We describe a globalization construction for the Rozansky–Witten model in the BV-BFV formalism for a source manifold with and without boundary in the classical and quantum case. After having introduced the necessary background, we define an AKSZ sigma model, which, upon globalization through notions of formal geometry extended appropriately to our case, is shown to reduce to the Rozansky–Witten model. The relations with other relevant constructions in the literature are discussed.
Moreover, we split the model as a $BF$-like theory and we construct a perturbative quantization of the model in the quantum BV-BFV framework. In this context, we are able to prove the modified differential Quantum Master Equation and the flatness of the quantum Grothendieck BFV operator. Additionally, we provide a construction of the BFV boundary operator in some cases.

§ INTRODUCTION

§.§ Overview and motivation

An important class of field theories in physics is represented by gauge theories. These are theories containing a redundant number of degrees of freedom which causes physical quantities to be invariant under certain local transformations, called gauge symmetries. Indeed the presence of gauge symmetries lead to challenging problems from the definition of path integral to the general problem of understanding the perturbative quantization of a gauge theory. Since the physical information about a classical field theory is encoded in the set of solutions of the Euler–Lagrange equations (the critical locus), a possible solution to deal with such problems is to consider the critical locus modulo the gauge symmetries. The fields are then constructed as functions on this quotient. However, this is not feasible since it turns out that these quotients are, in general, singular. Batalin and Vilkovisky introduced a method, which is known today as the BV formalism <cit.>, that employs symplectic (co)homological tools <cit.> to treat these field theories, in particular it overcomes difficulties connected to the singularity of the quotient by taking homological resolution of the critical locus. A crucial observation in the BV formalism is also that gauge-fixing then corresponds to the choice of a Lagrangian submanifold.
Another method developed around the same time is the BFV formalism by  Batalin, Fradkin and Vilkovisky <cit.>, which deals with gauge theories in the Hamiltonian setting, while the BV construction is formulated in the Lagrangian approach.

Recently, the study of gauge theories on spacetime manifolds with boundary lead Cattaneo, Mnev and Reshetikhin <cit.> to relate these two formulations in order to develop the BV-BFV formalism. Their idea was that, under certain conditions, BV theories in the bulk can induce a BFV theory on the boundary. This approach was successfully applied to a large number of physical theories such as e.g. electrodynamics, Yang-Mills theory, scalar field theory and $BF$-theories <cit.>. In particular, the AKSZ construction, developed in <cit.>, produces naturally a large variety of theories which satisfy automatically the BV-BFV axioms as it was shown in <cit.>. This is quite remarkable since many theories of interest are actually of AKSZ-type, such as e.g. Chern–Simons (CS) theory, $BF$-theory and the Poisson sigma model (PSM) <cit.>.

In <cit.>, a perturbative quantization scheme for gauge theories in the BV-BFV framework was introduced, which was called quantum BV-BFV formalism. The importance of this method relies on its compatibility with cutting and gluing in the sense of topological quantum field theories (TQFTs). The quantum BV-BFV formalism has been applied successfully in various physically relevant theories such as e.g. $BF$-theory and the PSM <cit.>, split CS theory <cit.> and CS theory <cit.>, the relational symplectic groupoid[The relational symplectic groupoid was first defined in <cit.>.] <cit.> and 2D Yang–Mills theory on manifolds with corners <cit.>.

An important effort has been spent to study TQFT within the quantum BV-BFV framework. Indeed, the method has been introduced to accomplish the goal of constructing perturbative topological invariants of manifolds with boundary compatible with cutting and gluing for topological field theories. During the years, two prominent TQFTs have been studied in detail: CS theory <cit.> in <cit.> and the PSM <cit.> in <cit.>.

In <cit.> a globalized version of the (quantum) BV-BFV formalism in the context of nonlinear split AKSZ sigma models on manifolds with and without boundary by using methods of formal geometry à la Bott <cit.>, Gelfand and Kazhdan <cit.> (see also <cit.> for an application of the globalization procedure for the PSM in the context of a closed source manifold) was developed. Their construction is able to detect changes of the quantum state when one modifies the constant map around which the perturbation is developed. This required them to formulate a “differential" version of the (modified) Classical Master Equation and the (modified) Quantum Master Equation, which are the two key equations in the BV(-BFV) formalism. As an example, this procedure was applied to the PSM on manifolds with boundary and extended to the case of corners in <cit.>.

In this paper, we continue the effort in analyzing TQFTs within the quantum BV-BFV formalism by studying the Rozansky–Witten (RW) theory.

The RW model is a topological sigma model with a source 3-dimensional manifold $\Sigma_3$, which was introduced by Rozansky and Witten in <cit.> through a topological twist of a 6-dimensional supersymmetric sigma model with target a hyperKähler manifold $M$. Of particular interest is the perturbative expansion of the RW partition function. Rozansky and Witten obtained this expansion as a combinatorial sum in terms of Feynman diagrams $\Gamma$, which are shown to be trivalent graphs:

\begin{equation}
Z_M(\Sigma_3)=\sum_\Gamma b_\Gamma(M)I_\Gamma(\Sigma_3),
\end{equation}

the $b_\Gamma (M)$ are complex valued functions on trivalent graphs constructed from the target manifold, while $I_\Gamma(\Sigma_3)$ contains the integral over the propagators of the theory and depends on the source manifold. There are evidences which suggest that $I_\Gamma(\Sigma_3)$ are the LMO invariants of Le, Murakami and Ohtsuki <cit.>. On the other hand, Rozansky and Witten showed that $b_\Gamma(M)$ satisfy the famous AS (which is reflected in the absence of tadpoles diagrams) and IHX relations. As a result, $b_\Gamma(M)$ constitute the Rozansky–Witten weight system for the graph homology, the space of linear combinations of equivalence classes of trivalent graphs (modulo the AS and IHX relations). This means that the RW weights can be used to construct new finite type topological invariants for 3-dimensional manifolds <cit.>.

The RW theory opened up a new branch of research which was undertaken by many mathematicians and physicists (e.g. <cit.>). Shortly after the original paper, Kontsevich understood that the RW invariants could be obtained by the characteristic classes of foliations and Gelfand-Fuks cohomology <cit.>. Inspired by the work of Kontsevich, Kapranov reformulated the weight system in cohomological terms (instead of using differential forms) in <cit.>. This idea relies on the fact that one can replace the Riemann curvature tensor by the Atiyah class <cit.>, which is the obstruction to the existence of a global holomorphic connection. As a consequence of Kontsevich's and Kapranov's approaches, the RW weights were understood to be invariant under the hyperKähler metric on $M$: in fact, the model could be constructed more generally with target a holomorphic symplectic manifold. In this way, the RW weights were also called RW invariants[The terminology is unfortunate as in reality the proper invariants should be the products of the weights with $I_\Gamma(\Sigma_3)$.] of $M$ (see <cit.> for a detailed exposition). On the other hand, the possibility to consider as target manifold a holomorphic symplectic manifold was later interpreted in the context of topological sigma models by Rozansky and Witten in the appendix of <cit.>.

In the last 20 years, the RW model has been the focus of intense research in order to formulate it as an extended TQFT (see <cit.>), in order to investigate its boundary conditions and defects <cit.>, and in order to construct its globalization formulation <cit.>.

§.§ Our contribution

The main contribution of this paper is to add the RW theory to the list of TQFTs which have been studied successfully within the globalized version of the quantum BV-BFV framework <cit.>. This will be a step towards the higher codimension quantization of RW theory, which will possibly lead to new insights towards the 3-dimensional correspondence between CS theory <cit.> and the Reshetikhin–Turaev construction <cit.> from the point of view of (perturbative) extended field theories described by Baez–Dolan <cit.> and Lurie <cit.>. Moreover this could also help in understanding (generalizations of a globalized version of the) Berezin–Toeplitz quantization (star product) <cit.> through field-theoretic methods using cutting and gluing similarly as it was done for Kontsevich's star product <cit.> in the case of the PSM in <cit.>.

We construct the BV-BFV extension of an AKSZ model having a 3-dimensional manifold $\Sigma_3$ (possibly with boundary) as source and a holomorphic symplectic manifold $M$ as target with holomorphic symplectic form $\Omega$. Following <cit.>, we define a formal holomorphic exponential map $\varphi$. This is used to linearize the space of fields of our model obtaining

\begin{equation}
\Tilde{\mathcal{F}}_{\Sigma_3, x}=\Omega^{\bullet}(\Sigma_3)\otimes T^{1,0}_{x}M,
\end{equation}

where $\Omega^{\bullet}(\Sigma_3)$ denotes the complex of de Rham forms on the source manifold and $T^{1,0}_{x}M$ is the holomorphic tangent space on the target. In order to vary the constant solution around which we perturb, we define a classical Grothendieck connection which can be seen as a complex extension of the Grothendieck connection used in <cit.>. In this way, we construct a formal global action for our model, i.e.

\begin{equation}
\Tilde{\Sc}\surg\coloneqq\varint_{\Sigma_3}\bigg(\frac{1}{2}\Omega_{ij}\hat{\mathbf{X}}^id\hat{\mathbf{X}}^j+\Big(\hat{R}^i\sur\Big)_j(x; \hat{\mathbf{X}})\Omega_{il}\hat{\mathbf{X}}^l dx^j+\Big(\hat{R}^i\sur\Big)_{\Bar{j}}(x; \hat{\mathbf{X}})\Omega_{il}\hat{\mathbf{X}}^l dx^{\Bar{j}}\bigg)
\end{equation}

with $\hat{\mathbf{X}}^i$ the coordinates of the spaces of fields $\Tilde{\mathcal{F}}_{\Sigma_3, x}$ organized as superfields, $x$ is the constant map over which we expand, $\Big(\hat{R}^i\sur\Big)_j$ and $\Big(\hat{R}^i\sur\Big)_{\Bar{j}}$ the components of the Grothendieck connection given by

\begin{equation}
\begin{split}
&R^i_j(x;y)dx^j:=-\bigg[\bigg(\frac{\partial \varphi}{\partial y}\bigg)^{-1}\bigg]^i_p\frac{\partial \varphi^p}{\partial x^j}dx^j,\\
&R^i_{\Bar{j}}(x;y)dx^{\Bar{j}}:=-\bigg[\bigg(\frac{\partial \varphi}{\partial y}\bigg)^{-1}\bigg]^i_p\frac{\partial \varphi^p}{\partial x^{\Bar{j}}}dx^{\Bar{j}},
\end{split}
\end{equation}

where $\{y^i\}$ are the generators of the fiber of $\reallywidehat{\Sym}^\bullet(T^{\vee1,0}M)$. The formal action is such that the differential Classical Master Equation (dCME) is satisfied, namely

\begin{equation}
\end{equation}

with $d_M=d_x+d_{\Bar{x}}$ the sum of holomorphic and antiholomorphic Dolbeault differentials on $M$. The dCME presented here is different from the one presented in e.g. <cit.> since there $d_M$ was the de Rham differential on the body of the target manifold.

The globalized model is then shown to be a globalization of the RW model <cit.>, which reduces to the RW model itself in the appropriate limits. Our globalization of the RW model is compared with other globalization constructions as the one developed in <cit.> for a closed source manifold by using Costello's approach <cit.> to derived geometry <cit.>, the procedure in <cit.> which extends the work of <cit.> to manifolds with boundary and the procedure in <cit.>. In general, our model is compatible with all these apparently different views. In particular, we give a detailed account of the similarities between our method and the one in <cit.>, thus confirming the claim in Remark 3.6 in <cit.> about the equivalence between Costello's approach and ours.

In order to quantize the theory according to the quantum BV-BFV formalism, we formulate a split version of our globalized RW model. Since the globalization is controlled by an $\Linf$-algebra, following <cit.> and inspired by the work of Cattaneo, Mnev and Wernli for CS theory <cit.>, we assume that we can split the $\Linf$-algebra in two isotropic subspaces. The action of the globalized split RW model is then

\begin{equation}
\Tilde{\Sc}\surgS=%\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}
\braket{\hat{\mathbf{B}}, D\hat{\mathbf{A}}}+\Big\langle\Big(\hat{R}\sur\Big)_j(x; \hat{\mathbf{A}}+\hat{\mathbf{B}})dx^j,\hat{\mathbf{A}}+\hat{\mathbf{B}}\Big\rangle+\Big\langle\Big(\hat{R}\sur\Big)_{\bar{j}}(x; \hat{\mathbf{A}}+\hat{\mathbf{B}})dx^{\bar{j}},\hat{\mathbf{A}}+\hat{\mathbf{B}}\Big\rangle,
\end{equation}

where $\braket{-,-}$ denotes the BV symplectic form on the space of fields $ℱ$ with values in the Dolbeault complex of $M$, $𝐀̂^i$ and $𝐁̂_i$ are the fields found from the splitting of the field $𝐗̂^i$, and $D$ denotes the superdifferential. Note that $d$ is the de Rham differential on the target, not on the source.

Finally, we quantize the globalized split RW model within the quantum BV-BFV formalism framework. Here, we obtained the following two theorems.
\begin{theoremn}[Flatness of the qGBFV operator (Theorem \ref{thm:flatness})]
The \emph{quantum Grothendieck BFV (qGBFV) operator} $\nabla_{\textup{G}}$ for the anomaly-free globalized split RW model squares to zero, i.e.
\begin{equation}
\label{flatness_GBFV}
\end{equation}
\begin{equation}
\nabla_{\textup{G}}=d_{M}-i\hbar \Delta_{\mathcal{V}_{\Sigma_3, x}}+\frac{i}{\hbar}\boldsymbol{\Omega}_{\partial \Sigma_3}=d_x+d_{\Bar{x}}-i\hbar \Delta_{\mathcal{V}_{\Sigma_3, x}}+\frac{i}{\hbar}\boldsymbol{\Omega}_{\partial \Sigma_3},
\end{equation}
with $d_M$ the sum of the holomorphic and antiholomorphic Dolbeault differentials on the target $M$, $\Delta_{\mathcal{V}_{\Sigma_3, x}}$ the BV Laplacian and $\boldsymbol{\Omega}_{\partial \Sigma_3}$ the full BFV boundary operator.
\end{theoremn}
\begin{theoremn}[mdQME for anomaly-free globalized split RW model (Theorem \ref{thm:mdQME})]
Consider the full covariant perturbative state $\hat{\psi}_{\Sigma_3,x}$ as a quantization of the anomaly-free globalized split RW model. Then
\begin{equation}
\label{mdqme_thm}
\bigg(d_M-i\hbar \Delta_{\mathcal{V}_{\Sigma_3, x}}+\frac{i}{\hbar}\boldsymbol{\Omega}_{\partial \Sigma_3}\bigg)\boldsymbol{\hat{\psi}}\surgR=0.
\end{equation}
\end{theoremn}
The proof of both the theorems is very similar to the ones exhibited in \cite{CMoW19} for non linear split AKSZ sigma models. Hence, we refer to \cite{CMoW19} when the procedure is the same whereas we remark when there are differences (which are related to the presence of the sum of the holomorphic and antiholomorphic Dolbeault differentials in the quantum Grothendieck BFV operator instead of the de Rham differential as in \cite{CMoW19}).

We provide an explicit expression for the BFV boundary operator up to one bulk vertices in the $𝔹$-representation by adapting to our case the degree counting techniques of \cite{CMoW19}. Unfortunately, due to some complications related to the number of Feynman rules, we are not able to provide an explicit expression of the BFV boundary operator in the $𝔹$-representation in the case of a higher number of bulk vertices. See \cite{Sac21} for a limited example of graphs that appear when there are three bulk vertices.

This paper is structured as follows:
\begin{itemize}
\item In Section \ref{sec:BV-BFV} we introduce the most important notions of the classical an quantum BV-BFV formalism. Moreover, we give an overview of \textit{AKSZ theories}.%, which are a particularly nice subclass of BV theories which often appear as suitable deformations of abelian $BF$ theories.
\item In Section \ref{sec:RW_model} we introduce the necessary preliminaries to understand the RW model.%, as well as to construct its globalization in the case of a manifold with boundary, which is the focus of this paper.
% In this regard, we reserve Appendix \ref{app:TQFT} to give some basics about topological quantum field theories (TQFTs) of which the RW model is a prominent example. \textcolor{red}{OK?}
\item In Section \ref{sec:Classical_Theory} we define an AKSZ model which upon globalization can be reduced to the RW model.
\item In Section \ref{sec:comp_orig_RW} we compare our construction to  the original construction by Rozansky and Witten.
\item In Section \ref{sec:comp_other} we compare our globalization construction with other globalization constructions of the RW model.
\item In Section \ref{sec:BF-like_formulation} we give a $BF$-like formulation by a splitting of the fields of the RW model in order to to be able to give a suitable description of its quantization.
\item In Section \ref{sec:pert_quant_RW} we quantize the globalized split RW model according to the quantum BV-BFV formalism introduced in Section \ref{sec:BV-BFV}.
\item In Section \ref{sec:mdQME} we introduce the \emph{quantum Grothendieck BFV operator} for the globalized split RW model, we prove that it is flat and, in the end, we use it to prove the modified differential Quantum Master Equation.
\item Finally, in Section \ref{sec:outlook} we present some possible future directions.
\end{itemize}

\textbf{Notation}
Throughout the whole paper, we will keep the following conventions:
\begin{itemize}
\item we will drop the wedge product wherever its presence would make the expressions too cumbersome;
% \item $\phi$: formal exponential map
% \item $\varphi$: holomorphic formal exponential map
\item we will employ the Einstein summation convention, meaning that expressions of the form $A^iB_i$ should be interpreted as $\sum_i A^iB_i$;
\item we will denote the dual of a vector space $V$ as $V^\vee$.
\end{itemize}

\textbf{Acknowledgements}
We thank A. S. Cattaneo, I. Contreras, P. Mnev and T. Willwacher for comments and remarks.
D. S. wants to thank P. Steffens for sending him his master thesis and for valuable comments.
This research was (partially) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, and by the SNF grant No. 200020\_192080.
This paper is based on the master thesis of D. S. \cite{Sac21}.

\section{The BV-BFV formalism}
\label{sec:BV-BFV}

\subsection{Classical BV-BFV formalism}
In this section, we will recall the classical BV-BFV formalism as in \cite{CMR14}. We also refer to \cite{Mnev2019} for an excellent introduction to the BV formalism and to \cite{CattMosh1} for a more detailed exposition on the BV-BFV formalism.
\begin{defn}[BV manifold]
\label{BVmfld}
A \textit{BV manifold} is a triple $(\F, \omega, \mathcal{S})$ where $\F$ is a supermanifold\footnote{See \cite{BerLei75} for an original reference and \cite{CattaneoSchaetz2011,CattMosh1} for a concise introduction.} with $\mathbb{Z}$-grading, $\omega$ an odd symplectic form of degree $-1$ and $\mathcal{S}$ an even function of degree zero on $\F$, satisfying the \textit{Classical Master
Equation} (CME):
\begin{equation}
\label{cms}
\end{equation}
We denote the odd Poisson bracket associated to $\omega$ with round brackets $(-,-)$. Usually we refer to $(-,-)$ as the \textit{BV bracket} (or \emph{anti bracket}).
\end{defn}
\begin{rmk}
%Let us devote some lines to a remark regarding the grading on $\F$.
Note that we have two different gradings, the $\mathbb{Z}_2$-grading from
the supermanifold structure and an additional $\mathbb{Z}$-grading. In a physics context, the $\mathbb{Z}$-grading corresponds to the ghost number $gh$ and the $\mathbb{Z}_2$-grading corresponds to the ``parity", which distinguishes bosonic and fermionic particles.
\end{rmk}

Equivalently, one may introduce the \textit{Hamiltonian vector field} $Q$ of $𝒮$, with ghost degree 1, defined by
\begin{equation}
\label{cms2}
\iota_{Q}\omega=\delta \mathcal{S}
\end{equation}
Furthermore, we require $Q$ to be a \textit{cohomological vector field}, i.e.
\begin{equation}
\label{Q}
\end{equation}
where $[-,-]$ denotes the Lie bracket of vector fields.

\begin{rmk}
Using symplectic geometry tools, one can show that Eq. \eqref{Q} and Eq. (\ref{cms2}) together imply $(S,S)=\text{constant}$, which for degree reasons reduces to Eq. (\ref{cms}).
This allows us then to write an equivalent definition of a BV manifold as a quadruple collection of data $(\F, \omega, \mathcal{S}, Q)$ satisfying Eq. \eqref{cms2} and Eq. (\ref{Q}).
\end{rmk}

\begin{defn}[BV theory] A $d$-dimensional \emph{BV theory} on a closed manifold $M$ is the association of a BV manifold to every closed $d$ manifold $M$:
\begin{equation}
M \mapsto (\F_M,\omega_M,\mathcal{S}_M,Q_M)
\end{equation}
\end{defn}
The natural question one might ask is how these definitions extend to the case of a manifold with boundary, which will also be the relevant case for our work.

\begin{defn}[BV extension] A BV theory $(\F_M,\omega_M,\mathcal{S}_M,Q_M)$ is a \textit{BV extension} of a local field theory $M \mapsto (F_M, S_M)$ if for all $d$-manifolds $M$, the degree zero part $(\F_M)_0$ of $\F_M$ satisfies $(\F_M)_0=F_M$ and $\mathcal{S}_M\vert_{(\mathcal{F}_M)_0}=S_M$. In addition we want $\omega_M, \mathcal{S}_M$ and $Q_M$ to be local.
\end{defn}
Extending the BV formalism to manifolds with boundary amounts to considering its Hamiltonian counterpart, namely the BFV formalism.
\begin{defn}[BFV manifold] A \emph{BFV manifold} is a triple $(\F^\partial, \omega^\partial,Q^\partial)$, where similarly as in Definition \ref{BVmfld}, $\F^\partial$ is a graded manifold, $\omega$ an even symplectic form of degree zero, and $Q^\partial$ a degree 1 cohomological vector field on $\F^\partial$. Moreover, if $\omega^\partial=\delta \alpha^\partial$, i.e. exact, the BFV manifold is called \textit{exact}.
\end{defn}
The result of merging BV and BFV formalism is encapsulated in the following definition:
\begin{defn}[BV-BFV manifold, \cite{CMR14}]
A \emph{BV-BFV manifold} over a given exact BFV manifold $\F^\partial=(\F^\partial,\omega^\partial=\delta\alpha^\partial,Q^\partial)$
is a quintuple
\begin{equation}
\F=(\F,\omega,\mathcal{S},Q,\pi)
\end{equation}
with $\pi:\F \rightarrow \F^\partial$ a surjective submersion obeying
\begin{equation}
\label{cme_boundary}
\iota_Q\omega=\delta \mathcal{S}+\pi^*\alpha^\partial
\end{equation}
and $Q^\partial=\delta \pi Q$, where $\delta \pi$ is the differential of $\pi$.
\end{defn}
\begin{rmk}
Note that if $\F^\partial$ is a point then $(\F,\omega,\mathcal{S})$ is a BV manifold.
\end{rmk}
We will adopt the short notation $π:→^∂$ for a BV-BFV manifold.

We can now formulate a generalization of a BV theory:
\begin{defn}[BV-BFV theory]
A $d$-dimensional \textit{BV-BFV theory} associates to every closed
$(d-1)$-dimensional manifold $\Sigma$ a BFV manifold $\F^\partial_\Sigma$, and to a $d$-dimensional manifold $M$ with
boundary $\partial M$ a BV-BFV manifold $\pi_M:\F_M \rightarrow \F^\partial_{\partial M}$.
\end{defn}
\begin{rmk}
For $Q$ a Hamiltonian vector field of $\mathcal{S}$, one can formally write
\[(\mathcal{S},\mathcal{S})=\iota_Q\iota_Q \omega = Q(\mathcal{S}).\] In the case of a BV-BFV theory for a manifold $M$ with boundary $\partial M$, we have
\begin{equation}
\end{equation}
This can be phrased equivalently as
\begin{equation}
\iota_Q\iota_Q\omega=2\pi^*\mathcal{S}^\partial,
\end{equation}
which we will refer to as the \textit{modified Classical Master Equation} (mCME).
\end{rmk}
An important as well as classical example of a BV-BFV theory are the \textit{$BF$-like theories}.
\begin{defn}[$BF$-like theory]A BV-BFV theory is called \emph{$BF$-like}  if
\begin{equation}
\begin{split}
\F_M&=\big(\Omega^\bullet(M) \otimes V[1]\big) \oplus \big(\Omega^\bullet (M)\otimes V^\vee[d-2]\big),\\
\mathcal{S}_M&=\varint_M\big(\braket{B,dA} + \mathcal{V}(A,B)\big),
\end{split}
\end{equation}
with $V$ a graded vector space, $\langle -, - \rangle$ a pairing between $V^\vee$ and $V$, and $\mathcal{V}$ a density-valued function of the fields $A$ and $B$, such that $\mathcal{S}_M$ satisfies the CME for $M$ without boundary.
\end{defn}
\begin{rmk}
\label{bv-bfv:rmk_bf_like}
Equivalently, by picking up a graded basis $e^i$ for $V$ and $e_i$ for $V^\vee$, we may define a $BF$-like theory as a BV-BFV theory with
\begin{equation}
\begin{split}
\F_M&=\big(\Omega^\bullet(M) \otimes V[k_i]\big) \oplus \big(\Omega^\bullet (M)\otimes V^\vee[d-k_i-1]\big),\\
\mathcal{S}_M&=\varint_M\big(B_idA^i + \mathcal{V}(A,B)\big).
\end{split}
\end{equation}
To pass from one definition to the other, it is sufficient to set $k_i=1-|e_i|$, where $|e_i|$ is the degree of $e_i$.
\end{rmk}

\subsection{Quantum BV-BFV formalism}
\label{sec_qbvbfv}
In this section, we introduce a perturbative quantization method for BV-BFV theories compatible with cutting and gluing. Originally, this procedure was proposed in \cite{CMR17} under the name of \textit{quantum BV-BFV formalism}. We start by defining what is a quantum BV-BFV theory and then we will explain how to produce such a theory by quantizing perturbatively a classical BV-BFV theory.

\begin{defn}[Quantum BV-BFV theory]
\label{def_Quantum BV-BFV theory}
Given a BV-BFV theory\footnote{The perturbative quantization scheme goes through if certain conditions are satisfied. In the following, we will be interested to $BF$-like theories for which this method works smoothly.}, a $d$-dimensional \textit{quantum BV-BFV theory} associates
\begin{itemize}
\item To every closed $(d-1)$-dimension manifold $\Sigma$ a graded $\mathbb{C}[\![\hbar]\!]$-module $\mathcal{H}_{\Sigma}$, called the \textit{space of states}. %It is produced as geometric quantization\footnote{For an introduction to geometric quantization see \cite{} \CIT Weinstein bates.} of the symplectic manifold $\mathcal{F}\bonN$.
\item To every $d$-dimensional manifold (possibly with boundary) $M$:
\begin{itemize}
\item a degree 1 coboundary operator $\Omega\surM$ on $\mathcal{H}\surM$ called \textit{quantum BFV operator}. We call $\mathcal{H}\surM$ \textit{space of boundary states}
\item a finite-dimensional BV manifold $\mathcal{V}_M$, called \textit{space of residual fields}.
\item a homogeneous element\footnote{Usually, the quantum state $\hat{\psi}_M$ will have degree 0. This is always the case when the gauge-fixing Lagrangian has degree 0, which is true for all the examples considered in this paper.}, the \textit{quantum state} $ \hat{\psi}_M\in \hat{\mathcal{H}}_M$. By denoting the space of half-densities on $\mathcal{V}_M$ as $\Dens^{\frac12}(\mathcal{V}_M)$, we define $\hat{\mathcal{H}}_M$ as $\hat{\mathcal{H}}_M\coloneqq\Dens^{\frac12}(\mathcal{V}_M)\otimes \mathcal{H}\surM$. It is a graded vector space endowed with two commuting boundary operators
\begin{equation}
\hat\Omega_{\partial M}\coloneqq\Id \otimes \Omega\surM\quad \text{and}\quad \hat\Delta_{\mathcal{V}_M}\coloneqq\Delta_{\mathcal{V}_M}\otimes\Id,
\end{equation}
where $\Delta_{\mathcal{V}_M}$ is the canonical BV Laplacian on half-densities on residual fields. However, by abuse of notation, we will still write $\Omega_{\partial M}$ whenever we actually mean $\hat{\Omega}_{\partial M}$. The same is done for the BV Laplacian.\\
We require the state to satisfy the \textit{modified Quantum Master Equation (mQME)}:
\begin{equation}
\label{ov:bv-bfv_mqme}
\end{equation}
\end{itemize}
%It arises as quantization of $S\bonN$.
\end{itemize}
\end{defn}

In the following, we will refer to a quantum BV-BFV theory with the shorthand notation
\[M\mapsto (\hat{\mathcal{H}}_M, \hat{\psi}_M,\Delta_{\mathcal{V}_M}, \Omega\surM).\]

\begin{rmk}
Since $\Delta_{\mathcal{V}_M}^2=0$, $\Omega_M$ and $\Delta_M$ endow $\hat{\mathcal{H}}_M$ with the structure of a bicomplex.
\end{rmk}

\begin{rmk}
Here we would like to precise the terminology used in Definition \ref{def_Quantum BV-BFV theory} by relating it to the literature. First of all, we call $\mathcal{H}_\Sigma$ space of fields because it is constructed by quantizing the symplectic manifold of boundary fields (as we will see below). An element of this space is thus called state. It is produced by integrating over bulk fields. However, following Wilson's ideas, it is useful to split the contribution of bulk fields into ``low energy" (or ``slow") fields, which we refer to as residual fields, and a complement (usually called ``high energy" or ``fluctuation" fields) on which we integrate over. Hence, our state will depend on both residual fields and boundary contribution. We have the following cases:
\begin{enumerate}
\item in absence of residual fields, $\hat{\psi}_M$ is referred as state in \cite{Wit89},
\item when $M$ is a cylinder, $\hat{\psi}_M$ is an evolution operator,
\item in absence of boundaries and residual fields, $\hat{\psi}_M$ is referred as partition function (see \eqref{ov.tft:part_funct_1}),
\item in the presence of both boundaries and residual fields, $\hat{\psi}_M$ will be a proper state only after we have integrated out the residual fields. We note that this is actually not always possible (see e.g. \cite{Mo20b} and references therein).
\end{enumerate}
Keeping in mind these possibilities, we still prefer to refer to $\psi_M$ as state. %\textcolor{red}{REMARK 2.25 PERTURBATIVE QUANTUM CMR PAGE 19}
\end{rmk}

\begin{defn}[Equivalence]
\label{bv-bfv:def_equiv}
Two quantum BV-BFV theories $(\hat{\mathcal{H}}_M, \hat{\psi}_M,\Delta_{\mathcal{V}_M}, \Omega\surM)$\\ and $(\hat{\mathcal{H'}}_M, \hat{\psi'}_M,\Delta_{\mathcal{V'}_M}, \Omega'\surM)$ are \textit{equivalent} if for every manifold $M$ with boundary $\partial M$ there is a quasi-isomorphism of bicomplexes
\begin{equation}
I_M: (\hat{\mathcal{H}}_M,\Delta_{\mathcal{V}_M}, \Omega\surM)\rightarrow (\hat{\mathcal{H'}}_M,\Delta_{\mathcal{V'}_M}, \Omega'\surM)
\end{equation}
such that $I_M(\hat{\psi}_M)=\hat{\psi'}_M$.
\end{defn}

\begin{defn}[Change of data]
\label{bv-bfv:def_change_data}
Two quantum BV-BFV theories $(\hat{\mathcal{H}}_M, \hat{\psi}_M,\Delta_{\mathcal{V}_M}, \Omega\surM)$ and $(\hat{\mathcal{H'}}_M, \hat\psi'_M,\Delta_{\mathcal{V'}_M}, \Omega'\surM)$ are related by change of data if there is an operator $\tau$ of degree $0$ on $\mathcal{H}\surM$ and an element $\chi\in\hat{\mathcal{H}}_M$ with $\deg(\chi)=\deg(\psi)-1$ such that
\begin{equation}
\begin{split}
\Omega'\surM&=[\Omega\surM, \tau],\\
\hat\psi'_M&=(\hbar^2\Delta_{\mathcal{V}_M}+\Omega\surM)\chi-\hat{\tau}\hat{\psi}_M,
\end{split}
\end{equation}
where $\hat{\tau}=\Id\otimes \tau$ is the extension of $\tau$ to $\hat{\mathcal{H}}_M$.
\end{defn}

\subsubsection{BV pushforward}
Let $(ℳ_1,ω_1)$ and $(ℳ_2,ω_2)$ two graded manifolds with odd symplectic forms $ω_1$ and $ω_2$ and canonical Laplacians $Δ_1$ and $Δ_2$, respectively. Consider $ℳ=ℳ_1 ×ℳ_2$ with symplectic form $ω=ω_1+ω_2$ and canonical Laplacian $Δ$. The space of half-densities on $ℳ$ factorizes as
\begin{equation}
\Dens^{\frac12}(\mathcal{M})=\Dens^{\frac12}(\mathcal{M}_1)\hat{\otimes}\Dens^{\frac12}(\mathcal{M}_2).
\end{equation}
If we do a BV integration in the second factor, over the Lagrangian submanifold $ℒ ⊂ℳ_2$, we are able to define a BV pushforward map on half-densities
\begin{equation}
\varint_{\mathcal{L}}: \Dens^{\frac12}(\mathcal{M})\xrightarrow{\Id\otimes \varint_{\mathcal{L}}}\Dens^{\frac12}(\mathcal{M}_1).
\end{equation}
This map is also called \textit{fiber BV integral}, its properties are defined by the following theorem.

\begin{thm}[Batalin--Vilkovisky--Schwarz]
\label{bv_push}
Let $(\mathcal{M}_1,\omega_1)$ and $(\mathcal{M}_2,\omega_2)$ two graded manifolds with odd symplectic forms $\omega_1$ and $\omega_2$ and canonical Laplacian $\Delta_1$ and $\Delta_2$, respectively. Consider $\mathcal{M}=\mathcal{M}_1 \times \mathcal{M}_2$ with product symplectic form $\omega$ and canonical Laplacian $\Delta$ and let $\mathcal{L}, \mathcal{L}' \subset \mathcal{M}_2$ be any two Lagrangian submanifolds which can be deformed into each-other. For any half-density $f \in \mathrm{Dens}^{\frac12}(\mathcal{M})$ one has:
\begin{enumerate}
\item $\varint_{\mathcal{L}}\Delta f=\Delta_1 \varint_{\mathcal{L}}f$
\item $\varint_{\mathcal{L}}f-\varint_{\mathcal{L'}}f=\Delta_1 \xi$ for some $\xi \in \mathrm{Dens}^{\frac12}(\mathcal{M}_1)$, if $\Delta f=0$.
\end{enumerate}
\end{thm}

\subsubsection{Summary}
Let us explain here how to construct a quantum BV-BFV theory. Consider a classical BV-BFV theory $π: ℱ_M→ℱ^∂_∂M$. Note that from now on we will assume $ℱ_M$ and $ℱ^∂_∂M$ to be vector spaces. This will be the case when we will quantize the \textit{globalized split RW theory}.

The main steps can be summarized as follows
\begin{enumerate}
\item[(i)]\textbf{(Geometric Quantization)} Given a $(d-1)$-manifold $\Sigma$, the BV-BFV theory associates to it a symplectic manifold $(\mathcal{F}^{\partial}_\Sigma, \omega^{\partial}_\Sigma, Q^{\partial}_\Sigma$). The idea here is to construct the space of states $\mathcal{H}_\Sigma$ and the quantum BFV operator $\Omega_\Sigma$ as a \textit{geometric quantization}\footnote{For an introduction to geometric quantization see e.g. \cite{BW97}.} of this symplectic vector space\footnote{In fact, for the case when $\Sigma$ is given by the boundary of another manifold, the BFV operator is constructed by the methods of \emph{deformation quantization} as it was also pointed out in \cite{Moshayedi2021}.}.
In order to accomplish such task, we require the data of a polarization $\mathcal{P}$ on this symplectic vector space, in particular, we consider real fibrating polarizations. Then, it is sufficient to split $\mathcal{F}^{\partial}_\Sigma$ into Lagrangian subspaces as
\begin{equation}
\mathcal{F}^{\partial}_\Sigma\cong \mathcal{B}^{\mathcal{P}}_{\Sigma}\times \mathcal{K}^{\mathcal{P}}_\Sigma,
\end{equation}
with $\mathcal{K}^{\mathcal{P}}_\Sigma$ thought as a Lagrangian distribution on $\mathcal{F}^{\partial}_\Sigma$ and $\mathcal{B}^{\mathcal{P}}_{\Sigma}$ identified with the leaf space of the polarization, i.e. $\mathcal{B}^{\mathcal{P}}_{\Sigma}=\mathcal{F}^{\partial}_\Sigma/\mathcal{P}$. If we assume the 1-form $\alpha^{\partial}_\Sigma$ to vanish along $\mathcal{P}$ and in the case of real polarization, the space of states $\mathcal{H}_{\Sigma}$ is modeled as a space of complex-valued functionals on $\mathcal{B}^{\mathcal{P}}_{\Sigma}$ (or more generally $\mathcal{H}_{\Sigma}$ is the space of polarized sections of the trivial ``prequantum" line bundle over $\mathcal{F}^{\partial}_{\Sigma}$). This means that the space of states is obtained as a geometric quantization of the space of boundary fields as we preannounced above. %more precisely $\mathcal{H}_{\Sigma}=\Dens^{\frac12}(\mathcal{B}^{\mathcal{P}}_{\Sigma}))$.
On the other hand, when $\alpha^{\partial}_\Sigma   \Big|_{\mathcal{P}}\neq 0$, we can use a gauge transformation and modify $\alpha^{\partial}_\Sigma$ by an exact term $\delta f^{\mathcal{P}}_\Sigma$, with $f^{\mathcal{P}}_\Sigma$ a local functional. Consequently, if we assume from now on $\Sigma=\partial M$, to preserve Eq. (\ref{cme_boundary}), we change $\Sc$ by a boundary term obtaining $\Sc^{\mathcal{P}}$. In this case, with $\Sc^{\mathcal{P}}$ and $\alpha^{\mathcal{P}}_{\partial M}$, we have a new BV-BFV manifold.

%The space of states $\mathcal{H}\surPn$ is constructed as a geometric quantization of the BFV theory (located on the boundary when we are aiming to quantize a manifold with boundary).
%    \item[i](Polarization) Choose a polarization $\mathcal{P}$ on $\mathcal{F}\surPn$, given by Lagrangian foliation with a smooth leaf space $\mathcal{B}\surPn$. It is . The space of states  is a graded vector space produced as

\item[(ii)] \textbf{(Extraction of boundary fields)} The aim is to split bulk and boundary field contributions in the space of fields $\mathcal{F}_M$. We proceed as follows: consider the projection $\mathcal{F}^{\partial}_{\partial M}\xrightarrow{p^{\mathcal{P}}_{\partial M}} B^{\mathcal{P}}_{\partial M}$, we have a surjective submersion
\begin{equation}
p^{\mathcal{P}}_{\partial M}\circ \pi:\mathcal{F}_M\rightarrow B^{\mathcal{P}}_{\partial M}.
\end{equation}
Assume we can choose a section $\sigma$ on $\mathcal{F}_M\rightarrow B^{\mathcal{P}}_{\partial M}$ such that we can split
\begin{equation}
\label{ov:bv-bfv_splitting}
\mathcal{F}_M\cong \sigma(B^{\mathcal{P}}_{\partial M}) \otimes \mathcal{Y}.
\end{equation}
The space $\sigma(B^{\mathcal{P}}_{\partial M})$ is a bulk extension of $B^{\mathcal{P}}_{\partial M}$ which we denote as $\tilde{B}^{\mathcal{P}}_{\partial M}$.
This splitting is subjected to the following assumption\footnote{In order to accomplish this assumption, we are forced to choose singular extensions of boundary fields. The boundary fields are thus extended by 0 to the bulk.}:
\begin{assump}
\label{ov:bv-bfv_ass1}
There is a weakly symplectic form $\omega_{\mathcal{Y}}$ on $\mathcal{Y}$ such that $\omega_M$ is the extension of $\omega_{\mathcal{Y}}$ to $\mathcal{F}_M$.
\end{assump}
%From now on, we suppress $\sigma$ from the notation.
In the splitting (\ref{ov:bv-bfv_splitting}), the space $\mathcal{Y}$ is a complement of $\Tilde{B}^{\mathcal{P}}_{\partial M}$ which is interpreted as the space of \textit{bulk fields} (while $\Tilde{B}^{\mathcal{P}}_{\partial M}$ is thought of as the space of \textit{boundary fields} extended to the bulk).
\item[(iii)] \textbf{(Construction of $\Omega_{\partial M}$)} As a result of the geometric quantization procedure, $\mathcal{H}_{\partial M}$ is a cochain complex. Following the same line of thought, we construct the coboundary operator $\Omega_{\partial M}$ as quantization of the boundary actions $\Sc^{\partial}_{\partial M}$. We can proceed as follows. Assume we have Darboux coordinates $(q,p)$ on $\mathcal{F}^{\partial}_{\partial M}$. In particular $q$ are coordinates on $\mathcal{B}^{\mathcal{P}}_{\partial M}$ and $p$ are coordinates on the fiber $p^{\mathcal{P}}_{\partial M}:\mathcal{F}^{\partial}_{\partial M}\rightarrow B^{\mathcal{P}}_{\partial M}$, which is still part of $\mathcal{Y}$. We define $\Omega_{\partial M}$ as the standard-ordering quantization of $\Sc^{\partial}_{\partial M}$:
\begin{equation}
\label{omega_std_ordering}
\Omega_{\partial M}\coloneqq \Sc^{\partial}_{\partial M}\bigg(q, -i\hbar \frac{\partial}{\partial q}\bigg),
\end{equation}
where all the derivatives are positioned on the right.
\item[(iv)] \textbf{(Choice of residual fields)} %In order to recognize which contributions in the bulk fields come from residual fields ("low-energy" fields) and fluctuations ("high-energy" fields),
We further split the bulk contributions in $\mathcal{Y}$ into residual fields and a complement $\mathcal{Y}'$, which represents the space of \textit{fluctuation fields} (also called ``high-energy" or ``fast" fields). This means, we choose a splitting
\begin{equation}
\mathcal{Y}\cong \mathcal{V}^{\mathcal{P}}_M\times \mathcal{Y}'
\end{equation}
which depends on the boundary polarization and satisfies the following assumption:
\begin{assump}
\label{ov:bv-bfv_ass2}
The following holds:
\begin{enumerate}
\item[(1)] $\mathcal{V}^{\mathcal{P}}_M$ and $\mathcal{Y}'$ are BV manifolds,
\item[(2)] $\mathcal{V}^{\mathcal{P}}_M$ is finite-dimensional,
\item[(3)] The symplectic form splits as $\omega_{\mathcal{Y}}=\omega_{\mathcal{V}^{\mathcal{P}}_M}+\omega_{\mathcal{Y}'}$.
\end{enumerate}
\end{assump}
Usually, the space $\mathcal{V}^{\mathcal{P}}_M$ is chosen as the space of solutions of $\delta \Sc^0_M=0$ modulo gauge transformations, where $\Sc^0_M$ is the quadratic part of the action $\Sc_M$. This is called minimal choice, and we refer to this space as the space of \textit{zero modes}. Other choices are possible and they are all related by the equivalence relations defined above (see Definition \ref{bv-bfv:def_equiv} and Definition \ref{bv-bfv:def_change_data}). Finally, we sum up the last two bullet points with the following definition:
\begin{defn}[Good splitting, \cite{CMoW19}]
\label{bv-bfv_good_split}
A splitting
\begin{equation}
\mathcal{F}_M\cong \mathcal{B}^{\mathcal{P}}_{\partial M}\times \mathcal{V}^{\mathcal{P}}_M\times \mathcal{Y}'
\end{equation}
is called \textit{good} if it satisfies Assumptions \ref{ov:bv-bfv_ass1} and \ref{ov:bv-bfv_ass2}.
\end{defn}

Given a good splitting, an element $\mathbf{X}$ of $\mathcal{F}_M$ is written accordingly as $\mathbf{X}=\mathbb{X}+\mathsf{x}+\xi$.
\item[(v)] \textbf{(The state)} When we have a good splitting, the gauge-fixing consists of choosing a Lagrangian $\mathcal{L}\subset \mathcal{Y}'$. Set $\mathcal{Z}_M=\mathcal{B}^{\mathcal{P}}_{\partial M}\times \mathcal{V}^{\mathcal{P}}_M$ (\textit{bundle of residual fields} over $\mathcal{B}^{\mathcal{P}}_{\partial M}$). Then, we define $\hat{\mathcal{H}}^{\mathcal{P}}_M\coloneqq\Dens^{\frac12}(\mathcal{Z}_M)= \Dens^{\frac12}(\mathcal{V}^{\mathcal{P}}_M)\times \Dens^{\frac12}(\mathcal{B}^{\mathcal{P}}_{\partial M})$ and the BV Laplacian $\hat{\Delta}_{\mathcal{V}^{\mathcal{P}}_M}\coloneqq\Id\otimes \Delta_{\mathcal{V}^{\mathcal{P}}_M}$ (as before the hat will be omitted in the following).
The state is then defined as a BV pushforward of the exponential of the bulk action
\begin{equation}
\label{ov.bv-bfv_state}
\hat{\psi}_M(\mathbb{X},\mathsf{x})\coloneqq\varint_{\xi\in \mathcal{L}}e^{\frac{i}{\hbar}\Sc_M(\mathbb{X}+\mathsf{x}+\xi)}.
\end{equation}
Moreover, if $\Delta_{\mathcal{Y}}\Sc^{\mathcal{P}}_M=0$, as a consequence of Theorem \ref{bv_push}:
\begin{itemize}
\item $\hat{\psi}_M$ is closed under the coboundary operator $\hbar^2\Delta_{\mathcal{V}^{\mathcal{P}}_M}+\Omega\surM$, i.e. Eq. (\ref{ov:bv-bfv_mqme}) holds,
\item the state does not change under smooth deformation of the gauge-fixing Lagrangian $\mathcal{L}$ used in the BV pushforward up to $(\hbar^2\Delta_{\mathcal{V}^{\mathcal{P}}_M}+\Omega\surM)$-exact terms.
\end{itemize}
% As a consequence of the previous,
% \begin{equation}
%     (\hbar^2\Delta_{\mathcal{V}_M}+\Omega\surM)\hat{\psi}_M=0
% \end{equation}
\item[(vi)] \textbf{(Perturbative expansion)} The procedure detailed so far is valid for finite-dimensional situations. However, the space of fields $\mathcal{F}_M$ is usually infinite-dimensional, since, for example, it can contain the de Rham complex of differential forms over $\partial M$. As a result, the integral in Eq. (\ref{ov.bv-bfv_state}) is ill-defined. To fix this problem, we \textit{define} the integral perturbatively, i.e. as formal power series in $\hbar$ with coefficients given by sums of Feynman diagrams. For the perturbative expansion to be well-defined, we need the following assumption to be satisfied:
\begin{assump}
\label{ov.bv-bfv_ass3}
The restriction of the action $\Sc^{\mathcal{P}}_M$ to $\mathcal{L}$ has isolated critical points.
\end{assump}
We note that this does not hold for every Lagrangian.

\begin{rmk}
It is important to highlight that, for Assumption \ref{ov.bv-bfv_ass3} to be satisfied, we need to choose carefully the residual fields. The problem here is represented by \textit{zero modes} $\mathcal{V}^0_M$, which can be present in the quadratic part of the bulk action. The zero modes are bulk fields configurations that are annihilated by the kinetic operator and correspond to the tangent directions to the Euler--Lagrange moduli space (solutions of $\delta \Sc^0_M=0$ modulo gauge transformation). Hence, their presence implies non-isolated critical points in the action: the perturbative expansion is obstructed. To solve this situation, we need the space of residual fields to at least contain the space of zero modes, i.e. $\mathcal{V}^0_M\subseteq \mathcal{V}_M$. In this way, we can obtain a good gauge-fixing Lagrangian, which satisfies Assumption \ref{ov.bv-bfv_ass3}. We call it \textit{minimal choice} (or \textit{minimal realization} of the state)\footnote{We have a non-minimal realization when $\mathcal{V}^0_M\subset \mathcal{V}_M$. In that case, we can pass from a non-minimal realization to a smaller one by BV pushforward, which can be interpreted as a sort of \textit{renormalization group flow} \cite{Ir18}.}, when $\mathcal{V}_M\cong\mathcal{V}^0_M$.
\end{rmk}

\noindent When we pass to the infinite case, another problem arises: the BV Laplacian is ill-defined. Therefore, every equation containing it is only \textit{formal}. In this regard, Theorem \ref{bv_push} has only been proven in the finite-dimensional setting. Hence, we can not conclude that the mQME is satisfied even if the action is formally annihilated by the Laplacian. The mQME has to be verified for each theory at the level of Feynman diagrams.
In this paper, we add the globalized RW theory to the class of $BF$-like theories for which the mQME has been proven in the infinite-dimensional perturbative setting. The proof relies on Stokes' Theorem for integrals over compactified configuration spaces.
\end{enumerate}

\subsection{Quantum states in $BF$-like theories}
\label{bv-bfv_sub_qs_bf_like}
In $BF$-like theories one can define the quantum state in a perturbative way using  Feynman graphs via integrals defined on the configuration space of these graphs. Two convenient choices of polarizations in $BF$-like theories are the $δ/δ𝐀$- and the $δ/δ𝐁$-polarization. Concretely, we fix a polarization  by splitting the boundary $∂M$ into two parts $∂_1 M$ and $∂_2M$, where we choose the polarization  $δ/δ𝐁$ on $∂_1M$ and $δ/δ𝐀$ on $∂_2M$. The associated space of leaves for the $A$-leaf and $B$-leaf are denoted by $𝔸 ∈ℬ_∂M^δ/δ𝐁$ and $𝔹 ∈ℬ_∂M^δ/δ𝐀 $ respectively.

For $BF$-like theories, the first splitting determined by the polarization is
\begin{equation}
\begin{split}
\mathcal{B}^{\mathcal{P}}_{\partial M}&=\big(\Omega^\bullet (\partial_1M)\otimes V[1]\big) \oplus \big(\Omega^\bullet(\partial_2M)\otimes V^{\vee}[d-2]\big), \\
\mathcal{Y}&=\big(\Omega^\bullet(M,\partial_1M) \otimes V[1]\big) \oplus \big(\Omega^\bullet(M,\partial_2M)\otimes V^{\vee}[d-2]\big).
\end{split}
\end{equation}
The minimal space of residual fields is
\begin{equation}
\mathcal{V}^{\mathcal{P}}_M \cong \big(H^\bullet (M,\partial_1M) \otimes V[1]\big) \oplus (H^\bullet \big(M,\partial_2M)\otimes V^{*}[d-2]\big)
\end{equation}
for $V$ some graded vector space. One way to get a good splitting is then by considering a splitting of the complex of de Rham forms with relative boundary conditions into a subspace $𝒱^𝒫_M$ isomorphic to cohomology and a complementary space $𝒴'$ in a way compatible with the symplectic structure. This can be done by using a Riemannian metric and embed the cohomology as harmonic forms.
As a result, the space of fields $_M$ splits as $_M=ℬ^𝒫_∂M ×𝒱^𝒫_M ×𝒴'$, where an element $(𝐀, 𝐁) ∈_M$ is given by
\begin{equation}
\begin{split}
\mathbf{A}&=\mathbb{A} + \underline{\mathbf{A}}=\mathbb{A} +\mathsf{a} + \alpha,\\
\mathbf{B}&=\mathbb{B}+\underline{\mathbf{B}}=\mathbb{B} + \mathsf{b} + \beta.
\end{split}
\end{equation}

There is one last ingredient that we need to introduce before defining the quantum state, namely the \textit{composite fields}. We denote them by square brackets $[  ]$, i.e. for a boundary field $𝔸$ we have $[𝔸^i_1 …𝔸^i_k]$. One can think of them as a \textit{regularization}  of higher functional derivatives, in the sense %of the higher functional derivative $\frac{\delta^k}{\delta\mathbb{A}^{i_1}\dots \delta \mathbb{A}^{i_k}}$.
that a higher functional $δ^k/δ𝔸^i_1…δ𝔸^i_k$ is replaced by a first order functional derivative $δ^k/[δ𝔸^i_1…δ𝔸^i_k]$. For further details see \cite{CMR17}.

%Practically this corresponds to introducing additional boundary vertices as in FIG.\textcolor{red}{REFER FIGURE}

\begin{defn}[Regular functional]
\label{regularfct}
A \textit{regular functional} on the space of base boundary fields is a linear combination of expressions of the form
\begin{equation}
\label{ov:bv-bfv.reg_func}
\begin{split}
\varint_{\mathrm{C}_{m_1}(\partial_1M)\times\mathrm{C}_{m_2}(\partial_2M)} L^{J^1_1\dots J^{l_1}_1\dots J^1_2\dots J^{l_2}_2\dots}_{I^1_1\dots I^{r_1}_1\dots I^1_2\dots I^{r_2}_2\dots}\wedge \pi^*_1\prod^{r_1}_{j=1}\bigg[\mathbb{A}^{I^j_1}\bigg]\wedge \dots&\wedge \pi^*_{m_1}\prod^{r_{m_1}}_{j=1}\bigg[\mathbb{A}^{I^j_{m_1}}\bigg]\wedge \dots \\
&\wedge \pi^*_1\prod^{l_1}_{j=1}\bigg[\mathbb{B}_{J^j_1}\bigg]\wedge \dots \wedge \pi^*_{m_1}\prod^{l_{m_2}}_{j=1}\bigg[\mathbb{B}_{J^j_{m_2}}\bigg],
\end{split}
\end{equation}
where $I^j_i$ and $J^j_i$ are (target) multi-indices and $L^{J^1_1\dots J^{l_1}_1\dots J^1_2\dots J^{l_2}_2\dots}_{I^1_1\dots I^{r_1}_1\dots I^1_2\dots I^{r_2}_2\dots}$ is a smooth differential form on the direct product of compactified configuration spaces  $\mathrm{C}_{m_1}(\partial_1M)\times \mathrm{C}_{m_2}(\partial_2M)$, which depends on the residual fields. A regular functional is called \textit{principal} if all multi-indices have length 1. For more details on configuration spaces and configuration space integrals we refer to \cite{Kontsevich1994,BC,CamposIdrissiLambrechtsWillwacher2018} (see also Remark \ref{rem:conf}).
\end{defn}

\begin{defn}[Full space of boundary states]
The \textit{full space of boundary states} $\mathcal{H}^{\mathcal{P}}_{\partial M}$ is given by the linear combination of regular functionals of the form (\ref{ov:bv-bfv.reg_func}).
\end{defn}
\begin{defn}[Principal space of boundary states] The \textit{principal space of boundary
states} $\mathcal{H}_{\partial M}^{\mathcal{P}, \text{princ}}$ is defined as the subspace of $H^{\mathcal {P}}_{\partial M}$
, where we only consider principal regular functionals.
\end{defn}
We use Feynman rules and graphs to define the state. Let us elaborate them in the BV-BFV setting (for perturbations of abelian $BF$-theory).
\begin{defn}[($BF$) Feynman graph]
A \textit{(BF) Feynman graph} is an oriented graph with three types of vertices $V(\Gamma)=V_{\text{bulk}}(\Gamma)\sqcup V_{\partial_1}\sqcup V_{\partial_2}$, called bulk vertices and type 1 and 2 boundary vertices, such that
\begin{itemize}
\item bulk vertices can have any valence,
\item type 1 boundary vertices carry any number of incoming half-edges (and no outgoing half-edges),
\item type 2 boundary vertices carry any number of outgoing half-edges (and no incoming half-edges),
\item multiple edges and loose half-edges (leaves) are allowed.
\end{itemize}
\end{defn}
A labeling of a Feynman graph is a function from the set of half-edges to ${1,…,V}$.
\begin{defn}[Principal graph]A Feynman graph is called \textit{principal} if all boundary vertices (type 1 and type 2) are univalent or zero valent.
\end{defn}
Let $Γ$ be a Feynman graph and $M$ a manifold with boundary $∂M=∂_1 M ⊔∂_2M$ and define
\begin{equation}
\mathrm{Conf}_\Gamma(M)\coloneqq \mathrm{Conf}_{V_{\text{bulk}}}(M) \times \mathrm{Conf}_{V_{\partial_1}}(\partial_1 M) \times \mathrm{Conf}_{V_{\partial_2}}(\partial_2 M).
\end{equation}
The Feynman rules are given by a map associating to a Feynman graph $Γ$ a differential form $ω_Γ∈Ω^∙(Conf_Γ(M))$.
\begin{defn}[($BF$) Feynman rules] Let $\Gamma$ be a labeled Feynman graph. We choose a configuration $\iota:V(\Gamma) \rightarrow \mathrm{Conf} (\Gamma)$, such that decompositions are respected. Then, we \textit{decorate} the graph according to the following rules, namely, the \textit{Feynman rules}:
\begin{itemize}
\item Bulk vertices in $M$ decorated by ``vertex tensors"
\begin{equation}
\mathcal{V}^{i_1\dots i_s}_{j_1\dots j_t} \coloneqq \frac{\partial^{s+t}}{\partial \underline{\mathbf{A}}^{i_1}\dots\partial \underline{\mathbf{A}}^{i_s} \partial \underline{\mathbf{B}}_{j_1}\dots \partial \underline{\mathbf{B}}_{j_t}} \bigg|_{\underline{\mathbf{A}}=\underline{\mathbf{B}}=0} \mathcal{V}(\underline{\mathbf{A}},\underline{\mathbf{B}}),
\end{equation}
where $s, t$ are the out- and in-valencies of the vertex and $i_1, \dots, i_s$ and $j_1, \dots, j_t$ are the labels of the out- and in- oriented half-edges and $\mathcal{V}(\underline{\mathbf{A}},\underline{\mathbf{B}})$ is the interaction term in a $BF$-like theory.

\item Boundary vertices $v \in V_{\partial_1}(\Gamma)$ with incoming half-edges labeled $i_1, \dots, i_k$ and no out-going  half-edges are decorated by a composite field $[\mathbb{A}^{i_1} \dots \mathbb{A}^{i_k}]$ evaluated at the point (vertex location) $\iota(v)$ on $\partial_1 M$.
\item Boundary vertices $v \in V_{\partial_2}$ on $\partial_2 M$ with outgoing half-edges labeled $j_1 \dots j_l$ and no in-going  half-edges are decorated by $[\mathbb{B}_{j_1} \dots \mathbb{B}_{j_l}]$ evaluated at the point on $\partial_2 M$.
\item Edges between vertices $v_1, v_2$ are decorated with the propagator $\eta (\iota(v_1),\iota(v_2))\cdot \delta^i_j$, with $\eta$ the propagator induced by $\mathcal{L} \subset \mathcal{Y}'$, the gauge-fixing Lagrangian.
\item Loose half-edges (leaves) attached to a vertex $v$ and labeled $i$ are decorated with the residual
fields $\mathsf{a}_i$ (for out-orientation), $\mathsf{b}^i$
(for in-orientation) evaluated at the point $\iota(v)$.
\end{itemize}

\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.25\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=1pt}]
\vertex[blob] (m) at (0,-2) {$\mathsf{a}$};
\vertex (a) at (0,0) {$x$} ;
\diagram*{
(a) -- [fermion, edge label' = $i$] (m)
\vertex [right=3em of m] {\(=\mathsf{a}^i(x)\)};
\end{feynman}
\end{tikzpicture}
\caption{}
\label{}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{0.25\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=0.5pt}]
\vertex[blob] (m) at (0,-2) {$\mathsf{b}$};
\vertex (a) at (0,0) {$x$} ;
\diagram*{
(m) -- [fermion, edge label = $i$] (a)
\vertex [right=3em of m] {\(=\mathsf{b}^i(x)\)};
\end{feynman}
\end{tikzpicture}
\caption{}
\label{}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{0.4\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=0.5pt}]
\vertex (a) at (0,0) {$x$};
\vertex (b) at (2,0) {$y$} ;
\diagram*{
(a) -- [fermion, edge label = $i\hspace{5mm} j$] (b)
\vertex [right=3em of b] {\(=\delta^i_j\eta(x,y)\)};
\end{feynman}
\end{tikzpicture}
\caption{}
\label{}
\end{subfigure}
\caption{Feynman rules for residual fields and propagator.}
\label{bv-bfv:fig_feyn_rules1}
\end{figure}

\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=0.5pt}]
%\vertex (m) at (0,0) {dot};
\vertex (a) at (-1,0);% {$\partial_2 \Sigma_3$};
\vertex (b) at (1,0);
\vertex (m1) at (0, 1);

\diagram*{
(a) -- m [dot] -- (b),
(m1) -- [fermion] m
%(m1) -- [fermion] (m)
\vertex [below=0.75em of m] {\(\mathbb{B}\)};
\vertex [left=0.25em of a] {\(\partial_2 \Sigma_3\)};
\end{feynman}
% \draw (m) circle (0.5);
\end{tikzpicture}
\caption{}
\label{}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=0.5pt}]
%\vertex (m) at (0,0) {dot};
\vertex (a) at (-1,0);% {$\partial_2 \Sigma_3$};
\vertex (b) at (1,0);
\vertex (m1) at (0, -1);

\diagram*{
(a) -- m [dot] -- (b),
m -- [fermion] (m1)
%(m1) -- [fermion] (m)
\vertex [above=0.75em of m] {\(\mathbb{A}\)};
\vertex [left=0.25em of a] {\(\partial_1 \Sigma_3\)};
\end{feynman}
% \draw (m) circle (0.5);
\end{tikzpicture}
\caption{}
\label{}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{0.42\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=0.5pt}]
\vertex (a) at (-1.2, 0.35);
\vertex (b) at (-0.3, 0.79);
\vertex (c) at (+1.2, 0.4);
\vertex (e) at (-1.2, -0.35);
\vertex (f) at (-0.3, -0.79);
\vertex (g) at (+1.2, -0.4);
%   \vertex (e) at (-0.8, -0.71);
%   \vertex (f) at (+0.5, -0.71);
\diagram*{
(a) -- [fermion] m [dot],
(b) -- [fermion] m [dot],
(c) -- [fermion] m [dot],
(e) -- [anti fermion] m [dot],
(f) -- [anti fermion] m [dot],
(g) -- [anti fermion] m [dot],
%   (e) -- m [dot],
%   (f) -- m [dot]
\vertex [right=7em of m] {\(=\mathcal{V}^{i_1\dots i_s}_{j_1\dots j_t}\)};

\vertex [above=0.2em of m, label=80:\(\dots\)] {};
%\vertex [above=0.2em of m, label=140:\(j_2\)] {};
\vertex [] at (-1.35,0.4) {\(j_1\)};
\vertex [] at (-0.45,1) {\(j_2\)};
\vertex [] at (1.4, 0.4) {\(j_t\)};
\vertex [] at (-1.35,-0.4) {\(i_1\)};
\vertex [] at (-0.45,-1) {\(i_2\)};
\vertex [] at (1.4, -0.4) {\(i_s\)};
\end{feynman}
\end{tikzpicture}
\caption{}
\label{}
\end{subfigure}%
\caption{Feynman rules for boundary fields and interaction vertices.}
\label{bv-bfv:fig_feyn_rules2}
\end{figure}

\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=0.5pt}]
%\vertex (m) at (0,0) {dot};
\vertex (a) at (-2,0);% {$\partial_2 \Sigma_3$};
\vertex (b) at (2,0);
\vertex (d) at (0.4, 0.75);
\vertex (e) at (1, 0.75);
\vertex (f) at (-1, 0.75);
\diagram*{
(a) -- m [dot] -- (b),
(d) -- [anti fermion] m,
(e) -- [anti fermion] m,
(f) -- [anti fermion] m
%(m1) -- [fermion] (m)
\vertex [below=0.75em of m] {\([\mathbb{A}^{i_1}\dots \mathbb{A}^{i_k}]\)};
\vertex [left=0.25em of a] {\(\partial_1 \Sigma_3\)};
\node at (-0.2, 0.5) {$\dots$};
\node at (-1.1, 0.6) {$i_k$};
\node at (0.5, 1) {$i_2$};
\node at (1.2, 0.9) {$i_1$};
\end{feynman}%
% \draw (m) circle (0.5);
\end{tikzpicture}
\caption{}
\label{fig:comp_fields}
\end{subfigure}%
\qquad \begin{subfigure}[b]{0.3\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=0.5pt}]
%\vertex (m) at (0,0) {dot};
\vertex (a) at (-2,0);% {$\partial_2 \Sigma_3$};
\vertex (b) at (2,0);
\vertex (d) at (0.4, 0.75);
\vertex (e) at (1, 0.75);
\vertex (f) at (-1, 0.75);
\diagram*{
(a) -- m [dot] -- (b),
(d) -- [fermion] m,
(e) -- [fermion] m,
(f) -- [fermion] m
%(m1) -- [fermion] (m)
\vertex [below=0.75em of m] {\([\mathbb{B}_{j_1}\dots \mathbb{B}_{j_l}]\)};
\vertex [left=0.25em of a] {\(\partial_2 \Sigma_3\)};
\node at (-0.2, 0.5) {$\dots$};
\node at (-1.1, 0.6) {$j_l$};
\node at (0.5, 1) {$j_2$};
\node at (1.2, 0.9) {$j_1$};
\end{feynman}
\end{tikzpicture}
\caption{}
\label{}
\end{subfigure}%
\caption{Feynman rules for the composite fields.}
\label{fig:comp_fields_old}
\end{figure}
The differential forms given by the decorations are denoted collectively by $\omega_d$. The differential form $\omega_\Gamma$ at $\iota$ is then defined by the product of all decorations and summing over all labels:
\begin{equation}
\omega_\Gamma=\sum_{\text{labellings of $\Gamma$}}\;\,\, \prod_{{\text{decorations $d$ of $\Gamma$}}} \omega_d.
\end{equation}
The Feynman rules are represented in Figs. \ref{bv-bfv:fig_feyn_rules1}, \ref{bv-bfv:fig_feyn_rules2} and \ref{fig:comp_fields_old}.
\end{defn}
\begin{rmk}[Configuration spaces]\label{rem:conf}
We will exploit the Fulton--MacPherson/Axelrod--Singer compactification of configuration spaces on manifolds with boundary (FMAS compactification \cite{FM94,AS94}). Axelrod and Singer proved via non-trivial analytic tools, that the propagator, a priori defined only on the open configuration space $\mathrm{Conf}_2 (M)$, extends to the compactification $\mathrm{C}_2(M)$. This implies that also $\omega_\Gamma$, for all Feynman graphs $\Gamma$, extends to the compactification $\mathrm{C}_\Gamma (M)$ of $\mathrm{Conf}_\Gamma(M)$. But, adding strata of lower codimension leaves the integrals unchanged, and hence this proves that the integrals appearing in Eq. \eqref{principalqs} below are finite. In addition, one can exploit the combinatorics of the stratification for various computations using Stokes' theorem.
\end{rmk}
\begin{defn}[Principal quantum state]
Let $M$ be a manifold (with boundary). Given a $BF$-like  BV-BFV theory $\pi_M:\F_M \rightarrow \F^\partial_{\partial M}$, a polarization $\mathcal{P}$ on $\F^\partial_{\partial M}$, a good splitting $\F_M=\mathcal{B}^{\mathcal{P}}_{\partial M} \times \mathcal{V}^{\mathcal{P}}_M \times \mathcal{Y}'$ and $\mathcal{L} \subset \mathcal{Y}'$, the gauge-fixing Lagrangian,  we can define the \textit{principal part of the quantum state} by the formal power series
\begin{equation}
\label{principalqs}
\hat{\psi}_{M}(\mathbb{A},\mathbb{B}; \mathsf{a},\mathsf{b}) \coloneqq T_{M}\exp\bigg(\frac{i}{\hbar}\sum_{\Gamma}\frac{(-i\hbar)^{\loops(\Gamma)}}{|\Aut(\Gamma)|}\varint_{\text{C}_\Gamma(M)}\omega_{\Gamma}(\mathbb{A}, \mathbb{B}; \mathsf{a}, \mathsf{b})
\bigg),
\end{equation}
where for an element $(\mathbf{A}, \mathbf{B}) \in \F_M$, we denote the split by
\begin{equation}
\begin{split}
\mathbf{A}&=\mathbb{A} + \mathsf{a} + \alpha,\\
\mathbf{B}&=\mathbb{B} + \mathsf{b} + \beta.
\end{split}
\end{equation}
The sum runs over all connected, oriented, principal $BF$ Feynman graphs $\Gamma$, $\Aut(\Gamma)$ denotes the set of all automorphisms of $\Gamma$, and $\loops(\Gamma)$ denotes the number of all loops of $\Gamma$. The coefficient $T_M$ is related to the Reidemeister torsion of $M$%ref
; its exact expression is not needed in our context.
\end{defn}
\begin{rmk}
The formal power series in  (\ref{principalqs}) is the definition of the formal perturbative expansion
of the BV integral
\begin{equation}
\hat{\psi}_M(\mathbb{A},\mathbb{B}; \mathsf{a},\mathsf{b})=\varint_{\mathcal{L}\subset \mathcal{Y}'}e^{\frac{i}{\hbar}\mathcal{S}_M(\mathbf{A},\mathbf{B})} \in \hat{\mathcal{H}}_M^{\mathcal{P}} \coloneqq \hat{\mathcal{H}}_{\partial M}^{\mathcal{P}} \otimes \Dens^{\frac12}(\mathcal{V}_M^{\mathcal{P}}).
\end{equation}
\end{rmk}
Given a good splitting (see Definition \ref{bv-bfv_good_split}), the action can be decomposed as (\cite{CMR17})
\begin{equation}
\Sc_M=\hat{\Sc}_M+\hat{\Sc}^{\, \text{pert}}_M+\Sc^{\, \text{res}}+\Sc^{\, \text{source}},
\end{equation}
\begin{equation}
\begin{split}
\hat{\Sc}_M&=\varint_{\Sigma_3}\beta_id\alpha^i,\\
\hat{\Sc}^{\, \text{pert}}_M&=\varint\sur \mathcal{V}(\mathsf{a}+\alpha, \mathsf{b}+\beta),\\
\Sc^{\, \text{res}}&= - \bigg(\varint_{\partial_2\Sigma_3}\mathbb{B}_i\mathsf{a}^i-\varint_{\partial_1\Sigma_3}\mathsf{b}_i\mathbb{A}^i\bigg),\\
\Sc^{\, \text{source}}&= - \bigg(\varint_{\partial_2\Sigma_3}\mathbb{B}_i\alpha^i-\varint_{\partial_1\Sigma_3}\beta_i\mathbb{A}^i\bigg).\\
\end{split}
\end{equation}
\begin{rmk}
We can rewrite

\begin{equation}
\hat{\psi}_M(\mathbb{A},\mathbb{B}; \mathsf{a},\mathsf{b})=T_{M}\big\langle e^{\frac{i}{\hbar}(\Sc^{\text{res}}+\Sc^{\text{source})}}\big\rangle=T_{M}e^{\frac{i}{\hbar}\Sc^{\text{eff}}},
\end{equation}
\end{rmk}
where $⟨ ⟩$ denotes the expectation value with respect to the bulk theory $+^pert$ and
\begin{equation}
\Sc^{\, \text{eff}}=- \bigg(\varint_{\partial_2M}\mathbb{B}_i\mathsf{a}^i-\varint_{\partial_1M}\mathsf{b}_i\mathbb{A}^i\bigg)+\varint_{\partial_2M\times\partial_1M}\pi^*_1\mathbb{B}_i\eta^i_{\; j} \pi^*_2\mathbb{A}^j.
\end{equation}
Note that the \textit{effective action} manifests as we sum over connected graphs.

We are now interested in constructing a product on the full state space using composite fields. We define the \textit{bullet product}:
\begin{equation}
\label{bulletproof}
\begin{split}
&\varint_{\partial_1M} u_i \wedge \mathbb{A}^i \bullet \varint_{\partial_1M} v_j \wedge \mathbb{A}^j \coloneqq \\
&(-1)^{\mid \mathbb{A}^i \mid (d-1+\mid v_j\mid)+\mid u_i\mid(d-1)} \bigg( \varint_{\mathrm{C}_2(\partial_1M)} \pi_1^*u_i \wedge \pi_2^*v_j \wedge \pi_1^*\mathbb{A}^i \wedge \pi_2^*\mathbb{A}^j+\varint_{\partial_1M}u_i \wedge v_j \wedge [\mathbb{A}^i \mathbb{A}^j] \bigg ),
\end{split}
\end{equation}
with $u,v$ smooth differential forms depending on the bulk and residual fields.
\begin{rmk}
Consider the operator $\varint_{\partial_1M} F^{ij} \frac{\delta^2}{\delta \mathbb{A}^i \delta \mathbb{A}^j}$. It can be interpreted as $\varint_{\partial_1M} F^{ij} \frac{\delta}{\delta[\mathbb{A}^i\mathbb{A}^j]}$, and therefore, we have
\begin{equation}
\varint_{\partial_1M}F^{ij}\frac{\delta^2}{\delta \mathbb{A}^i \delta \mathbb{A}^j}\bigg (\varint_{\partial_1M} u_i \wedge \mathbb{A}^i \bullet \varint_{\partial_1M}v_j \wedge \mathbb{A}^j \bigg )= \varint_{\partial_1M} u_iv_jF^{ij},
\end{equation}
which matches our prediction.
\end{rmk}

\begin{defn}[Full quantum state]
\label{full_quantum_state}
Let $M$ be a manifold (with boundary). Given a $BF$-like  BV-BFV theory $\pi_M:\F_M \rightarrow \F^\partial_{\partial M}$, a polarization $\mathcal{P}$ on $\F^\partial_{\partial M}$, a good splitting $\F_M=\mathcal{B}_{\partial M}^{\mathcal{P}} \times \mathcal{V}_M^{\mathcal{P}} \times \mathcal{Y}'$ and $\mathcal{L} \subset \mathcal{Y}'$, the gauge-fixing Lagrangian,  we can define the \textit{full quantum state} by the formal power series
\begin{equation}
\boldsymbol{\hat{\psi}}_M(\mathbb{A},\mathbb{B};\mathsf{a},\mathsf{b})=T_{M}\exp\bigg(\frac{i}{\hbar}\sum_{\Gamma}\frac{(-i\hbar)^{\text{loops}(\Gamma)}}{|\Aut(\Gamma)|}\varint_{\text{C}_\Gamma(M)}\omega_{\Gamma}(\mathbb{A}, \mathbb{B}; \mathsf{a}, \mathsf{b})
\bigg).
\end{equation}
\end{defn}

\begin{rmk}Exploiting the bullet product in \eqref{bulletproof}, we can write the full quantum state as the expectation value
\begin{equation}
\boldsymbol{\hat{\psi}}_M(\mathbb{A},\mathbb{B};\mathsf{a},\mathsf{b})=T_{M}\big\langle e^{\frac{i}{\hbar}(\Sc^{\text{res}}+\Sc^{\text{source}})}_{\bullet}\big\rangle=T_{M}e^{\frac{i}{\hbar}\Sc^{\text{eff}}}_{\bullet},
\end{equation}
with $e_\bullet$ the exponential with respect to the bullet product.
\end{rmk}

\subsubsection{The BFV boundary operator}
peOur next ingredient is the quantum BFV boundary operator for $BF$-like theories \cite{CMR17}. We will follow the same procedure as with the state, writing firstly its principle part and then extending to a regularization using the composite fields. One obtains the quantum BFV boundary operator via the quantization of the BFV action such that Theorem \ref{bv-bfv_thm_mqme} is satisfied.
\begin{defn}[Principal part of the BFV boundary operator] The \textit{principal part} of the BFV boundary operator is given by
\begin{equation}
\Omega^{\text{princ}}=\underbrace{\Omega_0^{\mathbb{A}}+\Omega_0^{\mathbb{B}}}_{\coloneqq \Omega_0}+\underbrace{\Omega_{\text{pert}}^{\mathbb{A}}+\Omega_{\text{pert}}^{\mathbb{B}}}_{\coloneqq \Omega_{\text{pert}}^{\text{princ}}},
\end{equation}
\begin{equation}
\begin{split}
\Omega_0^{\mathbb{A}} &\coloneqq (-1)^d i\hbar \varint_{\partial_1M} \bigg(d\mathbb{A} \frac{\delta}{\delta \mathbb{A}} \bigg),\\
\Omega_0^{\mathbb{B}} &\coloneqq (-1)^d i\hbar \varint_{\partial_2M} \bigg(d\mathbb{B} \frac{\delta}{\delta \mathbb{B}} \bigg),
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\Omega_{\text{pert}}^{\mathbb{A}}&\coloneqq \sum_{n,k \geq 0}\sum_{\Gamma_1'}\frac{(i \hbar)^{\text{loops($\Gamma_1')$}}}{\mid \Aut(\Gamma_1')\mid} \varint_{\partial_1M}\bigg(\sigma_{\Gamma_1'}\bigg)_{i_1\dots i_n}^{j_1\dots j_k}\wedge \mathbb{A}^{i_1}\wedge \dots \wedge \mathbb{A}^{i_n}\bigg((-1)^d i\hbar\frac{\delta}{\delta \mathbb{A}^{j_1}}\bigg)\dots \bigg((-1)^d i\hbar \frac{\delta}{\delta \mathbb{A}^{j_k}} \bigg),\\
\Omega_{\text{pert}}^{\mathbb{B}}&\coloneqq \sum_{n,k \geq 0}\sum_{\Gamma_2'}\frac{(i \hbar)^{\text{loops($\Gamma_2')$}}}{\mid \Aut(\Gamma_2')\mid} \varint_{\partial_2M}\bigg(\sigma_{\Gamma_2'}\bigg)_{i_1\dots i_n}^{j_1\dots j_k}\wedge \mathbb{B}_{j_1}\wedge \dots \wedge \mathbb{B}_{j_k}\bigg((-1)^d i\hbar\frac{\delta}{\delta \mathbb{B}_{i_1}}\bigg)\dots \bigg((-1)^d i\hbar \frac{\delta}{\delta \mathbb{B}_{i_n}} \bigg),
\end{split}
\end{equation}
where, for $\mathbb{F}_1=\mathbb{A}$, $\mathbb{F}_2=\mathbb{B}$ and $l \in \{1,2\}$, $\Gamma_l'$ runs over graphs with
\begin{itemize}
\item $n$ vertices on $\partial_lM$ of valence 1 with adjacent half-edges oriented inwards and decorated with boundary fields $\mathbb{F}_l^{i_1},\dots,\mathbb{F}_l^{i_n}$ all evaluated at the point of collapse $p\in \partial_lM$,
\item $k$ outward leaves if $l = 1$ and $k$ inward leaves if $l = 2$, decorated with variational derivatives
in boundary fields
\begin{equation}
(-1)^d i\hbar \frac{\delta}{\delta \mathbb{F}_l^{j_1}},\dots,(-1)^di\hbar\frac{\delta}{\delta\mathbb{F}_l^{j_k}}
\end{equation}
at the point of collapse,
\item  no outward leaves if $l = 2$ and no inward leaves if $l = 1$ (graphs with them do not contribute).
\end{itemize}
\end{defn}
The form $σ_Γ_l'$ is obtained as the integral over the compactification $C̃_Γ_l'(ℍ^d)$ of the  open configuration space modulo scaling and translation, with $ℍ^d$ the $d$-dimensional upper half-space:
\begin{equation}
\sigma_{\Gamma_l'}=\varint_{\tilde{\mathrm{C}}_{\Gamma_l'(\mathbb{H}^d)}} \omega_{\Gamma_l'},
\end{equation}
where $ω_Γ_l'$ is the product of limiting propagators at the point $p$ of collapse and vertex tensors.

Our goal now is to describe the BFV boundary operator with composite fields. For this, we introduce the following auxiliary concept.

Consider the regular functional in (\ref{regularfct}), where we get a term $L$ replaced by $dL$ plus all the terms corresponding to the boundary of the configuration space.  Since $L$ is smooth, its restriction to the boundary is smooth as well, and can be integrated on the fibers giving rise to a smooth form on the base configuration space. For example
\begin{equation}
\Omega_0 \varint_{\partial_1M}L_{IJ} \wedge [\mathbb{A}^I] \wedge [\mathbb{A}^{J}]= \pm i \hbar \varint_{\partial_1M}dL_{IJ}\wedge [\mathbb{A}^I]\wedge [\mathbb{A}^J],
\end{equation}
\begin{equation}
\Omega_0\varint_{\mathrm{C}_2(\partial_1M)}L_{IJK}\wedge \pi_1^*([\mathbb{A}^I]\wedge[\mathbb{A}^J]) \wedge \pi_2^*[\mathbb{A}^K]\pm i\hbar\varint_{\partial_1M}\underline{L_{IJK}}\wedge [\mathbb{A}^I]\wedge [\mathbb{A}^J] [\mathbb{A}^K],
\end{equation}
with $L_IJK=π_*^∂$, where $π^∂: ∂C_2(∂_1M)→∂_1M$ is the canonical projection. \newline
For any two regular functionals $S
_1$ and $S_2$ we can write
\begin{equation}
\Omega_0(S_1 \bullet S_2)=\Omega_0(S_1) \bullet S_2 \pm S_1 \bullet \Omega_0(S_2).
\end{equation}
The rest of the allowed generators are products of expressions in the following shape:
\begin{equation}
\varint_{\partial_1M}L^J_{I^1 \dots I^r}[\mathbb{A}^{I_1}]\wedge \dots \wedge [\mathbb{A}^{I_r}] \frac{\delta^{\mid J \mid}}{\delta [\mathbb{A}^I]},
\end{equation}
\begin{equation}
\varint_{\partial_2M}L^{J^1\dots J^l}_{I}[\mathbb{B}_{J_1}]\wedge \dots \wedge [\mathbb{B}_{J_l}] \frac{\delta^{\mid I \mid}}{\delta [\mathbb{B}_J]}.
\end{equation}
\begin{defn}[Full BFV boundary operator]
\label{full_bfv_def}
The \textit{full BFV boundary operator} is
\begin{equation}
\boldsymbol{\Omega}_{\partial M}\coloneqq\Omega_0 + \underbrace{\boldsymbol{\Omega}_{\text{pert}}^{\mathbb{A}} + \boldsymbol{\Omega}_{\text{pert}}^{\mathbb{B}}}_{\boldsymbol{\Omega}_{\text{pert}}},
\end{equation}
\begin{equation}
\begin{split}
\boldsymbol{\Omega}_{\text{pert}}^{\mathbb{A}}&\coloneqq \sum_{n,k \geq 0}\sum_{\Gamma_1'}\frac{(i \hbar)^{\text{loops($\Gamma_1')$}}}{\mid \Aut(\Gamma_1')\mid} \varint_{\partial_1M}\bigg(\sigma_{\Gamma_1'}\bigg)_{I_1\dots I_n}^{J_1\dots J_k}\wedge \mathbb{A}^{I_1}\wedge \dots \wedge \mathbb{A}^{I_n}\bigg((-1)^{kd} (i\hbar)^{k}\frac{\delta^{\mid J_1\mid + \dots +\mid J_k\mid}}{\delta [\mathbb{A}^{J_1} \dots \mathbb{A}^{J_k}]}\bigg),\\
\boldsymbol{\Omega}_{\text{pert}}^{\mathbb{B}}&\coloneqq \sum_{n,k \geq 0}\sum_{\Gamma_2'}\frac{(i \hbar)^{\text{loops($\Gamma_2')$}}}{\mid \Aut(\Gamma_2')\mid} \varint_{\partial_2M}\bigg(\sigma_{\Gamma_2'}\bigg)^{I_1\dots I_n}_{J_1\dots J_k}\wedge \mathbb{B}_{I_1}\wedge \dots \wedge \mathbb{B}_{I_n}\bigg((-1)^{kd} (i\hbar)^{k}\frac{\delta^{\mid J_1\mid + \dots +\mid J_k\mid}}{\delta [\mathbb{B}_{J_1} \dots \mathbb{B}_{J_k}]}\bigg),
\end{split}
\end{equation}
where, for $\mathbb{F}_1=\mathbb{A}$, $\mathbb{F}_2=\mathbb{B}$ and $l \in \{1,2\}$, $\Gamma_l'$ runs over graphs with
\begin{itemize}
\item $n$ vertices on $\partial_lM$, where vertex $s$ has valence $| I_s| \geq 1$, with adjacent half-edges oriented inwards and decorated with boundary fields $[\mathbb{F}_l^{I_1}],\dots,[\mathbb{F}_l^{I_n}]$ all evaluated at the point of collapse $p\in \partial_lM$,
\item $| J_1 |+ \dots+| J_k|$ outward leaves if $l = 1$ and $| J_1|+ \dots+| J_k |$ inward leaves if $l = 2$, decorated with variational derivatives
in boundary fields
\begin{equation}
(-1)^d i\hbar \frac{\delta}{\delta [\mathbb{F}_l^{J_1}]},\dots,(-1)^di\hbar\frac{\delta}{\delta[\mathbb{F}_l^{J_k}]}
\end{equation}
at the point of collapse,
\item  no outward leaves if $l = 2$ and no inward leaves if $l = 1$ (graphs with them do not contribute).
\end{itemize}
\end{defn}
Similarly as before, the form $σ_Γ_l' $
can be obtained as the integral over the compactified configuration space $C̃_Γ_l'(ℍ^d)$, given by
\begin{equation}
\sigma_{\Gamma_l'}=\varint_{\tilde{\mathrm{C}}_{\Gamma_l'(\mathbb{H}^d)}} \omega_{\Gamma_l'},
\end{equation}
where $ω_Γ_l'$ is the product of limiting propagators at the point $p$ of collapse and vertex tensors.

\begin{thm}[\cite{CMR17}]
\label{bv-bfv_thm_mqme}
Let $M$ be a smooth manifold (possibly with boundary). Then the following statements are satisfied:
\begin{enumerate}
\item  The full covariant state $\boldsymbol{\hat{\psi}}_M$ satisfies the \textit{modified Quantum Master Equation} (mQME):
\begin{equation}
(\hbar^2 \Delta_{\mathcal{V}_M} + \boldsymbol{\Omega}_{\partial M}) \boldsymbol{\hat{\psi}}_M=0.
\end{equation}
\item The full BFV boundary operator $\boldsymbol{\Omega}_{\partial M}$ squares to zero:
\begin{equation}
(\boldsymbol{\Omega}_{\partial M})^2=0.
\end{equation}
\item A change of propagator or residual fields leads to a theory related by change of data as in  Definition \ref{bv-bfv:def_change_data}.
\end{enumerate}
\end{thm}

\subsection{AKSZ theories}
\label{subsec:AKSZ}
In \cite{AKSZ97}, Alexandrov, Kontsevich, Schwarz, and Zaboronsky presented a class of local field theories that are compatible with the BV construction called \textit{AKSZ theories}. The compatibility here means that the constructed local actions are solutions to the CME. These theories thus form a subclass of BV theories. We describe here the essential concepts needed for the future sextions\footnote{Maybe it is appropriate here to give a short warning about the notation. In the previous chapters, we have mostly denoted by $M$ the source, whereas now we will denote the target by $M$. Moreover, before the letter $\Sigma$ was reserved for a manifold with one dimension less than the target. From now on it will be mostly used for the source.}.
\begin{defn}[Differential graded symplectic manifold]  A \textit{differential graded
symplectic manifold} of degree $k$ is a triple
\begin{equation}
\end{equation}
with $M$ a $\mathbb{Z}$-graded manifold, $\Theta_M \in \mathcal{C}^{\infty}(M)$ a function on $M$ of degree $k+1$, $d_M$ the de Rham differential on $M$ and $\omega_M \in \Omega^2(M)$, an exact symplectic form of degree $k$ with primitive 1-form $\alpha_M \in \Omega^1(M)$, such that
\begin{equation}
(\Theta_M, \Theta_M)_{\omega_M}=0,
\end{equation}
where $(-,-)_{\omega_M}$ is the odd Poisson bracket induced by the symplectic form $\omega_M$.
\end{defn}
\begin{rmk}
We denote by $Q_M$ the Hamiltonian vector field of $\Theta_M$. Then the quadruple $(M,Q_M,\Theta_M,\omega_M=d_M\alpha_M)$ is also called a \textit{Hamiltonian $Q$-manifold}.
\end{rmk}

\subsubsection{AKSZ sigma models}
Let $Σ_d$  be a $d$-dimensional compact, oriented manifold and let $T[1]Σ_d$ be its shifted tangent bundle. We fix a Hamiltonian $Q$-manifold
\begin{equation}
(M,Q_M,\Theta_M,\omega_M=d_M \alpha_M)
\end{equation}
of degree $d-1$ for $d ≥0$. The space of fields can be defined as the mapping space space of graded manifolds:
\begin{equation}
\F_{\Sigma_d} \coloneqq \Maps(T[1]\Sigma_d,M)
\end{equation}
where $$ denotes the mapping space. Our goal is to endow $_Σ_d$ with a $Q$-manifold structure, and to do this we consider the lifts of the de Rham differential $d_Σ_d$ on $Σ_d$ and of the cohomological vector field $Q_M$ on the target $M$ to the mapping space. Therefore, we get the following cohomological vector field
\begin{equation}
Q_{\Sigma_d} \coloneqq \hat{d}_{\Sigma_d}+\hat{Q}_M,
\end{equation}
with $d̂_Σ_d$ and $Q̂_M$ the corresponding lifts to the mapping space. We remark that we can see $d_Σ_d$ as the cohomological vector field on $T[1]Σ_d$. Consider the push-pull diagram
\begin{equation}
\F_{\Sigma_d} \xleftarrow{\text{p}} \F_{\Sigma_d} \times T[1]\Sigma_d \xrightarrow {\text{ev}} M,
\end{equation}
with $p$ and $ev$ the projection and evaluation map respectively. One can construct a \textit{transgression} map
\begin{equation}
\mathcal{T}_{\Sigma_d} \coloneqq \mathrm{p}_*\mathrm{ev}^*:\Omega^\bullet(M) \rightarrow \Omega^\bullet (\F_{\Sigma_d}).
\end{equation}
Note that map $p_*$ is given by fiber integration on $T[1]Σ_d$. As a next step, we will endow the space of fields with a symplectic structure $ω_Σ_d$ defined as:
\begin{equation}
\omega_{\Sigma_d} \coloneqq (-1)^d \mathcal{T}_{\Sigma_d}(\omega_M) \in \Omega^2(\F_{\Sigma_d}).
\end{equation}
Remarkably, we get a solution $𝒮_Σ_d$ of the CME, namely the BV action functional
\begin{equation}
\mathcal{S}_{\Sigma_d} \coloneqq \underbrace{ \iota_{{\hat{d}_{\Sigma_d}}}\mathcal{T}_{\Sigma_d}(\alpha_M)}_{\coloneqq \mathcal{S}_{\Sigma_d}^{\text{kin}}} \underbrace{ +\mathcal{T}_{\Sigma_d}(\Theta_M)}_{\coloneqq \mathcal{S}_{\Sigma_d}^{\text{target}}}\in \mathcal{C}^\infty(\F_{\Sigma_d}).
\end{equation}
We can indeed check that
\begin{equation}
\end{equation}

Note that the symplectic form $ω_Σ_d$ has degree $(d-1)-d=-1$ as predicted and the action $𝒮_Σ_d$ has degree 0. Hence, this setting induces a BV manifold $(_Σ_d, 𝒮_Σ_d, ω_Σ_d)$. Let ${x^μ}$ and ${u^i}$ $(1 ≤i ≤d)$, be  local coordinates on $M$  and $Σ_d$ respectively. We will denote the odd fiber  coordinates of degree $+1$ on $T[1]Σ_d$ by $θ^i=d_Σ_du^i$. For a field $𝐗 ∈_Σ_d$ we then have the following local expression
\begin{equation}
\begin{split}
\mathbf{X}^\mu(u,\theta)=\sum_{l=0}^d\,\, \underbrace{\sum_{1 \leq i_1<\dots<i_l\leq d}\mathbf{X}^\mu_{i_1\dots i_l}(u) \theta^{i_1} \wedge \dots\wedge \theta^{i_l}}_{\mathbf{X}^\mu_{(l)}(u,\theta)} \in \bigoplus_{l=0}^d\mathcal{C}^\infty(\Sigma_d) \otimes \bigwedge\nolimits^lT^\vee \Sigma_d.
\end{split}
\end{equation}
The functions $𝐗^μ_i_1…i_l ∈𝒞^∞(Σ_d)$ have degree $(x^μ)-l$ on $_Σ_d$. The symplectic form $ω_M$ and its primitive 1-form $α_M$ on $M$ are given by
\begin{equation}
\begin{split}
\alpha_M&=\alpha_\mu(x)d_Mx^\mu \in \Omega^1(M),\\
\omega_M&=\frac{1}{2}\omega_{\mu_1 \mu_2}(x)d_Mx^{\mu_1} d_Mx^{\mu_2} \in \Omega^2(M).
\end{split}
\end{equation}
Using the above equations, we locally get the following  expressions for the BV symplectic form, its primitive 1-form and the BV action functional:
\begin{equation}
\begin{split}
\alpha_{\Sigma_d}&=\varint_{\Sigma_d}\alpha_\mu(\mathbf{X})\delta\mathbf{X}^\mu \in \Omega^1(\F_{\Sigma_d}),\\
\omega_{\Sigma_d}&=(-1)^d\frac{1}{2}\varint_{\Sigma_d}\omega_{\mu_1\mu_2}(\mathbf{X})\delta \mathbf{X}^{\mu_1}  \delta \mathbf{X}^{\mu_2} \in \Omega^2(\F_{\Sigma_d}),\\
\mathcal{S}_{\Sigma_d}&=\varint_{\Sigma_d} \alpha_\mu(\mathbf{X})d_{\Sigma_d}\mathbf{X}^\mu + \varint_{\Sigma_d} \Theta_M(\mathbf{X}) \in \mathcal{C}^\infty(\F_{\Sigma_d}).
\end{split}
\end{equation}
We have denoted by $δ$ the de Rham differential on $_Σ_d$. Using Darboux coordinates on $M$, we can write
\begin{equation}
\omega_M=\frac{1}{2}\omega_{\mu_1\mu_2}d_Mx^{\mu_1} d_Mx^{\mu_2},
\end{equation}
with $ω_μ_1 μ_2$ constant, implying that $α_M=1/2x^μ_1ω_μ_1μ_2d_Mx^μ_2$. We get the BV symplectic form
\begin{equation}
\begin{split}
\omega_{\Sigma_d}&=\frac{1}{2} \varint_{T[1]\Sigma_d}\mu_{\Sigma_d}(\omega_{\mu_1\mu_2} \delta \mathbf{X}^{\mu_1} \delta\mathbf{X}^{\mu_2}) \\
&=\frac{1}{2}\varint_{\Sigma_d}(\omega_{\mu_1 \mu_2}\delta \mathbf{X}^{\mu_1} \delta \mathbf{X}^{\mu_2})^{\text{top}}.
\end{split}
\end{equation}
The (master) action is
\begin{equation}
\mathcal{S}_{\Sigma_d}=\varint_{T[1]\Sigma_d}\mu_{\Sigma_d}\bigg(\frac{1}{2} \mathbf{X}^{\mu_1} \omega_{\mu_1 \mu_2} D\mathbf{X}^{\mu 2}\bigg)+(-1)^d \varint_{T[1]\Sigma_d}\mu_
\end{equation}
with $μ_Σ_d$ a canonical measure on $T[1]Σ_d$ and $D=θ^jδ/δu_j$, the superdifferential n $T[1]Σ_d$.

\section{The Rozansky--Witten model}
\label{sec:RW_model}
The RW model is a 3-dimensional topological sigma model. It was originally discovered with target a hyperK{\"a}hler manifold in \cite{RW96} as a result of a topological twist of 3-dimensional $N=4$ super Yang-Mills theory. However, shortly after, Kapranov \cite{KA99} and Kontsevich \cite{Ko99} showed that the model required less structure than originally thought: the target manifold does not have to be hyperk{\"a}hler, but, more generally, it can have a holomorphic symplectic structure. Since we will focus on this latter case, here we will present how this generalization proposed by Kapranov and Kontsevich was understood in the context of topological sigma models by Rozansky and Witten. After the work of Kapranov and Kontsevich, Rozansky and Witten added an appendix to \cite{RW96}, where they explained how to extend their formulation of the model to the case of a holomorphic symplectic target manifold.

\begin{notat}
Except for the name of the manifolds, which we adapt to the notation we will use in Section \ref{sec:Classical_Theory}, the notation will be the same as in \cite{RW96}.
\end{notat}

\subsection{First definitions}
Let $Σ_3$ be the source 3-dimensional manifold and $(M, ω)$ the target holomorphic symplectic manifold. The fields are the following:
\begin{itemize}
\item \textit{bosonic fields} described by the smooth maps $\phi:\Sigma_3\rightarrow M$, in local coordinates we have $\phi^I(x^\mu)$ and $\Bar{\phi}^{\Bar{I}}(x^\mu)$,
\item \textit{fermionic (or Grassman) fields} $\eta\in\Gamma(\Sigma_3,\phi^*T^{0,1}M)$ and $\chi\in\Gamma(\Sigma_3,\Omega^1(M)\otimes\phi^*T^{1,0}M)$, with $T^{1,0}(M)$ and $T^{0,1}M$, the holomorphic and anti-holomorphic tangent bundle, respectively. In local coordinates we can write them as $\eta^{\Bar{I}}_\mu(x^\mu)$ and $\chi^I_\mu(x^\mu)$.
\end{itemize}

Consider a single fermionic symmetry on these fields, which we will denote by $Q$. Its action is:
\begin{alignat}{2}
&\delta\phi^I=0,\qquad  &&\delta\Bar{\phi}^{\Bar{I}}=\eta^{\Bar{I}},\\
&\delta\eta^{\bar{I}}=0,\qquad  &&\delta\chi_\mu^I=-\partial_\mu\phi^I.
\end{alignat}

To introduce the Lagrangian density of the theory, we add some extra structure to the target manifold. Let $Γ^I_JK$ be a symmetric connection in the holomorphic tangent bundle of $M$, i.e. $Γ^I_JK=Γ^I_KJ$. The $(1,1)$-part of the curvature, related to $Γ_JK^I$ represents the \textit{Atiyah class}\footnote{The Atiyah class is defined to be the obstruction to the existence of a global holomorphic connection.} of $M$ \cite{At57}:
\begin{equation}
\Riem{I}{J}{K}{\Bar{L}}=\frac{\partial \Chr{I}{J}{K}}{\partial \Bar{\phi}^{\Bar{L}}}.
\end{equation}

In \cite{RW96}, it is noted that the connection does not have to be compatible with the holomorphic symplectic form\footnote{This compatibility condition will be assumed for the RW model when we compare it with the model we will develop in Section \ref{sec:Classical_Theory} (see Section \ref{sec:comp_orig_RW}).} $ω_IJ$. We require $ω_IJ$ to be non-degenerate and closed, i.e.
\begin{equation}
\frac{\partial \omega_{IJ}}{\partial \Bar{\phi}^{\Bar{K}}}=0,\qquad \frac{\partial \omega_{IJ}}{\partial \phi^K}+\frac{\partial \omega_{KI}}{\partial \phi^J}+\frac{\partial \omega_{JK}}{\partial \phi^I}=0.
\end{equation}

Rozansky and Witten define a $Q$-invariant Lagrangian density $ℒ$ to be $ℒℒ_2+ℒ_1$. The Lagrangian $ℒ_2$ is given by
\begin{equation}
\mathscr{L}_2=\frac12\frac{1}{\sqrt{h}}\epsilon^{\mu\nu\rho}\bigg(\omega_{IJ}\chi_\mu^I\nabla_\nu\chi_\rho^J-\frac{1}{3}\omega_{IJ}R^J_{KL\Bar{M}}\chi_\mu^I\chi_\nu^K\chi_\rho^L\eta^{\Bar{M}}+\frac{1}{3}(\nabla_L\Omega_{IK})(\partial_\mu\phi^I)\chi_\nu^K\chi_\rho^L \bigg),
\end{equation}
where $∇_μ$ is a covariant derivative with respect to the pullback of the connection $IJK$:
\begin{equation}
\nabla_\mu\chi^I_\nu=\partial_\mu\chi^I_\nu+(\partial_\mu\phi^J)\Chr{I}{J}{K}\chi^K_\nu.
\end{equation}

In order to construct the $Q$-exact Lagrangian $ℒ_1$, we need to choose an Hermitian metric $g_ij$ on $M$. We define by
\begin{equation}
\Tilde{\Gamma}^{\Bar{I}}_{\Bar{J}\Bar{K}}\coloneqq\frac12g^{\Bar{I}L}\bigg(\frac{\partial g_{L\Bar{J}}}{\partial \Bar{\phi}^{\Bar{K}}}+\frac{\partial g_{L\Bar{K}}}{\partial \Bar{\phi}^{\Bar{J}}}\bigg),\quad \Tilde{T}^{\Bar{I}}_{\Bar{J}\Bar{K}}\coloneqq\frac12g^{\Bar{I}L}\bigg(\frac{\partial g_{L\Bar{J}}}{\partial \Bar{\phi}^{\Bar{K}}}-\frac{\partial g_{L\Bar{K}}}{\partial \Bar{\phi}^{\Bar{J}}}\bigg),
\end{equation}
the \textit{symmetric connection} and the \textit{torsion} associated with $g_IJ$. Then $ℒ_1$ is defined by
\begin{equation}
\mathscr{L}_1\coloneqq\Bar{Q}\bigg(g_{I\Bar{J}}\chi_\mu^I(\partial_\mu\Bar{\phi}^{\Bar{J}})\bigg)=g_{I\Bar{J}}\partial_\mu\phi^I\partial_\mu\Bar{\phi}^{\Bar{J}}+g_{I\Bar{J}}\chi^I_\mu\Tilde{\nabla}_\mu\eta^{\Bar{J}},
\end{equation}
where $∇_μ$ is a covariant derivative with respect to the connection  $Γ^I_JK+T^I_JK$, i.e. we have
\begin{equation}
\Tilde{\nabla}_\mu\eta^{\Bar{I}}=\partial_\mu \eta^{\Bar{I}}+(\partial_\mu\Bar{\phi}^{\Bar{J}})(\Tilde{\Gamma}^{\Bar{I}}_{\Bar{J}\Bar{K}}+\Tilde{T}^{\Bar{I}}_{\Bar{J}\Bar{K}})\eta^{\Bar{K}}.
\end{equation}
Moreover, if $g_IJ$ is a K{\"a}hler metric, then $T^I_JK=0$.

\subsection{Perturbative quantization}
\label{sec:pert_exp}
The partition function of the RW model is
\begin{equation}
Z_{M}(\Sigma_3)\coloneqq\varint e^{\frac{i}{\hbar}S} \mathscr{D}[\phi^i]\mathscr{D}[\eta^I]\mathscr{D}[\chi^I_\mu],
\end{equation}
where $S_Σ_3 ℒ$ and $𝒟$ is a formal measure.

As mentioned in \cite{RW96} (see also \cite{Th99,HT99}), in order to do a perturbative expansion around critical points of the action (which are constant maps from $Σ_3$ to $M$), we need to deal with the \textit{zero modes}:
\begin{itemize}
\item \textit{bosonic zero modes}: they are constant modes of $\phi$;
\item \textit{fermionic zero modes}: here we should distinguish two cases
\begin{itemize}
\item if $M$ is a rational homology sphere (i.e. the first Betti number $b_1=0$), the fermionic zero modes are the constant modes of $\eta$. There are $2n$ zero modes if $\dim M=4n$.
\item if $M$ is not a rational homology sphere but the first Betti number $b_1>0$, then there are $2nb_1$ zero modes of $\chi_\mu$ additionally.
\end{itemize}
\end{itemize}

Taking into account the zero modes, one can decompose $ϕ^i=ϕ^i_0+ϕ^i_⊥$, where $ϕ^i_0$ are the constant maps and $ϕ^i_⊥$ are required to be orthogonal to $ϕ^i_0$. Similarly, the $η^I$ are also decomposed as $η^I=η^I_0+η^I_⊥$, where $η^I_0$ are harmonic 0-forms with coefficients in the fiber $V_ϕ_0$ of the $(n)$-bundle $V→X$ and $η^I_⊥$ are orthogonal to the harmonic part.

For our purposes, we will only consider the Lagrangian $ℒ_2$, in light of these decompositions we can rewrite it as
\begin{equation}
\label{RW:L_2}
\mathscr{L}_2=\frac12\bigg(\omega_{IJ}(\phi_0)\chi^Id\chi^J+\frac13\Riem{}{IJ}{K}{\Bar{L}}(\phi_0)\chi^I\chi^J\chi^K\eta^{\Bar{L}}_0+\frac{1}{3}(\nabla\Omega_{IK})(\partial\phi^I_\perp)\chi^K\chi^L\bigg).
\end{equation}

As a result of an analysis on the absorption of fermionic zero modes by the Feynman diagrams, Rozansky and Witten concluded that only diagrams with trivalent vertices contribute. Moreover, these trivalent vertices have to be exactly $2n$ to saturate the $2n$ zero modes of $η$. They call these diagrams ``minimal". The Lagrangian $L_2$ contains the following vertex with the needed properties
\begin{equation}
\end{equation}
Here we should think of $η_0$ as a ``coupling constant", in fact, we should focus on the order of the $η^I_0$ during the perturbative expansion.

Since all the fields $η$ are used to absorb zero modes, we only need the propagators for the fields $ϕ^i$ and $χ^I_mu$. According to \cite{RW96}, these are
\begin{equation}
\label{propagators_RW}
\begin{split}
&\braket{\phi^i(x_1),\phi^j(x_2)}=-\hbar g^{ij}G^{\phi}(x_1,x_2),
\end{split}
\end{equation}
with $G^χ_μν(x_1,x_2)$ and $G^ϕ(x_1,x_2)$ Green's functions. We refer to \cite{RW96} for a detailed description of the Green's functions.

The Feynman diagrams participating in the calculation of the partition function depend only on the dimension of the target manifold $M$ and on the first Betti number $b_1$ of the source 3-manifold $Σ_3$. The former causes the number of vertices of the graphs to be equal to $2n$. The latter has consequences on the valence of the vertices. We have the following cases:
\begin{itemize}
\item($b_1=0$) There are no $\chi$ zero modes to absorb. Hence, all the Feynman diagrams are closed graphs with $2n$ trivalent vertices. This is the case when $\Sigma_3$ is a rational homotopy sphere.
\item($b_1=1$) There are $2n$ $\chi$ zero modes coming from a harmonic 1-form. As a consequence, each vertex absorbs exactly one zero mode of $\chi$, and thus all the Feynman diagrams are closed graphs with $2n$ bivalent vertices.
\item($b_1=2$) There are $4n$ $\chi$ zero modes coming from a two harmonic 1-forms on $\Sigma_3$. As a consequence, each vertex absorbs exactly two zero modes of $\chi$, one for each harmonic 1-form, and thus all the Feynman diagrams are closed graphs with $2n$ univalent vertices.
\item($b_1=3$) There are $6n$ $\chi$ zero modes coming from three harmonic 1-forms on $\Sigma_3$. As a consequence, each vertex absorbs exactly three zero modes of $\chi$, one for each harmonic 1-form, and thus all the Feynman diagrams are a collection of $2n$ totally disconnected vertices with no edges.
\item($b_1\geq 4$) The $\chi$ zero modes become too many and they can not be integrated out by the $\chi$ present in the vertices  (at most three), so the RW partition function vanishes.
\end{itemize}

Let us denote by $Γ_n,m$ the set of all closed graphs with $2n$ $m$-valent vertices and as $Z_M, Γ(Σ_3; ϕ^i_0)$ the sum of all the contributions of the minimal Feynman diagrams corresponding to a given graph $Γ$. The total contribution of Feynman diagrams is
\begin{equation}
Z_{M}(\Sigma_3; \phi^i_0)=\sum_{\Gamma\in\Gamma_{n,3-b_1(\Sigma_3)}}Z_{M, \Gamma}(\Sigma_3; \phi^i_0),
\end{equation}
where each $Z_M, Γ(Σ_3; ϕ^i_0)$ can actually be written as a sum of two factors
\begin{equation}
Z_{M, \Gamma}(\Sigma_3; \phi^i_0)=\text{W}_\Gamma(M;\phi_0^i)\sum_a I_{\Gamma,a}(\Sigma_3).
\end{equation}
We will explain each factor one-by-one. First, $I_Γ,a(Σ_3)$ includes the integral over $Σ_3$ of the propagators $G^χ_μν(x_1,x_2)$ and $G^ϕ(x_1,x_2)$ as well as $χ$ zero modes. The sum is over all the possible ways to contract the fields of $Γ$ with the propagators (\ref{propagators_RW}). On the other hand, the factor $W_Γ(M;ϕ^i_0)$ is a product of tensors $IJKL$ coming from the vertices $V_1$ and $V_2$, which are contracted by the $ω^IJ$ contained in the propagators. After antisymmetrizing over the anti-holomorphic indices (coming from the zero modes' contributions), we obtain a $∂$-closed $(0,2n)$-form on $M$. In other words, we have a map
\begin{equation}
\Gamma_{n,3}\rightarrow H^{0,2n}(M),
\end{equation}
where $H^0,2n(X)$ is the Dolbeault cohomology. This corresponds to a weight system, the \textit{Rozansky--Witten weight system}. By definition a function on $Γ_n,3$ is called a weight if it satisfies the AS and IHX relations (see also \cite{B95}).

The AS relation means that $Γ_n,3$ is antisymmetric under the permutation of legs at a vertex. For RW, this is not valid on the nose since the curvature tensor is completely symmetric. However, we can prove the vanishing of tadpole diagrams (i.e. diagrams with a loop centered at a vertex), which is consistent with the AS relation. The proof follows simply because the loop is constructed by contracting two indices of the symmetric tensor $IJKLη^L_0$ with $ω^IJ$, which is antisymmetric.

On the other hand, the IHX relation means that the sum over all possible (three) ways of collapsing a propagator such that we obtain a graph with a four-valent vertex, while the other vertices are trivalent, vanishes (see Fig. \ref{fig:IHX}).

\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.33\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}[every blob={/tikz/fill=white!30,/tikz/inner sep=0.5pt}]
\diagram*{
a -- m [dot] -- b,
c -- e [dot] -- d,
m -- e
\end{feynman}
\draw [dashed] (1.75,-0.5) arc[x radius=0.75, y radius=0.75, start angle=0, end angle=360];
\node[] at (3, -0.45) {$+$};
\end{tikzpicture}
\end{subfigure}\hspace{-1.5cm}
\begin{subfigure}[b]{0.33\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}
\vertex (c) at (-0.6,0.25);
\node[dot] (m) at (-0.6, -0.5);
\node[dot] (e) at (0.6, -0.5);
\vertex (b) at (-0.6, -1.25);
\vertex (a) at (0.6, -1.25);
\vertex (f) at (0.6, 0.25);
\diagram*{
(c) -- (m) -- (b),
(f) -- (e) -- (a),
(m) -- (e)
\end{feynman}
\draw [dashed] (0.85,-0.5) arc[x radius=0.85, y radius=0.85, start angle=0, end angle=360];
\node at (1.75,-0.5) {$+$};
\end{tikzpicture}
\end{subfigure}\hspace{-1.75cm}
\begin{subfigure}[b]{0.33\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}
\vertex (c) at (-0.75,-1);
\node[dot] (m) at (-0.375, -0.75);
\node[dot] (e) at (0.375, -0.75);
\vertex (b) at (0.75, -1);
\vertex (a) at (-0.75, 0);
\vertex (f) at (0.75, 0);
\diagram*{
(c) -- (m) -- (f),
(b) -- (e) -- (a),
(m) -- (e)
\end{feynman}
\draw [dashed] (0.75,-0.5) arc[x radius=0.75, y radius=0.75, start angle=0, end angle=360];
\node at (1.75,-0.5) {$=0$};
\end{tikzpicture}
\end{subfigure}
\caption{The IHX relation. It holds whenever the graphs are identical outside the dashed circle.} \label{fig:IHX}
\end{figure}
Explicitly, the sum of the three contributions is equal to the expression
\begin{equation}
\eta^{\Bar{L}}_0\eta^{\Bar{L'}}_0\omega^{IJ}\bigg(\Riem{}{IJ}{K}{\Bar{L}}\Riem{}{I'J'}{K'}{\Bar{L'}}+\Riem{}{IJ}{K'}{\Bar{L}}\Riem{}{I'J'}{K}{\Bar{L'}}+\Riem{}{IJ}{J'}{\Bar{L}}\Riem{}{I'K}{K'}{\Bar{L'}}\bigg)- \Bar{L}\leftrightarrow \Bar{L'},
\end{equation}
with the notation $L↔L'$ we mean that we are subtracting the same quantity with the indices $L$ and $L'$ switched, in this way the expression vanishes. In other words, the IHX relation follows as a result of the Bianchi identity for the curvature tensor $R$. The validity of the IHX relation ensures we are obtaining topological invariants of 3-manifolds in the perturbative expansion of the partition function \cite{Sa04}.

At this point, we can take the product of the $(0,2n)$-form which is the image of the graph $Γ$ with the $(2n,0)$-form $ω^n∈H^2n,0(M)$ and integrate the resulting $(2n,2n)$-form over $M$. In this way we obtain the weights $b_Γ(M)$, which are numbers called \textit{Rozansky--Witten invariants} studied by Sawon in \cite{Sa04}. More explicitly, we have
\begin{equation}
b_\Gamma(M)= \frac{1}{(2\pi)^{2n}} \varint_M \text{W}_\Gamma(M;\phi^i_0)\omega^n.
\end{equation}

Finally, the RW partition function is shown to be \cite{RW96}
\begin{equation}
Z_{M}(\Sigma_3)=\Big|H_1(\Sigma_3, \mathbb{Z})\Big|'\sum_{\Gamma\in\Gamma_{n,3-b_1(\Sigma_3)}}b_\Gamma(M)\sum_aI_{\Gamma,a},
\end{equation}
where $|H_1(Σ_3, ℤ)|'$ is the number of torsion elements in $H_1(Σ_3, ℤ)$ (see \cite{FG91}).

\subsection{Comparison with Chern--Simons theory}
In this section, we are going to explore briefly the similarities between CS theory and the RW theory as exhibited in \cite{RW96}. The main message is that RW is a kind of ``Grassmann odd version" of CS theory. Let us make this more precise.

Recall the CS Lagrangian
\begin{equation}
\mathscr{L}_{\text{CS}}=\Tr\bigg(AdA+\frac{2}{3}A^3\bigg).
\end{equation}
Let us compare it with the RW Lagrangian in \eqref{RW:L_2}. As we can see from Table \ref{Tab:RW_CS} (where we denote by $T_a$ the generators of the Lie algebra and $f_abc$ the structure constants), there is almost a direct match. We are using the word ``almost" because the symmetry properties of the various objects in the table are reversed: $T_aT_b$ is symmetric in its arguments while the holomorphic symplectic form $ω_IJ$ is antisymmetric, $f_abc$ is totally antisymmetric whereas $IJKLη^L_0$ is totally symmetric. However, this should not come as a surprise since by definition $A^a$ is an anti-commuting object, while $χ^I$ is commuting.
\begin{table}[h]
\centering
\begin{tabular}{ |cc| }
\toprule
CS & RW \\
\midrule
$A^a$ & $\chi^I$ \\
$\Tr T_aT_b$ & $\omega_{IJ}$ \\
$f_{abc}$ & $\Riem{}{IJ}{K}{L}\eta^L_0$ \\
\bottomrule
\end{tabular}
\caption{Comparison between CS theory and RW theory.}
\label{Tab:RW_CS}
\end{table}

By doing the associations in the table, the vertex in CS is the same as the vertex in RW. It follows that the diagrams of the two theories coincide. Consequently, the partition function differs only for the weight factors since for RW they are proportional to the curvature tensor of $M$ rather than the structure constants of a Lie group.

Other similarities come at the level of gauge-fixing. We refer the interested reader to \cite{RW96,HT99} for a detailed discussion.

% Moreover the space of fields of the two theories are very similar. In both cases the structure is
% \begin{equation}
%     \Tilde{\mathcal{F}}\surg=\Omega^{\bullet}(\Sigma_3)\otimes \mathfrak{g}_{M}[1].
% \end{equation}
% The difference lies in the term $\mathfrak{g}_{M}[1]$, which is a Lie Algebra in CS and a curved $\Linf$-algebra for RW. \textcolor{red}{REMOVE?}

\begin{rmk}
\label{rmk_diff_rw_cs}
There is an important difference between CS and RW theories. In the RW model, the vertex carries an odd Grassmann odd harmonic mode $\eta^I_0$, hence it can never appear more than $2n$ times in any diagram. This corresponds to a natural \textit{cut-off} of the perturbative expansion of the RW model.
\end{rmk}

\section{Classical formal globalization}
\label{sec:Classical_Theory}
% After having established the tools needed in the previous sections, we can finally define a model which, when globalized, reduces to the original RW model.

% The aforementioned model is a very simple AKSZ theory for a 3-dimensional source manifold (with or without boundary) and a holomorphic symplectic target manifold.  Exploiting tools of formal geometry (see Appendix \ref{app:formal_geometry} for a brief introduction on the subject), we globalize the model. In particular, the construction of the globalized action is done along the same lines as in \cite{CMoW19} and in \cite{QZ10,KQZ13} (nonetheless in this last case the authors are not using formal geometry as we will see)\footnote{At this point, our model is not a split $BF$-like theory, hence our globalization construction does not mirror perfectly the procedure in \cite{CMoW19}. Thus, in Section \ref{sec:BF-like_formulation}, we split the model and consider it as a $BF$-like theory, such that the two constructions become more similar even if some differences remain.}. We show that the globalized model reduces to the original RW model for a holomorphic symplectic manifold as defined in the appendix of \cite{RW96}. Moreover, we compare our results with other globalization constructions detailed in \cite{QZ10,KQZ13} and in \cite{CLL17,Ste17}, respectively. As regards the first, we explain how our model satisfies the claim in \cite{KQZ13} that an extension of their work could be obtained by formal geometry\footnote{In particular, they also argue that an extension of their construction could be obtained by using a Fedosov connection. This is actually provided in \cite{CLL17}.}. On the other hand, \cite{CLL17} is based on derived geometry (and, in particular, on Costello's ``version" of derived geometry \cite{Co11a,Co11b}, see Appendix \ref{app:derived_geometry} for a short introduction to both), which allows to globalize a theory (in this case RW theory) in a different, but equivalent way with respect to formal geometry as remarked in \cite{CMoW19}. However, in \cite{CLL17} they focus on the case of a closed source manifold whereas our construction considers the case of a manifold with boundary. The derived geometry extension to the "boundary case" has been studied e.g. in \cite{Ste17}. Hence, while we show that the methods in \cite{CLL17} and ours produce the same results, we refer to \cite{Ste17} to highlight that the equivalence extends also in the case with boundary.

% As noticed originally in \cite{RW96}, the RW model has clear similarities with CS theory \cite{AS91,AS94}. Hence, to obtain a $BF$-like formulation of our form of RW theory, we exploit the strategy used in \cite{CMnW17} to split CS theory (for a more detailed exposition see \cite{We18}) following similar ideas as in \cite{Ste17}. This rearrangement is needed in order to quantize the model according to the quantum BV-BFV model formalism.

%\subsection{Construction of the model}
The idea is to construct a 3-dimensional topological sigma model, which, when globalized, reduces to the original RW model. In particular, we are interested in the formulation of the RW model with target a holomorphic symplectic manifold\footnote{This construction reduces to the RW model considered in the bulk of \cite{RW96}, when we consider as target a hyperK{\"a}hler manifold.} (i.e. a complex symplectic manifold with a holomorphic symplectic form, see also the appendix in \cite{RW96}). Hence, let $M$ be a holomorphic symplectic manifold endowed with coordinates $X^i$ and $X^i$, and with a holomorphic symplectic form $Ω=Ω_ijδX^iδX^j$, i.e. a closed, non degenerate $(2,0)$ form. Moreover, consider a 3-dimensional manifold $Σ_3$ and construct an AKSZ sigma model\footnote{See Section \ref{subsec:AKSZ} for an introduction.} with source $T[1]Σ_3$ and target $M$. In this case, the space of maps is
\begin{equation}
\mathcal{F}\sur\coloneqq \Maps\bigg(T[1]\Sigma_3, M\bigg).
\end{equation}

On the source manifold, we choose bosonic coordinates ${u}$ (ghost degree 0) on $Σ_3$ and fermionic odd coordinates ${θ}$ (ghost degree 1) on the fibers of $T[1]Σ_3$. Moreover, by picking up local coordinates $X^i$ on $M$, maps in $ℱ$ can be described by a superfield $𝐗$, whose components are chosen as:
\begin{equation}
\mathbf{X}^i=X^i(u)+\theta^\mu X^i_\mu(u)+\frac{1}{2!}\theta^\mu\theta^\nu X^i_{\mu\nu}(u)+\frac{1}{3!}\theta^\mu\theta^\nu\theta^\chi X^i_{\mu\nu\chi}(u),
\end{equation}
where $X^i$ is a 0-form, $X^i_μ$ is a 1-form etc. %We will use the notation $X^i$ for $X^i$, $X^i_{(1)}$ for $X^i_\mu d\xi^\mu$, etc.
To these maps maps $X^i, X^i_μ, …$ we assign ghost degrees such that the ghost degree of $𝐗$ is equal to the one of $X^i$ (that is 0), for example $X^i_μν$ has form degree 2 and ghost degree $-2$.

% We can sum up the conventions on the degrees in Table \ref{class:Tab_degrees}.

% \begin{table}[h!]
%     \centering
%     \begin{tabular}{|lcc|}
%     \toprule
%          & Form degree &  Ghost degree \\
%          \midrule
%     $X^i_0$ & 0     &  0 \\
%     $X^i_{(1)}$ & 1     &  $-1$ \\
%     $X^i_{(2)}$ & 2     &  $-2$\\
%     $X^i_{(3)}$ & 3     &  $-3$\\
%     \bottomrule
%     \end{tabular}
%     \caption{Form degree and ghost degree for each component of the superfields $\mathbf{X}$.}
%     \label{class:Tab_degrees}
% \end{table}

Now, we can define a symplectic form for the space of fields. Since it should have ghost degree $-1$, we assign ghost degree\footnote{More precisely, we add a formal parameter $q$ of ghost degree 2 in front of $\Omega_{ij}$. The parameter is immediately suppressed from the notation.} 2 to $Ω_ij$ (in this way the target manifold has degree 2 and the AKSZ construction can be done without problems) and define
\begin{equation}
\omega\sur=\varint_{T[1]\Sigma_3}\mu\sur\bigg(\frac12\Omega_{ij}\delta \mathbf{X}^i\delta \mathbf{X}^j\bigg),
\end{equation}
where by $δ$ we denote the de Rham differential on the space of fields. Since we have a canonical Berezinian $μ_Σ_3$ on $T[1]Σ_3$ of degree $-3$, the symplectic form has degree $-1$ as we desired. Hence, the space of fields is equipped with an odd Poisson bracket $(-,-)$.%, which we call the \textit{BV bracket} (or \textit{anti bracket}).
We have an associated AKSZ action given by
\begin{equation}
\Sc\sur=\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg(\frac{1}{2}\Omega_{ij}\mathbf{X}^iD\mathbf{X}^j\bigg),
\end{equation}
where $D=θ^μ∂/∂u^μ$ is the differential on $T[1]Σ_3$. When $Σ_3$ is a closed manifold, the action $$ satisfies the CME
\begin{equation}
\end{equation}
Equivalently, we can introduce a cohomological Hamiltonian vector field $Q$ on $ℱ$ defined by
\begin{equation}
\iota_{Q\sur}\omega\sur=\delta S\sur.
\end{equation}
This vector field has the following form
\begin{equation}
Q\sur=\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg( D\mathbf{X}^i\frac{\delta}{\delta \mathbf{X}^i}\bigg).
\end{equation}

The above can be restated by saying that $(ℱ, ω, )$ is a BV manifold.

In the presence of boundaries, the model can be extended to a BV-BFV theory by associating the BV-BFV manifold
\begin{equation}
(\mathcal{F}\bon, \omega\bon=\delta \alpha\bon, \Sc\bon, Q\bon)
\end{equation}
over the BV manifold $(ℱ_Σ_3, ω_Σ_3, Q_Σ_3)$, with the following set of data
\begin{equation}
\begin{split}
&\mathcal{F}\bon=\Maps (T[1]\partial \Sigma_3, X), \\
&\Sc\bon=\varint_{T[1]\partial\Sigma_3}\mu_{\partial \Sigma_3}\bigg(\frac{1}{2}\Omega_{ij}\mathbf{X}^iD\mathbf{X}^j\bigg),\\
&\alpha\bon=\varint_{T[1]\partial\Sigma_3}\mu_{\partial \Sigma_3}\bigg(\frac12\Omega_{ij}\mathbf{X}^i\delta \mathbf{X}^j\bigg),\\
&Q\bon=\varint_{T[1]\partial\Sigma_3}\mu_{\partial \Sigma_3}\bigg(D\mathbf{X}^i\frac{\delta}{\delta \mathbf{X}^i}\bigg),
\end{split}
\end{equation}
with $μ_∂Σ_3$ the Berezinian on the boundary $∂Σ_3$ of degree $-2$. The data is such that
\begin{equation}
\iota_{Q\sur}\omega\sur=\delta \Sc\sur+\pi^*\alpha\bon.
\end{equation}

\begin{rmk}
%The model described is a simple example of AKSZ construction where the target manifold has a 0 Hamiltonian. However, in principle, one could add an Hamiltonian term.
A possible modification of the model consists in coupling the target manifold with $\mathfrak{g}^{\vee}[1]\otimes \mathfrak{g}[1]$ or $\mathfrak{g}[1]$, forming thus the ``$BF$-RW" model and the ``CS-RW" model \cite{KQZ13}, respectively. In this way, after globalization, one should get an extension of the results obtained by K{\"a}llén, Qiu and Zabzine \cite{KQZ13}.
\end{rmk}

%Moreover $\iota_{Q\bon}\omega\bon=\delta \bon$ and $\iota_{Q\bon}\iota_{Q\bon}\omega\bon=2\pi^*_{\Sigma}S\bon$. \fix

\subsection{Globalization}
\label{sec:globalization}
In the last section, we introduced a very simple AKSZ sigma model. Here we globalize that construction using methods of formal geometry \cite{GK71,Bo11} (see Appendix \ref{app:formal_geometry} for an introduction) following \cite{CMoW19}. First, we expand around critical points of the kinetic part of the action. The Euler--Lagrange equations for our model are simply $d𝐗^i=0$, which means that the component of $𝐗^i$ of ghost degree 0 is a constant map: we denote it by $x^i$ and we think of it as a \textit{background field} \cite{Moshayedi2021}. Moreover, since we want to vary $x$ itself, we lift the fields as the pullback of a formal exponential map at $x$. We also note that the fields $𝐗^i$ are just \emph{spectators}, which means that they do not contribute to the action, hence we can think of taking constant maps also in the antiholomorphic direction.

% The next remark describes the role of $x$:
% \begin{rmk}
% In general, recall that classical solution of AKSZ sigma models are given by differential graded maps:
% \begin{equation}
%     (T[1]\Sigma_3, d\sur)\rightarrow (M, Q\sur),
% \end{equation}
% where in our case $Q\sur=0$. As explained in \cite{Mo20}, if we consider the moduli space of classical solutions $\mathcal{M}_{\text{cl}}$ given by constant maps $x: T[1]\Sigma_3\rightarrow M$, we obtain an isomorphism $\mathcal{M}_{\text{cl}}\cong M$. These constant solutions can be thought of as \textit{background fields}.
% \end{rmk}

The above allows to linearize the space of fields $ℱ$ by working in the formal neighbourhoods of the constant map $x∈M$. We define the following \textit{holomorphic formal exponential map}
\begin{align}
\begin{split}
\varphi:\ T^{1,0}M &\rightarrow M\\
(x,y)&\mapsto \varphi^{i}(x,y)=x^i+y^i+\frac{1}{2}\varphi^i_{jk}(x^i,x^{\Bar{i}})y^jy^k+\dots
\end{split}
\end{align}
\begin{rmk}
We think about the holomorphic formal exponential map here defined as an extension to the complex case of the formal exponential map used in e.g. \cite{CF01}. This notion should correspond to the ``canonical coordinates'' introduced in \cite{BCOV94} and the holomorphic exponential map applied by Kapranov to the RW case in \cite{KA99}.
\end{rmk}

The formal exponential map lifts $ℱ$ to
\begin{align}
\begin{split}
\Tilde{\varphi}_{x}:\ \Tilde{\mathcal{F}}_{\Sigma_3, x}:=\Maps (T[1]\Sigma_3, T^{1,0}_{x}M)&\rightarrow \Maps (T[1]\Sigma_3, M)\\
\hat{\mathbf{X}}&\mapsto \mathbf{X}
\end{split}
\end{align}
which is given by precomposition with $φ^-1_x$, i.e. $ℱ_Σ_3, x=φ^-1_x∘ℱ$ and $𝐗=φ_x(𝐗̂)$. Now, since the target is linear, we can write the space of fields as
\begin{equation}
\Tilde{\mathcal{F}}_{\Sigma_3, x}=\Omega^{\bullet}(\Sigma_3)\otimes T^{1,0}_{x}M.
\end{equation}
% where we used the isomorphism
% \begin{equation}
%     \mathcal{C}^{\infty}(T[1]\Sigma_3)\cong \Omega^\bullet(\Sigma_3).
% \end{equation}
Consequently, we lift the BV action, the BV 2-form and the primitive 1-form obtaining:
\begin{equation}
\begin{split}
&\Sc\surg:=\mathrm{T}\Tilde{\varphi}_{x}^*\Sc\sur=\varint_{T[1]\Sigma_3}\mu\sur\, \bigg(\frac12\Omega_{ij}\hat{\mathbf{X}}^iD\hat{\mathbf{X}}^j\bigg),\\
&\omega\surg\coloneqq\Tilde{\varphi}^*_{x}\omega\sur=\varint_{T[1]\Sigma_3}\mu\sur\,\bigg(\frac12\Omega_{ij}\delta \hat{\mathbf{X}}^i\delta \hat{\mathbf{X}}^j\bigg),\\
&\alpha^{\partial}_{\partial \Sigma_3, x}\coloneqq \Tilde{\varphi}^*_{x}\alpha\bon=\varint_{T[1]\partial\Sigma_3}\mu_{\partial\Sigma_3}\,\bigg(\frac12\Omega_{ij}\hat{\mathbf{X}}^i\delta \hat{\mathbf{X}}^j\bigg),
\end{split}
\end{equation}
%where $\hat{\mathbf{X}}$ is the lift of $\mathbf{X}$  as the pullback of the
%formal "holomorphic" exponential map at a constant field ($\varphi_{x}$).
where $T$ denotes the Taylor expansion around the fiber coordinates ${y}$ at zero.
This set of data satisfies the mCME for any $x∈M$:
\begin{equation}
\iota_{Q\surg}\omega\surg=\delta \Sc\surg+\pi^*\alpha^{\partial}_{\partial \Sigma_3, x},
\end{equation}
with $Q=_T[1]Σ_3μ_Σ_3  (D𝐗̂^iδ/δ𝐗̂^i)$. Hence, we have a BV-BFV manifold associated to the space of fields $ℱ$.

The next remark introduces an important ingredient to write down the globalized action.
\begin{rmk}
\label{rmk_R}
The constant map $x:T[1]\Sigma_3\rightarrow M$ in $\mathcal{F}\sur$ can be thought of as an element in $M$. Hence, we have a natural inclusion $M\hookrightarrow \mathcal{F}\sur$.  We exploit this fact by defining, for a constant field $x$ and $\mathbf{X}\in \mathcal{F}\sur$, a 1-form:
\begin{equation}
R\sur=\Big(R\sur\Big)_j(x;\mathbf{X})dx^j+\Big(R\sur\Big)_{\Bar{j}}(x;\mathbf{X})dx^{\Bar{j}}\in \Omega^1\Big(M, \Der\Big(\reallywidehat{\Sym}^\bullet(T^{\vee1,0}M)\Big)\Big).
\end{equation}
% on $M$ with values in differential operators on $\mathcal{F}\sur$, i.e. $R\sur\in \Omega^1\Big(M, \Der\Big(\reallywidehat{\Sym}^\bullet(T^{\vee1,0}M)\Big)\Big)$.
As before, we lift this 1-form to $\Tilde{\mathcal{F}}\surg$. This lift, denoted by $\hat{R}\sur$, is locally written as:
\begin{equation}
\hat{R}\sur=\Big(\hat{R}\sur\Big)_j(x;\hat{\mathbf{X}})dx^j+\Big(\hat{R}\sur\Big)_{\Bar{j}}(x;\mathbf{X})dx^{\Bar{j}}.
\end{equation}
\end{rmk}

\subsection{Variation of the classical background}
\label{class:sec:var_class_back}
So far, the classical background $x$ has been fixed. However, our aim is to vary $x$ and construct a global formulation of the action. Hence, we understand the collection ${}_x∈M$ as a map $$ to be given by $: x↦S$ and we compute how it changes over $M$. In order to accomplish this task, inspired by \cite{CMoW19,CMoW20, KQZ13, BCM12}, choosing a background field $x∈M$, we define
\begin{equation}
\label{class:glob_terms}
\Sc\surgR\coloneqq\varint_{\Sigma_3}\bigg(\Big(\hat{R}^i\sur\Big)_j(x; \hat{\mathbf{X}})\Omega_{il}\hat{\mathbf{X}}^l dx^j+\Big(\hat{R}^i\sur\Big)_{\Bar{j}}(x; \hat{\mathbf{X}})\Omega_{il}\hat{\mathbf{X}}^l dx^{\Bar{j}}\bigg)=\mathcal{S}_R+\Sc_{\bar{R}}.
\end{equation}
The integrand is a well defined term of degree 3, since we assigned degree 2 to the symplectic form and $R̂$ is a 1-form on $M$. After integration, $$ is then of total degree 0.

The term $R̂$ has been introduced in Remark \ref{rmk_R}. However, its connection with the globalization procedure is not clear. To explain it, we introduce the \textit{classical Grothendieck connection} adapted to our case (see Appendix \cite{CF01}).

\begin{defn}[Classical Grothendieck connection]
Given a holomorphic formal exponential map $\varphi$, we can define the associated \textit{classical Grothendieck connection} on $\reallywidehat{\Sym}^\bullet(T^{\vee1,0}M)$, given by $\Gr \coloneqq d_M+R$, where $d_M$ is the sum of the holomorphic and antiholomorphic Dolbeault differentials on $M$ and $R\in \Omega^1\Big(M, \Der\Big(\reallywidehat{\Sym}^\bullet(T^{\vee1,0}M)\Big)\Big)$. By using local coordinates $\{x\}$ on the basis and $\{y\}$ on the fibers, we have $R=R_j(x;y)dx^j+R_{\bar{j}}(x;y)dx^{\bar{j}}$, where $R_j=R^i_j(x;y)\frac{\partial}{\partial y}$ and $R_{\bar{j}}=R^i_{\bar{j}}(x;y)\frac{\partial}{\partial y}$ with
\begin{equation}
\begin{split}
&R^i_j(x;y)dx^j:=-\bigg[\bigg(\frac{\partial \varphi}{\partial y}\bigg)^{-1}\bigg]^i_p\frac{\partial \varphi^p}{\partial x^j}dx^j,\\
&R^i_{\Bar{j}}(x;y)dx^{\Bar{j}}:=-\bigg[\bigg(\frac{\partial \varphi}{\partial y}\bigg)^{-1}\bigg]^i_p\frac{\partial \varphi^p}{\partial x^{\Bar{j}}}dx^{\Bar{j}}.
\end{split}
\end{equation}
\end{defn}

Note that $R^i_j(x;y)$ and $R^i_j̅(x;y)$ are formal power series in the second argument, namely
\begin{equation}
\begin{split}
R^i_j(x;y)&=\sum^{\infty}_{k=0} R^i_{j;j_1,\dots,j_k}(x)y^{j_1}\dots y^{j_k},\\
R^i_{\bar{j}}(x;y)&=\sum^{\infty}_{k=0} R^i_{{\bar{j}};j_1,\dots,j_k}(x)y^{j_1}\dots y^{j_k}.
\end{split}
\end{equation}

\begin{rmk}
\label{class:RMK_Grothendieck_prop}
The classical Grothendieck connection has a couple of important properties:
\begin{itemize}
\item It is flat, which can be rephrased by saying that the following equation is satisfied
\begin{equation}
\label{class:flatnessR}
\end{equation}
\item A section $\sigma$ is closed under $\mathcal{D}_{\text{G}}$ i.e. $\mathcal{D}_{\text{G}}\sigma=0$ if and only if $\sigma=\mathrm{T}\varphi^*_xf$, where $f\in \mathcal{C}^\infty(M)$.
\end{itemize}
In more down-to-Earth terms, the second property says that the classical Grothendieck connection selects those sections which are global.
\end{rmk}

Finally, we can clarify the relation between $R̂$ and the Grothendieck connection. The components $(R̂^i)_j(x;𝐗̂)$ and $(R̂^i)_j̅(x;𝐗̂)$ are given by the components of the classical Grothendieck connection $R^i_j(x;y)$ and $R^i_j̅(x;y)$ evaluated in the second argument at $𝐗̂$.

Having set up all the necessary tools, we can compute how $$ varies when we change the background $x∈M$. On a closed manifold, we have
\begin{equation}
\label{dx_eq}
d_M \hat{\Sc}\sur=-(\Sc\surgR, \hat{\Sc}\sur),
\end{equation}
which follows from the Grothendieck connection and that $=Tφ^*_x$.

The above identities can be collected in a nicer way via  the following definition.

\begin{defn}(Formal global action) The \textit{formal global action} for the model is defined by
\begin{equation}
\label{glob_action}
\begin{split}
\Tilde{\Sc}\surg&\coloneqq\varint_{\Sigma_3}\bigg(\frac{1}{2}\Omega_{ij}\hat{\mathbf{X}}^id\hat{\mathbf{X}}^j+\Big(\hat{R}^i\sur\Big)_j(x; \hat{\mathbf{X}})\Omega_{il}\hat{\mathbf{X}}^l dx^j+\Big(\hat{R}^i\sur\Big)_{\Bar{j}}(x; \hat{\mathbf{X}})\Omega_{il}\hat{\mathbf{X}}^l dx^{\Bar{j}}\bigg)\\
\end{split}
\end{equation}
\end{defn}

By using the formal global action, the \textit{differential Classical Master Equation} (dCME) is satisfied %\fix
\begin{equation}
\label{class:dCME}
\end{equation}

\begin{rmk}
Note that $\Tilde{\Sc}\surg$ is an inhomogeneous form over $M$, where $\hat{S}\sur$ is a 0-form and $S\surgR$ is a 1-form. Therefore, Eq. \eqref{class:dCME} has a 0-form, a 1-form and a 2-form part. Specifically, the 0-form part
\begin{equation}
\end{equation}
is the usual CME. The 1-form part:
\begin{equation}
\end{equation}
means that $\hat{\Sc}\sur$ is a global object (see Remark \ref{class:RMK_Grothendieck_prop}). The 2-form part
\begin{equation}
\end{equation}
means that the operator $\mathcal{D}_G$ is flat connection (see Eq. (\ref{class:flatnessR})). Explicitly, we have
\begin{align}
&d_x\Sc_R+\frac12(\Sc_R,\Sc_R)=0 \label{dcme_1},\\
&d_{\bar{x}}\Sc_R+\frac12(\Sc_{\bar{R}},\Sc_R)=0 \label{dcme_3},\\
&d_{\bar{x}}\Sc_{\bar{R}}+\frac12(\Sc_{\bar{R}},\Sc_{\bar{R}})=0. \label{dcme_4}
\end{align}
\end{rmk}

% where
% \begin{equation}
%     \begin{split}
%         \Tilde{d}_x=\varint\sur-d\hat{\mathbf{X}}\frac{\delta}{\delta \hat{\mathbf{X}}}&+\Omega^{pq}\frac{\delta Y^i_j}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^l dx^j \frac{\delta}{\delta \hat{\mathbf{X}}^q}+
%     \Omega^{pq}Y^i_j\Omega_{ip}dx^j\frac{\delta}{\delta \hat{\mathbf{X}}^q}\\
%     &+\Omega^{pq}\frac{\delta Y^i_{\Bar{j}}}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^l dx^{\Bar{j}} \frac{\delta}{\delta \hat{\mathbf{X}}^q}+
%     \Omega^{pq}Y^i_{\Bar{j}}\Omega_{ip}dx^{\Bar{j}}\frac{\delta}{\delta \hat{\mathbf{X}}^q}
%     \end{split}
% \end{equation}
% \begin{proof}

% \begin{equation*}
%     \begin{split}
%         \Tilde{d}_x\Tilde{S}\surg=\varint\sur &-d\hat{\mathbf{X}}^q\Omega_{qs}d\hat{\mathbf{X}}^s
%     +\cancel{d\hat{\mathbf{X}}^q\frac{\delta Y^i_j}{\delta \hat{\mathbf{X}}^q}\Omega_{il}\hat{\mathbf{X}}^ldx^j}
%     +\cancel{d\hat{\mathbf{X}}^qY^i_j\Omega_{iq}dx^j}\\
%     &+\Omega^{qp}\frac{\delta Y^i_j}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^ldx^j\frac{\delta Y^m_n}{\delta \hat{\mathbf{X}}^q}\Omega_{mr}\hat{\mathbf{X}}^rdx^n
%     +\cancel{\Omega^{qp}\frac{\delta Y^i_j}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^ldx^jd\hat{\mathbf{X}}^q}\\
%     &+\cancel{\Omega^{qp}\frac{\delta Y^i_j}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^ldx^jY^m_n\Omega_{mq}dx^n}
%     +\cancel{\Omega^{qp}Y^i_j\Omega_{ip}dx^j\frac{\delta Y^m_n}{\delta \hat{\mathbf{X}}^q}\Omega_{mr}\hat{\mathbf{X}}^rdx^n}\\
%     &+\cancel{\Omega^{qp}Y^i_j\Omega_{ip}dx^j\Omega_{qn}d\hat{\mathbf{X}}^n}
%     +\Omega^{qp}Y^i_j\Omega_{ip}dx^jY^m_n\Omega_{mq}dx^n\\
%     &+\cancel{d\hat{\mathbf{X}}^q\frac{\delta Y^i_{\Bar{j}}}{\delta \hat{\mathbf{X}}^q}\Omega_{il}\hat{\mathbf{X}}^ldx^{\Bar{j}}}
%     +\cancel{d\hat{\mathbf{X}}^qY^i_{\Bar{j}}\Omega_{iq}dx^{\Bar{j}}}\\
%     &+\Omega^{qp}\frac{\delta Y^i_{\Bar{j}}}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^ldx^{\Bar{j}}\frac{\delta Y^m_{\Bar{n}}}{\delta \hat{\mathbf{X}}^q}\Omega_{mr}\hat{\mathbf{X}}^rdx^{\Bar{n}}
%     +\cancel{\Omega^{qp}\frac{\delta Y^i_{\Bar{j}}}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^ldx^{\Bar{j}}d\hat{\mathbf{X}}^q}\\
%     &+\cancel{\Omega^{qp}\frac{\delta Y^i_{\Bar{j}}}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^ldx^{\Bar{j}}Y^m_{\Bar{n}}\Omega_{mq}dx^{\Bar{n}}}
%     +\cancel{\Omega^{qp}Y^i_{\Bar{j}}\Omega_{ip}dx^{\Bar{j}}\frac{\delta Y^m_{\Bar{n}}}{\delta \hat{\mathbf{X}}^q}\Omega_{mr}\hat{\mathbf{X}}^rdx^{\Bar{n}}}\\
%     &+\cancel{\Omega^{qp}Y^i_{\Bar{j}}\Omega_{ip}dx^{\Bar{j}}\Omega_{qn}d\hat{\mathbf{X}}^n}
%     +\Omega^{qp}Y^i_{\Bar{j}}\Omega_{ip}dx^{\Bar{j}}Y^m_{\Bar{n}}\Omega_{mq}dx^{\Bar{n}}\\
%     &=-\frac{1}{2}(S\surg, S\surg)
%     \end{split}
% \end{equation*}

% \end{proof}

Let $Σ_3$ be (again) a manifold with boundary. The BV-BFV theory on $ℱ$ furnishes the cohomological vector field $Q$. Moreover, by using the lift of $R̂$, we can define
\begin{equation}
\Tilde{Q}\surg=Q\surg+\hat{R}\sur.
\end{equation}
Then, the \textit{modified differential Classical Master Equation} (mdCME) is satisfied:
\begin{equation}
\label{mdcme}
\iota_{\Tilde{Q}\surg}\omega\surg=\delta \Tilde{\Sc}\surg+\pi^*\alpha^{\partial}_{\partial \Sigma_3, x},
\end{equation}
\begin{equation}
\begin{split}
\Tilde{Q}\surg=\varint\sur\bigg(-d\hat{\mathbf{X}}\frac{\delta}{\delta \hat{\mathbf{X}}}&-\Omega^{pq}\frac{\delta \Big(\hat{R}^i\sur\Big)_j(x; \hat{\mathbf{X}})}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^l  dx^j\frac{\delta}{\delta \hat{\mathbf{X}}^q}-
\Big(\hat{R}^i\sur\Big)_j(x; \hat{\mathbf{X}})dx^j\Omega_{ip}\frac{\delta}{\delta \hat{\mathbf{X}}^p}\\
&-\Omega^{pq}\frac{\delta \Big(\hat{R}^i\sur\Big)_{\bar{j}}(x; \hat{\mathbf{X}})}{\delta \hat{\mathbf{X}}^p}\Omega_{il}\hat{\mathbf{X}}^l  dx^{\Bar{j}}\frac{\delta}{\delta \hat{\mathbf{X}}^q}-
\Big(\hat{R}^i\sur\Big)_{\bar{j}}(x; \hat{\mathbf{X}})dx^{\Bar{j}}\Omega_{ip}\frac{\delta}{\delta \hat{\mathbf{X}}^p}\bigg).
\end{split}
\end{equation}

In preparation for the comparisons we will draw in the following section, we redefine the components $(R̂^i)_j$ and $(R̂^i)_j̅$ by a multiplicative factor $1/k!$  as
\begin{equation}
\label{redef_R}
\begin{split}
\Big(\hat{R}^i\sur\Big)_j(x;\hat{\mathbf{X}})&=\sum^{\infty}_{k=0} \frac{1}{(k+1)!}\hat{R}^i_{j;j_1,\dots,j_k}(x)\hat{\mathbf{X}}^{j_1}\dots \hat{\mathbf{X}}^{j_k},\\
\Big(\hat{R}^i\sur\Big)_{\bar{j}}(x;\hat{\mathbf{X}})&=\sum^{\infty}_{k=0}\frac{1}{(k+1)!} \hat{R}^i_{{\bar{j}};j_1,\dots,j_k}(x)\hat{\mathbf{X}}^{j_1}\dots \hat{\mathbf{X}}^{j_k}.
\end{split}
\end{equation}
% \begin{equation}
%     S\surg:=\text{T}\Tilde{\varphi}_{x}S\sur=\varint_{T[1]\Sigma_3}
%     \frac{1}{2}
%     \hat{\mathbf{X}}^i D\hat{\mathbf{X}}^j+Y^i_j(\hat{\mathbf{X}})\Omega_{il}\hat{\mathbf{X}}^l dx^j+Y^i_{\Bar{j}}(\hat{\mathbf{X}})\Omega_{il}\hat{\mathbf{X}}^l dx^{\Bar{j}}
% \end{equation}

\section{Comparison with the original Rozansky--Witten model}
\label{sec:comp_orig_RW}

In this section, we show that the globalized model we have just constructed reduces to the RW model (Section \ref{sec:RW_model}) and, moreover it provides a globalization of the former.

In order to compare effectively these models, we need to be more explicit about the terms involved in the classical Grothendieck connection. First, we discuss the choice of holomorphic formal exponential map in more detail. Since our target is a symplectic manifold, we choose the formal exponential map which preserves the symplectic form considered in \cite{QZ15} and we adapt it to our case, i.e.
\begin{equation}
\label{comp:exp_map_zq}
\varphi^{i}=x^i+y^i-\frac{1}{2}\Chr{i}{j}{k}y^jy^k+\bigg\{-\frac{1}{6}\partial_c\Chr{i}{j}{k}+\frac{1}{3}\Chr{i}{m}{c}\Chr{m}{j}{k}-\frac{1}{24}\Riem{i}{c}{j}{k}\bigg\}y^cy^jy^k+O(y^4),
\end{equation}
where $icjk=(Ω^-1)^biR^ a_bc  kΩ_aj$.

The Grothendieck connection is then
\begin{equation}
\label{Grothendieck_con}
\Gr=dx^i\frac{\partial}{\partial x^i}+dx^{\bar{i}}\frac{\partial}{\partial x^{\bar{i}}}+
dx^j\Big(R\sur\Big)_j +dx^{\Bar{j}}\Big(R\sur\Big)_{\Bar{j}},
\end{equation}
where the third term on the right hand side was computed in \cite{QZ15},
\begin{equation}
\Big(\hat{R}^i\sur\Big)_{j} dx^j=-\bigg[\bigg(\frac{\partial \varphi}{\partial y}\bigg)^{-1}\bigg]^i_p\frac{\partial \varphi^p}{\partial x^j}dx^j=-\bigg[dx^j\bigg(\delta^i_j+\Chr{i}{k}{j}y^k-\bigg(\frac{1}{8}R^{\hspace{1.5mm} i}_{j\; ks}+\frac{1}{4}R^{\hspace{2.5mm} i}_{jk\; s}\bigg)y^ky^s\bigg)\frac{\partial}{\partial y^i}+\dots\bigg],
\end{equation}
whereas the fourth term is
\begin{equation}
\begin{split}
\Big(\hat{R}^i\sur\Big)_{\bar{j}}dx^{\Bar{j}}&=-\bigg[\bigg(\frac{\partial \varphi}{\partial y}\bigg)^{-1}\bigg]^i_p\frac{\partial \varphi^p}{\partial x^{\Bar{j}}}dx^{\Bar{j}} =\bigg[-\frac{1}{2}\Chr{p}{ab}{,\Bar{j}}y^ay^bdx^{\Bar{j}}\bigg][\delta^i_p+\dots]       =-\bigg[-\frac{1}{2}\Chr{i}{ab}{,\Bar{j}}y^ay^bdx^{\Bar{j}}+\dots\bigg]\\
&=\Riem{i}{a}{b}{\bar{j}}y^ay^bdx^{\bar{j}}-\dots .
\end{split}
\end{equation}

Considering the terms coming from the classical Grothendieck connection and the redefinition \eqref{redef_R}, we can re-write the formal global action \eqref{glob_action} as
\begin{equation}
\label{class:expl_S_global}
\Tilde{\Sc}\surg=\varint\sur\bigg(\frac12 \Omega_{ij}\hat{\mathbf{X}}^id\hat{\mathbf{X}}^j-\frac12\Chr{i}{j}{k}\hat{\mathbf{X}}^k\Omega_{il}\hat{\mathbf{X}}^ldx^j-\delta^i_j\Omega_{il}\hat{\mathbf{X}}^ldx^j+\dots+\frac{1}{3!}\Riem{i}{k}{s}{\bar{j}}\hat{\mathbf{X}}^k\hat{\mathbf{X}}^s\Omega_{il}\hat{\mathbf{X}}^ldx^{\bar{j}}+\dots\bigg).
\end{equation}

For convenience, we recall the RW action \cite{RW96}
\begin{equation}
\label{class:S_RW_original}
S_{\text{RW}}=\varint\sur\frac12\frac{1}{\sqrt{h}}\epsilon^{\mu\nu\rho}\bigg(\Omega_{IJ}\chi_\mu^I\nabla_\nu\chi_\rho^J-\frac13\Omega_{IJ}R^J_{KL\Bar{M}}\chi_\mu^I\chi_\nu^K\chi_\rho^L\eta^{\Bar{M}}+\frac13(\nabla_L\Omega_{IK})(\partial_\mu\phi^I_\perp)\chi_\nu^K\chi_\rho^L \bigg).
\end{equation}
If we assume that the connection is compatible with the symplectic form, the third term in the RW action \eqref{class:S_RW_original} drops. We are left with the first two terms. By associating $𝐗̂^i↔χ^I$ and $dx^j̅↔η^M̅$, we can sum up the comparison in Table \ref{class:Tab.model_RW}.
\begin{table}[h!]
\centering
\begin{tabular}{|lcc|}
\toprule
& Kinetic term &  Interaction term \\
\midrule
Original RW model & $\frac12\frac{1}{\sqrt{h}}\epsilon^{\mu\nu\rho}\Omega_{IJ}\chi_\mu^I\nabla_\nu\chi_\rho^J$     &  $-\frac{1}{3!}\Omega_{IJ}R^J_{KL\Bar{M}}\chi_\mu^I\chi_\nu^K\chi_\rho^L\eta^{\Bar{M}}$ \\
Our model & $\frac12 \Omega_{ij}\hat{\mathbf{X}}^id\hat{\mathbf{X}}^j-\frac12\Chr{i}{j}{k}\hat{\mathbf{X}}^k\Omega_{il}\hat{\mathbf{X}}^ldx^j$     &  $\frac{1}{3!}\Riem{i}{k}{s}{\bar{j}}\hat{\mathbf{X}}^k\hat{\mathbf{X}}^s\Omega_{il}\hat{\mathbf{X}}^ldx^{\bar{j}}$ \\
\bottomrule
\end{tabular}
\caption{Comparison between kinetic term and interaction term for the RW theory and our model.}
\label{class:Tab.model_RW}
\end{table}

The sign discrepancy comes from having defined the connection as $∇=d-Γ$ which gives a negative sign in front of the $ijk$ (see Eq. \eqref{comp:exp_map_zq}).

Moreover, when the curvature misses the $(2,0)$-part (which could happen when we have a Hermitian metric), the remaining terms in our model are just the perturbative expansion of $iksj̅$ around $x$. If we cut off the expansion at the first order, we are left with the original RW model.

\section{Comparison with other globalization constructions}
\label{sec:comp_other}
In the next sections, we are going to compare our globalization model with other three constructions: the first by \cite{CLL17}
uses tools of derived geometry to linearize the space of fields in the neighbourhood of a constant map as well as the Fedosov connection \cite{Fe49}, the second \cite{Ste17} is an extension to a manifold with boundary of the first procedure, while the third \cite{QZ09,QZ10,KQZ13} uses an approach similar to ours.

\subsection{Comparison with the CLL construction}
\label{comp:sec:summary}
We compare our model with the formulation of the RW model constructed in \cite{CLL17} in the setting of derived geometry (see Appendix \ref{app:derived_geometry}).
% We start by recalling briefly the procedure explained in \cite{CLL17}, we refer to the original article for further details and for the proves of the Lemmata as well as Propositions we present.