation laws, see e.g. [22, 23, 24]. This time it is convenient to
work in the metric instead of the vierbein formalism we used above. So, let
$M$ be a four-dimensional spacetime manifold endowed with a pseudo-Riemannian
metric $g_{\mu\nu}$ and let $\sqrt{-g}d^{4}x$ denote the canonical volume form
on $(M,g)$. The main idea of unimodular gravity, which goes back to Einstein,
is to impose the algebraic constraint

$\sqrt{-g}=1$ (5.35)

on the metric tensor. Physically, one may regard this as the partial fixing of
a reference frame. The unimodularity condition (5.35) breaks the group of
spacetime diffeomorphisms – the gauge group of general relativity – to the
subgroup of volume preserving diffeomorphisms. If $\xi$ is a vector field
generating an infinitesimal gauge transformation

$\delta_{\xi}g_{\mu\nu}=L_{\xi}g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}\,,$
(5.36)

then the unimodularity condition requires that

$\delta_{\xi}\sqrt{-g}=L_{\xi}\sqrt{-g}=\sqrt{-g}\nabla_{\mu}\xi^{\mu}=0\qquad\Rightarrow\quad\nabla_{\mu}\xi^{\mu}=0\,.$
(5.37)

Hence, the gauge parameter $\xi$ appears to be constrained by a differential
equation. One can solve this equation for $\xi^{\mu}$ in terms of an arbitrary
bivector $\xi^{\mu\nu}=-\xi^{\nu\mu}$ as $\xi^{\mu}=\nabla_{\nu}\xi^{\nu\mu}$.
Then the volume preserving gauge transformations take the form

$\delta_{\xi}g_{\mu\nu}=\nabla_{\mu}\nabla^{\lambda}\xi_{\lambda\nu}+\nabla_{\nu}\nabla^{\lambda}\xi_{\lambda\mu}\,.$
(5.38)

Unlike (5.36), these transformations appear to be reducible. Indeed, using the
symmetry properties of the Riemann and Ricci tensors, one can see that the
shift

$\xi^{\mu\nu}\rightarrow\xi^{\mu\nu}+\nabla_{\lambda}\xi^{\lambda\mu\nu}$
(5.39)

does not affect the r.h.s. of Eq. (5.38) for an arbitrary $3$-vector
$\xi^{\lambda\mu\nu}$; hence, one may regard (5.39) as gauge symmetry for
gauge symmetry. The latter transformation, in its turn, is invariant under
similar redefinitions

$\xi^{\lambda\mu\nu}\rightarrow\xi^{\lambda\mu\nu}+\nabla_{\sigma}\xi^{\sigma\lambda\mu\nu}\,,$
(5.40)

with $\xi^{\sigma\lambda\mu\nu}$ being an arbitrary $4$-vector. At this step
the sequence of reducibility relations stops by the reason of dimension. Now,
to put unimodular gravity into the standard BV formalism, one just promotes
the gauge parameters $\xi^{\mu\nu\cdots}$ to the corresponding ghost fields
$C^{\mu\nu\cdots}$ and introduces the conjugate antifields888For an
alternative BV formulation of unimodular gravity we refer to [25].. The full
field content is represented in Table 2 below.

| $C^{*}_{\sigma\lambda\mu\nu}$ | $C^{\ast}_{\lambda\mu\nu}$ | $C^{\ast}_{\mu\nu}$ | $g^{\ast}_{\mu\nu}$ | $g^{\mu\nu}$ | $C^{\mu\nu}$ | $C^{\lambda\mu\nu}$ | $C^{\sigma\lambda\mu\nu}$
---|---|---|---|---|---|---|---|---
$\mathrm{gh}$ | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3
Table 2: Fields and antifields of unimodular gravity.

Notice that all the ghost and antighost fields are totally skew-symmetric
tensors and the antifield $g^{\ast}_{\mu\nu}=g^{\ast}_{\nu\mu}$ is supposed to
be traceless, i.e.,

$g^{\mu\nu}g^{\ast}_{\mu\nu}=0\,.$ (5.41)

A relevant presymplectic structure on the fields and antifields is obtained by
restricting the canonical symplectic form

$\omega=\big{(}\delta g^{\ast}_{\mu\nu}\wedge\delta g^{\mu\nu}+\delta
C^{\ast}_{\mu\nu}\wedge\delta C^{\mu\nu}+\delta
C^{\ast}_{\lambda\mu\nu}\wedge\delta C^{\lambda\mu\nu}+\delta
C^{\ast}_{\sigma\lambda\mu\nu}\wedge\delta
C^{\sigma\lambda\mu\nu}\big{)}\sqrt{-g}d^{4}x$ (5.42)

onto the subspace of fields obeying the constraints (5.35) and (5.41). Since
the bracket of constraints defined by $\omega$ is clearly nonzero,

$\big{\\{}d^{4}x(\sqrt{-g}-1),d^{4}x\sqrt{-g}g^{\mu\nu}g^{\ast}_{\mu\nu}\big{\\}}=2\sqrt{-g}d^{4}x>0\,,$
(5.43)

the restricted form is likewise symplectic.

Now, the (minimal) master action for unimodular gravity can be written as

$S=\int
d^{4}x\sqrt{-g}\Big{[}R+g^{\ast}_{\mu\nu}\nabla^{\mu}\nabla_{\lambda}C^{\lambda\nu}+C^{\ast}_{\mu\nu}\nabla_{\lambda}C^{\lambda\mu\nu}+C^{\ast}_{\mu\nu\lambda}\nabla_{\sigma}C^{\sigma\mu\nu\lambda}\Big{]}\,.$
(5.44)

Although the first term looks like the conventional Einstein–Hilbert action,
one should keep in mind the algebraic constraint (5.35) enforced on the
metric. Evaluating the variation of the action (5.44) under the variation of
metric subject to the condition $g^{\mu\nu}\delta g_{\mu\nu}=0$, one can see
that that the corresponding equations of motion are fully equivalent to the
Einstein equations with cosmological constant:

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{1}{4}\Lambda g_{\mu\nu}\,.$ (5.45)

The only difference is that the constant $\Lambda$ is now a constant of
integration rather than a parameter in the action functional. It follows from
Eq. (5.45) that $R=\Lambda$, i.e., the scalar curvature $R$ represents a zero-
degree conservation law. One may wonder about a global symmetry this
conservation law comes from. The answer is almost obvious: the action is
invariant under the transformations

$C^{\mu\nu\lambda\sigma}\rightarrow
C^{\mu\nu\lambda\sigma}+\frac{\kappa}{\sqrt{-g}}\epsilon^{\mu\nu\lambda\sigma}\,,$
(5.46)

where $\epsilon^{\mu\nu\lambda\sigma}$ is the Levi–Civita symbol and $\kappa$
is a constant parameter of ghost number $3$. By the Noëther theorem this gives
immediately the conserved current

$J_{0}=C^{\ast}_{\mu\nu\lambda}dx^{\mu}\wedge dx^{\nu}\wedge
dx^{\lambda}\,,\qquad\mathrm{gh}(J_{0})=-3\,.$ (5.47)

Although this current on its own has no physical interpretation, it gives rise
to a sequence of lower-degree conservation laws. Indeed, applying the BRST
differential generated by the master Lagrangian (5.44) and the symplectic
structure (5.42), we find

$\begin{array}[]{lll}\delta_{Q}J_{0}=dJ_{1}\,,&J_{1}=C^{\ast}_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}\,,&\mathrm{gh}(J_{1})=-2\,,\\\\[8.53581pt]
\delta_{Q}J_{1}=dJ_{2}\,,&J_{2}=\nabla^{\mu}g^{\ast}_{\mu\nu}dx^{\nu}\,,&\mathrm{gh}(J_{2})=-1\,,\\\\[8.53581pt]
\delta_{Q}J_{2}=dJ_{3}\,,&J_{3}=-\frac{1}{4}R\,,&\mathrm{gh}(J_{3})=0\,.\end{array}$
(5.48)

The last nontrivial current of ghost number zero is proportional to the scalar
curvature we have discussed above.

As one more source of examples of nonlinear gauge theories with lower-degree
conservation laws, we would like to mention Higher Spin Gravity. As shown in
the recent paper [26], the higher-spin extension of four-dimensional gravity
enjoys an infinite number of nontrivial conservation laws of degrees zero and
two.

## Acknowledgments

The second author is grateful to Glenn Barnich for useful discussions. This
study was supported by the Tomsk State University Development Programme
(Priority-2030) and by the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS”.

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