# Presymplectic BV-AKSZ formulation of Conformal Gravity

Ivan Dneprov and Maxim Grigoriev  Lebedev Physical Institute,
Leninsky Ave. 53, 119991 Moscow, Russia
Institute for Theoretical and Mathematical Physics,
Lomonosov Moscow State University, 119991 Moscow, Russia

###### Abstract

We elaborate on the presymplectic BV-AKSZ approach to local gauge theories and
apply it to conformal gravity. More specifically, we identify a compatible
presymplectic structure on the minimal model of the total BRST complex of this
theory and show that together with the BRST differential it determines a full-
scale BV formulation for a specific frame-like action which seems to be
previously unknown. Remarkably, the underlying frame-like description requires
no artificial off-shell constraints. Instead, the action becomes equivalent to
the usual conformal gravity one, upon gauging away all the variables belonging
to the kernel of the presymplectic structure. Finally, we show how the
presymplectic BV-AKSZ approach extends to generic gauge theories.

###### Contents

  1. 1 Introduction
  2. 2 Presymplectic BV-AKSZ formulation
    1. 2.1 AKSZ sigma models
    2. 2.2 Presymplectic BV-AKSZ formulation
    3. 2.3 Target space and the total BRST complex
  3. 3 Presymplectic AKSZ form of conformal gravity
    1. 3.1 Conformal gravity
    2. 3.2 Frame-like formulation
    3. 3.3 Jet-bundle BRST complex for CGR
    4. 3.4 Minimal model of the BRST complex for CGR
    5. 3.5 Presymplectic structure
    6. 3.6 Symplectic quotient and BV formulation
  4. 4 General presymplectic BV-AKSZ formalism
    1. 4.1 Gauge PDEs
    2. 4.2 Vertical forms
    3. 4.3 Presymplectic gauge PDEs
    4. 4.4 Symplectic quotient and BV formulation
    5. 4.5 Example: Maxwell
  5. 5 Conclusion
  6. A Tensor conventions
  7. B Invariance of the presymplectic structure
  8. C The kernel of the presymplectic structure
  9. D Metric elimination
  10. E Presymplectic representation of the standard  ”gauge” formulation

##  1 Introduction

It is well known that AKSZ sigma models [1] with finite dimensional target
spaces are topological (= no local degrees of freedom). However, if one
replaces the target space symplectic structure with a possibly degenerate
presymplectic one, one arrives at an elegant AKSZ-like representation [2] for
the familiar first-order Lagrangians and makes manifest the graded geometry
underlying the frame-like formulations of various models, including gravity
and higher spin fields.

It was later realised [3] that the target space of a presymplectic AKSZ model
arises as an equivalent reformulation of the total BRST complex of the local
gauge theory in question, while the presymplectic structure can be seen as a
BRST extension of the well-known on-shell presymplectic structure [4, 5, 6],
alias the presymplectic current. Moreover, the presymplectic current itself
can be seen [7] as the descent completion of the Batalin-Vilkovisky (BV)
symplectic structure.

Recently, it has become clear [8] that not only the Lagrangian, but also the
entire BV formulation [9, 10] of a given local gauge field theory, can be
encoded in the presymplectic AKSZ formulation.111Mention also an alternative
AKSZ approach [11] to general relativity based on space+time decomposition. In
particular, the target space presymplectic structure defines a presymplectic
structure on the space of AKSZ fields. Taking the symplectic quotient results
in the BV symplectic structure which, together with the BV-BRST differential,
determines the BV master action. This allows the presymplectic BV-AKSZ
approach to be considered as a rather flexible and general extension of the
usual BV-BRST framework. In this approach a Lagrangian local gauge theory is
described in terms of fields and gauge parameters that can be subject to
differential constraints. For instance, the formulation of a Lagrangian system
in terms of its stationary surface (equation manifold in the PDE theory
terminology) is a typical example of such a description.

When applied to Lagrangian gauge theories extending gravity, this approach
gives an interesting interpretation of the theories (and their underlying
geometry, e.g. Cartan geometry) in terms of the differential graded
(pre)symplectic geometry. It is also worth mentioning the recent application
of presymplectic AKSZ approach to constructing Lagrangian description of
interacting higher spin theories [12].

In this work we apply the full-scale presymplectic AKSZ approach to conformal
gravity. The theory has been extensively studied [13, 14, 15, 16, 17, 18], and
has interesting properties, for instance it is renormalizable222Though the
renormalization breaks the Weyl invariance and the theory is anyway non-
unitary. and admits supersymmetric extension, see e.g. [14]. We concentrate on
the minimal version (also known as minimal model) of the jet-bundle BV-BRST
complex of the theory, which can be obtained via an additional reduction of
the complex initially put forward in [19, 20, 21] (see also [22]), and propose
a presymplectic structure therein which defines a full-scale BV-AKSZ
description of the theory. This gives a new formulation of conformal gravity
in terms of the connection of the conformal algebra, where the field content
and gauge symmetries are fully determined by the underlying presymplectic
$Q$-manifold and there is no need for artificial constraints like the familiar
torsion-free constraint of the formulation [13] (see also [23] for the
discussion of frame-like formulations of conformal gravity). In this respect
our formulation can be considered natural and arising from first principles.

The graded geometry structures identified in this work give a new perspective
on the underlying conformal geometry and more generally Cartan geometry. The
formulation itself can be useful in the analysis of the associated QFT, as it
naturally comprises the respective BV formulation and the local BRST
cohomology complex.

Finally, we present an extension of the presymplectic BV-AKSZ formulation to
the case of not necessarily diffeomorphism-invariant systems, where the
underlying $Q$-manifold is not a product of the AKSZ source and the target,
and therefore one has to resort to more general objects known as gauge PDEs
[24] (see also [25, 26] for earlier but less geometrical description). Gauge
PDEs can be described as $Q$-bundles [27] over the shifted tangent bundle over
the space-time.

The paper is organized as follows: in Section 2 we revisit the presymplectic
BV-AKSZ formulation of diffeomorphism-invariant local gauge theories. In
Section 3 we briefly review conformal gravity and its well-known frame-like
(gauge) formulation and propose a specific presymplectic structure on its BV-
BRST complex, which yields the proper presymplectic BV-AKSZ representation of
the system. Section 4 is devoted to a generalization of the approach to not
necessarily diffeomorphism-invariant systems. Technical details are relagated
to the Appendix. In addition, Appendix E contains an alternative proposal for
the symplectic structure which is based on the well-known frame-like (gauge)
formulation of conformal gravity.

##  2 Presymplectic BV-AKSZ formulation

###  2.1 AKSZ sigma models

The AKSZ approach [1] has been initially formulated for Lagrangian theories.
Its extension to gauge systems at the level of equations of motion is
straightforward and amounts to forgetting the symplectic structure and
rephrasing everything in terms of the BRST differential rather than BV master
action and it is instructive to start with this more general case.

By AKSZ sigma model at the level of equations of motion we mean a pair of
$\mathbb{Z}$-graded $Q$-manifolds, one of which, $(T[1]X,d_{X})$, where
$d_{X}$ is the de Rham differential of $X$ seen as a $Q$ structure on $T[1]X$,
is regarded as source (i.e. $X$ is a genuine space-time) and another one
$(F,q)$ as the target. The $\mathbb{Z}$-grading is denoted as $\mathrm{gh}($)
in what follows. In particular, for $(T[1]X,d_{X})$ the ghost degree is just a
form-degree.

Fields of the model are maps from the source to the target. It is convenient
to introduce a trivial fiber bundle $(E,Q)=(T[1]X,d_{X})\times(F,q)$ (here we
mean product of $Q$-manifolds, i.e. $Q=q+d_{X}$ in the adapted coordinates and
regard fields as section $T[1]X\to E$. Recall that such maps are by definition
homomorphisms of the algebras of functions and, in contrast to supermaps, they
preserve the degree. The generalization to the case where such a bundle is
trivial only locally (but still $Q=q+d_{X}$) is straightforward. More general
situation is discussed in Section 4.

If $\sigma:T[1]X\to E$ denotes a field configuration then the equations of
motion read as

$d_{X}\circ\sigma^{*}=\sigma^{*}\circ Q\,.$ (2.1)

The infinitesimal gauge variation of the configuration $\sigma$ is given by

$\delta\sigma^{*}=d_{X}\circ\xi_{\sigma}^{*}+\xi_{\sigma}^{*}\circ Q\,$ (2.2)

where $\xi_{\sigma}^{*}:\mathcal{C}^{\infty}(E)\to\mathcal{C}^{\infty}(T[1]X)$
is a gauge parameter map, which has a degree $-1$ and satisfies

$\xi_{\sigma}^{*}(fg)=(\xi_{\sigma}^{*}(f))\sigma^{*}(g)+(-1)^{|f|}\sigma^{*}(f)\xi_{\sigma}^{*}(g)\,.$
(2.3)

In a similar way one defines gauge for gauge symmetries. The above is just a
coordinate-free reformulation of the equations of motion and gauge symmetries
encoded in the BRST differential of the AKSZ sigma models. Details can be
found in [24].

A remarkable feature of the AKSZ construction is that the full-scale BV
formulation (at the level of equations of motion) is also encoded in the AKSZ
data. More precisely, the space of fields, ghosts, and antifields is given by
super-maps from $T[1]X$ to $F$ (super-sections of $E$) so that the space of
sections is recovered as the degree $0$ component (the body) of the space of
super-sections. The coordinates (fields and antifields) on the space of
supermaps can be introduced as:

$\displaystyle\widehat{\sigma}^{*}(\psi^{A})={\overset{0}{\psi}}{}^{A}(x)+{\overset{1}{\psi}}{}^{A}_{\mu}(x)\theta^{\mu}+{\overset{2}{\psi}}{}^{A}_{\mu\nu}(x)\theta^{\mu}\theta^{\nu}+\ldots\,,$
(2.4)

where $\widehat{\sigma}^{*}$ is a pullback map induced by super-section
$\widehat{\sigma}$ and we made use of local coordinates $\psi^{A}$ on $M$ and
the adapted local coordinates $x^{\mu},\theta^{\mu}$ induced by local
coordinates $x^{\mu}$ on $X$. Note that form-degree $k$ components
${\overset{k}{\psi}}{}^{A}_{\mu_{1}\ldots\mu_{k}}$ carry ghost degree
$\mathrm{gh}(\psi^{A})-k$. The space of maps is recovered by setting to zero
all the coordinates of nonvanishing degree.

In the field-theoretical terms the BRST differential of the AKSZ sigma model
can be written as:

$s=\int
d^{n}xd^{n}\theta\big{(}q^{A}(\psi(x,\theta))-d_{X}\psi^{A}(x,\theta)\big{)}\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{{\delta}}{\delta\psi^{A}(x,\theta)}$}\kern
1.0pt\,,$ (2.5)

where $n=\dim{X}$.

If in addition to $Q$-structure the target space is equipped with the
compatible symplectic structure $\omega$ of degree $\mathrm{gh}(\omega)=n-1$,
i.e. $\omega$ is nondegenerate, $d\omega=0$ and $L_{q}\omega=0$ then the
theory described by the AKSZ model is Lagrangian. More specifically,
introducing $\mathcal{L}$ via $i_{q}\omega+d\mathcal{L}=0$ the action can be
written as:

$S[\sigma]=\int_{T[1]X}[(\sigma^{*}\,\chi)(d_{X})+\sigma^{*}(\mathcal{L})]\,,$
(2.6)

where $\sigma$ is a section: $T[1]X\to E$ and we have introduced one form
presymplectic potential $\chi$ such that $d\chi=\omega$. Here
$(\sigma^{*}\,\chi)(d_{X})$ denotes evaluation of a 1-form $\sigma^{*}\,\chi$
on the vector field $d_{X}$.

It turns out that just like the BRST differential the BV master action can
also be immediately written just in terms of $q,\omega,d_{X}$. More precisely,
it is given by

$S_{BV}[\widehat{\sigma}]=\int_{T[1]X}[(\widehat{\sigma}^{*}\,\chi)(d_{X})+\widehat{\sigma}^{*}(\mathcal{L})]\,,$
(2.7)

i.e. has the same form as the classical action but with $\sigma$ replaced with
$\widehat{\sigma}$. It satisfies the master equation with respect to the odd
Poisson bracket of degree $1$ induced by $\omega$ on the space of super-
sections [1].

If the target space is finite-dimensional and $n>1$ the AKSZ sigma model is
topological.333For $n=1$ AKSZ sigma-models describe constrained Hamiltonian
systems with trivial Hamiltonians [28] , typical example of which are
(spinning) relativistic particles. It is debatable whether it is natural to
call such models topological. However, generic diffeomorphism-invariant gauge
theories can be represented as AKSZ sigma models at the price of considering
infinite-dimensional targets. For instance taking as $M$ a fiber of the jet-
bundle of BV-BRST formulation for a diff-invariant gauge system and as $q$ the
restriction of its BV-BRST differential to the fiber (in the diff-invariant
case the BV-BRST formulation can be locally encoded in the fiber, see e.g.
[29, 30, 25]) the respective AKSZ sigma model is equivalent to the initial
gauge theory [25] (see also [26, 31] for the Lagrangian version). Various
equivalent (AKSZ) formulations can be arrived at by equivalent reductions of
the target $Q$-manifold.

###  2.2 Presymplectic BV-AKSZ formulation

It turns out that relaxing the nondegeneracy condition for $\omega$ gives a
natural generalization of AKSZ sigma models. By presymplectic AKSZ sigma model
we mean the same data $(T[1]X,d_{X})$ and $(F,q)$, where $F$ is equipped with
not necessarily invertible 2-form $\omega$ satisfying
$\mathrm{gh}(\omega)=n-1$, $L_{q}\omega=0$, $d\omega=0$.

An important difference with the usual AKSZ is that we do not require that the
underlying gauge system is necessarily determined (at the level of equations
of motion) by the BRST differential $s$ given by (2.5). Instead, the gauge
system is determined by the action functional (2.6), where some of the fields
(on which the action doesn’t really depend) are gauged away. It turns out that
a variety of gauge theories can be represented in this way [2]. Now we give a
systematic exposition of BV-AKSZ formulation following [8].

To make the definition of a presymplectic BV-AKSZ formulation more precise let
us restrict for the moment to a coordinate patch $U\subset X$ and use the
adapted coordinates $x^{\mu},\theta^{\mu},\psi^{A}$. Then it is convenient to
represent the space of supermaps $Smaps(T[1]U,F)$ as $Smaps(U,\bar{F})$, where
$\bar{F}=Smaps(\mathbb{R}^{n}[1],F)$. A generic supermap from $\bar{F}$ is
given by:

$\widehat{\sigma}^{*}(\psi^{A})=\sum_{l=0}^{n}\frac{1}{l!}{\overset{l}{\psi}}{}^{A}_{\mu_{1}\ldots\mu_{l}}\theta^{\mu_{1}}\ldots\theta^{\mu_{l}}\,,$
(2.8)

so that as coordinates on $\bar{F}$ one can take
${\overset{l}{\psi}}{}^{A}_{\mu_{1}\ldots\mu_{l}}$, $l=0,\ldots n$,
$n=\dim{X}$. In what follows we use $\Psi^{I}$ as a collective notation for
the coordinates of $\bar{F}$. Applying the same construction to all the
coordinate charts of $X$ results in the locally-trivial fiber bundle $\bar{E}$
over $X$ with a typical fiber $\bar{F}$. 444Note that although $E$ is assumed
trivial, $\bar{E}$ can be only locally-trivial if $TX$ is not globally-
trivial.

The 2-form $\omega$ on $F$ defines a $2$ form $\bar{\omega}$ on $\bar{F}$ in
the standard way:

$\bar{\omega}=\int
d^{n}\theta\,\omega_{AB}(\psi(\theta))\,d\psi^{A}(\theta)d\psi^{B}(\theta)\,,$
(2.9)

where $d\psi^{A}(\theta)$ can be seen as a generating function for the
differentials $d\Psi^{I}$. Note that this form transforms as a density on $U$
under the change of $x^{\mu}$-coordinates. It defines a density on $X$ with
values in the 2-forms on the fiber, which is defined globally.555Another
possibility to work with an invariantly-defined object is to define an $n+2$
form $\omega^{\bar{E}}=(dx)^{n}\bar{\omega}$ on the total space of $\bar{E}$.

By construction $\bar{\omega}$ is closed and $\mathrm{gh}(\bar{\omega})=-1$.
In addition we assume that $\bar{\omega}$ is regular. This may involve taking
as $\bar{F}$ only those supermaps where the rank of $\bar{\omega}$ is constant
and maximal (this is exactly what one naturally does [8] in the case of GR
and, as we are going to see, in the case of conformal gravity). A regular
closed two form defines an involutive distribution of its zero vectors, i.e.
vector fields satisfying $i_{V}\bar{\omega}=0$. It is the standard fact that
the symplectic quotient (the space of the integrable submanifolds, which
exists, at least locally) is naturally a symplectic manifold which we denote
$\bar{G}$ in what follows. Moreover, it can be realised (again, at least
locally) as a submanifold transversal to the kernel distribution in which case
the symplectic structure coincides with the restriction of $\bar{\omega}$ to
the submanifold. In this way we have arrived at the space of fields equipped
with the odd symplectic structure of ghost degree $-1$, just like in the case
of BV formalism.

To have a complete BV formulation, in addition to the symplectic structure
$\bar{\omega}$ on the space of fields we need a master-action satisfying
master equation. To analyse this it is convenient to consider a jet bundle
$J^{\infty}(\bar{E})$, where $\bar{E}\to X$ is the introduced above fiber
bundle with the typical fiber $\bar{F}$. The BV-like symplectic structure is
then introduced in the standard way as an $(n,2)$ form

$\omega^{pBV}=(dx)^{n}\bar{\omega}_{IJ}\mathrm{d_{v}}\Psi^{I}\mathrm{d_{v}}\Psi^{J}\,,$
(2.10)

where we used standard local coordinates on $J^{\infty}(\bar{E})$, induced by
trivialisation $\bar{E}|_{U}=T[1]U\times\bar{F}$. Note that this symplectic
structure is defined globally on $\bar{E}$. More invariant description of the
underlying geometry and its generalization is presented in section 4.

When written in terms of $J^{\infty}(\bar{E})$ the integrand (over $X$) of the
BV-like action (2.7) reads as

$\begin{gathered}L^{BV}=K+\bar{\mathcal{L}}\,,\\\ K=(dx)^{n}\int
d^{n}\theta\chi_{A}(\psi(\theta))\theta^{\mu}D_{\mu}\psi^{A}(\theta)\,,\qquad\bar{\mathcal{L}}=(dx)^{n}\int
d^{n}\theta\mathcal{L}(\psi(\theta))\,,\end{gathered}$ (2.11)

where $D_{\mu}$ denote total derivatives on $J^{\infty}(\bar{E})$. It turns
out that the restriction of $L^{BV}$ to the jet sub-bundle associated to the
symplectic quotient $\bar{G}$ realized as a submanifold of $\bar{F}$ satisfies
all the conditions for the BV master action.

To see this let us first introduce a natural prolongation $\bar{q}$ of the
vector field $q$ from $F$ to $\bar{F}$:

$\bar{q}(\psi^{A}(\theta))=(q\psi^{A})(\theta).$ (2.12)

Note that $\bar{q}$ is a Hamiltonian vector field on $\bar{F}$ with
Hamiltonian $\bar{\mathcal{L}}$, i.e.
$i_{\bar{q}}\bar{\omega}-d\bar{\mathcal{L}}=0$. Now we can introduce a
vertical evolutionary vector field $s$:

$s=D^{F}+\bar{q}^{pr}\,,$ (2.13)

where $\bar{q}^{pr}$ denotes a natural prolongation of $\bar{q}$ to $\bar{E}$
defined by requiring $[\bar{q}^{pr}{,}\,D_{\mu}]=0$ and $D^{F}$ is a vertical
evolutionary vector field defined through
$D^{F}\psi^{A}(\theta)=-\theta^{\mu}D_{\mu}(\psi^{A}(\theta))$. Of course, $s$
is just a BRST differential (2.5) written in terms of $J^{\infty}(\bar{E})$.

We have the following relations [8]:

$i_{D^{F}}\omega^{pBV}+\mathrm{d_{v}}K=\mathrm{d_{h}}(\cdot)\,,\qquad
i_{D^{F}}i_{D^{F}}\omega^{pBV}=\mathrm{d_{h}}(\cdot)\,,$ (2.14)

as well as

$\qquad i_{D^{F}}i_{\bar{q}^{pr}}\omega^{pBV}=\mathrm{d_{h}}(\cdot)\,.$ (2.15)

which can be directly checked and amount to:

$i_{s}\omega^{pBV}+\mathrm{d_{v}}L^{BV}=\mathrm{d_{h}}(\cdot)\,,\qquad
i_{s}i_{s}\omega^{pBV}=\mathrm{d_{h}}(\cdot)\,.$ (2.16)

The above relations remain intact when restricted to $J^{\infty}(\bar{G}\times
X)$ understood as a jet subbundle of $\bar{E}$ (recall that we assumed that
the symplectic quotient $\bar{G}$ is realised as a submanifold of $\bar{F}$),
where by restriction of $s$ we mean its projection induced by the projection
$\bar{F}\to\bar{G}$ (recall that there is a natural projection to the
symplectic quotient). If by some abuse of notations $s$ denotes its projection
to $J^{\infty}(\bar{G}\times X)$ the restricted relations read as:

$i_{s}\omega^{BV}+\mathrm{d_{v}}L^{BV}=\mathrm{d_{h}}(\cdot)\,,\qquad
i_{s}i_{s}\omega^{BV}=\mathrm{d_{h}}(\cdot)\,,$ (2.17)

where $\omega^{BV}$ is the restriction of $\omega^{BV}$ to
$J^{\infty}(\bar{G}\times X)\subset J^{\infty}(\bar{E})$. Because $\bar{G}$ is
symplectic the above relation imply the classical master equation satisfied by
$L^{BV}$.

To summarize, we have shown how a presymplectic BV-AKSZ system gives rise to a
genuine BV gauge system. The only subtle point is whether a resulting BV
master action is a proper solution to the master equation. This has to be
addressed separately.

###  2.3 Target space and the total BRST complex

A natural question is what is the origin of the target space $Q$-manifold. It
turns out that it is directly related to the total BRST complex of the system.

Suppose we start with a jet-bundle BV formulation. The underlying bundle is
$\mathcal{E}\to X$, i.e. coordinates on its fibres are fields, ghosts, ghosts-
for ghosts, …, as well as antifields (For the moment we do not assume the
system to be Lagrangian and hence in general there is no natural pairing
between fields and antifields). The jet-space BRST complex is given by local
functions on $J^{\infty}(\mathcal{E})$ with the differential being the BRST
differential $s$, i.e. a degree $1$ vertical evolutionary vector field on
$J^{\infty}(\mathcal{E})$.

If the system in question is diffeomorphism-invariant (i.e. space-time
reparmetrizations are among the gauge symmetries encoded in $s$), differential
$s$ does not depend on $x$ and hence its restriction to a fiber define a
$Q$-structure (which we keep denoting by $s$) on a typical fiber $\mathcal{F}$
of $J^{\infty}(\mathcal{E})$. Moreover, locally on $X$, $Q$-manifold
$(\mathcal{F},s)$ is equivalent to the total BRST complex of the system, i.e.
the complex of horizontal forms on $J^{\infty}(\mathcal{E})$ with the
differential being $\widetilde{s}=\mathrm{d_{h}}+s$ and the degree being ghost
degree plus form degree.

It is known [25] that in diff-invariant case the AKSZ sigma model (at the
level of equations of motion) with the target $(\mathcal{F},S)$ (or its
equivalent reduction) and source $T[1]X$ gives an equivalent formulation of
the gauge system in question. Furthermore, the compatible presymplectic
structure $\omega$ on $F$ can be traced back [8] (see also [7]) to the initial
BV symplectic structure extended to a cocycle of the total BRST differential
$\widetilde{s}$. More detailed and general exposition of the relation is given
in Section 4.

##  3 Presymplectic AKSZ form of conformal gravity

###  3.1 Conformal gravity

Conformal gravity is a well-known theory of gravity with an additional gauge
invariance compared to GR: local Weyl invariace. The action for such theory is
given by

$S[g]=\int d^{4}x\sqrt{-g}W_{\mu\nu\rho\lambda}W^{\mu\nu\rho\lambda}\,,$ (3.1)

where $W_{\mu\nu\rho\lambda}$ is the trace-free part of the Riemann curvature
$R_{\mu\nu\rho\lambda}$. The indices are raised and lowered by the metric
$g_{\mu\nu}$. Weyl invariance implies that the action only depends on the
conformal class of the metric $[g_{\mu\nu}]=g_{\mu\nu}/\sim$, where
$g_{\mu\nu}\sim g^{\prime}_{\mu\nu}=e^{2\phi}g_{\mu\nu}$. In field theoretical
terms, the infinitesimal gauge symmetries of the theory are given by

$\delta
g_{\mu\nu}=\xi^{\rho}\partial_{\rho}g_{\mu\nu}+g_{\mu\rho}\partial_{\nu}\xi^{\rho}+g_{\rho\nu}\partial_{\mu}\xi^{\rho}+2\omega
g_{\mu\nu}\,,$ (3.2)

where $\xi^{\mu}$ and $\omega$ are parameters of the diffeomorphisms and Weyl
transformations respectively.

The equations of motion determined by $S[g]$ read as:

$B_{\mu\nu}=0\,,\qquad\text{where}\qquad B_{\mu\nu}\equiv
P^{\rho\lambda}W_{\mu\rho\nu\lambda}+\nabla^{\rho}\nabla_{\rho}P_{\mu\nu}-\nabla^{\rho}\nabla_{\mu}P_{\nu\rho}$
(3.3)

and $P_{\mu\nu}$ is the Schouten tensor (for more details and precise
definitions see Appendix A). Tensor $B_{\mu\nu}$ is known as Bach tensor and
it is Weyl invariant so that, as they should be, EOMs are diffeomorphism and
Weyl invariant.

###  3.2 Frame-like formulation

Just like in the case of Einstein gravity where Poincaré algebra is the
maximal symmetry of a vacuum solution and the theory itself can be
reformulated in terms of the Poincaré algebra connection, the maximal symmetry
of a solution to conformal gravity is the conformal algebra $o(d,2)$ and the
theory also admits a formulation in terms of the $o(4,2)$-connection. Although
we don’t really employ it in what follows, we now briefly review the
formulation initially put forward in [13]. For further details and pedagogical
presentation see e.g. [32].

The usual basis in $o(4,2)$ given by translations $P_{a}$, special conformal
transformations $K_{a}$, Lorentz transformations $J_{ab}$, and dilation $D$,
where the commutation relation read as:

$\begin{gathered}[J_{ab}{,}\,J_{cd}]=\eta_{bc}J_{ad}+\eta_{ad}J_{bc}-\eta_{ac}J_{bd}-\eta_{bd}J_{ac}\,,\\\
[J_{ab},P_{c}]=\eta_{bc}P_{a}-\eta_{ac}P_{b},\qquad[J_{ab},K_{c}]=\eta_{bc}K_{a}-\eta_{ac}K_{b}\,,\\\
[P_{a},D]=P_{a},\qquad[K_{a},D]=-K_{a}\,,\\\
[K_{a},P_{b}]=-2(\eta_{ab}D+J_{ab})\,.\end{gathered}$ (3.4)

Here $\eta_{ab}$ denotes a constant Minkowski metric with almost positive
signature.

In this basis the components of the connection 1-form $A$ and its curvature
$\Omega$ are introduced as:

$A=e^{a}P_{a}+\omega^{ab}J_{ab}+f^{a}K_{a}+\lambda D$ (3.5)

and

$\Omega=T^{a}P_{a}+R^{ab}J_{ab}+S^{a}K_{a}+\Lambda D\,.$ (3.6)

Here and in what follows we systematically omit the wedge product symbol.

The action of the conformal gravity in terms of a connection of the conformal
group can be written in the following form [13]:

$S[e,f,\omega,\lambda]=\int R_{ab}R_{cd}\epsilon^{abcd}\,,$ (3.7)

where

$R_{ab}={\overset{(0)}{R}}_{ab}+(e_{a}f_{b}-e_{b}f_{a})$ (3.8)

is the Lorenz sector part of the full conformal curvature,

$\stackrel{{\scriptstyle(0)}}{{R}}_{ab}=d\omega_{ab}+\omega_{a}{}^{c}\omega_{cb}$
(3.9)

is the curvature of the Lorentz-subalgebra component of the connection, and
$\epsilon^{abcd}$ is an invariant totally-antisymmetric tensor with
$\epsilon^{0123}=1$.

Here we do not discuss global space-time geometry and hence consider
Lagrangians modulo total derivatives. In particular the Gauss-Bonnet term
${\overset{(0)}{R}}_{ab}\stackrel{{\scriptstyle(0)}}{{R}}_{cd}\epsilon^{abcd}$
can be omitted and the action can be rewritten as:

$\begin{gathered}S=\int(2\stackrel{{\scriptstyle(0)}}{{R}}_{ab}(e_{c}f_{d}-e_{d}f_{c})+(e_{a}f_{b}-e_{b}f_{a})(e_{c}f_{d}-e_{d}f_{c})\epsilon^{abcd}=\\\
=\int(4\stackrel{{\scriptstyle(0)}}{{R}}_{ab}e_{c}f_{d}+4e_{a}f_{b}e_{c}f_{d})\epsilon^{abcd}=\\\
=4\int(d\omega_{ab}e_{c}f_{d}+\omega_{a}{}^{k}\omega_{kb}e_{c}f_{d}+e_{a}f_{b}e_{c}f_{d})\epsilon^{abcd}\,.\end{gathered}$
(3.10)

It turns out [13] that action (3.10) is equivalent through elimination of the
auxiliary fields to the standard conformal gravity action in the metric
formulation (3.1), provided one in addition imposes the torsion constraint

$T_{a}\equiv de_{a}+\omega_{a}{}^{b}e_{b}+e_{a}\lambda=0.$ (3.11)

This constraint is in fact algebraic (allows to uniquely express $\omega_{ab}$
in terms of $e_{a}$ and $\lambda$) and hence the system remains Lagrangian.
Now we briefly recall how this works. Technical details can be found in
Appendix A.

Let us consider first equations for $f_{d}$:

$\epsilon^{abcd}R_{ab}e_{c}=0.$ (3.12)

The curvature 2-form can be expanded as $R_{ab}=R_{abcd}e^{c}e^{d}$. Then
equation (3.12) implies $R^{c}{}_{bcd}=0$ which in turn gives:

$f=-\frac{1}{6}\stackrel{{\scriptstyle(0)}}{{R}}$ (3.13)

and

$f_{b,\mu}=-\frac{1}{2}(e^{d}{}_{,\mu}\stackrel{{\scriptstyle(0)}}{{R^{c}{}_{bcd}}}-\frac{1}{6}e_{b,\mu}\stackrel{{\scriptstyle(0)}}{{R}})\,,$
(3.14)

where $f=f^{a}{}_{\mu}e_{a}{}^{\mu}$ and
$\stackrel{{\scriptstyle(0)}}{{R}}=\stackrel{{\scriptstyle(0)}}{{R}}{}^{a}{}_{bac}\eta^{bc}$
). One concludes that on-shell $f_{a,\mu}$ is an auxiliary field which
coincides with the Schouten tensor (expressed through the frame field
$e^{a}{}_{\mu}$) and can be eliminated.

Taking into account $T_{a}=0$, the Bianchi identity $d\Omega+[A{,}\,\Omega]=0$
in the sector of translations give:

$e_{c}R^{c}{}_{a}+e_{a}\Lambda=0$ (3.15)

which, in turn, implies (see Appendix A for details) that the curvature in the
sector of dilations vanishes

$\Lambda=d\lambda+e_{a}f^{a}=0\,.$ (3.16)

Let us turn our attention to gauge transformations. Consider the following
transformations:

$\delta A=d\alpha+[A,\alpha]\,,$ (3.17)

under which $\Omega$ transforms as:

$\delta\Omega=[\Omega,\alpha]\,.$ (3.18)

Note that in general these are not gauge symmetries of the action. It turns
out that for $\alpha=b^{a}K_{a}$, i.e. for $\alpha$ non-vansihing only in the
sector of ”special conformal transformations”, this transformation is a gauge
symmetry of the action thanks to $T^{a}=0$. These gauge transformations are
algebraic in the sector of $\lambda$ (dilation part of the connection):

$\delta\lambda=-e^{a}b_{a}\,.$ (3.19)

It follows $\lambda$ is pure gauge and can always be set to zero by a gauge
transformation. In this gauge the torsion free condition (3.11) implies that
$\omega$ is just the Levi-Civita connection.

To summarize: action (3.7) supplemented with the torsion constraint (3.11) is
equivalent to the Weyl action through the elimination of generalized auxiliary
fields (auxiliary fields + algebraically pure gauge fields). The equivalent
reduction boils down to the following steps:

  1. 1.

Elimination of the auxiliary field $f^{a}{}_{\mu}$

  2. 2.

Algebraically solving zero torsion condition for the Lorentz connection,
$\omega=\omega(e,\lambda)$

  3. 3.

Using gauge symmetry (3.17) with $\alpha=b^{a}K_{a}$, which is algebraic in
the sector of $\lambda_{\mu}$, to set $\lambda_{\mu}=0$. In this way we
arrived to action (3.1) modulo a topological term but expressed in terms of
the frame field $e_{\mu}^{a}$ rather than metric.

  4. 4.

Using gauge symmetry (3.17) with $\alpha=k^{a}J_{ab}$ to eliminate all the
components of the frame field but the metric,666This step is analogous to the
elimination of the metric from the CGR BRST complex and strictly speaking
might require assumptions. See Appendix D. giving the initial (3.1) in terms
of the metric.

###  3.3 Jet-bundle BRST complex for CGR

As we recalled in Section 2.3 the target space of the AKSZ formulation of a
given diffeomorphism-invariant system originates from a fiber of the jet-
bundle BV-BRST formulation of the system. Now we recall this formulation for
CGR in the metric-like form and construct a particularly useful minimal model
for the respective total BRST complex.

Let us restrict to local analysis and introduce the respective structures
using local coordinates. The bundle $\mathcal{E}$ is that of the metric
$g_{\mu\nu}(x)$, ghosts $\xi^{\mu}(x)$ associated to the diffeomorphism
parameters and ghost $\lambda(x)$, associated to the parameter of Weyl
rescalings. The associated coordinates on the jet-bundle
$J^{\infty}(\mathcal{E})$ read as:

$x^{\mu}\,\quad
g_{\mu\nu|\alpha...}\,,\quad\xi^{\mu}{}_{|\alpha...}\,,\quad\lambda_{|\alpha...}\,.$
(3.20)

In these coordinates total derivatives $D_{\rho}$ have the standard form, e.g.
$D_{\rho}g_{\mu\nu}=g_{\mu\nu|\rho}$.

The gauge part $\gamma$ of the BRST differential is determined by

$\begin{gathered}\gamma
g_{\mu\nu}=\xi^{\rho}g_{\mu\nu|{\rho}}+\xi^{\rho}{}_{|\mu}g_{\rho\nu}+\xi^{\rho}{}_{|\nu}g_{\rho\mu}-2\lambda
g_{\mu\nu}\,,\\\ \gamma\xi^{\mu}=\xi^{\rho}\xi^{\mu}{}_{|\rho}\,,\\\
\gamma\lambda=\xi^{\rho}\lambda_{|\rho}\,,\end{gathered}$ (3.21)

together with the conditions $[D_{\mu}{,}\,\gamma]$ and $\gamma x^{\mu}=0$
(i.e. it is a vertical and evolutionary vector field).

The total BRST complex is given by the local horizontal forms on
$J^{\infty}(\mathcal{E})$ (which, as before, we identify with functions on
$J^{\infty}(\mathcal{E})$ pulled back to $T[1]X$. In local coordinates these
are simply functions of new variables $\theta^{\mu}$, i.e. coordinate on the
fibers of $T[1]X$, and coordinates (3.20). The total BRST differential reads
as

$\widetilde{\gamma}=\gamma+\mathrm{d_{h}}\,,\qquad\mathrm{d_{h}}=\theta^{\mu}D_{\mu}$
(3.22)

Its cohomology in form degree $n+k$, $k\,{\geqslant}\,0$ are known to be
locally isomorphic to the cohomology of $\gamma$ in the space of local
functionals.

The above complex is the total BRST complex for off-shell CGR, i.e. CGR
without equations of motion imposed. Its on-shell version is obtained by
replacing $J^{\infty}(\mathcal{E})$ by its subbundle $\mathcal{G}$ determined
by the equations of motion $B_{\mu\nu}=0$ and their prolongations
$D_{(\alpha)}B_{\mu\nu}=0$. A possible alternative is to introduce antifields
and to work in terms of the complete BRST differential $s=\gamma+\delta$.

Because CGR is diff invariant the total BRST complex is locally equivalent to
that of local functions on the typical fiber $\mathcal{F}$ of $\mathcal{G}$
equipped with $\gamma$ restricted to the fiber. Moreover, the AKSZ sigma model
with the source $T[1]X$ and target $(\mathcal{F},\gamma)$ is locally
equivalent [25] to the CGR defined at the level of equations of motion.

###  3.4 Minimal model of the BRST complex for CGR

We are interested in the presymplectic BV-AKSZ formulation that is natural in
the sense that it involves a minimal number of fields (among all equivalent
formulation of this type). In particular, the resulting formulation can be
regarded as a canonical one (of course there remains a freedom of field
redefinition). To this end it is natural to consider as a target space a
minimal $Q$-manifold that is equivalent to $(\mathcal{F},\gamma)$ in the sense
of natural equivalence of $Q$-manifolds. It is known [25] that AKSZ sigma
models (at the level of equations of motion) based on equivalent target
$Q$-manifolds are equivalent in the sense of elimination of generalized
auxiliary fields. Moreover, minimal $Q$-manifold corresponds to a frame-like
formulation of the underlying model. Minimal models for various local gauge
theories were discussed from different perspectives in e.g. [33, 30, 25, 34]
but the idea and somewhat implicit version appeared already in [35, 36, 37].

Let us recall the notion of equivalence for $Q$-manifolds. Here we follow the
approach developed in [38, 25, 24]. First of all we need the following:

######  Definition 3.1.

A contractible $Q$-manifold is a $Q$-manifold of the form $(T[1]V,d_{V})$,
where $V$ is a graded linear space understood as a graded manifold and $d_{V}$
is a canonical differential (De Rham) on $T[1]V$.

For our present purposes it is enough to consider a rather restrictive version
of equivalence: a $Q$-manifold $(N,q)$ is called an equivalent reduction of
$(M,Q)$ if $(M,Q)=(N,q)\times(T[1]V,d_{V})$, i.e. $M$ is a product $N\times
T[1]V$ and $Q$ is a product $Q$-structure. This generates the equivalence
relation for $Q$-manifolds.

It follows from the definition of the equivalent reduction that there exists
independent functions $w^{a},v^{a}$ on $M$ such that $Qw^{a}=v^{a}$ and
$(N,q)$ can be identified as a $Q$-submanifold determined by $w^{a}=0$ and
$v^{a}=0$. Indeed, as $w^{a},v^{a}$ one can take linear coordinates on $T[1]V$
pulled back to $M$ by the canonical projection $M\to T[1]V$. Moreover, one can
introduce a local coordinate system $\phi^{i},w^{a},v^{a}$ on $M$ where

$Q\phi^{i}=q^{i}(\phi)\,,\qquad Qw^{a}=v^{a}\,.$ (3.23)

In this form it is clear why the equivalent reduction from $(M,Q)$ to $(N,q)$
is often refereed to as elimination of contractible pairs (in this form it has
been extensively used in the context of local BRST cohomology for a long time,
see e.g. [39, 29, 30, 40, 41]).

A particular equivalent reduction of the $Q$-manifold $(\mathcal{F},\gamma)$
in the case of off-shell CGR has been obtained by N. Boulanger in [19, 20].
More precisely, it was shown that all the derivatives of the metric except for
Weyl curvature and its independent covariant derivatives can be taken as
$w^{a}$ variables while the respective $v^{a}$ variables parameterize all the
derivatives of ghosts of order 2 and higher. As coordinates $\phi^{i}$ one
takes: $\xi^{\mu},\xi_{\mu}{}^{\nu}$ related to $\xi^{\mu}$ ghost and its 1-st
derivatives, $\lambda,\kappa_{a}$ related to $\lambda$ ghost and its 1-st
derivatives, and $W_{\mu\nu\rho\sigma|\alpha\ldots}$ parameterizing Weyl
tensor and its independent covariant derivatives. The action of $\gamma$ on
the ghost degree $1$ variables is given by

$\begin{gathered}{\gamma}\xi_{\mu}=\xi^{\rho}{\xi}_{\rho\mu}\,,\\\
{\gamma}{\xi}_{\mu}{}^{\nu}={\xi}_{\mu}{}^{\rho}{\xi}_{\rho}{}^{\nu}+P^{\lambda\nu}_{\mu\rho}\kappa_{\lambda}\xi^{\rho}+\frac{1}{2}\xi^{\rho}\xi^{\lambda}W_{\mu}{}^{\nu}{}_{\rho\lambda}\,,\\\
{\gamma}\kappa_{\nu}={\xi}_{\nu}{}^{\rho}\kappa_{\rho}+\frac{1}{2}\xi^{\rho}\xi^{\lambda}C_{\nu\rho\lambda}\,,\\\
{\gamma}\lambda=\kappa_{\rho}\xi^{\rho}\,,\end{gathered}$ (3.24)

where
$P^{\lambda\nu}_{\mu\rho}=-g^{\lambda\nu}g_{\mu\rho}+\delta^{\lambda}_{\mu}\delta^{\nu}_{\rho}+\delta^{\lambda}_{\rho}\delta^{\nu}_{\mu}$,
and denotes the Cotton tensor which can be expressed in our coordinates as
$C_{\mu\nu\rho}=W^{\alpha}{}_{\mu\nu\rho|\alpha}$. In what follows we will
also denote
$C_{\mu\nu\rho|\alpha_{1}...}:=W^{\alpha}{}_{\mu\nu\rho|\alpha\alpha_{1}...}$
. The action of $\gamma$ on the metric components reads as

${\gamma}g_{\mu\nu}=-2\lambda
g_{\mu\nu}+{\xi}_{\mu}{}^{\rho}g_{\rho\nu}+{\xi}_{\nu}{}^{\rho}g_{a\rho}$
(3.25)

and on Weyl tensor and its covariant derivatives $\gamma$ acts in the
following way:

${\gamma}W=\xi^{\mu}R_{\mu}W+{\xi}^{\mu}{}_{\nu}\Delta^{\nu}{}_{\mu}W+\kappa_{\mu}\Gamma^{\mu}W\,,$
(3.26)

where

$\displaystyle R_{\beta}W^{\mu}{}_{\nu\rho\sigma|\alpha_{1}...\alpha_{k}}$
$\displaystyle=W^{\mu}{}_{\nu\rho\sigma|\alpha_{1}...\alpha_{k}\beta}\,,$
(3.27)
$\displaystyle\Delta^{\beta}{}_{\gamma}W^{\mu}{}_{\nu\rho\sigma|\alpha_{1}...\alpha_{k}}$
$\displaystyle=-\delta^{\mu}_{\gamma}W^{\beta}{}_{\nu\rho\sigma|\alpha_{1}...\alpha_{k}}+\delta^{\beta}_{\nu}W^{\mu}{}_{\gamma\rho\sigma|\alpha_{1}...\alpha_{k}}+\ldots\,.$

The action of $\Gamma^{\mu}$ can be determined from the nilpotency of
${\gamma}$ and can be found in [19]. For example, ${\gamma}^{2}\kappa_{a}=0$
implies the relation $\Gamma^{\kappa}C_{\mu\nu\rho}=W^{\kappa}{}_{\mu\nu\rho}$
which we need in what follows. We refer to [19] for further details.

For our purposes we need to reduce the above $Q$-manifold even further. It
turns out that under some assumption on allowed class of metric configurations
one can eliminate the metric together with the symmetric component of
$\xi_{\mu\nu}$. Indeed,

${\gamma}g_{\mu\nu}=-2\lambda
g_{\mu\nu}+{\xi}_{\mu}{}^{\rho}g_{\rho\nu}+{\xi}_{\nu}{}^{\rho}g_{a\rho}=-2\lambda
g_{\mu\nu}+2{\xi}_{(\mu\nu)}\,.$ (3.28)

We relegate the discussion of the conditions under which this gives an
equivalent reduction (for instance this the case if the metric is Euclidean)
to Appendix D and simply assume that these are fulfilled.

In a suitable coordinate system $x^{a}$ we decompose
$g_{ab}=\eta_{ab}+h_{ab}$, with $\eta_{ab}$ being standard Minkowski metric
and find

$\gamma h_{ab}=-2\lambda
g_{ab}+2\xi_{(ab)}\,,\qquad\xi_{(ab)}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}(\xi_{a}{}^{c}g_{cb}+\xi_{b}{}^{c}g_{ac})\,.$
(3.29)

So that $h_{ab}$ can be taken as $w^{a}$ variables in the equivalent
reduction. The reduced $Q$-manifold is obtained by setting $h_{ab}=\gamma
h_{ab}=0$, giving

$g_{ab}=\eta_{ab}\,,\qquad\xi_{(ab)}=\lambda\eta_{ab}\,.$ (3.30)

In what follows we use the antysymmetric matrix
$\rho_{ab}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}(\xi_{ab}-\xi_{ba})$ to
parametrize the remaining components of $\xi_{ab}$.

To summarize, the reduced $Q$-manifold $(F,q)$ is coordinatized by degree $1$
coordinates $\xi^{a},\rho^{ab},\kappa_{b},\lambda$ and degree $0$ coordinates
$W_{a(b}{}^{c}{}_{d|j\ldots)}$ (though it is very convenient to work with the
overcomplete set of $W$-coordinates $W_{abcd|j\ldots}$ as we keep doing). The
action of the reduced $\gamma$ (denoted by $q$ below) on the ghost degree $1$
coordinates can be easily obtained from (3.24) and is given explicitly by
777By setting to zero $W$ and $C$ one arrives at the Chevalley-Eilenberg
differential for conformal algebra, however in a slightly rescaled basis
comparing to (3.4):

$\begin{gathered}q\xi^{a}=\rho^{a}{}_{c}\xi^{c}+\xi^{a}\lambda\,,\\\
q\rho^{a}{}_{b}=\rho^{a}{}_{c}\rho^{c}{}_{b}+(\xi^{a}\kappa_{b}-\xi_{b}\kappa^{a})+\frac{1}{2}\xi^{c}\xi^{d}W^{a}{}_{bcd}\,,\\\
q\kappa_{b}=\kappa_{c}\rho^{c}{}_{b}+\lambda\kappa_{b}+\frac{1}{2}\xi^{c}\xi^{d}C_{bcd}\,,\\\
q\lambda=\kappa_{c}\xi^{c}\,.\end{gathered}$ (3.31)

The action of $q$ on $W$-coordinates is also easily inferred from (3.26). In
particular, for the Weyl and Cotton tensors we explicitly get:

$\displaystyle qW^{a}{}_{bcd}=$
$\displaystyle~{}\xi^{k}W^{a}{}_{bcd|k}-\rho_{k}{}^{a}W^{k}{}_{bcd}+$ (3.32)
$\displaystyle~{}\rho_{b}{}^{k}W^{a}{}_{kcd}+\rho_{c}{}^{k}W^{a}{}_{bkd}+\rho_{d}{}^{k}W^{a}{}_{bck}+2\lambda
W^{a}{}_{bcd}\,,$ $\displaystyle qC_{abc}=$
$\displaystyle~{}\xi^{k}C_{abc|k}+\rho_{a}{}^{k}C_{kbc}+$
$\displaystyle~{}\rho_{b}{}^{k}C_{akc}+\rho_{c}{}^{k}C_{abk}+3\lambda
C_{abk}+\kappa_{k}W^{k}{}_{abc}\,.$

It should be stressed that the second equation is not written in independent
coordinates which are the properly symmetrized Weyl ”derivatives”.

Furthermore, we need to switch to the on-shell version of this $Q$-manifold,
where the Bach equation and its differential consequences are taken into
account. The elimination of contractible pairs of $\gamma$ is not affected by
this restriction as the equations do not affect ghosts and gauge non-invariant
quantities. In particular, all the variables entering (3.31) and (3.32) remain
independent upon restriction to the surface of equations except for
$C_{abc|k}$ whose traces vanish on the surface. Using the restriction of these
coordinates to the surface as the coordinates therein one concludes that the
above relations stay true. In this way we have arrived at the $Q$-manifold
$(F,q)$ which is going to serve as a target space of the presymplectic AKSZ
formulation of CGR.

###  3.5 Presymplectic structure

As was discussed above the minimal $Q$-manifold $(F,q)$ of CGR should admit a
compatible presymplectic structure of ghost degree $n-1$, which originates
from the descent completion of the BV symplectic structure. More precisely,
the minimal $Q$-manifold can be seen as a $Q$-submanifold of total BRST
complex of CGR and the presymplectic structure can be obtained as the
restriction of this descent completion to the submanifold.888In this setup,
this is just an explicit realization of the homotopy transfer.

Instead of computing the descent completion and its transfer to the minimal
$Q$-manifold $(F,q)$, which can be quite involved technically, we try to find
such a structure by considering a generic ansatz of the closed 2-form of ghost
degree $3$ and not involving higher Weyl tensors. By some trial and error one
finds that the correct presymplectic structure should involve Weyl tensor and
Cotton tensors and the remaining ambiguity is fixed by $q$-invariance. The end
result reads explicitly as:

$\begin{gathered}\omega=\omega_{W}-2\omega_{C}\,,\\\
\omega_{W}=d(\rho_{ab})d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})\,,\qquad\omega_{C}=d(\xi_{a})d(C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})\,.\end{gathered}$
(3.33)

######  Proposition 3.2.

Presymplectic structure (3.33) on $F$ satisfies: $d\omega=0$, $L_{q}\omega=0$,
and $\mathrm{gh}(\omega)=n-1$ and hence together with the homological vector
field $q$ it defines a presymplectic AKSZ model.

The first and the third properties are obvious. Let us give here main points
of the proof of $L_{q}\omega=0$ (details are relegated to the Appendix B). It
is enough to show that $i_{q}\omega=d\mathcal{L}$ for some $\mathcal{L}$. We
have the following relations:

$i_{q}\omega_{W}=d(\rho_{al}\rho^{l}{}_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+(\xi_{a}\kappa_{b}-\xi_{b}\kappa_{a})d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+\\\
+\frac{1}{2}W_{abij}\xi^{i}\xi^{j}d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+d(\rho_{ab})\xi^{j}W^{abnm}{}_{|j}\epsilon_{nmpk}\xi^{p}\xi^{k}\,.$
(3.34)

It is nearly $dH_{W}$ with $H_{W}$ given by

$H_{W}=\rho_{al}\rho^{l}{}_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}+2\xi_{a}\kappa_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}+\\\
+W_{abij}\xi^{i}\xi^{j}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}\,,$ (3.35)

with the discrepancy being

$i_{q}\omega_{W}-dH_{W}={-2d(\xi_{a})\kappa_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}}+\\\
+{2\xi_{a}d(\kappa_{b})W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}}+{d(\rho_{ab})\xi^{j}W^{abnm}{}_{|j}\epsilon_{nmpk}\xi^{p}\xi^{k}}\,.$
(3.36)

As for the second term $\omega_{C}$ we find:

$\begin{gathered}i_{q}\omega_{C}=\rho_{an}\xi^{n}d(C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})+d(\xi^{a})\rho_{a}{}^{n}C_{nbc}\epsilon^{bcpk}\xi_{p}\xi_{k}+\\\
+\xi_{a}\lambda d(C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})+d(\xi_{a})\lambda
C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k}+\\\
+d(\xi_{a})\xi^{d}C_{abc|d}\epsilon^{bcpk}\xi_{p}\xi_{k}+d(\xi^{a})\kappa_{k}W^{k}{}_{abc}\epsilon^{bcpk}\xi_{p}\xi_{k}\,.\end{gathered}$
(3.37)

Introducing $H_{C}$ via

$H_{C}=\rho_{an}\xi^{n}C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k}+\xi_{a}\lambda
C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k}$ (3.38)

we find

$\begin{gathered}i_{q}\omega_{C}-dH_{C}={-d(\rho_{an})\xi^{n}C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k}}+{\xi_{a}d(\lambda)C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k}}+\\\
+{d(\xi_{a})\xi^{d}C_{abc|d}\epsilon^{bcpk}\xi_{p}\xi_{k}}{-d(\xi^{a})\kappa^{k}W_{akbc}\epsilon^{bcpk}\xi_{p}\xi_{k}}\,,\end{gathered}$
(3.39)

so that

$i_{q}(\omega_{W}-2\omega_{C})=d\mathcal{L}\,,$ (3.40)

where the ”Hamiltonian” is given by:

$\mathcal{L}=H_{W}-2H_{C}=\\\
=\rho_{al}\rho^{l}{}_{b}\mathcal{L}=W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}+2\xi_{a}\kappa_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}-2\rho_{an}\xi^{n}C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k}-\\\
-2\xi_{a}\lambda
C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k}+W_{abij}\xi^{i}\xi^{j}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}\,.$
(3.41)

Note that we have changed the sign convention for $\mathcal{L}$ with respect
to Section 2 to comply with the standard conventions for curvatures. This is
equivalent to changing the sign of $q$ from the very beginning and hence does
not affect the BV master equation.

We are now ready to spell out an explict form of the action. Introducing
component fields parameterizing $\sigma:T[1]X\to F$ via

$\begin{gathered}\sigma^{*}(\xi^{a})=e^{a}_{\mu}(x)\theta^{\mu}\,,\quad\sigma^{*}(\rho^{ab})=\omega^{ab}_{\mu}(x)\theta^{\mu}\,,\\\
\sigma^{*}(\kappa^{a})=f^{a}_{\mu}(x)\theta^{\mu}\,,\quad\sigma^{*}(\lambda)=v_{\mu}(x)\theta^{\mu}\,,\\\
\sigma^{*}(W_{abcd})=W_{abcd}(x)\,,\quad\sigma^{*}(C_{abc})=C_{abc}(x)\,,\quad\end{gathered}$
(3.42)

where by some abuse of notations we sometimes used the same letters to denote
some target space coordinates and the respective fields. It can be convenient
to treat $e,\omega,W,C$ as differential forms on $X$. With this identification
the action (2.6) can be written as:

$S[e,\omega,W,C]=\int_{X}\big{[}(d\omega_{ab}+\omega_{ac}\omega^{c}{}_{b}+e_{a}f_{b}-e_{b}f_{a})W^{abnm}\epsilon_{nmpk}e^{p}e^{k}+\\\
+W_{abcd}e^{c}e^{d}W^{abnm}\epsilon_{nmpk}e^{p}e^{k}-2(de_{a}+\omega_{ad}e^{d}+e_{a}v)C^{a}{}_{bc}\epsilon^{bcpk}e_{p}e_{k}\big{]}\,,$
(3.43)

where we omit the wedge product symbol. Note that before the symplectic
reduction has been done the action is to be considered as that depending on
all the degree zero fields. However here, we only spelled explicitly those
fields on which it depends nontrivially. In particular, thanks to the symmetry
properties of $W,C$, fields $v,f$ do not contribute (as we are going to see
this is in agreement with the fact that
$\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda}$}\kern
1.0pt}{\frac{\partial}{\partial\lambda}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\lambda}$}\kern
1.0pt}$ and
$\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\kappa}$}\kern
1.0pt}{\frac{\partial}{\partial\kappa}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\kappa}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\kappa}$}\kern
1.0pt}$ are in the kernel of the presymplectic structure).

To make the structure of the action more explicit define

$\begin{gathered}R_{ab}=d\omega_{ab}+\omega_{ac}\omega^{c}{}_{b}+(e_{a}f_{b}-e_{b}f_{a})\,,\qquad
T_{a}=de_{a}+\omega_{ad}\xi^{d}+e_{a}v\,,\\\
*W^{ab}=W^{abnm}\epsilon_{nmpk}e^{p}e^{k}\,,\qquad*C^{a}=C^{a}{}_{bc}\epsilon^{bcpk}e_{p}e_{k}\,.\end{gathered}$
(3.44)

In these terms the action reads as:

$S=\int_{X}\big{[}R_{ab}\wedge*W^{ab}+W_{ab}\wedge*W^{ab}-2T_{a}\wedge*C^{a}\big{]}\,.$
(3.45)

###  3.6 Symplectic quotient and BV formulation

In order to give a consistent interpretation to the gauge theory determined by
(3.43) and to construct the respective BV description we need to identify a
kernel distribution for the symplectic structure $\bar{\omega}$, induced on
the (open domain of) the space of supermaps $Smaps(\mathbb{R}^{4}[1],F)$. In
the case at hand the manifold $\bar{F}$ is coordinatized by the degree zero
coordinates introduced in (3.42) together with the remaining coordinates
introduced as e.g.:

$\widehat{\sigma}^{*}(\xi^{a})=\sum_{l=0}^{4}\frac{1}{l!}{\overset{l}{\xi}}{}^{a}_{\mu_{1}\ldots\mu_{l}}\theta^{\mu_{1}}\ldots\theta^{\mu_{l}}\,,$
(3.46)

with the identification ${\overset{1}{\xi}}{}^{a}_{\mu}=e^{a}_{\mu}$. The
ghost degree of the coefficients is determined by
$\mathrm{gh}(\theta^{\mu})=1$. We then take $\bar{F}$ to be the space of
supermaps with nondegenerate component $e^{a}_{\mu}$ because as we are going
to see these coordinates correspond to the frame field which must be
invertible.999This step is completely analogous to the respective
considerations [8] in the case of Einstein gravity.

We have the following

######  Proposition 3.3.

The presymplectic structure $\bar{\omega}$ is regular (i.e. has a constant
rank) on $\bar{F}$.

The idea of the proof is taken from [8] and amounts to identifying vector
fields $X_{\alpha}$ on $F$ such that they are in the kernel of $\omega$ and
their prolongations to $\bar{F}$ generate the kernel at any point of the body
of $\bar{F}$. In terms of local coordinates the prolongation of a vector field
$X$ on $F$ can be defined in local coordinates through:

$\bar{X}\psi^{A}(\theta)=X^{A}(\psi(\theta))\,,\qquad
X^{A}(\psi)=X\psi^{A}\,,$ (3.47)

where in the left hand side $\bar{X}$ acts on the coordinates on $\bar{F}$,
which are packed in the generating function $\psi^{A}(\theta)$. It is easy to
check that $i_{X}\omega=0$ implies $i_{\bar{X}}\bar{\omega}=0$.

The rank of $\bar{\omega}$ can’t decrease in a neigbourhood of a point.
Because the rank of the distribution generated by $\bar{X}_{\alpha}$ also can
not decrease one concludes that the rank is constant and the kernel
distribution is generated by $\bar{X}_{\alpha}$. Explicit form of $X_{\alpha}$
and the proof that their prolongations exhaust the kernel at any point of the
body of $\bar{F}$ is given in the Appendix C.

Because $\bar{\omega}$ is regular the general construction of Section 2.2
applies and we end up with the first order BV formulation whose field-
antifield space is $\bar{G}$ which is a symplectic quotient of $\bar{F}$. This
BV formulation has (3.43) as a classical action. To make sure we are indeed
dealing with a specific BV formulation of the conformal gravity we need to
make sure that (i) (3.43) reduced to the symplectic quotient is equivalent to
the conformal gravity and that (ii) the gauge generators encoded in the BV
action take all the gauge symmetries of system into account.

(i) Because (3.43) depends on fields of ghost degree zero only it is enough to
analyze the reduction only in the sector of degree zero variables. First of
all we disregard all the fields associated to higher order $W$ as (3.43) and
$\bar{\omega}$ do not involve them and hence they can be set to zero. The same
applies to $v_{\mu}$ and $f^{a}{}_{\mu}$. Among $X_{\alpha}$ the only vector
fields that affect the remaining ghost ghost degree 0 variables are

$\begin{gathered}X^{(trace)}=\xi^{a}\frac{\partial}{\partial\xi_{a}}-2W^{abnm}\frac{\partial}{\partial
W^{abnm}}-3C^{abc}\frac{\partial}{\partial C^{abc}}\,,\\\
Y^{(antisym)}_{d}=\epsilon_{abcd}\xi^{c}\frac{\partial}{\partial\rho_{ab}}+W_{dnm}{}^{a}\epsilon^{nmbc}\frac{\partial}{\partial
C^{abc}}\,,\\\
Y^{(trace)}_{d}=\xi^{c}\delta^{ab}_{cd}\frac{\partial}{\partial\rho_{ab}}-W^{adbc}\frac{\partial}{\partial
C^{abc}}\,.\end{gathered}$ (3.48)

The action of their prolongations on $e_{a\mu},\omega_{ab,\mu}$ read as:

$\begin{gathered}\bar{X}^{(trace)}e^{a}{}_{\mu}=e^{a}{}_{\mu}\,,\\\
\bar{Y}^{(antisym)}_{d}\omega_{ab,\mu}=\epsilon_{abcd}e^{c}{}_{\mu}\,,\qquad\bar{Y}^{(trace)}_{d}\omega_{ab,\mu}=(e_{a,\mu}\eta_{bc}-e_{b,\mu}\eta_{ac})\,.\end{gathered}$
(3.49)

It follows that by choosing the embedding of $\bar{G}\subset\bar{F}$ one can
assume that the determinant of $e^{a}_{\mu}$ as well as totally antisymmetric
and the trace components of $\omega_{ab,\mu}e_{c}{}^{\mu}$ can be set to
whatever we want. We use this to set $det(e)=1$ and to set the respective
componets of the torsion $T_{a}=de_{a}+\omega_{ab}e^{b}$ to zero. The
remaining irreducible component is associated to a hook-type Young tableaux
and is set to zero by the equation of motion obtained by varying (3.43) with
respect to $C_{abc}$. Eliminating the remaining component of $\omega$ together
with $C_{abc}$ sets $\omega$ to the Levi-Civita connection
$\omega_{ab,\mu}(e)$. Finally, equations for $W_{abcd}$ can be solved with
respect to $W_{abcd}$ itself and its elimination gives:

$S[e,\omega(e)]=\int W_{ab}\wedge*W^{ab}\,,$ (3.50)

i.e. the standard action of GGR written in terms of the frame field. The
standard argument then implies that all the components of $e^{a}_{\mu}$ which
do not contribute to the metric can be gauge-fixed by local Lorentz
transformations (which are algebraic) and the action can be equivalently
rewritten in terms of the metric, giving the proof that we indeed obtained CGR
action from the presymplectic AKSZ.

(ii) To check that the BV formulation indeed takes into account all the gauge
symmetries of the classical action let us analyze the spectrum of ghost
variables. Ghosts $\lambda,\kappa^{a}$ are in the kernel and do not survive on
$\bar{G}$. At the same time $\xi^{a},\rho^{ab}$ are not affected by the kernel
and hence their restriction to the symplectic quotient remain independent
coordinates. Their associated gauge transformations are precisely the
diffeomorphisms and the local Lorentz transformations. Note that we do not see
ghosts associated to Weyl transformations because this invariance is taken
into account in this formulation through the kernel of the presymplectic
structure and we fixed it when passing to the symplectic quotient. Note also
that there are no ghosts pertaining to the special conformal sector since they
are also in the kernel. We then conclude that we have indeed reconstructed a
first order BV formulation of CGR.

A substantial difference between the presymplectic AKSZ formulation of CGR and
the Einstein gravity is that in the case of Einstein gravity one can avoid
considering the entire minimal model as a target space and truncate it to just
a Poincaré algebra with shifted degree, see [8] for more details. The
analogous truncation doesn’t work in the case of CGR. However, one can still
construct a presymplectic AKSZ formulation based on conformal algebra and
closely related to the frame-like formulation from [13] but this requires
imposing additional constraints by hands. This possibility is discussed in
Appendix E.

Let us mention that all the coordinates on $(F,q)$ entering the action are
irreducible tensors of $O(4,2)$. Moreover, each field can be decomposed into a
sum of its self-dual and anti-self-dual components (this would require
complexified fields or changing a spacetime signature) which are irreducible
under $o(4.2)$. In particular the action can be easily rewritten using double-
spinor notations in which the decomposition into the self-dual and anti-self-
dual components is manifest. By setting to zero e.g. anti-self-dual component
of $W_{abcd}$ and dropping $W^{2}$-term one arrives to an equivalent
representation of the chiral (self-dual) conformal gravity [42], see also [43,
44, 45].101010We are grateful to E. Skvortsov for attracting our attention to
the self-dual version of CGR.

Let us also note that the above presymplectic formulation should admit
extension to the case of conformal supergravity, where the underlying algebra
is the superconformal algebra [13]. This superalgebra has eight fermionic
generators $Q_{\alpha},\Sigma_{\alpha}$ alongside with the 15 conformal
generators $J,P,K,D$ and the axial charge generator $U$. We denote by
$R(\cdot)$ the component of curvature associated to the respective generator,
i.e. $R(P_{a})=T_{a}=de_{a}+\omega_{ab}e^{b}+e_{a}\lambda$. Then we introduce
auxiliary fields associated to the components of the curvature that are
nonzero in the respective minimal model. In the CGR case these were
$W_{ab}(J_{cd})=W_{abcd}$ and $W_{ab}(K_{c})=C_{abc}$. In the case of
supergravity the nonvanishing curvatures can be infered from [14] so that one
in addition has the following auxiliaries: $W(Q)$, $W(\Sigma)$, $W(U)$. By
inspecting our presymplectic action (3.45) it is clear that it has a natural
extension of the same structure. More precisely, it reads schematically as

$R(J)\wedge*W(J)+R(U)\wedge*W(U)+\\\
+R(P)\wedge*W(K)+R(Q)\wedge*W(\Sigma)+R(\Sigma)\wedge*W(Q)+\\\
+W(J)\wedge*W(J)+W(U)\wedge*W(U)+W(Q)\wedge*W(\Sigma)\,,$

where the pairing is determined by the invariant inner product on the
superconformal algebra. Of course, determining such an action is equivalent to
determining an underlying presymplectic form. We expect that the action of
this structure should do the job but refrain from making a precise statement
because it might involve extra constraints and gauge-fixings.

##  4 General presymplectic BV-AKSZ formalism

Our discussion in Section 2 was limited to the case of diffeomorphism-
invariant systems for which AKSZ representation (at the level of EOMs) is
available. Now we explain how the presymplectic BV-AKSZ formalism extends to
the general situation of not-necessarily diffeomorphism-invariant systems.

###  4.1 Gauge PDEs

If we are only interested in the equations of motion and gauge symmetries (and
hence disregard the Lagrangian) the data of a local gauge theory can be
encoded into the following geometric object, known as gauge PDE [24] (see also
[25, 26, 31] for the earlier and less-geometrical version):

######  Definition 4.1.

Gauge PDE $(E,T[1]X,Q)$ is a $\mathbb{Z}$-graded fiber bundle $\pi:E\to T[1]X$
whose total space is equipped with a homological vector field $Q$ of degree
$1$ satisfying: $Q\circ\pi^{*}=\pi^{*}\circ d_{X}$ and $d_{X}$ is the de Rham
differential on $X$ seen as a homological vector field on $T[1]X$. Moreover,
$(E,Q)$ should be equivalent to a non-negatively graded gauge PDE.

One also often imposes some extra conditions of a technical nature to exclude
pathological examples. We assume that $X$ is a real manifold (space-time
manifold) while $E$ (and $T[1]X$) is a $\mathbb{Z}$-graded one with the degree
denoted by $\mathrm{gh}(\cdot)$. Note that gauge PDEs are special cases of
$Q$-bundles [27]. However, gauge PDEs are necessarily $\mathbb{Z}$-graded and
are typically infinite-dimensional. Further details and references can be
found in [24].

Just like in the case of AKSZ sigma models discussed in Section 2, solutions
of a gauge PDE are $Q$-sections, i.e. sections satisfying $\sigma^{*}\circ
Q=d_{X}\circ\sigma^{*}$, cf. (2.1). Infinitesimal gauge transformation of
sections is also defined in exactly the same way through the relation (2.2).

The notion of gauge PDE is rather flexible. For instance, the standard BV
formulation at the level of equations of motion is reproduced by taking as $E$
a jet-bundle associated to the bundle of BV fields (fields, ghosts,
antifields) seen as a bundle over $T[1]X$ (i.e. functions on $E$ are local
horizontal forms on the jet-bundle) and $Q=\mathrm{d_{h}}+s$, where
$\mathrm{d_{h}}$ is a canonical horizontal differential on the jet-bundle and
$s$ is the BV-BRST differential. It was shown in [38] that, at least locally,
this gauge PDE gives an equivalent (in the sense of elimination of generalized
auxiliary fields) description of the same gauge system.

Another standard example is PDE. Let $E_{X}\to X$ be an equation manifold seen
as a bundle over the space-time $X$. The Cartan distribution gives rise to a
homological vector field $\mathrm{d_{h}}$ on $E$ which is $E_{X}$ pulled back
to $T[1]X$. In a suitable local coordinate system $x^{a},\theta^{a},\psi^{A}$,
where $x^{a},\theta^{a}$ are adapted local coordinates on $T[1]X$ , induced by
local coordinates $x^{a}$ on $X$, and $\psi^{A}$ originate from local
coordinates on a typical fiber, one takes:

$\mathrm{d_{h}}=\theta^{a}D_{a}\,,\qquad
D_{a}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern 1.0pt}{\frac{\partial}{\partial
x^{a}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern
1.0pt}+\Gamma_{a}^{A}(x,\psi)\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}{\frac{\partial}{\partial\psi^{A}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}\,,$ (4.1)

where $D_{a}$ are components of the total derivative on the equation manifold.
Note that this is a generic form of the $Q$-structure on $E$ if the typical
fiber is a real manifold (so that $\mathrm{gh}(\psi^{A})=0$). It is easy to
see that in these coordinates solutions are covariantly-constant sections in
agreement with the standard picture. Mention also that a closely related
representation of PDEs is known under the name of unfolded formulation [37,
46].

From the gauge PDE perspective AKSZ sigma models discussed in Section 2
correspond to trivial (as a $Q$-bundle) $Q$-bundles
$(E,Q)=(F,q)\times(T[1]X,d_{X})$, where $(F,q)$ is a fiber.

###  4.2 Vertical forms

The Lagrangian (or partially Lagrangian) systems can be described as gauge
PDEs with extra structures. To define these we first need to recall a
particular realization of vertical forms.

Let $\pi:E\to\mathscr{X}$ be a generic fiber bundle. The exterior algebra of
local forms on $E$ contains the ideal $\mathcal{I}$ generated by forms of the
form $\pi^{*}\alpha$, where $\alpha\in\bigwedge^{k}(\mathscr{X})$, $k>0$. The
algebra of vertical forms is the quotient of $\bigwedge^{\bullet}(E)$ by the
ideal. Note that vertical zero-forms are just zero-forms as the ideal does not
contain zero-forms. It is convenient to work with vertical forms in terms of
the representatives as we do in what follows.

The De Rham differential $d$ is well defined on the equivalence classes.
Understood as an operation on vertical forms it is usually refereed to as a
vertical differential. Suppose that $Y$ is a vector field on $E$, which is
related by the projection to a vector field $y$ on $\mathscr{X}$, i.e.
$Y\circ\pi^{*}=\pi^{*}\circ y$, which in turn implies
$L_{Y}\circ\pi^{*}=\pi^{*}\circ L_{y}$. It is easy to check that $L_{Y}$ is
well defined on the equivalence classes and hence defines a Lie derivative on
vertical forms. Despite $i_{Y}$ is not well-defined on equivalence classes
(unless $Y$ is vertical) one can still employ the Cartan formula
$L_{Y}=i_{Y}d+(-1)^{|Y|}di_{Y}$ for representatives.

Given a local trivialization of $E$ over $U\subset\mathscr{X}$, i.e.
$E|_{U}=U\times F$, each equivalence class has a unique representative that
vanishes on any horizontal vector. In the adapted coordinates
$\psi^{A},X^{\alpha}$, with $X^{\alpha}$ being coordinates on the base, this
representative does not involve basis differential $dX^{\alpha}$. In terms of
such representatives and the adapted coordinates the vertical differential can
be explicitly written as
$d\psi^{A}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}{\frac{\partial}{\partial\psi^{A}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}$. It is then clear that the vertical differential is locally acyclic in
positive form-degree.

###  4.3 Presymplectic gauge PDEs

A presymplectic BV-AKSZ system is a gauge PDE $(E,T[1]X,Q)$ equipped with a
compatible vertical presymplectic structure $[\omega]$, i.e. a 2-form $\omega$
satisfying $\mathrm{gh}(\omega)=n-1$, $d\omega\in\mathcal{I}$,
$L_{Q}\omega=\mathcal{I}$, where $n=\dim{X}$. In addition we assume the
following technical conditions: $i_{Q}d\omega\in\mathcal{I}$ and
$i_{Q}L_{Q}\omega\in\mathcal{I}$. Here and in what follows we work in terms of
representatives.

Given a presymplectic gauge PDE one can consider a natural Lagrangian system
associated to it. More precisely, we first introduce covariant Hamiltonian
$\mathcal{L}$ as a solution to111111In the case of constrained mechanical
systems $\mathcal{L}=\Omega-\theta H$, where $H$ is a Hamiltonian and $\Omega$
is the BRST charge. In this context the formulation in question and
$\mathcal{L}$, in particular, was proposed in [28] (see also [47, 48] where
$\mathcal{L}$ appeared earlier in a slightly different context).

$i_{Q}\omega+d\mathcal{L}\in\mathcal{I}\,.$ (4.2)

To see that $\mathcal{L}$ exists observe that $di_{Q}\omega\in\mathcal{I}$
thanks to $L_{Q}\omega\in\mathcal{I}$ and $i_{Q}d\omega\in\mathcal{I}$. Then
acyclicity of $d$ on vertical forms implies the existence of $\mathcal{L}$
(for $n>1$ it exists globally). Note that $\mathcal{L}$ is a zero form and
hence is defined uniquely modulo functions of the form $\pi^{*}(h)$,
$h\in\mathcal{C}^{\infty}(T[1]X)$. As we are going to see this ambiguity
doesn’t affect the equations of motion.

If we pick a symplectic potential $\chi$ satisfying
$\omega-d\chi\in\mathcal{I}$ this data defines a natural action functional on
sections $T[1]X\to E$:

$S[\sigma]=\int_{T[1]X}[(\sigma^{*}\,\chi)(d_{X})+\sigma^{*}(\mathcal{L})]\,.$
(4.3)

This action is a generalization of the presymplectic version [2] of the
conventional AKSZ action discussed in Section 2. At the same time, in the case
where $(E,T[1]X,Q)$ is a PDE, it coincides with the so-called intrinsic action
from [3] (see also [49]). In the later case $\omega$ can be identified with a
compatible presymplectic current (see [4, 5, 50, 51, 6] for further details)
defined on the equation manifold.

Let us check that $S[\sigma]$ does not depend on the choice of a
representative $\chi$ of the equivalence class $[\chi]$. If instead of $\chi$
we took another representative $\chi^{\prime}=\chi+\epsilon$, where
$\epsilon\in\mathcal{I}$ so that it can be written in local coordinates as
$\epsilon=dx^{a}\alpha_{a}+d\theta^{b}\beta_{b}$ we find:

$(\sigma^{*}(\epsilon))(d_{X})=\theta^{a}\sigma^{*}(\alpha_{a})\,.$ (4.4)

At the same time, the variation $\mathcal{L}_{\epsilon}$ of $\mathcal{L}$ is
determined by $i_{Q}d\epsilon+d\mathcal{L}_{\epsilon}\in\mathcal{I}$. Using
$L_{Q}\epsilon\in\mathcal{I}$ one finds
$di_{Q}\epsilon+d\mathcal{L}_{\epsilon}\in\mathcal{I}$ and hence
$\mathcal{L}_{\epsilon}+i_{Q}\epsilon=\pi^{*}(h)$. A variation of the form
$\pi^{*}(h)$, $h\in\mathcal{C}^{\infty}(T[1]X),\mathrm{gh}(h)=n$ is locally
$d_{X}$-exact and hence it preserves the equivalence class of the Lagrangian.
Furthermore,

$\sigma^{*}(i_{Q}\epsilon)=\sigma^{*}(\theta^{a}\alpha_{a})=\theta^{a}\sigma^{*}(\alpha_{a})\,,$
(4.5)

where we again use adapted coordinate system $x^{a},\theta^{a},\psi^{A}$ on
$E$. Note that in this coordinates $Qx^{a}=\theta^{a}$, $Q\theta^{a}=0$. It
follows that the action is unchanged under $\chi\to\chi^{\prime}$ if we
disregard the variations of the form $\pi^{*}(h)$, which do not affect
equations of motion.

In this way we have arrived at a natural Lagrangian system associated to a
presymplectic BV-AKSZ system. However, in general this system contains an
infinite amount of fields and the interpretation of the action is to be
clarified. In the case where $(E,T[1]X,Q)$ is a usual PDE (seen as a gauge
PDE) the consistent interpretation can be arrived at by disregarding those
fields which are in the kernel of the symplectic structure [52]. In this way
one can often (though not always) describe Lagrangian systems in terms of the
intrinsic geometry of their equation manifolds. Just like in the case of diff-
invariant systems discussed in Section 2 a more refined interpretation can be
given by passing to the symplectic quotient of the induced symplectic
structure on the space of super-sections of $E$.

###  4.4 Symplectic quotient and BV formulation

It is convenient to identify super-sections as the space of section of a
bundle $\bar{E}$ over $X$, whose fiber over $p\in X$ is the space of supermaps
from $(T_{p}X)[1]$ to $F$.

To be more explicit we now work locally using a trivialization
$E|_{T[1]U}=T[1]U\times F$ over $U\subset X$. The fiber $\bar{F}$ of
$\bar{E}|_{U}$ is a space of supermaps from $\mathbb{R}^{n}[1]$ to $F$.
Coordinates on $\bar{F}$ are introduced as in (2.8), where we keep denoting
coordinates on $F$ by $\psi^{A}$ and use $\Psi^{I}$ to denote coordinates on
$\bar{F}$.

It turns out that the presymplectic structure $\omega$ on $E$ gives rise to an
$n+2$-form on $\bar{E}$ of the following structure:

$\bar{\omega}^{\bar{E}}=(dx)^{n}\bar{\omega}_{IJ}(\Psi,x)d\Psi^{I}d\Psi^{J}\,,$
(4.6)

where
$\bar{\omega}_{IJ}=\bar{\omega}(\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\Psi^{I}}$}\kern
1.0pt}{\frac{\partial}{\partial\Psi^{I}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\Psi^{I}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\Psi^{I}}$}\kern
1.0pt},\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\Psi^{J}}$}\kern
1.0pt}{\frac{\partial}{\partial\Psi^{J}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\Psi^{J}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\Psi^{J}}$}\kern
1.0pt})$, with $\bar{\omega}$ on each fiber given by

$\bar{\omega}=\int
d^{n}\theta\omega_{AB}(\psi^{A}(\theta),x,\theta)d\psi^{A}(\theta)\,d\psi^{B}(\theta)\,.$
(4.7)

In other words on a given fiber the form is still the same (2.9) tensored with
the coordinate volume form $(dx)^{n}$. It is easy to see that
$d\bar{\omega}=0$. An alternative language would be to identify $\bar{\omega}$
as a vertical 2-form with values in densities on $X$.121212A more geometrical
way to define $\bar{\omega}$ is to introduce super-jet bundle of $E\to T[1]X$
and pullback $\omega$ to the super-jet bundle. In a suitable coordinate system
this can be seen as a term in $\bar{\omega}$ proportional to $(\theta)^{n}$.

We assume that the $n+2$ form $\bar{\omega}$ is regular (in the sense that the
rank of $\bar{\omega}_{IJ}$ is constant) and consider the vertical kernel of
$\bar{\omega}$. This consists of vertical vector fields $W$ such that
$i_{W}\bar{\omega}=0$. Thanks to $d\bar{\omega}=0$ the vertical kernel is an
involutive distribution and one can (at least locally) pass to the quotient
space. Because the kernel distribution is vertical the quotient is again a
bundle over $X$ (its fibers are simply the symplectic quotients of the
symplectic form $\bar{\omega}_{IJ}d\Psi^{i}d\Psi^{J}$ on $\bar{F}$). Form
$\bar{\omega}$ induces a vertically-nondegenerate $n+2$-form on the quotient.
This gives us a fiber-bundle equipped with the $n+2$-form satisfying all the
conditions of the BV $n+2$-form. It has a natural lift to an $(n,2)$-form on
$J^{\infty}(\bar{E})$.

By repeating the arguments given in Section 2.2 in this more general setting
one can show that BV-like action (2.7) still induces the BV action on the
symplectic quotient and it satisfies the BV master equation modulo boundary
terms.

To summarize: under some regularity assumptions a presymplectic gauge PDE in a
natural way encodes a Lagrangian BV system. This system is not necessarily
equivalent (at the level of equations of motion) to the starting point gauge
PDE. Neveretheless, for a given Lagrangian BV system one can always find a
presymplecic gauge PDE such that the BV system it encodes is equivalent to the
initial BV system. More precisely, this can be done by using the Lagrangian
parent formulation [26] as explained in [3, 49]. Note, however, that there
exist equivalent gauge PDEs such that one of them admits a compatible
presymplectic structure leading to an equivalent BV system while another one
does not. The simplest example is provided by the Fierz-Pauli Lagrangian
describing massive spin-2 field. It turns out [49] that the respective
equation manifold does not admit a compatible presymplectic structure that
reproduces an equivalent Lagrangian.

###  4.5 Example: Maxwell

We now illustrate the construction using the example of Maxwell theory. More
precisely we take the minimal model of the respective gauge PDE (the
respective total space $Q$-manifold is known in the literature for quite some
time, see e.g. [30] and refs. therein. The solutions of this gauge PDE satisfy
the so-called unfolded form of Maxwell equations, see e.g. [53]). The
coordinates on the total space of $E\to T[1]X$ are

$\begin{gathered}x^{a},\theta^{a},\quad C\,,\quad\mathrm{gh}(C)=1\\\
F_{a|b},\quad F_{a|bc},\quad\ldots\quad F_{a|b_{1}\ldots
b_{l}}\,,\quad\ldots\,,\quad\mathrm{gh}(F_{a|b\ldots})=0\,,\end{gathered}$
(4.8)

where tensors $F_{a|b_{1}\ldots b_{l}}$ are symmetric in the second group of
indices, satisfy the Young symmetry condition $F_{(a|b_{1}\ldots b_{l})}=0$,
and are totally traceless. The action of $Q$ is given by

$\begin{gathered}Qx^{a}=\theta^{a}\,,\quad
QC=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}\theta^{a}\theta^{b}F_{a|b}\,,\\\
QF_{a|b}=\theta^{c}(F_{a|bc}+\ldots)\,,\quad
QF_{a|bd}=\theta^{c}(F_{a|bdc}+\ldots)\,,\quad\ldots\end{gathered}$ (4.9)

where $\ldots$ denote terms maintaining the Young symmetry properties. It is
clear that we have a $Q$-bundle structure as $Q\circ\pi^{*}=\pi^{*}\circ
d_{X}$, with
$d_{X}=\theta^{a}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern 1.0pt}{\frac{\partial}{\partial
x^{a}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern 1.0pt}$ and $\pi^{*}(x^{a})=x^{a}$ and
$\pi^{*}(\theta^{a})=\theta^{a}$ in these coordinates.

The compatible presymplectic structure and the respective potential can be
taken as [2]:

$\omega=dCdF^{a|b}(\theta)^{(n-2)}_{ab}\,,\qquad\chi=dCF^{a|b}(\theta)^{(n-2)}_{ab}\,,$
(4.10)

where $(\theta)^{(n-k)}_{a_{1}\ldots
a_{k}}=\frac{1}{(n-k)!}\epsilon_{a_{1}\ldots a_{k}b_{1}\ldots
b_{n-k}}\theta^{b_{1}}\ldots\theta^{b_{n-k}}$. Note that in checking
$L_{Q}\omega\in\mathcal{I}$ one has to use the ”equations of motion”, i.e.
that $F_{a|bc}$ are totally traceless. Recall that ideal $\mathcal{I}$ is
generated by $dx^{a},d\theta^{a}$.

The covariant Hamiltonian $\mathcal{L}$ defined through
$i_{Q}\omega+d\mathcal{L}\in\mathcal{I}$ is given by

$\mathcal{L}=-\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}F_{a|b}F^{a|b}(\theta)^{(n)}$
(4.11)

Introducing component fields parameterizing section $\sigma:T[1]X\to E$
according to

$\sigma^{*}(C)=A_{b}(x)\theta^{b}\,,\qquad\sigma^{*}(F_{a|b})=F_{ab}(x)\,,\qquad\ldots$
(4.12)

the action takes the form

$S[\sigma]=\int
d^{n}x\big{[}F^{ab}(\partial_{a}A_{b}-\partial_{b}A_{a})-\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}F^{ab}F_{ab}\big{]}\,,$ (4.13)

which is a standard covariant 1-st order action of the Maxwell field.

The induced symplectic structure $\bar{\omega}$ on the fiber $\bar{F}$ is
given by

$\bar{\omega}=d{\overset{0}{C}}\,d{\overset{2}{F}}{}^{a|b}_{ab}+2dA_{a}\,d{\overset{1}{F}}{}^{a|b}_{b}+d{\overset{2}{C}}{}_{ab}\,d{\overset{0}{F}}{}^{a|b}\,,$
(4.14)

where the relevant components of supersection are introduced via:

$\displaystyle\widehat{\sigma}^{*}(C)$
$\displaystyle={{\overset{0}{C}}(x)}+{A_{a}(x)}\theta^{a}+{\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}{{{\overset{2}{C}}_{ab}(x)}}\theta^{a}\theta^{b}\ldots$
(4.15) $\displaystyle\widehat{\sigma}^{*}(F^{a|b})$
$\displaystyle={{F}{}^{a|b}(x)}+{{\overset{1}{F}}{}^{a|b}_{c}(x)}\theta^{c}+\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}{{\overset{2}{F}}{}^{a|b}_{cd}(x)}\theta^{c}\theta^{d}+\ldots\,.$

It follows that all the fields except for

${\overset{0}{C}},\quad C^{*}={\overset{2}{F}}{}^{a|b}_{ab},\quad A_{a},\quad
A_{*}^{a}={\overset{1}{F}}{}^{a|b}_{b},\quad{F}{}^{a|b},\quad
F^{*}_{a|b}={\overset{2}{C}}{}_{ab}$ (4.16)

are in the kernel of $\bar{\omega}$ and can be set to zero to obtain the
symplectic quotient. The ghost degree of the remaining fields is given by

$\begin{gathered}\mathrm{gh}(A_{a})=\mathrm{gh}(F^{a|b})=0\,,\qquad\mathrm{gh}(C)=1\,,\\\
\mathrm{gh}(A^{A}_{*})=\mathrm{gh}(F^{*}_{a|b})=-1\,,\qquad\mathrm{gh}(C^{*})=-1\,.\end{gathered}$
(4.17)

By inspecting the degree of the remaining fields and the above symplectic
structure one concludes that we have arrived at the proper field-antifield
space of the BV formulation of the Maxwell field. It is straightforward to
check that the BV action (2.7) specialised to this case explicitly gives a
proper solution of the master equation extending the classical action (4.13).
Extension to the YM theory is also straightforward.

As a final remark mention that in 4 dimensions it is not difficult to find a
presymplectic BV-AKSZ formulation of the self-dual version of the system. More
precisely, all $F$-coordinates can be decomposed into the self dual $F^{+}$
and anti-self-dual $F^{-}$ components. It is easy to check that replacing $F$
with $F^{+}$ in the relation
$QC=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}\theta^{a}\theta^{b}F_{a|b}$ and
replacing $F$ with $F^{-}$ in the expression for the presymplectic structure
results in the presymplectic AKSZ formulation of the self-dual Maxwell theory.
Nonabelian extension is straightforward and results in the SDYM theory [54].

##  5 Conclusion

In this work we have proposed the presymplectic BV-AKSZ formulation of the
conformal gravity in 4 dimensions. Among possible formulation of this type the
present one is minimal in the sense that it is based on the minimal model of
the respective total BRST complex and hence it involves a minimal number of
fields among possible AKSZ-type formulations. In particular, fields of the
system are irreducible Lorentz tensors. In a certain sense the proposed
formulation is a ”canonical” first-order frame like-formulation. At the same
time this does not mean that other formulations can not be more useful. For
instance an interesting issue is to extend the present formulation to the one
where $o(4,2)$-symmetry is realised in a manifest way.

The proposed formulation could be also useful in describing less symmetric
models such as CGR with the extra $R^{2}$ counterterm added. Although such
system is not Weyl invariant and a minimal presymplectic formulation should
involve only Poincaré algebra one can try to keep the entire conformal algebra
at the price of introducing the appropriate compensator field. This is very
much in spirit of tractor description of Riemannian geometry, see e.g. [55].

Another interesting open problem is the extension of the approach to the case
of conformal higher spin fields [14] (see also [56, 57, 58, 59] for a more
recent discussion from various perspectives) for which the interacting local
theory is known [60, 61] (see also [62, 63]) and present a substantial
interest from various perspectives. It would also be desirable to make the
relation with the tractor description of conformal geometry [64, 55] more
precise.

In the present work we also extend the approach to not necessarily
diffeomorphism-invariant theories. In this case the AKSZ framework is not
enough and one needs to resort to so called gauge PDEs of which the usual AKSZ
setup is a particular case. Another extreme case is the usual geometrical
description of PDE in terms of the equation manifold (for a modern review see
e.g. [65]). In this case the presymplectic BV-AKSZ formulation reduces to the
so-called intrinsic Lagrangian approach [3, 49]. As an illustration of the
general situation we consider a minimal model of the Maxwell system endowed
with a compatible presymplectic structure and explicitly demonstrate how the
full-scale BV description emerges from it.

## Acknowledgments

ID acknowledges useful discussions with V. Gritzaenko and A. Yan. MG wishes to
thank N. Boulanger, A. Sharapov, E. Skvortsov and especially A. Kotov for
fruitful discussions. The work was supported by the Russian Science Foundation
grant 18-72-10123. Part of this work was done when authors participated in the
thematic program ”Higher Structures and Field Theory” at the Erwin Schrödinger
International Institute for Mathematics and Physics, Vienna, Austria. This
work was partially completed while M.G. participated in the workshop ”Higher
Spin Gravity and its Applications” supported by the Asia Pacific Center for
Theoretical Physics, Pohang, Korea.

##  Appendix A Tensor conventions

In this section we describe our conventions for the tensors emerging in the
context of CGR and give missing details on its frame-like description.

The Bach tensor is defined as

$B_{\mu\nu}=P^{\rho\lambda}W_{\mu\rho\nu\lambda}+\nabla^{\rho}\nabla_{\rho}P_{\mu\nu}-\nabla^{\rho}\nabla_{\mu}P_{\nu\rho}\,,$
(A.1)

where $\nabla^{\mu}$ is the metric-compatible covariant derivative,
$P_{\mu\nu}$ is the Schouten tensor given by

$P_{\mu\nu}=\frac{1}{2}(\stackrel{{\scriptstyle(0)}}{{R}}{}^{\rho}{}_{\mu\rho\mu}-\frac{\stackrel{{\scriptstyle(0)}}{{R}}{}^{\rho\sigma}{}_{\rho\sigma}}{6}g_{\mu\nu})\,,$
(A.2)

$\stackrel{{\scriptstyle(0)}}{{R}}{}^{\rho}{}_{\mu\nu\sigma}$ is the Riemann
curvature, and Weyl tensor is defined as:

$W_{\mu\nu}{}^{\rho\sigma}=\stackrel{{\scriptstyle(0)}}{{R}}{}_{\mu\nu}{}^{\rho\sigma}-4P_{[\mu}^{[\rho}\delta_{\nu]}^{\sigma]}\,.$
(A.3)

Now we give missing details on the derivation of the metric-like action (3.1)
from the frame-like action (3.10). Starting with the equation of motion
$\epsilon^{abcd}R_{ab}e_{c}=0$ for $f^{a}$, introducing tangent components
$R_{abcd}$ via $R_{ab}=R_{abcd}e^{c}e^{d}$, and wedging with $e^{f}$ one finds

$(R^{ab}{}_{km})\delta^{k}_{[a}\delta^{m}_{b}\delta_{d]}^{f}=0\,,$ (A.4)

which implies

$2R^{ab}{}_{ab}\delta^{f}_{d}+4R^{fb}{}_{bd}=0\,.$ (A.5)

Contracting $f,d$ one gets $R^{ab}{}_{ab}=0$ and

$R^{c}{}_{bcd}=0\,.$ (A.6)

Explicitly one has

$R^{c}{}_{bcd}=\stackrel{{\scriptstyle(0)}}{{R^{c}{}_{bcd}}}+(4f_{d,b}-e_{d,a}f_{a,b}-e_{a,b}f_{d,a}+e_{d,b}f)=0\,,$
(A.7)

where $f=f^{a}_{a}$. It follows

$f=-\frac{1}{6}\stackrel{{\scriptstyle(0)}}{{R}}$ (A.8)

and

$f_{b,\mu}=-\frac{1}{2}(e^{d}{}_{\mu}\stackrel{{\scriptstyle(0)}}{{R^{c}{}_{bcd}}}-\frac{1}{6}e_{b,\mu}\stackrel{{\scriptstyle(0)}}{{R}})$
(A.9)

so that $f_{\mu}^{b}$ is indeed a Schouten tensor on shell.

Now we turn to the properties of the Weyl and Cotton tensors and their Hodge
duals, which we need in the proof of the invariance of the presymplectic
structure. The Hodge duals are defined as:

$C^{*}_{abc}=C_{a}{}^{pk}\epsilon_{pkbc}\,,\qquad
W^{*}_{abcd}=W_{ab}{}^{pk}\epsilon_{pkcd}\,.$ (A.10)

The inverse relations read:

$\displaystyle C^{*}_{abc}\epsilon^{bcnm}$
$\displaystyle=C_{a}{}^{pk}\epsilon_{pkbc}\epsilon^{bcnm}=4C_{a}{}^{nm}\,,$
(A.11) $\displaystyle W^{*}_{abcd}\epsilon^{cdnm}$
$\displaystyle=W_{ab}{}^{pk}\epsilon_{pkcd}\epsilon^{cdnm}=4W_{ab}{}^{nm}\,.$

Every property of Weyl and Cotton tensors has its counterpart in terms of the
duals.

  1. 1.

Trace-free and Young symmetry

$\begin{gathered}\eta^{ac}C_{abc}=0={C^{*}_{apk}\epsilon^{pkba}}\,,\\\
\epsilon^{abcd}C_{abc}=0=C^{*}_{a}{}^{pk}\epsilon_{pkbc}\epsilon^{abcd}=2C^{*}_{a}{}^{ad}\,,\end{gathered}$
(A.12)
$\begin{gathered}\eta^{ac}W_{abcd}=0={W^{*}_{abpk}\epsilon^{pkad}}\,,\\\
\epsilon^{abcn}W_{abcd}=0=W^{*}_{ab}{}^{pk}\epsilon_{pkcd}\epsilon^{abcn}=2W^{*}_{ab}{}^{ab}\delta^{n}_{d}+8W^{*}_{ad}{}^{an}\,.\end{gathered}$
(A.13)

  2. 2.

Bach equation

$C_{abc|}{}^{b}=0={C^{*}_{a}{}^{pk|b}\epsilon_{pkbc}}\,.$ (A.14)

  3. 3.

Definition of $C$ in terms of higher Weyl-tensors:

$W_{abcd}{}^{|a}=W_{bcda}{}^{|a}-W_{bdca}{}^{|a}=C_{bcd}\,.$ (A.15)

  4. 4.

The Weyl tensor enjoys the following relation:

$W_{ab}{}^{ef}\epsilon_{cdef}=W^{ef}{}_{cd}\epsilon_{abef}\,.$ (A.16)

##  Appendix B Invariance of the presymplectic structure

Here we explicitly demonstrate that the presymplectic structure (3.33) is
$q$-invariant.

Equation (3.34) is obtained in the following way:

$i_{q}\omega_{W}=\rho_{al}\rho^{l}{}_{b}d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+(\xi_{a}\kappa_{b}-\xi_{b}\kappa_{a})d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+\\\
+\frac{1}{2}W_{abij}\xi^{i}\xi^{j}d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+\\\
+d\rho_{ab}(\xi^{j}W^{abnm}{}_{|j}+\rho^{a}{}_{j}W^{jbnm}+\rho^{b}{}_{j}W^{ajnm}+\rho^{n}{}_{j}W^{abjm}+\\\
+\rho^{m}{}_{j}W^{abnj}+2\lambda W^{abnm})\epsilon_{nmpk}\xi^{p}\xi^{k}+\\\
+2d\rho_{ab}W^{abnm}\epsilon_{nmpk}(\rho^{p}{}_{j}\xi^{j}\xi^{k}+\xi^{p}\lambda\xi^{k})=\\\
=d(\rho_{al}\rho^{l}{}_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+(\xi_{a}\kappa_{b}-\xi_{b}\kappa_{a})d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+\\\
+\frac{1}{2}W_{abij}\xi^{i}\xi^{j}d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+d(\rho_{ab})\xi^{j}W^{abnm}{}_{|j}\epsilon_{nmpk}\xi^{p}\xi^{k}\,,$
(B.1)

where we made use of:

$\rho_{al}\rho^{l}{}_{b}d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+d\rho_{ab}(\rho^{a}{}_{j}W^{jbnm}+\rho^{b}{}_{j}W^{ajnm})\epsilon_{nmpk}\xi^{p}\xi^{k}=\\\
=d(\rho_{al}\rho^{l}{}_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})\,.$ (B.2)

For the second piece $\omega_{C}$ of the presymplectic structure we have

$i_{q}\omega_{C}=\rho_{an}\xi^{n}d(C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})+\xi_{a}\lambda
d(C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})+\\\
+d(\xi_{a})(\xi^{d}D_{d}C_{abc}+\rho_{a}{}^{n}C_{nbc}+\rho_{b}{}^{n}C_{anc}+\rho_{c}{}^{n}C_{abn}+3\lambda
C_{abc}+\kappa_{n}W^{n}{}_{abc})\epsilon^{bcpk}\xi_{p}\xi_{k}+\\\
+2d(\xi_{a})C^{a}{}_{bc}\epsilon^{bcpk}(\rho_{p}{}^{l}\xi_{l}\xi_{k}+\xi_{p}\lambda\xi_{k})\,,$
(B.3)

where the following cancellations can be observed:

$3d(\xi_{a})\lambda
C_{abc}\epsilon^{bcpk}\xi_{p}\xi_{k}+2d(\xi_{a})C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\lambda\xi_{k}=d(\xi_{a})\lambda
C_{abc}\epsilon^{bcpk}\xi_{p}\xi_{k}$ (B.4)

and

$d(\xi_{a})(\rho_{b}{}^{n}C_{anc}+\rho_{c}{}^{n}C_{abn})\epsilon^{bcpk}\xi_{p}\xi_{k}+2d(\xi_{a})C^{a}{}_{bc}\epsilon^{bcpk}\rho_{p}{}^{l}\xi_{l}\xi_{k}=0\,.$
(B.5)

Taking them into account we indeed arrive at (3.37). To see that terms from
equations (3.36),(3.39) give (3.40) we use properties (A.13), (A.12), (A.14),
(A.15), (A.16). The nontrivial calculation is to show that:

${d(\rho^{bc})\xi^{a}\xi^{p}\xi^{k}W^{*}_{bcpk|a}=-2d(\rho^{bc})\xi_{c}\xi^{p}\xi^{k}C^{*}_{bpk}}.$
(B.6)

To prove it we start with (A.15):

$\begin{gathered}W_{abcd}{}^{|a}=W_{bcda}{}^{|a}-W_{bdca}{}^{|a}=C_{bcd}\,,\\\
W_{cdba}{}^{|a}-W_{cbda}{}^{|a}=C_{cdb}\,,\\\
W_{dbca}{}^{|a}-W_{dcba}{}^{|a}=C_{dbc}\,,\end{gathered}$ (B.7)
$W_{bcda}{}^{|a}=\frac{1}{2}(C_{bcd}+C_{cdb}-C_{dbc})=(C_{bcd}+C_{cdb})\,,$
(B.8)
$\begin{gathered}W^{*}_{bc}{}^{pk|a}\epsilon_{pkda}=(C^{*}_{b}{}^{pk}\epsilon_{pkcd}+C^{*}_{c}{}^{pk}\epsilon_{pkdb})\,.\end{gathered}$
(B.9)

Multiply this equation by $\epsilon^{dnmr}$ to get

$\delta^{nmr}_{pka}W^{*}_{bc}{}^{pk|a}=(-C^{*}_{b}{}^{pk}\delta^{nmr}_{pkc}+C^{*}_{c}{}^{pk}\delta^{nmr}_{pkb})\,,$
(B.10)
$\xi^{a}\xi^{p}\xi^{k}W^{*}_{bcpk|a}=(-\xi_{c}\xi^{p}\xi^{k}C^{*}_{bpk}+\xi_{b}\xi^{p}\xi^{k}C^{*}_{cpk})\,.$
(B.11)

Contracting this equation with $d\rho^{bc}$ yields the desired relation.

We conclude that

$i_{q}(\omega_{W}-2\omega_{C})=d(\rho_{al}\rho^{l}{}_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+d(2\xi_{a}\kappa_{b}W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})+\\\
+d(-2\rho_{an}\xi^{n}C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})+d(-2\xi_{a}\lambda
C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})+\\\
+\frac{1}{2}W_{abij}\xi^{i}\xi^{j}d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})\,.$
(B.12)

##  Appendix C The kernel of the presymplectic structure

Here we study the kernel of the presymplectic structure (3.33) and show the
properties needed in the proof of Proposition 3.3. We first identify
explicitly vector fields on $F$ which are in the kernel of $\omega$ and whose
prolongations generate the kernel distribution for $\bar{\omega}$ on
$\bar{F}$. It is convenient to group them as follows:

  * •

Vector fields along $\rho^{ab}$ which we group according to the homogeneity in
$\xi$. We have:

$Y^{(4)}_{ab}=\xi^{(4)}\frac{\partial}{\partial\rho_{ab}},\qquad
Y^{(3)}_{l}=\xi^{(3)}_{d}\frac{\partial}{\partial\rho_{dl}}\,.$ (C.1)

Next there are fields which are second order on $\xi$:

$Y^{(2ant)}_{ld}=\epsilon_{abcd}\xi^{c}\xi_{l}\frac{\partial}{\partial\rho_{ab}}\,,\qquad
Y^{(2trace)}_{ld}=\xi^{c}\xi_{l}\delta^{ab}_{cd}\frac{\partial}{\partial\rho_{ab}}\,.$
(C.2)

Note that these are linearly dependent as
$Y^{(2ant)}_{ld}\epsilon^{ldkp}=Y^{(2trace)}{}^{kp}$. The linear in $\xi$ ones
are:

$\displaystyle Y^{(ant)}_{d}$
$\displaystyle=\epsilon_{abcd}\xi^{c}\frac{\partial}{\partial\rho_{ab}}+W_{dnm}{}^{a}\epsilon^{nmbc}\frac{\partial}{\partial
C^{abc}}\,,$ (C.3) $\displaystyle Y^{(trace)}_{d}$
$\displaystyle=\xi^{c}\delta^{ab}_{cd}\frac{\partial}{\partial\rho_{ab}}-W^{adbc}\frac{\partial}{\partial
C^{abc}}\,.$

To see that $Y^{(ant)}_{d}$ belong to the kernel we observe that:

$i_{\epsilon_{abcd}\xi^{c}\frac{\partial}{\partial\rho_{ab}}}\omega_{W}=\epsilon_{abcd}\xi^{c}d(W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k})=\\\
=4\xi^{c}W_{cdab}d(\xi^{a}\xi^{b})=8W_{dcba}d(\xi^{a})\xi^{c}\xi^{b}\,.$ (C.4)

This term cancels with

$i_{Y^{(ant)}_{d}}(-2\omega_{C})=-2(d(\xi^{a})W_{dnma}\epsilon^{nmbc}\epsilon_{bcpk}\xi^{p}\xi^{k})\,.$
(C.5)

The considerations for $Y^{(trace)}_{d}$, $Y^{(2ant)}_{ld}$, and
$Y^{(2trace)}_{ld}$ are pretty much the same.

  * •

Vector fields along $\xi^{a}$. We also group them according to the homogeneity
in $\xi$:

$X^{(4)}_{a}=\xi^{(4)}\frac{\partial}{\partial\xi_{a}}\,,\qquad
X^{(3)}=\xi^{(3)}_{a}\frac{\partial}{\partial\xi_{a}}\,.$ (C.6)

Second order ones :

$\begin{gathered}X^{(2ant)}_{d}=\epsilon_{abcd}\xi^{b}\xi^{c}\frac{\partial}{\partial\xi_{a}}-\xi_{d}W^{ab}{}_{pk}\epsilon^{pknm}\frac{\partial}{\partial
W^{abnm}}-\\\ -4\xi_{d}C^{a}{}_{pk}\epsilon^{pkbc}\frac{\partial}{\partial
C^{abc}}-4\xi^{a}C_{pkd}\epsilon^{bcpk}\frac{\partial}{\partial C^{abc}}\,,\\\
X^{(2trace)}_{d}=\xi_{b}\xi_{c}\delta^{bc}_{ad}\frac{\partial}{\partial\xi_{a}}-\xi_{d}W^{abnm}\frac{\partial}{\partial
W^{abnm}}-\\\ -6\xi_{d}C^{abc}\frac{\partial}{\partial
C^{abc}}\,.\end{gathered}$ (C.7)

To see that they are in the kernel we observe the following relations:

$i_{\epsilon_{aljd}\xi^{l}\xi^{j}\frac{\partial}{\partial\xi_{a}}}\omega_{W}=2d(\rho_{ab})W^{abnm}\epsilon_{nmpk}\epsilon^{pljd}\xi_{l}\xi_{j}\xi^{k}=\\\
=2d(\rho_{ab})W^{abnm}\delta^{ljd}_{nmk}\xi_{l}\xi_{j}\xi^{k}=4d(\rho_{ab})W^{abnm}\xi_{n}\xi_{m}\xi^{d}\,,$
(C.8)
$i_{\epsilon_{aljd}\xi^{l}\xi^{j}\frac{\partial}{\partial\xi_{a}}}\omega_{C}=\epsilon_{aljd}\xi^{l}\xi^{j}d(C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})+2d(\xi_{a})C^{a}{}_{bc}\epsilon^{bcpk}\epsilon_{pljd}\xi^{l}\xi^{j}\xi_{k}=\\\
=2\delta_{ad}^{pk}d(C^{a}{}_{bc})\epsilon^{bcpk}+2\delta_{aljd}^{bcpk}\xi^{l}\xi^{j}C^{a}{}_{bc}d(\xi_{p})\xi_{k}+2d(\xi_{a})C^{a}{}_{bc}\delta^{bck}_{ljd}\xi^{l}\xi^{j}\xi_{k}=\\\
=4\xi^{b}\xi^{c}C^{a}{}_{bc}d(\xi_{a})\xi_{d}+4\xi_{a}C^{a}{}_{bd}\xi^{b}\xi^{k}d(\xi_{k})\,.$
(C.9)

And finally the first order one is given by:

$X^{(trace)}=\xi^{a}\frac{\partial}{\partial\xi_{a}}-2W^{abnm}\frac{\partial}{\partial
W^{abnm}}-3C^{abc}\frac{\partial}{\partial C^{abc}}\,.$ (C.10)

The proof that it is the kernel is based on:

$i_{\xi^{a}\frac{\partial}{\partial\xi_{a}}}\omega=2d(\rho_{ab})W^{abnm}\epsilon_{nmpk}\xi^{p}\xi^{k}-\\\
-2(\xi_{a}d(C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})+2d(\xi_{a})C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}\xi_{k})\,.$
(C.11)

The first term is canceled by $i_{-2W^{abnm}\frac{\partial}{\partial
W^{abnm}}}\omega_{W}$. Then
$\xi_{a}d(C^{a}{}_{bc})\epsilon^{bcpk}\xi_{p}\xi_{k}=0$ due to Bianchi
identity. In terms of
$\stackrel{{\scriptstyle*}}{{C^{apk}}}=C^{a}{}_{bc}\epsilon^{bcpk}$ the second
term gives:

$\xi_{a}C^{a}{}_{bc}\epsilon^{bcpk}d(\xi_{p})\xi_{k}-\xi_{a}C^{a}{}_{bc}\epsilon^{bcpk}\xi_{p}d(\xi_{k})=\stackrel{{\scriptstyle*}}{{C^{apk}}}(\xi_{a}d(\xi_{p})\xi_{k}-\xi_{a}\xi_{p}d(\xi_{k}))=\\\
=\stackrel{{\scriptstyle*}}{{C^{apk}}}\xi_{a}d(\xi_{p})\xi_{k}+\stackrel{{\scriptstyle*}}{{C^{apk}}}\xi_{p}d(\xi_{k})\xi_{a}=(\stackrel{{\scriptstyle*}}{{C^{apk}}}+\stackrel{{\scriptstyle*}}{{C^{kap}}})(\xi_{a}d(\xi_{p})\xi_{k})=\\\
=(-\stackrel{{\scriptstyle*}}{{C^{pka}}})(\xi_{a}d(\xi_{p})\xi_{k})=\stackrel{{\scriptstyle*}}{{C^{pka}}}(d(\xi_{p})\xi_{k}\xi_{a})\,.$
(C.12)

Together with the third terms in (C.11) they can be taken care of by the
$-3C^{abc}\frac{\partial}{\partial C^{abc}}$.

  * •

Finally we have two more independent kernel vector fields

$L=\frac{\partial}{\partial\lambda}\,,\qquad
K=\frac{\partial}{\partial\kappa}\,,$ (C.13)

simply because neither $d\kappa^{a}$ nor $d\lambda$ enter (3.33).

Now we consider the symplectic structure $\bar{\omega}$ on $\bar{F}$ at a
point $p$, where all the coordinates but $e^{a}_{\mu}$ are set to zero. At
this point $\bar{\omega}$ reads as:

$\begin{gathered}\bar{\omega}_{p}=(dw_{ab}d{\overset{2}{W}}{}^{abcd}{}_{,cd}+d\omega_{ab,c}d{\overset{1}{W}}{}^{abcd}{}_{,d}+d\omega_{ab,cd}dW^{abcd})-\\\
-2(d\epsilon_{a}d{\overset{2}{C}}{}^{acd}{}_{,cd}+de_{a,c}d{\overset{1}{C}}{}^{acd}{}_{,d}+de_{a,cd}dC^{acd})\,,\end{gathered}$
(C.14)

in a suitable coordinate system. By inspecting the prolongation of the above
kernel vector fields $X,Y,K,L$ at this point one finds that their linear span
coincides with the kernel. The argument given in the main text then implies
that their prolongations generate the kernel distribution of $\bar{\omega}$.

##  Appendix D Metric elimination

Here we discuss the assumptions under which the metric can be eliminated. As
we discuss in the main text the BRST complex of CGR as presented in [19]
contain ghost degree $1$ elements ${\xi}_{\mu}{}^{\nu}$ and the metric. Thanks
to

$\gamma g_{\mu\nu}=-2\lambda
g_{\mu\nu}+\xi_{\mu}{}^{\rho}g_{\rho\nu}+{\xi}_{\nu}{}^{\rho}g_{\mu\rho}=-2\lambda
g_{\mu\nu}+{\xi}_{(\mu\nu)}\,,$ (D.1)

$g_{\mu\nu}$ and ${\xi}_{(\mu\nu)}$ can be eliminated at least locally.

However, $g_{\mu\nu}$ are not unconstrained coordinates in the sense of
definition 3.1 because the determinant of the metric can not be zero.
Moreover, the space is further restricted to the space of symmetric matrices
of definite signature.

It turns our that in the case of Riemannian metric (i.e. Euclidean signature)
the metric can be eliminated. To see this we employ Cholesky decomposition
which states that every symmetric, positive-definite matrix can be uniquely
decomposed as

$g=LL^{T}\,,$ (D.2)

where L is a lower-triangular matrix with positive elements on the diagonal.
The space of such matrices is diffeomorphic to $R^{n(n+1)/2}$ and the concrete
diffeomorphism can be obtained by taking the coordinates on $R^{n(n+1)/2}$ to
be the entries of a lower-triangular matrix $A$ and taking $L=e^{A}$. Entries
of the matrix $A$ then become the free coordinates parameterizing the space of
metrics and form contractible pairs with the symmetric part of $\xi_{\mu\nu}$.

In the case of pseudo-Euclidean metrics there might be no such decomposition.
We will make use of the following theorem [66] Th. 4.1.1, 4.1.2

######  Theorem 1.

If all the leading principal minors of $A\in R^{n\times n}$ are nonvanishing
then there exist a unique lower triangular matrix $L$ with positive diagonal
values and a unique diagonal matrix $D=diag(d_{1},...,d_{n}),d_{k}=\pm 1$ such
that $A=LDL^{T}$.

A metric in general coordinates might not satisfy Theorem 1. For example take
the metric on a plane in $z$ coordinates: $ds^{2}=dzd\bar{z}$. If we restrict
to such metrics that admit such factorization the elimination can still be
performed. This subspace of metrics includes important cases, for example all
metrics close to Minkowsky one.

##  Appendix E Presymplectic representation of the standard
”gauge” formulation

The simple approach one might try to use to find a presymplectic AKSZ
formulation of CRR is to consider just the conformal algebra with shifted
grading $g[1]$ as the target space with the canonical Chevalie-Eilenberg
differential $q_{CE}$ on it. The presymplectic potential and structure can be
read off the action (3.10) presented in [13]

$\begin{gathered}\chi_{G}=(d\rho_{ab}\xi_{c}\kappa_{d})\epsilon^{abcd}=(\xi_{a}\kappa_{b}d\rho_{cd})\epsilon^{abcd}\,,\\\
\omega_{G}=d\chi=(d\xi_{a}\kappa_{b}d\rho_{cd}-\xi_{a}d\kappa_{b}d\rho_{cd})\epsilon^{abcd}.\end{gathered}$
(E.1)

This structure is invariant under $q_{CE}$ and induces the following action:

$S=\int(d\omega_{ab}e_{c}f_{d}+\omega_{a}{}^{k}\omega_{kb}e_{c}f_{d}+e_{a}f_{b}e_{c}f_{d})\epsilon^{abcd}.$
(E.2)

This procedure reproduces the action (3.10) but for it to be equivalent to the
conformal gravity in metric formulation one has to additionally impose the
zero torsion condition.

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