% In \cite{CLL17}, Chan, Leung and Li provide a BV construction for the RW model by using tools of derived geometry. As we have seen before, to construct a BV manifold we need to provide three ingredients: a space of fields, a 0 graded action functional and a $-1$ graded symplectic form. Let us resume how we can obtain these elements in their context.
Let $Σ_3$ be a closed $3$ dimensional manifold and $M$ be a holomorphic symplectic manifold with a non-degenerate holomorphic $2$-form $ω$.

To determine fields we use the language of $$-spaces (see \cite{Co11a,Co11b} for an introduction) and we define the the space of fields as
\begin{equation}
\Maps(\MdR,M_{\Bar{\partial}}),
\end{equation}
where $$ is the elliptic ringed space equipped with a sheaf of differential forms over $Σ_3$, i.e. %$\mathcal{A}_{\Sigma_3}=
$Ω^∙(Σ_3)$ and $M_∂=(M,_M)$ is a sheaf of $$-algebras, where $_M=Ω^∙,∙(M)⊗T^1,0M[-1]$ with $T^1,0M$ the holomorphic tangent bundle.
Since the critical points of the action functional are constant maps from $Σ_3$ to $M$, we are going to study $(,M_∂)$ in the neighbourhood of a constant map $x∈(,)$, namely
\begin{equation}
\mathcal{F}_{\text{CLL}}\coloneqq \reallywidehat{\Maps}(\MdR,\Xd)=\Omega^{\bullet}(\Sigma_3)\otimes \g_M[1],
\end{equation}
with $(,)$ defined as in \cite{Co11a,Co11b}.

Having specified the space of fields, the shifted symplectic structure is given by
\begin{equation}
\label{class:sympl_cll}
\begin{split}
\braket{-,-}:\ &\FDg\otimes_{\Omega^{\bullet,\bullet}(M)}\FDg \rightarrow \Omega^{\bullet,\bullet}(M)[-1]\\
&\braket{\alpha\otimes g_1,\beta\otimes g_2}:=\underbrace{\omega(g_1,g_2)}_{\text{sympl. struct. on $M$}}\,\,\,\varint\sur\alpha\wedge \beta,
\end{split}
\end{equation}
where $Ω^∙,∙(M)=Γ(⋀^∙T^∨M)$ is a section of the cotangent bundle.
% In particular, since our action functional will be a local functional, we consider the subspace $\mathcal{C}^{\infty}_{\text{loc}}(\FDg)=\reallywidehat{\Sym}^\bullet_{\Omega^\bullet(M)}(\mathrm{Jet}_{\FDg^{\vee}})[\![\hbar ]\!]\otimes \text{Dens}(\Sigma_3)\subset \mathcal{C}^{\infty}(\FDg)$, where $\text{Dens}(\Sigma_3)$ is the line bundle of densities on $\Sigma_3$.

Since $C^∙(_M):=^∙_Ω^∙,∙(M)(^∨_M[1])=Ω^∙,∙(M)⊗_𝒞^∞(M)^∙_𝒞^∞(M)(T^∨1,0M)$, to construct the action functional and to find our $L_∞$-algebra we can use a procedure similar to the Fedosov's construction of a connection on a symplectic manifold \cite{F94}.
Let us denote the sections of the \textit{holomorphic Weyl bundle} on $M$ by
\begin{equation}
\mathcal{W}=\Omega^{\bullet,\bullet}(M)\otimes_{\mathcal{C}^{\infty}(M)}\CS[\![\hbar ]\!]
\end{equation}
where $[[ħ]]$ is the completed symmetric algebra over $T^∨1,0M$, the holomorphic cotangent bundle which has a local basis ${y^i}$ with respect to the local holomorphic coordinates ${x^i}$.  We call the sub-bundle $Ω^p,q(M)⊗^r(T^∨1,0M)$ of $$ its $(p,q,r)$ component, in particular we refer to $r$ as \textit{weight}. To $ħ$ is assigned a weight of 2.

% On the quantum Weyl bundle, we have a product induced by the inverse of the holomorphic symplectic form. In local coordinates we have
% \begin{equation}
%     \begin{split}
%         \alpha\circ\beta&=\exp\bigg(\frac{\hbar}{2}\Omega^{i_1j_1}\dots\Omega^{i_kj_k}\frac{\partial^k}{\partial y^{i_1}\dots \partial y^{i_k}}\frac{\partial^k}{\partial y^{j_1}\dots \partial y^{j_k}}\bigg)\alpha(z)\beta(w)\bigg|_{z=w}\\
%         &=\sum_{k\geq 1}\frac{1}{k!}\bigg(\frac{\hbar}{2}\bigg)^k\Omega^{i_1j_1}\dots\Omega^{i_kj_k}\frac{\partial^k\alpha}{\partial y^{i_1}\dots \partial y^{i_k}}\frac{\partial^k\beta}{\partial y^{j_1}\dots \partial y^{j_k}},
%     \end{split}
% \end{equation}
% where the product depends only on $y$ since we have a holomorphic symplectic form.
% Note that it is independent on the basis chosen on $T^{\vee1,0}M$ and it is associative. There is an associative bracket on $\W$
% \begin{equation}
%         [\alpha,\beta]_{\W}=\alpha\circ\beta-(-1)^{s+t}\beta\circ \alpha,
% \end{equation}
% where $\alpha\in\Omega^{p,q}(M)\otimes_{\mathcal{C}^\infty(M)}\Sym^r(T^{\vee1,0}M)$ and $\beta\in\Omega^{s,t}(M)\otimes_{\mathcal{C}^\infty(M)}\Sym^r(T^{\vee1,0}M)$.

% On the quantum Weyl bundle, we define some operators which mirror the ones constructed by Fedosov in \cite{F94} and satisfy  similar identities:
% \begin{lmm}[\cite{CLL17}]
% Let
% \begin{equation}
%     \begin{split}
%         \delta&=dx^i\wedge\frac{\partial }{\partial y^i},\\ \delta^*&=y^i\cdot\iota_{\partial_{x^i}},\\
%         \delta^{-1}&=\frac{1}{p+r}\delta^*,
%     \end{split}
% \end{equation}
% be operators on $\W$ and $\pi_0$ be the projection from the whole Weyl bundle $\W$ onto its $(0,\bullet,0)$ component (i.e. the 0 holomorphic component and any antiholomorphic component for $\Omega^{\bullet,\bullet}(M)$, the 0 component of $\reallywidehat{\Sym}^\bullet(T^{\vee1,0}M)$). We have the following identities
% \begin{equation}
%     \begin{split}
%         \delta\circ\delta^{-1}+\delta^{-1}\circ\delta&=\Id+\pi_0,\\
%         \delta^2&=(\delta^{-1})^2=0,\\
%         \delta&=\frac{1}{\hbar}[\omega_{ij}dx^i\otimes y^j,-]_{\W}.
%     \end{split}
% \end{equation}
% \end{lmm}

% \begin{rmk}
% $\W$ is an algebra (of differential forms). The multiplication is defined by external product of differential forms $dx^i,\ dx^{\Bar{i}}$ and Weyl product of polynomial in $y^i$ ($dx$ commutes with $y$).
% \end{rmk}

% On $\W$ we can construct a connection. To do so, first, we pick a connection $\nabla$ on the holomorphic tangent bundle $T^{1,0}M$ (and naturally induced on $T^{\vee1,0}M$) satisfying the following conditions
% \begin{enumerate}
%     \item $\nabla$ is compatible with the complex structure, which means that if we project the covariant derivative on the (0,1) component we get the \textit{Dolbeault differential}:
%     \begin{equation}
%         \Pi_{0,1}\nabla=\Bar{\partial},
%     \end{equation}
%     where $\Pi_{0,1}$ denotes the projection onto the $(0,1)$ component.
%     \item $\nabla$ is torsion free.
%     % \begin{equation}
%     %     T^{\nabla}(X,Y)=\nabla Y-\nabla X+[X,Y]
%     % \end{equation}
%     \item $\nabla$ is compatible with the holomorphic symplectic form $\omega$, i.e. $\nabla \omega=0$.
%     % \begin{equation}
%     % \label{comp_omega_hol}
%     %     \nabla\omega(X,Y)=0\Leftrightarrow Z(\omega(X,Y))=\omega(\nabla_ZX,Y)+\omega(X,\nabla_Z Y)\quad \forall X,Y,Z
%     % \end{equation}
% \end{enumerate}

% The connection is then extended to $\W$ and it is denoted again as $\nabla$.

% In Darboux local coordinates, we can write $\nabla$ as
% \begin{equation}
% \label{defnabla2}
%     \nabla \alpha=d\alpha-\frac{1}{\hbar}[\Chr{}{}{},\alpha],
% \end{equation}
% where
% \begin{equation}
%     \Chr{}{}{}=\frac{1}{2}\big[\Chr{}{ij}{k}y^iy^jdx^k+ \Chr{}{ij}{\bar{k}}y^iy^jdx^{\Bar{k}}\big]\in \Gamma\Big(\Sym^2(T^{\vee 1,0}M))\otimes \Lambda^{1,0}(TM))\oplus \Sym^2(T^{\vee 1,0}M))\otimes \Lambda^{0,1}(TM))\Big)
% \end{equation}
% is a local $1$-form with values in $\Sym^2(T^{\vee 1,0}M)$.
%  %$\Chr{}{ij}{k}$ defined as before and $d=\partial+\Bar{\partial}$, where $\partial$ and $\Bar{\partial}$ are Dolbeaut operators.

% \begin{rmk}
% With respect to the original definition in \cite{CLL17}, we have defined $\Gamma$ as $-\Gamma_{\text{original}}$ to compare with the formal exponential map in \eqref{comp:exp_map_zq}.
% \end{rmk}

% Following \cite{F94}, we can define the following section of $\W$
% \begin{equation}
%     R:=\frac{1}{4}\Riem{}{n}{q}{jr}dx^q\wedge dx^n\otimes y^jy^r+\frac{1}{2}\Riem{}{k}{\Bar{q}}{jr}d\Bar{x}^qdx^k\otimes y^j y^r.
% \end{equation}
% The connection is not flat; indeed we have
% \begin{lmm}[\cite{CLL17}]
% \begin{equation}
%     \nabla^2=\frac{1}{\hbar}[R,-]_{\W}.
% \end{equation}
% \end{lmm}

% However, the connection can be made flat by adding a correction as explained in the following Proposition.
\begin{prp}[\cite{CLL17}]
\label{prp_flatness}
There is a connection on the holomorphic Weyl bundle of the following form
\begin{equation}
\label{Fedosov_conn}
\mathcal{D}_\mathrm{F}=\nabla-\delta+\frac{1}{\hbar}[I,-]_{\W},
\end{equation}
which is flat modulo $\hbar$ and $[-,-]_{\mathcal{W}}$ is defined as in \cite{CLL17}. Here $I$ is a $1$-form valued section of the Weyl bundle of weight $\geq 3$, i.e. $I\in \bigoplus_{r\geq 3}\Gamma(\Sym^r(T^{\vee 1,0}M)\otimes T^\vee M)$, $\nabla$ is the extension to $\mathcal{W}$ of a connection on $T^{1,0}M$ which is compatible with the complex structure as well as with the holomorphic symplectic form and torsion free, $\delta=dx^i\wedge \frac{\partial}{\partial y^i}$ is an operator on $\mathcal{W}$.
\end{prp}

The connection $$ is called \textit{Fedosov connection} and it provides the $L_∞$-structure on $𝔤_M$. In these terms
% Having specified the $\Linf$-structure, we can describe the action functional. With this goal in mind, let us consider the natural map
% \begin{equation}
%     \rho: \mathcal{W}\rightarrow \mathcal{C}^{\infty}_\mathrm{loc}(\FDg)
% \end{equation}
% which works as follows. Given a section $I$ of $\Sym^k(T^{\vee1,0}M)$, we can associate a(n) ($\Omega^{\bullet}$-valued) functional on $\mathfrak{g}_M[1]$ using the natural pairing between $T^{1,0}M$ and $T^{\vee1,0}M$. This functional can be further extended to $\mathcal{F}_{\text{CLL}}$ via integration of differential forms over $\Sigma_3$. Explicitly, we have
% \begin{equation}
%     \rho(I): \mathcal{F}_{\text{CLL}}\rightarrow \Omega^{\bullet}(M),\quad \alpha \mapsto \frac{1}{k!}\varint\sur I(\alpha,\dots, \alpha),
% \end{equation}
% with $\alpha\in \mathcal{F}_{\text{CLL}}$.
% The action is then defined as
% \begin{equation}
% \label{class:dg_action}
%     \Sc_{\text{CLL}}(\alpha)\coloneqq\varint\sur \frac{1}{2}\omega(d\sur\alpha+\nabla\alpha,\alpha)+\rho(I-\delta^{-1}(\omega))(\alpha),
% \end{equation}
% where $d\sur$ is the de Rham differential on $\Sigma_3$, $\omega$ is the holomorphic symplectic used to pair factors in $T^{1,0}M$, and $\alpha\in \mathcal{F}_{\text{CLL}}$.
the action can be written as
\begin{equation}
\label{class:dg_action_2}
\Sc_{\text{CLL}}=\frac{1}{2}\braket{d\sur\alpha,\alpha}+\sum^{\infty}_{k=0}\frac{1}{(k+1)!}\braket{\ell_k(\alpha^{\otimes k}),\alpha}
\end{equation}
with $α∈ℱ_CLL$, $⟨-,-|$⟩ defined as in (<ref>) and $\ell_k$ are the higher brackets in the $\Linf$-algebra, and $d_{\Sigma_3}$ the de Rham differential on the source $\Sigma_3$. We can read $\ell_0$ from the Fedosov connection in (<ref>), i.e.

\begin{equation}
\ell_0=-dx^i\frac{\partial}{\partial y^i}
\end{equation}

The $\Linf$-products $\ell_1$ and $\ell_2$ are computed in the next section, when we compare the Fedosov connection with the classical Grothendieck connection.

The action in (<ref>) satisfies the CME $(\Sc_{\text{CLL}},\Sc_{\text{CLL}})=0$ (see <cit.>). Moreover, in <cit.> it was observed that this construction is the formal version of the original RW model in the case when the $(2,0)$-part of the curvature is zero (see <cit.>).

§.§.§ Comparison between the Fedosov connection and the classical Grothendieck connection

The sufficient condition for the flatness of $\Fe$ (see the proof of Proposition <ref> in <cit.>)  implies that $I$ satisfies

\begin{equation}
\label{eqI}
I=\delta^{-1}(R+\nabla I)+\frac{1}{\hbar}\delta^{-1}I^2,
\end{equation}

where $\delta^{-1}=y^i\cdot \iota_{\partial_{x^i}}$ (up to a normalization factor) is another operator on $\mathcal{W}$ and $R$ is the curvature tensor.

Since $I$ is a $1$-form valued section of $\W$, we can decompose it into its holomorphic and antiholomorphic component respectively. In particular the antiholomoprhic part component is the Taylor expansion of the Atiyah class as noted in <cit.>. In the case $R^{2,0}=0$, the $\Linf$-algebra is fully encoded by Taylor expansion of the Atiyah class as first noted by Kapranov in <cit.>.

Since the operator $\delta^{-1}$ increases the weight by $1$, while $\nabla$ preserves the weight and $I$ has at least weight $3$, we can find a solution of the above equation with the following leading term (cubic term)[See also <cit.>.]:

\begin{equation}
\label{deltaR}
\delta^{-1}R=\frac18\big[-\Chr{}{ij}{k,r}+\Chr{}{si}{r}\Chr{}{pj}{k}\Omega^{sp}\big]y^i y^j y^r dx^k+\frac16\Riem{}{\Bar{k}r}{i}{j} y^i y^j y^r dx^{\Bar{k}}=\delta^{-1}R_t+\delta^{-1}\bar{R}.
\end{equation}

Since the Fedosov connection requires the computation of $\frac{1}{\hbar}[I,-]_{\W}$, we compute this commutator for the leading order term of $I$, which is the cubic term we have just found. For the first term on the right hand side of Eq. (<ref>) we have

\begin{equation}
\begin{split}
\frac{1}{\hbar}[\delta^{-1}R_t,-]_{\mathcal{W}}&=\bigg[\frac{1}{8}\bigg(-\Chr{}{rj}{k,q}+\Chr{}{s}{rq}\Chr{}{p}{jk}\Omega^{sp}\bigg)+\frac{1}{4}\bigg(-\Chr{}{qj}{k,r}+\Chr{}{sq}{k}\Chr{}{pj}{r}\Omega^{sp}\bigg)\bigg]\Omega^{qi}y^j y^r dx^k\frac{\partial}{\partial y^i}\\
&\hspace{10cm} \times\Omega^{qi}y^j y^r dx^k\frac{\partial}{\partial y^i}\\
&=\bigg[\frac18\Riem{\hspace{1.75mm} i}{k\ }{r}{j}+\frac14\Riem{\hspace{2.9mm}i}{k}{r\, }{j}\bigg]y^j y^r dx^k\frac{\partial}{\partial y^i}.
\end{split}
\end{equation}

For the second term we have

\begin{equation}
\frac{1}{\hbar}\big[\delta^{-1}\bar{R},-\big]_{\W}=\frac12\Chr{i}{j}{k,\Bar{r}}y^j y^k dz^{\Bar{r}}\frac{\partial}{\partial y^i}.
\end{equation}

After renaming some indices, the Fedosov connection is then

\begin{equation}
\label{class:Fed_conn_expl}
\Fe=d_x+d_{\bar{x}}-dx^j\frac{\partial}{\partial y^j}-dx^j\Chr{i}{k}{j}y^k\frac{\partial}{\partial y^i}+dx^j\bigg(\frac18R^{\hspace{1.5mm} i}_{j\; ks}+\frac14R^{\hspace{2.5mm} i}_{jk\; s}\bigg)y^ky^s\frac{\partial}{\partial y^i}+\frac12dx^{\Bar{j}}\Chr{i}{ks}{,\Bar{j}}y^ky^s\frac{\partial}{\partial y^i}+\dots.
\end{equation}

More explicitly,

\begin{equation}
\label{class:linf_prod}
\begin{split}
\ell_1&=-dx^j\Chr{i}{k}{j}y^k\frac{\partial}{\partial y^i},\\
\ell_2&=dx^j\bigg(\frac18R^{\hspace{1.5mm} i}_{j\; ks}+\frac14R^{\hspace{2.5mm} i}_{jk\; s}\bigg)y^ky^s\frac{\partial}{\partial y^i}+\frac12dx^{\Bar{j}}\Chr{i}{ks}{,\Bar{j}}y^ky^s\frac{\partial}{\partial y^i}.
\end{split}
\end{equation}

The first terms in the Fedosov connection, explicitly written in (<ref>) coincide with the first terms for the classical Grothendieck connection (<ref>). Furthermore, by substituting the explicit expressions of $\ell_1$ and $\ell_2$ in the action $\Sc_{\text{CLL}}$ (<ref>), we can see that it coincides with the action $\Tilde{\Sc}\surg$ (<ref>).

§.§.§ Comparison between the CLL space of fields and globalization space of fields

By rephrasing the argument of <cit.> to our context, we can extend the classical Grothendieck connection $\Gr$ to the complex

\begin{equation}
\label{extensioncomplex}
\Gamma\bigg(\bigwedge\nolimits^{\bullet}T^\vee M\otimes   \CS \bigg).
\end{equation}

which is the algebra of functions on the formal graded manifold

\begin{equation}
T[1]M\bigoplus T^{1,0}M.
\end{equation}

This graded manifold is turned into a differential graded manifold by the classical Grothendieck connection $\Gr$. Moreover, since $\Gr$ vanishes on the body of the graded manifold, we can linearize at $x\in M$ and we get

\begin{equation}
T_x[1]M\bigoplus T_x^{1,0}M.
\end{equation}

On this graded manifold, we have a curved $\Linf$-structure (which is the same as $\mathfrak{g}_M[1]$) and Eq. (<ref>) can be interpreted as the Chevalley–Eilenberg complex of the aforementioned $\Linf$-algebra. Then, the space of fields for the globalized theory can be rewritten as

\begin{equation}
\label{extendedsof}
\Tilde{\mathcal{F}}\surg=\Omega^{\bullet}(\Sigma_3)\otimes \Omega^{\bullet,\bullet}(M)\otimes T^{1,0}_xM
\end{equation}

which is the same as $\FDg$ by linearizing at $x\in M$ the holomorphic tangent bundle as $\mathcal{D}_\text{F}$ vanishes on $M$.

The idea that the classical Grothendieck connection and the Fedosov connection coincide is not new, in particular see <cit.> and <cit.>.

Finally note that in <cit.> the source manifold $\Sigma_3$ was considered to be a closed manifold. As explained above (see Section <ref>) our construction is valid also when $\partial\Sigma_3\neq \emptyset$. In the next section, we tackle this last setting by comparing our approach with <cit.>, where the derived geometric framework was implemented for manifolds with boundary.

§.§ Comparison with Steffens' construction

In <cit.>, Steffens applied the same derived geometry approach we have seen in the last section to what he calls AKSZ theories of Chern–Simons type: CS theory and RW theories. In particular, his BV formulation of the RW model is completely analogue to the one in <cit.>: same space of fields, $\Linf$-algebra, action, etc.
However, he takes a step further. He proves a formal AKSZ theorem <cit.> in the context of derived geometry. His RW model is then shown to be an AKSZ theory by attaching degree 2 to the holomorphic symplectic form (as we did ourselves in Section <ref>). Consequently, he provides a BV-BFV formulation for the RW model. The BFV action found in <cit.> is analogous to the action in (<ref>) in one dimension less (as it is customary with AKSZ theories). Even if the $\Linf$ products are not explicit in his construction, by using the ones in (<ref>), his BV-BFV formulation of the RW model is visibly  identical to ours.

§.§ Comparison with the (K)QZ construction

Let $\Sigma_3$ be a 3-dimensional manifold and $M$ a hyperKähler manifold with holomorphic symplectic form $\Omega$. Consider the symplectic graded manifold $\mathcal{M}\coloneqq T^{\vee0,1}[2]T^{\vee0,1}[1]M$ constructed out of $M$. It has the following coordinates: $X^i, X^{\Bar{i}}$ of degree 0 parametrizing $M$, $V^{\Bar{i}}$ of degree 1 parametrizing the fiber $T^{0,1}M$  and dual coordinates $P_{\Bar{i}}, Q_{\Bar{i}}$ of degree 2  and 1, respectively. The symplectic form is

\begin{equation}
\label{class:QZ_sympl_form}
\omega_{\mathcal{M}}=dP_{\Bar{i}} \wedge dX^{\Bar{i}}+dQ_{\Bar{i}}\wedge dV^{\Bar{i}}+\frac{1}{2}\Omega_{ij}dX^i\wedge dX^j.
\end{equation}

In order to have a ghost degree 2 symplectic form, the authors assign degree 2 to $\Omega$. With this setup, in <cit.>, Källén, Qiu and Zabzine construct an AKSZ model

\begin{align}
\mathcal{F}_{\text{QZ}}&\coloneqq  \Maps(T[1]\Sigma_3, T^{\vee0,1}[2]T^{\vee0,1}[1]M)\\
\label{comp:action_qz}
\Sc_{\text{QZ}}&=\varint_{T[1]\Sigma_3}d^3zd^3\theta\bigg(\mathbf{P}_{\Bar{i}}D\mathbf{X}^{\Bar{i}}+\mathbf{Q}_{\Bar{i}}D\mathbf{V}^{\Bar{i}}+\frac{1}{2}\Omega_{ij}\mathbf{X}^iD\mathbf{X}^j+\mathbf{P}_{\Bar{i}}\mathbf{V}^{\Bar{i}}\bigg)
\end{align}

endowed with a cohomological vector field

\begin{equation}
Q=\varint_{T[1]\Sigma_3}d^3zd^3\theta \bigg(D\mathbf{P}_{\Bar{i}}\frac{\partial}{\partial \mathbf{P}_{\Bar{i}}}+D\mathbf{Q}_{\Bar{i}}\frac{\partial}{\partial \mathbf{Q}_{\Bar{i}}}+D\mathbf{V}^{\Bar{i}}\frac{\partial}{\partial \mathbf{V}^{\Bar{i}}}+    D\mathbf{X}^i\frac{\partial}{\partial \mathbf{X}^i}+D\mathbf{X}^{\Bar{i}}\frac{\partial}{\partial \mathbf{X}^{\Bar{i}}}+\mathbf{P}_{\Bar{i}}\frac{\partial}{\partial \mathbf{Q}_{\Bar{i}}}+\mathbf{V}^{\Bar{i}}\frac{\partial}{\partial \mathbf{X}^{\Bar{i}}}\bigg),
\end{equation}

where to the source manifold $T[1]\Sigma_3$, we assign coordinates $\{z^i\}$ of ghost degree 0 and coordinates $\{\theta^i\}$ of degree 1.

With a suitable gauge-fixing consisting on a particular choice of Lagrangian submanifolds, the action $\Sc_{\text{QZ}}$ reduces to the RW model up to a factor of $\hbar$ (see <cit.>):

\begin{equation}
\Sc_{\text{QZ}}\bigg|_{\text{GF}}=\frac{1}{2}\varint d^3z \bigg(\Omega_{ij} X^i_{(1)}\wedge d^{\nabla} X^{j}_{(1)}-\frac{1}{3}R_{k\Bar{k}j}^i X^k_{(1)}\wedge \Omega_{li}X^l_{(1)}\wedge X^j_{(1)}V^{\bar{k}}_{(0)}\bigg),
\end{equation}

with $d^{\nabla}X^i_{(1)}=dX^i_{(1)}+\Chr{i}{j}{k}dX^j_{(0)}X^k_{(1)}$. Note that the only fields left are the even scalar $X^i_{(0)}$, the odd 1-form $X^i_{(1)}$ and the odd scalar $V^{\bar{k}}_{(0)}$. A quick glance to our expression for the RW model in (<ref>) (assume again the $(2,0)$ part of the curvature is zero as well as we cut off the perturbative expansion of the $(1,1)$ part at the $\Riem{i}{k}{s}{\bar{j}}$) suggests the association $V^{\bar{k}}_{(0)}\Leftrightarrow dx^{\bar{k}}$.
We will comment more on this later.

By expanding $\mathbf{X}^i$ through the geodesic exponential map and by pulling back $\omega_{\mathcal{M}}$ as well as $S_{\text{QZ}}$ through it, the authors find

\begin{align}
\exp^*\omega_{\mathcal{M}}&=dP_{\Bar{i}} \wedge dX^{\Bar{i}}+dQ_{\Bar{i}}\wedge dV^{\Bar{i}}+\frac{1}{2}\Omega_{ij}(x)dy^i\wedge dy^j-\delta X^{\bar{i}}\delta \Theta_{\bar{i}}\\
\exp^*\Sc_{\text{QZ}}\bigg|_{\tilde{\mathbf{P}}}&=\varint_{T[1]\Sigma_3}d^3zd^3\theta \bigg(\tilde{{\mathbf{P}}}_{\Bar{i}}D \mathbf{X}^{\Bar{i}}+\mathbf{Q}_{\Bar{i}}D\mathbf{V}^{\Bar{i}}+\frac{1}{2}\Omega_{ij}\mathbf{y}^i D\mathbf{y}^j-\Tilde{\mathbf{P}}_{\Bar{i}}\mathbf{V}^{\Bar{i}}+\Theta_{\Bar{i}}(x;\mathbf{y})\mathbf{V}^{\Bar{i}}\bigg)
\end{align}

where $\Theta_{\Bar{i}}$ is of degree 2 and given by

\begin{equation}
\label{class:qz_theta}
\Theta_{\Bar{i}}(x;y)=\sum^{\infty}_{n=3}\frac{1}{n!}\nabla_{l_4}\dots \nabla_{l_n}R^{\hspace{3mm}k}_{\Bar{i}l_1\ l_3}\Omega_{kl_2}(x)y^{l_1}\dots y^{l_n}%:=\sum^{\infty}_{n=3}\frac{1}{n!}Y^{\quad k}_{\Bar{i}l_1\ \dots l_n}\Omega_{kl_2}y^{l_1}\dots y^{l_n}
\end{equation}

and $\Tilde{P}_{\Bar{i}}:=P_{\Bar{i}}+\Theta_{\Bar{i}}$.

After removing the spectator fields (see <cit.>), the action becomes

\begin{equation}
\varint_{T[1]\Sigma_3}d^3zd^3\theta\bigg(\frac12\Omega_{ij}\mathbf{y}^iD\mathbf{y}^j+\Theta_{\bar{i}}(x;\mathbf{y})\mathbf{V}^{\bar{i}}\bigg),
\end{equation}

which further reduces to

\begin{equation}
\label{class:qz_action}
\varint_{T[1]\Sigma_3}d^3zd^3\theta\bigg(\frac12\Omega_{ij}\mathbf{y}^iD\mathbf{y}^j+\Theta_{\bar{i}}(x;\mathbf{y})V^{\bar{i}}_{(0)}\bigg)
\end{equation}

for degree reasons ($V^{\bar{i}}_{(0)}$ is an odd scalar). This action fails the CME by a $\bar{\partial}$-exact term due to $\Theta$ satisfying the Maurer–Cartan equation

\begin{equation}
\label{class:theta_MC}
\bar{\partial}_{[\bar{i}}\Theta_{\bar{j}]}=-(\Theta_{\bar{i}}, \Theta_{\bar{j}}),
\end{equation}

where $\bar{\partial}$ is the Dolbeault differential and $[\bar{i}\bar{j}]$ denotes antisymmetrization over the indices $\bar{i}$ and $\bar{j}$.

The hyperKähler structure is then relaxed. A new connection which still preserves $\Omega$ (crucial for the perturbative approach through the exponential map above) is found. However, since the connection is not Hermitian, the curvature of $\Gamma$ exhibits also a $(2,0)$-component. This complicates the exponential map which can not be worked out at all orders as in (<ref>). In <cit.>, the authors argue that a solution to this problem should originate from principles related to the globalization issues discussed in <cit.> and the application of Fedosov connection in order to deal with perturbation theory on curved manifold <cit.>. In the realm of this paper, we furnish an affirmative answer to both their ideas. In particular, as we have seen in Section <ref>, the Fedosov connection allowed to compute the terms in the $\Linf$-algebra and thus to work out the exponential map. In Section <ref>, we have seen the Grothendieck connection to accomplish the same in the context of formal geometry.

We can compare the procedure above with our globalization construction by associating $V^{\bar{i}}_{(0)}$ with $dx^{\bar{i}}$. First, note that $\Big(R\sur\Big)_{\bar{i}}$ in Eq. (<ref>) matches with the second term in Eq. (<ref>). Second, the action in (<ref>) coincides with our globalized action in (<ref>) if we “forget" the $(2,0)$-part of the curvature. In particular, by associating $\bar{\partial}$ with $dx^{\bar{i}}\frac{\partial}{\partial x^{\bar{i}}}$, we can interpret the failure of (<ref>) to satisfy the CME due to the term (<ref>) as a consequence of the action satisfying the $(1,1)$-part of the dCME (Eq. (<ref>)).

We reserve the last remark of the section to precise the association between $V^{\bar{i}}_{(0)}$ and $dx^{\bar{i}}$ as well as their “meaning" as we promised in Remark <ref>.

As we have seen above, $V^{\bar{i}}_{(0)}$ and $dx^{\bar{i}}$ arise in two different contexts: the first is an odd scalar coordinate parametrizing the fibers of $T^{0,1}M$, while the second is introduced through the classical Grothendieck connection as well as the perturbative expansion.

Nevertheless, the association makes sense considering that $V^{\bar{i}}_{(0)}$ is interpreted as an odd harmonic zero mode in <cit.>. In fact, recall from Section <ref>, $x$ is the zero mode obtained from the Euler–Lagrange equation $D\mathbf{X}=0$. If we enlarge the complex (see Eq. (<ref>)), the space of fields becomes (<ref>) meaning that $dx^{\bar{i}}\in T^{\vee 0,1}M$, i.e. an odd zero mode. This association was pointed out first by Qiu and Zabzine in <cit.>.

The presence of these quantities has been known in the literature since the early days of the RW model and has deep consequences. Since they are odd, there  can be as many as the dimension of $M$. As such, the perturbative expansion can not be infinite, but it can only stop at a certain order. This is a crucial difference between the CS and the RW theory, which was originally spotted in <cit.> and attributed to the need for the RW theory to saturate the zero modes. According to Kontsevich in <cit.>, as a result the RW model can be understood as an AKSZ model with “parameters" (these parameters are $V^{\bar{i}}_{(0)}$ or $dx^{\bar{i}}$). In the same article, he presented a different perspective on this subject by pointing out that the RW invariants come from characteristic classes of holomorphic connections.

§ $BF$-LIKE FORMULATION OF THE ROZANSKY–WITTEN MODEL

In order to quantize our globalized version of the RW model in the quantum BV-BFV framework <cit.>, we need to formulate the model as a $BF$-like theory. This can be done by exploiting the similarities between the RW theory and the CS theory. These similarities have also been crucial in the construction of <cit.>. There it was argued that RW could be split following a similar approach to the one of Cattaneo, Mnev and Wernli for the CS theory in <cit.> (see also <cit.> for a more detailed exposition).

As shown in <cit.> (see Eq. (<ref>)), we have a pairing on $ \Tilde{\mathcal{F}}\surg$ given by the BV symplectic form which can be defined on homogeneous elements $\hat{\mathbf{Y}}\otimes g_1$ and $\hat{\mathbf{Z}}\otimes g_2$ as

\begin{equation}
\begin{split}
\braket{-,-}:\ & \Tilde{\mathcal{F}}\surg\otimes  \Tilde{\mathcal{F}}\surg \rightarrow \Omega^{\bullet,\bullet}(M),\\
&\braket{\hat{\mathbf{Y}}\otimes g_1,\hat{\mathbf{Z}}\otimes g_2}:=\underbrace{\Omega(g_1,g_2)}_{\text{sympl. struct. on $M$}}\,\,\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg(\hat{\mathbf{Y}}\wedge \hat{\mathbf{Z}}\bigg).%=\varint_{T[1]\Sigma_3}\mu\bigg[\Omega_{ij}\hat{\mathbf{X}}^i\wedge \hat{\mathbf{X}}^j\bigg].
\end{split}
\end{equation}

By expanding $\hat{\mathbf{X}}\in \Tilde{\mathcal{F}}\surg$ as $\hat{\mathbf{X}}=\hat{\mathbf{X}}^ie_i$, we have

\begin{equation}
\braket{\hat{\mathbf{X}},\hat{\mathbf{X}}}=\Omega(e_i,e_j)\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg(\hat{\mathbf{X}}^i\wedge \hat{\mathbf{X}}^j\bigg)=\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg(\Omega_{ij}\hat{\mathbf{X}}^i\wedge \hat{\mathbf{X}}^j\bigg).
\end{equation}

We can rewrite the globalized action (<ref>) in the same way as in <cit.> (see the action in (<ref>)), we have

\begin{equation}
\Tilde{\Sc}\surg=\frac12\Big\langle\hat{\mathbf{X}}, D\hat{\mathbf{X}}\Big\rangle+\Big\langle\Big(\hat{R}\sur\Big)_j(x; \hat{\mathbf{X}})dx^j,\hat{\mathbf{X}}\Big\rangle+\Big\langle\Big(\hat{R}\sur\Big)_{\bar{j}}(x; \hat{\mathbf{X}})dx^{\bar{j}},\hat{\mathbf{X}}\Big\rangle,
\end{equation}

\begin{equation}
\label{class:coeff}
\begin{split}
\Big(\hat{R}\sur\Big)_j(x,\hat{\mathbf{X}})&=\sum^{\infty}_{k=0}\frac{1}{(k+1)!}\Big(\hat{R}_k\Big)_j(\hat{\mathbf{X}}^{\otimes k}),\\
\Big(\hat{R}\sur\Big)_{\bar{j}}(x,\hat{\mathbf{X}})&=\sum^{\infty}_{k=2}\frac{1}{(k+1)!}\Big(\hat{R}_k\Big)_{\bar{j}}(\hat{\mathbf{X}}^{\otimes k}).
\end{split}
\end{equation}

Now, similarly to the approach in <cit.>, we assume that we can split the $\Linf$-algebra as

\begin{equation}
\mathfrak{g}[1]=\Omega^{\bullet,\bullet}(M)\otimes T^{\vee1,0}M=\Omega^{\bullet,\bullet}(M)\otimes V\oplus \Omega^{\bullet,\bullet}(M)\otimes W,
\end{equation}

with $V$ and $W$ two isotropic subspaces. We identify $W\cong V^\vee$ via the pairing (in particular thanks to the holomorphic symplectic form). Consequently, the superfield splits as $\hat{\mathbf{X}}=\hat{\mathbf{A}}+\hat{\mathbf{B}}=\hat{\mathbf{A}}^i\xi_i+\xi^i\hat{\mathbf{B}}_i$ with $\xi_i\in V$ and $\xi^i\in W$. Concerning the assignment of degrees, we make the following choices. Since $\Omega$ has ghost degree $2$ (and as such $\Omega^{-1}$ has ghost degree $-2$), we assign total degree 0 to $\mathbf{A}^i$ and $\xi_i$, total degree $2$ to $B_i$ and total degree $-2$ to $\xi^i$. We refer to Table <ref> for an explanation of the ghost degrees for the components of the superfields $\hat{\mathbf{A}}^i$ and $\hat{\mathbf{B}}_i$. Then $\hat{\mathbf{A}}^i\oplus \hat{\mathbf{B}}_i\in \Omega^{\bullet}(\Sigma_3)\oplus\Omega^{\bullet}(\Sigma_3)[2]$, which is a $BF$-like theory.

Form degree      Ghost degree

$A^i_{(0)}$     0          0

$A^i_{(1)}$     1          $-1$

$A^i_{(2)}$     2          $-2$

$A^i_{(3)}$     3          $-3$

$B_{(0)i}$     0          2

$B_{(1)i}$     1          1

$B_{(2)i}$     2          $0$

$B_{(3)i}$     3          $-2$

Explanation for the form degree and ghost degree for the components of the superfields $\hat{\mathbf{A}}^i$ and $\hat{\mathbf{B}}_i$.

As explained in <cit.>, the splitting of the target $T^{\vee1,0}_xM$ into two transversal holomorphic Lagrangian subbundles is not possible when $M$ is a K3 surface. Instead, it is possible when $M=T^\vee Y$, with $Y$ any complex manifold. In this case $M$ with the standard holomorphic symplectic form will have a vertical as well as a horizontal polarization.

To sum up, the  space of fields is split as

\begin{equation}
\label{space_of_fields_gs}
\Tilde{\mathcal{F}}\surgS=\Omega^{\bullet}(\Sigma_3)\otimes\Omega^{\bullet,\bullet}(M)\otimes V\oplus \Omega^{\bullet}(\Sigma_3)[2]\otimes\Omega^{\bullet,\bullet}(M)\otimes W.
\end{equation}

The globalized split RW action is defined as

\begin{equation}
\label{split_global_action}
\begin{split}
\Tilde{\Sc}\surgS&\coloneqq%\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}
\Big\langle\hat{\mathbf{B}}, D\hat{\mathbf{A}}\Big\rangle+\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{split}
\end{equation}

\begin{equation}
\label{class:R_1}
\begin{split}
\bigg\langle\Big(\hat{R}\sur\Big)_j(x,\hat{\mathbf{A}}+\hat{\mathbf{B}})dx^{j},\hat{\mathbf{A}}+\hat{\mathbf{B}}\bigg\rangle&=\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg(\sum^{\infty}_{k=0}\frac{1}{(k+1)!}\Big(\hat{R}_k\Big)^i_j((\Omega_{il}\hat{\mathbf{A}}^l+\hat{\mathbf{B}}_i)^{\otimes k})(\hat{\mathbf{A}}+\hat{\mathbf{B}})dx^{j}\bigg)\\
\end{split}
\end{equation}

\begin{equation}
\label{class:R_2}
\begin{split}
\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&=\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg(\sum^{\infty}_{k=2}\frac{1}{(k+1)!}\Big(\hat{R}_k\Big)^i_{\bar{j}}((\Omega_{il}\hat{\mathbf{A}}^l+\hat{\mathbf{B}}_i)^{\otimes k})(\hat{\mathbf{A}}+\hat{\mathbf{B}})dx^{\bar{j}}\bigg)\\
\end{split}
\end{equation}

We call the model associated with the action (<ref>), globalized split RW model.

We present in Table <ref> the explicit expression as well as total degree for the component of $\Big(\hat{R}_k\Big)_j$ and $\Big(\hat{R}_k\Big)_{\bar{j}}$ in (<ref>) and (<ref>), respectively.

Operator         Explicit expression      Total degree

$(\hat{R}_0\big)^i_{j}\Omega_{il}$     -$\delta^{i}_j\Omega_{il}$          2

$(\hat{R}_0\big)^i_{j}$     $-\delta^i_j$          0

$\Big(\hat{R}_1\Big)^i_{j;s}\Omega_{il}$     $-\Chr{i}{s}{j}\Omega_{il}$          2

$\Big(\hat{R}_1\Big)^{is}_{j}\Omega_{il}$     $-\Chr{i}{q}{j}\Omega_{il}\Big(\Omega^{-1}\Big)^{qs}$          0

$\Big(\hat{R}_1\Big)^{is}_{j}$     $-\Chr{i}{j}{q}\Big(\Omega^{-1}\Big)^{qs}$           $-2$

$\Big(\hat{R}_2\Big)^i_{j;sm}\Omega_{il}$     $\bigg(\frac18R^{\hspace{1.5mm} i}_{j\; ms}+\frac14R^{\hspace{4mm} i}_{jm\; s}\bigg)\Omega_{il}$        2

$\Big(\hat{R}_2\Big)^{im}_{j;s}\Omega_{il}$     $\bigg(\frac18R^{\hspace{1.5mm} i}_{j\; ps}+\frac14R^{\hspace{3mm} i}_{jp\; s}\bigg)\Omega_{il}\Big(\Omega^{-1}\Big)^{pm}$         0

$\Big(\hat{R}_2\Big)^{ism}_{j}\Omega_{il}$     $\bigg(\frac18R^{\hspace{1.5mm} i}_{j\; pn}+\frac14R^{\hspace{3mm} i}_{jp\; n}\bigg)\Omega_{il}\Big(\Omega^{-1}\Big)^{ns}\Big(\Omega^{-1}\Big)^{pm}$        $-2$

$\Big(\hat{R}_2\Big)^{ism}_{j}$     $\bigg(\frac18R^{\hspace{1.5mm} m}_{j\hspace{2.5mm} kl}+\frac14R^{\hspace{2.5mm} m}_{jk\hspace{2.5mm} l}\bigg)\Big(\Omega^{-1}  \Big)^{ls}\Big(\Omega^{-1}\Big)^{ki}$     $-4$

$\Big(\hat{R}_2\Big)^i_{\bar{j};sm}\Omega_{il}$     $\bigg(\frac12\Riem{i}{\bar{j}}{m}{s}\bigg)\Omega_{il}$        2

$\Big(\hat{R}_2\Big)^{im}_{\bar{j};s}\Omega_{il}$     $\bigg(\frac12\Riem{i}{\bar{j}}{n}{s}\bigg)\Omega_{il}\Big(\Omega^{-1}\Big)^{nm}$         0

$\Big(\hat{R}_2\Big)^{ism}_{\bar{j}}\Omega_{il}$     $\bigg(\frac12\Riem{i}{\bar{j}}{n}{p}\bigg)\Omega_{il}\Big(\Omega^{-1}\Big)^{ns}\Big(\Omega^{-1}\Big)^{pm}$        $-2$

$\Big(\hat{R}_2\Big)^{ism}_{\bar{j}}$     $\bigg(\frac12\Riem{m}{\bar{j}}{k}{l}\bigg)\Big(\Omega^{-1}  \Big)^{ls}\Big(\Omega^{-1}\Big)^{ki}$     $ -4$

Explicit expression and total degree of the coefficients in (<ref>) and (<ref>).

If $\Sigma_3$ is a closed manifold, the globalized split RW action satisfies the dCME:

\[
\]

with $d_M=d_x+d_{\bar{x}}$ the sum of the holomorphic and antiholomorphic Dolbeault differentials on the target manifold $M$.
In the presence of boundary, the globalized split action satisfies the mdCME:

\begin{equation}
\label{iotasplit}
\iota_{\Tilde{Q}\surgS}\omega\surgS=\delta \tilde{\Sc}\surgS+\pi^*\alpha\surgSB,
\end{equation}

\begin{align}
\begin{split}
\Tilde{Q}\surgS{}&=\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg( -D\hat{\mathbf{A}}^i\frac{\delta}{\delta \hat{\mathbf{A}}^i}-D\hat{\mathbf{B}}_i\frac{\delta}{\delta \hat{\mathbf{B}}_i}+\sum^{\infty}_{k=0}\frac{1}{k!}\Big(\hat{R}_k\Big)^i_j((\hat{\mathbf{A}}+\hat{\mathbf{B}})^{\otimes k})dx^{j}\frac{\delta}{\delta \hat{\mathbf{A}}^i}\\
&\quad-\sum^{\infty}_{k=0}\frac{1}{k!}\Big(\hat{R}_{k}\Big)^l_j((\hat{\mathbf{A}}+\hat{\mathbf{B}})^{\otimes k})dx^{j}\Omega_{li}\frac{\delta}{\delta \hat{\mathbf{B}}_i}+\sum^{\infty}_{k=0}\frac{1}{k!}\Big(\hat{R}_k\Big)^i_{\bar{j}}((\hat{\mathbf{A}}+\hat{\mathbf{B}})^{\otimes k})dx^{{\bar{j}}}\frac{\delta}{\delta \hat{\mathbf{A}}^i},\\
&\quad-\sum^{\infty}_{k=0}\frac{1}{k!}\Big(\hat{R}_{k}\Big)^l _{\bar{j}}((\hat{\mathbf{A}}+\hat{\mathbf{B}})^{\otimes k})dx^{{\bar{j}}}\Omega_{li}\frac{\delta}{\delta  \hat{\mathbf{B}}_i}\bigg),
\end{split}
\end{align}

\begin{align}
\omega\surgS{}&=\varint_{T[1]\Sigma_3}\mu_{\Sigma_3}\bigg( \delta\hat{\mathbf{B}}_i\delta \hat{\mathbf{A}}^i\bigg),\\
\alpha\surgSB{}&=\varint_{T[1]\partial \Sigma_3}\mu_{\partial \Sigma_3}\bigg(\hat{\mathbf{B}}_i\delta \hat{\mathbf{A}}^i\bigg).
\end{align}

§ PERTURBATIVE QUANTIZATION OF THE GLOBALIZED SPLIT ROZANSKY–WITTEN MODEL

In the last section, we have formulated our globalized RW model as a $BF$-like theory. This allows us to quantize perturbatively the newly constructed globalized split RW model according to the Quantum BV-BFV framework <cit.> (see Section <ref> for an introduction). The quantization of the kinetic part of the action is analogous to the example of section 3 in <cit.>, since the theory reduces to the abelian $BF$ theory.. Hence we will be rather quick in the exposition referring to <cit.> for further details. We will focus our attention to the interacting part of the action (in our case this is actually just the globalization term), which has a rich, as well as complicated, structure. In particular, we will draw some comparison with the PSM, which has been considered in <cit.>.

§.§ Polarization

The recipe to perturbatively quantize a $BF$-like theory according to the quantum BV-BFV formalism starts by requiring the data of a polarization.

Following the result of Section <ref>, in the globalized split RW theory, the space of boundary fields splits as

\begin{equation}
\label{space_boundary_fields_sg}
\Tilde{\mathcal{F}}\surgSB=\Omega^\bullet(\partial\Sigma_3)\otimes\Omega^{\bullet,\bullet}(M)\otimes V\oplus\Omega^\bullet(\partial\Sigma_3)[2]\otimes\Omega^{\bullet,\bullet}(M)\otimes W.
\end{equation}

Since we split $T^{1,0}M$ into isotropic subspaces, by the isotropy condition the subspaces are, in particular, Lagrangian. Therefore, either of them can be used as a base or fiber of the polarization.

From now on we will drop the hat from the notation of the “globalized" superfields (e.g. $\hat{\mathbf{A}}^i$). Moreover, we will denote the coordinates on the base of the polarization by $\mathbb{A}^i$ or $\mathbb{B}^i$ and refer to this choice as $\mathbb{A}$- or $\mathbb{B}$-representation.

Let us choose a decomposition of the boundary $\partial \Sigma_3=\partial_1\Sigma_3 \sqcup\partial_2\Sigma_3$, where $\partial_1\Sigma_3$ and $\partial_2\Sigma_3$ are two compact manifolds. Here, we can define a polarization $\mathcal{P}$ by choosing the $\mathbb{A}$-representation on $\partial_1\Sigma_3$ and the $\mathbb{B}$-representation on $\partial_2\Sigma_3$.
The space of leaves of the associated foliations are  $\mathcal{B}_1:=\Omega^{\bullet}(\partial_1\Sigma_3)$ and $\mathcal{B}_2:=\Omega^{\bullet}(\partial_2\Sigma_3)[2]$, respectively. The space of boundary fields is $\mathcal{B}\bonP=\mathcal{B}_1\times \mathcal{B}_2\ni (\mathbb{A}^i, \mathbb{B}_i)$.

The BFV 1-form

\begin{equation}
\alpha\surgSBP=\varint_{\partial_1 \Sigma_3} \mathbf{B}_i\delta \mathbf{A}^i-\varint_{\partial_2 \Sigma_3} \delta \mathbf{B}_i \mathbf{A}^i
\end{equation}

and the quadratic part of the action (<ref>) is

\begin{equation}
\hat{\Sc}\surgSP=\varint_{\Sigma_3}\mathbf{B}_id\mathbf{A}^i-\varint_{\partial_2\Sigma_3}\mathbf{B}_i\mathbf{A}^i.
\end{equation}

§.§ Extraction of boundary fields

We split the space of fields as

\begin{align}
\begin{split}
\tilde{\mathcal{F}}\surgS &\rightarrow \tilde{\mathcal{B}}\bonP\oplus \mathcal{Y}\\ \label{quantiz:split_fields_1}
(\mathbf{A}^i, \mathbf{B}_i)&\mapsto (\Tilde{\mathbb{A}}^i,\tilde{\mathbb{B}}_i)\oplus (\underline{\mathbf{A}}^i,\underline{\mathbf{B}}_i),
\end{split}
\end{align}

where $\tilde{\mathcal{B}}\bonP$ denotes the bulk extension of $\mathcal{B}\bonP$ to $\tilde{\mathcal{F}}\surgS$ with $\Tilde{{\mathbb{A}}}^i$ and $\Tilde{{\mathbb{B}}}_i$ the extensions of the boundary fields $\mathbb{A}^i$ and $\mathbb{B}$ to the bulk space of fields $\Tilde{\mathcal{F}}\surgS$; $\underline{\mathbf{A}}^i$ and $\underline{\mathbf{B}}_i$ are the bulk fields, which are required to restrict to zero on $\partial_1\Sigma_3$ and $\partial_2\Sigma_3$, respectively. Here, the extensions are chosen to be singular: $\Tilde{\mathbb{A}}^i$ and $\tilde{\mathbb{B}}_i$ are required to restrict to zero outside the boundary (a choice pointed out first in <cit.>). The action reduces to

\begin{equation}
\hat{\Sc}\surgSP=\varint\sur \underline{\mathbf{B}}_id\underline{\mathbf{A}}^i-\bigg(\varint_{\partial_2\Sigma_3} \mathbb{B}_i\underline{\mathbf{A}}^i-\varint_{\partial_1\Sigma_3}\underline{\mathbf{B}}_i\mathbb{A}^i\bigg).
\end{equation}

§.§ Construction of O

At this point, we can construct the coboundary operator $\Omega_0$ by canonical quantization: we consider the boundary action and we replace any $\underline{\mathbf{B}}_i$ by $(-i\hbar \frac{\delta}{\delta \mathbb{A}^i})$ on $\partial_1\Sigma_3$, any $\underline{\mathbf{A}}^i$ by $(-i\hbar \frac{\delta}{\delta \mathbb{B}}_i)$ on $\partial_2\Sigma_3$. We obtain

\begin{equation}
\Omega_0=-i\hbar \bigg(\varint_{\partial_2\Sigma_3}d\mathbb{B}_i\frac{\delta}{\delta \mathbb{B}_i}+\varint_{\partial_1\Sigma_3}d\mathbb{A}^i\frac{\delta}{\delta \mathbb{A}^i}
\bigg).
\end{equation}

§.§ Choice of residual fields

The bulk contribution in the space of fields $\mathcal{Y}$ is further split into the space of residual fields $\mathcal{V}_{\Sigma_3}$ and a complement, the space of fluctuations fields $\mathcal{Y}'$, namely

𝒴       →𝒱_Σ_3⊕𝒴'

(𝐀^i,    𝐁_i)      ↦(𝖺^i,𝖻_i)⊕(α^i,β_i),

where $\mathsf{a}^i$ and $\mathsf{b}_i$ are the residual fields, whereas $\alpha^i$ and $\beta_i$ are the fluctuations. Note that the fluctuation $\alpha^i$ is required to restrict to zero on $\partial_1\Sigma_3$ while  $\beta_i$ is required to restrict to zero on $\partial_2\Sigma_3$.
In our case, the minimal space of residual fields is

\begin{equation}
\mathcal{V}_{\Sigma_3}=H^{\bullet}(\Sigma_3, \partial_1\Sigma_3)[0]\oplus H^{\bullet}(\Sigma_3, \partial_2\Sigma_3)[2]\ni (\mathsf{a}^i,\mathsf{b}_i).
\end{equation}

Here we can also define the BV Laplacian. To do it, pick a basis $\{[\chi_i]\}$ of $H^{\bullet}(\Sigma_3, \partial_1\Sigma_3)$ and its dual basis $\{[\chi^i]\}$ of $H^{\bullet}(\Sigma_3, \partial_2\Sigma_3)$ with representatives $\chi_i$ in $\Omega^{\bullet}(\Sigma_3, \partial_1\Sigma_3)$ and $\chi^i$ in $\Omega^{\bullet}(\Sigma_3, \partial_2\Sigma_3)$, with $\varint_{\Sigma_3}\chi_i\chi^j=\delta^j_i$. We can write the residual fields in a basis as

\begin{equation}
\begin{split}
\mathsf{a}^i&=\sum_k(z^{k}\chi_k)^i,\\
\mathsf{b}_i&=\sum_k (z^+_k\chi^{k})_i,
\end{split}
\end{equation}

where $\{z^k,z^+_k\}$ are canonical coordinates on $\mathcal{V}_{\Sigma_3}$ with BV symplectic form

\begin{equation}
\omega_{\mathcal{V}_{\Sigma_3}}=\sum_i(-1)^{\deg z^k}\delta z^+_k\delta z^k.
\end{equation}

Finally, the BV Laplacian on $\mathcal{V}_{\Sigma_3}$ is

\begin{equation}
\Delta_{\mathcal{V}_{\Sigma_3}}=\sum_i(-1)^{\deg z^k}\frac{\partial}{\partial z^k}\frac{\partial}{\partial z^+_k}.
\end{equation}

§.§ Gauge-fixing and propagator

We now have to fix a Lagrangian subspace $\mathcal{L}$ of $\mathcal{Y}'$. In the case of abelian BF theory, in <cit.>, the authors proved that such Lagrangian can be obtained from a contracting triple $(\iota,p,K)$
for the complex $\Omega^\bullet_{\underline{D}}(\Sigma_3)$.

In particular, the integral kernel of $K$ is the propagator, which we call $\eta$.

Since $K$ is actually the inverse of an elliptic operator (as shown in <cit.>), the propagator is singular on the diagonal of $\Sigma_3\times \Sigma_3$. Hence, we will define it as follows. Let

\begin{equation}
\mathrm{Conf}_2(\Sigma_3)=\{(x_1,x_2)\in \Sigma_3\mid x_1\neq x_2\},
\end{equation}

and let $\iota_{\mathfrak{D}}$ be the inclusion of

\begin{equation}
\mathfrak{D}\coloneqq\{x_1\times x_2\in (\partial_1\Sigma_3\times \Sigma_3)\cup(\Sigma_3\times \partial_2\Sigma_3)\mid x_1\neq x_2\}
\end{equation}

into $\mathrm{Conf}_2(\Sigma_3)$. Then the propagator is the 2-form $\eta\in \Omega^2(\mathrm{Conf}_2(\Sigma_3),\mathfrak{D})$, where

\begin{equation}
\Omega^\bullet(\mathrm{Conf}_2(\Sigma_3),\mathfrak{D})=\{\gamma\in \Omega^\bullet(\mathrm{Conf}_2(\Sigma_3))\mid \iota^{*}_{\mathfrak{D}}\gamma=0\}.
\end{equation}

\begin{equation}
\label{quantiz:prop}
\eta(x_1,x_2)=\frac{1}{T_{\Sigma_{3}}}\frac{1}{i\hbar}\varint_{\mathcal{L}}e^{\frac{i}{\hbar}\hat{\Sc}\surgSP}\pi^*_1\alpha^i(x_1)\pi^*_2\beta_i(x_2),
\end{equation}

with $\pi_1,\pi_2$ the projections from $M\times M$ to its first and second factor.
The coefficient $T_{\Sigma_3}$ is related to the Reidemeister torsion on $\Sigma_3$ as shown in <cit.>. However, its precise nature is irrelevant for the purposes of the present paper.

§.§ The quantum state

We can sum up the splittings we have made so far as

\begin{align}
\begin{split}
\tilde{\mathcal{F}}\surgS &\rightarrow \mathcal{B}\bonP\times \mathcal{V}^{\mathcal{P}}_{\Sigma_3}\times \mathcal{Y}'\\
(\mathbf{A}^i, \mathbf{B}_i)&\mapsto (\mathbb{A}^i,\mathbb{B}_i)+ (\mathsf{a}^i,\mathsf{b}_i)+ (\alpha^i,\beta_i).
\end{split}
\end{align}

As a result of the procedure detailed in <cit.>, this is referred to as good splitting.

According to the splitting of the space of fields, the action decomposes as

\begin{equation}
\Sc\surgSP=\hat{\Sc}\surgSP+\hat{\Sc}^{\, \text{pert}}+\Sc^{\, \text{res}}+\Sc^{\, \text{source}},
\end{equation}

\begin{align}
\hat{\Sc}\surgSP&=\varint_{\Sigma_3}\beta_id\alpha^i,\\
\hat{\Sc}^{\, \text{pert}}&=\varint\sur \mathcal{V}(\underline{\mathbf{A}}, \underline{\mathbf{B}}),\\
\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{align}

where $\hat{\Sc}^{\, \text{pert}}$ is an interacting term made up by a density-valued function $\mathcal{V}$ which depends on the fields but not on their derivatives (by assumption).

The state is given by:

\begin{equation}
\begin{split}
\hat{\psi}_{\Sigma_3}(\mathbb{A}, \mathbb{B}, \mathsf{a}, \mathsf{b})&=\varint_{(\alpha, \beta)\in \mathcal{L}}e^{\frac{i}{\hbar}\Sc\surgSP (\mathbb{A}+\mathsf{a}+\alpha,\mathbb{B}+\mathsf{b}+\beta)}\mathscr{D}[\alpha]\mathscr{D}[\beta]\\
&=e^{\frac{i}{\hbar}\Sc^{\, \text{res}}}\varint_{\mathcal{L}}e^{\frac{i}{\hbar}\hat{\Sc}\surgSP}e^{\frac{i}{\hbar}\hat{\Sc}^{\, \text{pert}}}e^{\frac{i}{\hbar}\Sc^{\text{source}}},
\end{split}
\end{equation}

where we denote by $\mathscr{D}$ a formal measure on $\mathcal{L}$.
The idea here is to compute the integral through a perturbative expansion, hence let us expand the exponentials as

\begin{equation}
\begin{split}
\hat{\psi}_{\Sigma_3}(\mathbb{A}, \mathbb{B}, \mathsf{a}, \mathsf{b})&=\sum_{k,l,m}\frac{1}{k!l!m!}(-1)^{k+m}\bigg(\varint_{\partial_2\Sigma_3}\mathbb{B}_i\mathsf{a}^i-\varint_{\partial_1\Sigma_3}\mathsf{b}_i\mathbb{A}^i\bigg)^k\times\\
&\quad\times\varint_{\mathcal{L}}e^{\frac{i}{\hbar}\hat{\Sc}\surgSP}\bigg(\varint\sur \mathcal{V}(\underline{\mathbf{A}}, \underline{\mathbf{B}})\bigg)^l
\bigg(\varint_{\partial_2\Sigma_3}\mathbb{B}_i\alpha^i-\varint_{\partial_1\Sigma_3}\beta_i\mathbb{A}^i\bigg)^m.
\end{split}
\end{equation}

In the globalized split RW model, the interaction term is actually given by the globalization terms (the second and third terms in the action <ref>).

After having expanded the globalization terms in in residual fields and in fluctuations, the integration over $\mathcal{L}$ can be solved by using the Wick theorem.

§.§ Feynman rules

In this section, we are going to introduce the Feynman rules needed to define precisely the quantum state of our theory.

Since our aim is to prove the mdQME for the globalized split RW model, we will need to take care of the quantum Grothendieck BFV operator. This is a coboundary operator in which higher functional derivatives may appear (and as we will see they will indeed be present). As explained in <cit.>, higher functional derivatives requires a sort of “regularization". This is provided by the composite fields, which we denote by square brackets $[\enspace]$ (e.g. for the boundary field $\mathbb{B}$, we will write $[\mathbb{B}_{i_1}\dots \mathbb{B}_{i_k}]$)[See Section  for a short introduction.].

A globalized split RW 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

* bulk vertices can have any valence,

* type 1 boundary vertices carry any number of incoming half-edges (and no outgoing half-edges),

* type 2 boundary vertices carry any number of outgoing half-edges (and no incoming half-edges),

* multiple edges and loose half-edges (leaves) are allowed.

A labeling of a Feynman graph is a function from the set of half-edges to $\{1,\dots,\dim V\}$.

In our case our source manifold $\Sigma_3$ has boundary $\partial \Sigma_3=\partial_1 \Sigma_3 \sqcup \partial_2 \Sigma_3$, let $\Gamma$ be a Feynman graph and define

\begin{equation}
\mathrm{Conf}_\Gamma(\Sigma_3)\coloneqq \mathrm{Conf}_{V_{\text{bulk}}}(\Sigma_3) \times \mathrm{Conf}_{V_{\partial_1}}(\partial_1 \Sigma_3) \times \mathrm{Conf}_{V_{\partial_2}}(\partial_2 \Sigma_3).
\end{equation}

The Feynman rules are given by a map associating to a Feynman graph $\Gamma$ a differential form $\omega_\Gamma \in \Omega^\bullet(\mathrm{Conf}_\Gamma(\Sigma_3))$.

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 decorate the graph according to the following rules, namely, the Feynman rules:

* Bulk vertices in $\Sigma_3$ decorated by “globalized vertex tensors"

\begin{equation}
\begin{split}
\Big(\hat{R}_k\Big)^{i_1\dots i_s}_{j;j_1\dots j_t}dx^j &\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} \Big(\hat{R}_k\Big)^i_j((\underline{\mathbf{A}}+\underline{\mathbf{B}})^{\otimes k})(\Omega_{il}\underline{\mathbf{A}}^l+\underline{\mathbf{B}}_i)dx^{j}\\
\Big(\hat{R}_k\Big)^{i_1\dots i_s}_{{\Bar{j}};j_1\dots j_t} dx^{\Bar{j}}&\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} \Big(\hat{R}_k\Big)^i_{\Bar{j}}((\underline{\mathbf{A}}+\underline{\mathbf{B}})^{\otimes k})(\Omega_{il}\underline{\mathbf{A}}^l+\underline{\mathbf{B}}_i)dx^{\Bar{j}}
\end{split}
\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 (respectively in-)oriented half-edges.

* 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 $\partial_1 \Sigma_3$.

* Boundary vertices $v \in V_{\partial_2}$ on $\partial_2 M$ with outgoing half-edges labeled $j_1, \dots, j_l$ are decorated by $[\mathbb{B}_{j_1} \dots \mathbb{B}_{j_l}]$ evaluated at the point on $\partial_2 \Sigma_3$.

* 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.

* 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 $\iota(v)$.

The Feynman Rules are represented in Figs. <ref>, <ref> and <ref>.

[every blob=/tikz/fill=white!30,/tikz/inner sep=1pt]
[blob] (m) at (0,-2) $\mathsf{a}$;
(a) at (0,0) $x$ ;
(a) – [fermion, edge label' = $i$] (m)
[right=3em of m] \(=\mathsf{a}^i(x)\);

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
[blob] (m) at (0,-2) $\mathsf{b}$;
(a) at (0,0) $x$ ;
(m) – [fermion, edge label = $i$] (a)
[right=3em of m] \(=\mathsf{b}_i(x)\);

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(a) at (0,0) $x$;
(b) at (2,0) $y$ ;
(a) – [fermion, edge label = $i\hspace{5mm} j$] (b)
[right=3em of b] \(=\delta^i_j\eta(x,y)\);

Feynman rules for residual fields and propagator

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]

(a) at (-1,0);
(b) at (1,0);
(m1) at (0, 1);

(a) – m [dot] – (b),
(m1) – [fermion] m

[below=0.75em of m] \(\mathbb{B}\);
[left=0.25em of a] \(\partial_2 \Sigma_3\);

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]

(a) at (-1,0);
(b) at (1,0);
(m1) at (0, -1);

(a) – m [dot] – (b),
m – [fermion] (m1)

[above=0.75em of m] \(\mathbb{A}\);
[left=0.25em of a] \(\partial_1 \Sigma_3\);

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(a) at (-1.2, 0.35);
(b) at (-0.3, 0.79);
(c) at (+1.2, 0.4);
(e) at (-1.2, -0.35);
(f) at (-0.3, -0.79);
(g) at (+1.2, -0.4);

(a) – [] m [dot],
(b) – [] m [dot],
(c) – [] m [dot],
(e) – [] m [dot],
(f) – [] m [dot],
(g) – [] m [dot],

[right=7em of m] \(\quad=\Big(\hat{R}_k\Big)^{i_1\dots i_s}_{j;j_1\dots j_t}dx^j\);

[above=0.2em of m, label=80:\(\dots\)] ;

[] at (-1.35,0.4) \(j_1\);
[] at (-0.45,1) \(j_2\);
[] at (1.4, 0.4) \(j_t\);
[] at (-1.35,-0.4) \(i_1\);
[] at (-0.45,-1) \(i_2\);
[] at (1.4, -0.4) \(i_s\);

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(a) at (-1.2, 0.35);
(b) at (-0.3, 0.79);
(c) at (+1.2, 0.4);
(e) at (-1.2, -0.35);
(f) at (-0.3, -0.79);
(g) at (+1.2, -0.4);

(a) – [] m [dot,red],
(b) – [] m [dot,red],
(c) – [] m [dot,red],
(e) – [] m [dot,red],
(f) – [] m [dot,red],
(g) – [] m [dot,red],

[right=7em of m] \(\quad=\Big(\hat{R}_k\Big)^{i_1\dots i_s}_{{\Bar{j}};j_1\dots j_t}dx^{\Bar{j}}\);

[above=0.2em of m, label=80:\(\dots\)] ;

[] at (-1.35,0.4) \(j_1\);
[] at (-0.45,1) \(j_2\);
[] at (1.4, 0.4) \(j_t\);
[] at (-1.35,-0.4) \(i_1\);
[] at (-0.45,-1) \(i_2\);
[] at (1.4, -0.4) \(i_s\);

Feynman rules for boundary fields and interaction vertices: we denote with a black dot the vertices arising from the $(2,0)$ part of the curvature (i.e. the terms corresponding to the term $\Sc_R$ in the action) and with a red dot the ones coming from the $(1,1)$ part (i.e. the terms corresponding to the term $\Sc_{\Bar{R}}$ in the action). Informally, we will call the first type of vertices “black" vertex and the second one “red" vertex.

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]

(a) at (-2,0);
(b) at (2,0);
(d) at (0.4, 0.75);
(e) at (1, 0.75);
(f) at (-1, 0.75);
(a) – m [dot] – (b),
(d) – [anti fermion] m,
(e) – [anti fermion] m,
(f) – [anti fermion] m

[below=0.75em of m] \([\mathbb{A}^{i_1}\dots \mathbb{A}^{i_k}]\);
[left=0.25em of a] \(\partial_1 \Sigma_3\);
at (-0.2, 0.5) $\dots$;
at (-1.1, 0.6) $i_k$;
at (0.5, 1) $i_2$;
at (1.2, 0.9) $i_1$;

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]

(a) at (-2,0);
(b) at (2,0);
(d) at (0.4, 0.75);
(e) at (1, 0.75);
(f) at (-1, 0.75);
(a) – m [dot] – (b),
(d) – [fermion] m,
(e) – [fermion] m,
(f) – [fermion] m

[below=0.75em of m] \([\mathbb{B}_{j_1}\dots \mathbb{B}_{j_l}]\);
[left=0.25em of a] \(\partial_2 \Sigma_3\);
at (-0.2, 0.5) $\dots$;
at (-1.1, 0.6) $j_l$;
at (0.5, 1) $j_2$;
at (1.2, 0.9) $j_1$;

Feynman rules for the composite fields.

The full covariant quantum state for globalized split RW theory is defined analogously as in <cit.>.

Let $\Sigma_3$ be a 3-dimensional manifold with boundary. Consider the data of a globalized split RW theory which consists of the globalized split space of fields $\tilde{\F}\surgS$ as in (<ref>), the globalized split space of boundary fields $\tilde{\F}\surgSB$ as in (<ref>), a polarization $\mathcal{P}$ on $\tilde{\F}\surgSB$  a good splitting $\tilde{\F}\surgS=\mathcal{B}_{\partial \Sigma_3}^{\mathcal{P}} \times \mathcal{V}_{\Sigma_3}^{\mathcal{P}} \times \mathcal{Y}'$ and $\mathcal{L} \subset \mathcal{Y}'$, the gauge-fixing Lagrangian. We can define the full quantum state for the globalized split RW theory by the formal power series

\begin{equation}
\boldsymbol{\hat{\psi}}\surgR(\mathbb{A},\mathbb{B};\mathsf{a},\mathsf{b})=T_{\Sigma_3}\exp\bigg(\frac{i}{\hbar}\sum_{\Gamma}\frac{(-i\hbar)^{\text{loops}(\Gamma)}}{|\text{Aut}(\Gamma)|}\varint_{\text{C}_\Gamma(\Sigma_3)}\omega_{\Gamma}(\mathbb{A}, \mathbb{B}; \mathsf{a}, \mathsf{b})
\bigg).
\end{equation}

§ PROOF OF THE MODIFIED DIFFERENTIAL QUANTUM MASTER EQUATION

In the BV-BFV formalism on manifolds with boundary we expect the mQME to hold. This is a condition which requires the quantum state to be closed under a certain coboundary operator (see <cit.>). However, in the context of a globalized AKSZ theory, this condition becomes more complicated. The new condition is called modified differential Quantum Master Equation (mdQME). We refer to <cit.> for a discussion of the classical and quantum aspects of this condition. An extension for this discussion for manifolds with boundary was provided in<cit.>. Finally, in <cit.> the mdQME for anomaly-free, unimodular split AKSZ theories was proven, and later on in <cit.> for the globalized PSM.

Our aim in this section is to prove the mdQME for the globalized split RW model, namely

\begin{equation}
\nabla_{\text{G}}\boldsymbol{\hat{\psi}}\surgR=0,
\end{equation}

where $\nabla_{\text{G}}$ is the quantum Grothendieck BFV (qGBFV) operator  and $\boldsymbol{\hat{\psi}}\surgR$ is the full covariant quantum state for the globalized split RW theory.
As we will see, the proof follows almost verbatim from the proof of the mdQME in <cit.>. Before addressing the proof, we focus on the qGBFV operator and we discuss the construction of the full BFV boundary operator.

§.§ The quantum Grothendieck BFV operator

Inspired by <cit.>, we define the qGBFV operator for the globalized split RW model as

\begin{equation}
\nabla_\mathrm{G}\coloneqq\bigg(d_x+d_{\Bar{x}}-i\hbar \Delta_{\mathcal{V}_{\Sigma_3, x}}+\frac{i}{\hbar}\boldsymbol{\Omega}_{\partial \Sigma_3}\bigg),
\end{equation}

with $\boldsymbol{\Omega}_{\partial \Sigma_3}$ the full BFV boundary operator

\begin{equation}
\boldsymbol{\Omega}_{\partial \Sigma_3}=\boldsymbol{\Omega}^{\mathbb{A}}_{\partial\Sigma_3}+\boldsymbol{\Omega}^{\mathbb{B}}_{\partial\Sigma_3}=\Omega^{\mathbb{A}}_{0}+\boldsymbol{\Omega}^{\mathbb{A}}_{\text{pert}}+\Omega^{\mathbb{B}}_0+\boldsymbol{\Omega}^{\mathbb{B}}_{\text{pert}},%+\underbrace{\Omega^{\mathbb{
\end{equation}

\begin{equation}
\begin{split}
\Omega^{\mathbb{A}}_0&=-i\hbar \varint_{\partial_1\Sigma_3}d\mathbb{A}^i\frac{\delta}{\delta \mathbb{A}^i},\\
\Omega^{\mathbb{B}}_0&=-i\hbar \varint_{\partial_2\Sigma_3}d\mathbb{B}_i\frac{\delta}{\delta \mathbb{B}_i}.
\end{split}
\end{equation}

and $\boldsymbol{\Omega}^{\mathbb{A}}_{\text{pert}}$ and $\boldsymbol{\Omega}^{\mathbb{B}}_{\text{pert}}$ are given by Feynman diagrams collapsing to the boundary in the $\mathbb{A}$-representation and $\mathbb{B}$-representation, respectively.

Note that $\nabla_{\text{G}}$ and $\Omega_{\partial \Sigma_3}$ are inhomogeneous forms on the holomorphic symplectic manifold $M$ since the globalized term in the action is a 1-form on $M$. Explicitly, for example in the $\mathbb{B}$-representation, we can decompose the $\boldsymbol{\Omega}^{\mathbb{B}}_{\mathrm{pert}}$ as

\begin{equation}
\boldsymbol{\Omega}^{\mathbb{B}}_{\mathrm{pert}}=\underbrace{\boldsymbol{\Omega}^{\mathbb{B}}_{1,0}+\boldsymbol{\Omega}^{\mathbb{B}}_{0,1}}_{\coloneqq  \boldsymbol{\Omega}^{\mathbb{B}}_{(1)}}+\underbrace{\boldsymbol{\Omega}^{\mathbb{B}}_{2,0}+\boldsymbol{\Omega}^{\mathbb{B}}_{1,1}+\boldsymbol{\Omega}^{\mathbb{B}}_{0,2}}_{\coloneqq\boldsymbol{\Omega}^{\mathbb{B}}_{(2)}}+\dots.
\end{equation}

and similarly in the $\mathbb{A}$-representation.

In the next section, we proceed to give an explicit expression for the BFV boundary operator in the $\mathbb{B}$ and $\mathbb{A}$ representation. We start with the former.

§.§ BFV boundary operator in the $\mathbb{B}$-representation

Let us remind the reader about the general form of the BFV boundary operator in the $\mathbb{B}$-representation for a split AKSZ theory <cit.>:

\begin{equation}
\boldsymbol{\Omega}_{\text{pert}}^{\mathbb{B}}\coloneqq \sum_{n,k \geq 0}\sum_{\Gamma}\frac{(i \hbar)^{\text{loops($\Gamma)$}}}{\mid \Aut(\Gamma)\mid} \varint_{\partial_2M}\bigg(\sigma_{\Gamma}\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{equation}

In order to find an explicit expression for the BFV boundary operator, we adopt the strategy in <cit.> to find the BFV boundary operator in the $\mathbb{E}$-representation for the PSM. Their idea was to use the degree counting. Indeed, in general, the form $\sigma_{\Gamma}$ is obtained as the integral over the compactification $\tilde{\mathrm{C}}_{\Gamma}(\mathbb{H}^d)$ of the  open configuration space modulo scaling and translation, with $\mathbb{H}^d$ the $d$-dimensional upper half-space:

\begin{equation}
\label{mdQME:integral}
\sigma_{\Gamma}=\varint_{\tilde{\mathrm{C}}_{\Gamma(\mathbb{H}^d)}} \omega_{\Gamma},
\end{equation}

where $\omega_{\Gamma}$ is the product of limiting propagators at the point $p$ of collapse and vertex tensors. Note that in order for the integral (<ref>) not to vanish the form degree of $\omega_\Gamma$ has to be the same as the dimension of $\tilde{\mathrm{C}}_{\Gamma}(\mathbb{H}^d)$. This gives constraints to the number of points in the bulk as well as points in the boundary admitted. We will apply this degree counting to our case, where, since we have $d=3$, the dimension of the compactified configuration space $\tilde{\mathrm{C}}_{\Gamma}(\mathbb{H}^3)$ is $\dim\tilde{\mathrm{C}}_{\Gamma}(\mathbb{H}^3)=3n+2m-3$, with $n$ the number of bulk vertices and $m$ the number of boundary vertices in $\Gamma$.

By using this procedure, in <cit.> it was possible to find an explicit expression for the BFV boundary operator in the $\mathbb{E}$-representation for the PSM. As we will see, for us this is not possible. One could say that the cause is the nature of the RW model reflected in a dramatic increment in the number of Feynman rules as we go on in the $k$-index for the globalized terms in the action (see Eq. (<ref>)).
To see this in practice, let us show explicitly the Feynman rules for the terms in (<ref>) and (<ref>), which we sum up in Table <ref>.

Vertex         Feynman rule      Total degree     Name

$(\hat{R}_0\big)^i_{j}$      15 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0, -0.75);

(d) – [fermion] m [dot]

0     I

$(\hat{R}_0\big)^i_{j}$      15 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0, -0.75);

(d) – [anti fermion] m [dot]

2     II

$\Big(\hat{R}_1\Big)^i_{j;s}$      20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0, -0.75);
(c) at (0, 0.75);

(d) – [fermion] m [dot] – [anti fermion] (c)

0     III

$\Big(\hat{R}_1\Big)^{is}_{j}$     20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0, -0.75);
(c) at (0, 0.75);

(d) – [anti fermion] m [dot] – [anti fermion] (c)

2     IV

$\Big(\hat{R}_1\Big)^{is}_{j}$     20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0, -0.75);
(c) at (0, 0.75);

(d) – [anti fermion] m [dot] – [fermion] (c)

4     V

$\Big(\hat{R}_2\Big)^i_{j;sm}$      20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0.5,  0.5);
(c) at (-0.5, 0.5);
(a) at (0, -0.7);

(d) – [fermion] m [dot] – [anti fermion] (c);
m – [anti fermion] (a)

0     VI

20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0.5,  0.5);
(c) at (-0.5, 0.5);
(a) at (0, -0.7);

(d) – [anti fermion] m [dot] – [anti fermion] (c);
m – [anti fermion] (a)

2     VII

Vertex         Feynman rule      Total degree     Name

$\Big(\hat{R}_2\Big)^{ism}_{j}$     20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0.5,  0.5);
(c) at (-0.5, 0.5);
(a) at (0, -0.7);

(d) – [anti fermion] m [dot] – [fermion] (c);
m – [anti fermion] (a)

4     VIII

20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0.5,  0.5);
(c) at (-0.5, 0.5);
(a) at (0, -0.7);

(d) – [anti fermion] m [dot] – [fermion] (c);
m – [fermion] (a)

6    IX

$\Big(\hat{R}_2\Big)^i_{\bar{j};sm}$     20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0.5,  0.5);
(c) at (-0.5, 0.5);
(a) at (0, -0.7);

(d) – [fermion] m [dot, red] – [anti fermion] (c);
m – [anti fermion] (a)

0    X

$\Big(\hat{R}_2\Big)^{im}_{\bar{j};s}$     20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0.5,  0.5);
(c) at (-0.5, 0.5);
(a) at (0, -0.7);

(d) – [anti fermion] m [dot, red] – [anti fermion] (c);
m – [anti fermion] (a)

2    XI

$\Big(\hat{R}_2\Big)^{ism}_{\bar{j}}$    20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0.5,  0.5);
(c) at (-0.5, 0.5);
(a) at (0, -0.7);

(d) – [anti fermion] m [dot, red] – [fermion] (c);
m – [anti fermion] (a)

4    XII

20 pt-
[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0.5,  0.5);
(c) at (-0.5, 0.5);
(a) at (0, -0.7);

(d) – [anti fermion] m [dot, red] – [fermion] (c);
m – [fermion] (a)

6    XIII

Feynman rules for the globalization terms in the action (<ref>).

Notice how the structure of the Feynman rules repeats similarly at each order (e.g. for $R_0$ we have 2 Feynman rules with degrees 0 and 2, respectively, while for $R_1$ we have 3 graphs with degrees 0, 2, 4). Hence, it is easy to understand how this works for higher order terms.
From there, one can notice that we have two types of vertices:

* vertices which are 1-forms in $dx^{i}$: we will denote them by a black dot ($\bullet$) and refer to them as black vertices;

* vertices which are 1-forms in $dx^{{\bar{i}}}$: we will denote them by a red dot ($\bullet$) and refer to them as red vertices.

In our computations, we will limit ourselves to the Feynman rules in Table <ref>, these are already enough to get a feeling about what is going on and even understand the behaviour of higher order terms, when possible. By using the names in the Table, since $n=\textrm{I}+\textrm{II}+\textrm{III}+\textrm{IV}+\textrm{V}+\textrm{VI}+\textrm{VII}+\textrm{VIII}+\textrm{IX}+\textrm{X}+\textrm{XI}+\textrm{XII}+\textrm{XIII}$ is the sum of the vertices, the degree counting produces the following equation

\begin{equation}
\begin{split}
\textrm{I}+\textrm{II}+\textrm{III}+\textrm{IV}+\textrm{V}+\textrm{VI}+\textrm{VII}+\textrm{VIII}&+\textrm{IX}+\textrm{X}+\textrm{XI}+\textrm{XII}+\textrm{XIII}+2m-3=\\
\end{split}
\end{equation}

where on the right hand side we are taking into account that in the $\mathbb{B}$-representation, the arrows leaving the globalization vertex have to stay inside the collapsing subgraph. If this is not the case, by the boundary conditions on the propagator <cit.>, the result would be zero.

First, let us focus on the black vertices (i.e. vertices I–IX). The equation reduces to

\begin{equation}
\label{degree counting eq}
\end{equation}

The Feynman diagrams contributing to the BFV boundary operator are those whose vertices solve the equation (<ref>). Hence, let us solve the equation case-by-case.
Up to one bulk vertex, with the Feynman rules I–IX we have one diagram (see Fig. <ref>).

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (0, 0.75);
(a) at (-1.25, -0.75);
(b) at (1.25, -0.75);
[dot] (x) at (0, -0.75);
(d) – [fermion] m [dot],
(a) – (b),
m –[fermion] (x)
[left=0.25em of a] \(\partial_2 \Sigma_3\);

(-0.9,-0.75) rectangle (0.9,0.9);
(0,-0.65) circle(0.9);

First graph with a single black vertex contributing to the BFV boundary operator.

From Fig. <ref>, we notice that in order to have a degree 1 operator which satisfies the degree counting for higher order terms we need vertices with an even number of heads and tails. We show the first higher order contributions in Fig. <ref>, while a general diagram contributing to the BFV operator
is exhibited in Fig. <ref>.

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (-0.75, 0.75);
(c) at (0.75, 0.75);
(a) at (-1.25, -0.75);
(b) at (1.25, -0.75);
[dot] (x) at (-0.5, -0.75);
[dot] (y) at (0.5, -0.75);
(d) – [fermion] m [dot],
(c) – [fermion] m,
(a) – (b),
m –[fermion] (x),
m –[fermion] (y)
[left=0.25em of a] \(\partial_2 \Sigma_3\);

(-0.9,-0.75) rectangle (0.9,0.9);
(0,-0.65) circle(0.9);

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (-0.75, 0.75);
(c) at (0.75, 0.75);
(e) at (0, 0.75);
(a) at (-1.25, -0.75);
(b) at (1.25, -0.75);
[dot] (x) at (-0.5, -0.75);
[dot] (y) at (+0.5, -0.75);
[dot] (z) at (0, -0.75);
(d) – [fermion] m [dot],
(c) – [fermion] m,
(e) – [fermion] m,
m  – [fermion] (x),
m  – [fermion] (y),
m  – [fermion] (z),
(a) – (b)
[left=0.25em of a] \(\partial_2 \Sigma_3\);
[left=0.25em of a] \(\partial_2 \Sigma_3\);

(-0.9,-0.75) rectangle (0.9,0.9);
(0,-0.65) circle(0.9);

Second and third graph with a single black vertex contributing to the BFV boundary operator.

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (-1.4, 0.7);
(c) at (0.9, 0.95);
(e) at (-0.8, 0.95);
(a) at (-1.65, -0.95);
(b) at (1.65, -0.95);
[dot] (x) at (-0.9, -0.95);
[dot] (y) at (+0.85, -0.95);
[dot] (z) at (-0.4, -0.95);
(d) – [fermion] m [dot],
(c) – [fermion] m,
(e) – [fermion] m,
m  – [fermion] (x),
m  – [fermion] (y),
m  – [fermion] (z),
(a) – (b)

[left=0.25em of a] \(\partial_2 \Sigma_3\);
[left=0.25em of a] \(\partial_2 \Sigma_3\);

(-1.2,-0.95) arc (180:0:1.2);
at (0,0.65) …;
at (0.2,-0.75) …;

A general Feynman diagram contributing to the BFV operator in the $\mathbb{B}$-representation up to one black bulk vertex.

Concerning the red vertices, the graphs contributing to the BFV operator up to one bulk vertex will start to appear from the vertices associated to the term $(R_3)_{\bar{j}}dx^{\bar{j}}$ (coming from the thid term in the action <ref>). Taking this into account, the general term of the diagrams with a red vertex is shown in Fig. <ref>.

[every blob=/tikz/fill=white!30,/tikz/inner sep=0.5pt]
(d) at (-1.4, 0.7);
(c) at (0.9, 0.95);
(e) at (-0.8, 0.95);
(a) at (-1.65, -0.95);
(b) at (1.65, -0.95);
[dot] (x) at (-0.9, -0.95);
[dot] (y) at (+0.85, -0.95);
[dot] (z) at (-0.4, -0.95);
(d) – [fermion] m [red, dot],
(c) – [fermion] m,
(e) – [fermion] m,
m  – [fermion] (x),
m  – [fermion] (y),
m  – [fermion] (z),
(a) – (b)

[left=0.25em of a] \(\partial_2 \Sigma_3\);
[left=0.25em of a] \(\partial_2 \Sigma_3\);

(-1.2,-0.95) arc (180:0:1.2);
at (0,0.65) …;
at (0.2,-0.75) …;

A general Feynman diagram contributing to the BFV operator in the $\mathbb{B}$-representation up to one red bulk vertex. In particular, the graph with a total number of 4 arrows (2 entering and 2 leaving the red vertex) is the first non-zero contribution.

These considerations prove the following proposition.

Consider the globalized split RW model, in the $\mathbb{B}$-representation, the first contribution to $\boldsymbol{\Omega}^{\mathbb{B}}_{\mathrm{pert}}$ is given by $\boldsymbol{\Omega}^{\mathbb{B}}_{(1)}= \boldsymbol{\Omega}^{\mathbb{B}}_{1,0}+ \boldsymbol{\Omega}^{\mathbb{B}}_{0,1}$ with

\begin{equation}
\begin{split}
\boldsymbol{\Omega}^{\mathbb{B}}_{1,0}&=\sum_{\substack{k\geq1,S_1, \dots S_k\\ i_1,\dots, i_k, j_1, \dots, j_k}}\frac{(-i\hbar)^k}{(k+S_1+\dots+S_k)!}\varint_{\partial_2\Sigma_3}\Big(\hat{R}_{2k-1}\Big)^{i_1\dots i_k}_{j;j_1\dots j_k} dx^j[\mathbb{B}_{i_1}\mathbb{B}_{S_1}]\dots \\
&\hspace{8cm}\times[\mathbb{B}_{i_k}\mathbb{B}_{S_k}] \frac{\delta^{|j_1+\dots+j_k|+|S_1|+\dots+|S_k|}}{\delta [\mathbb{B}_{j_1}\dots \mathbb{B}_{j_k}][\delta \mathbb{B}_{S_1}]\dots [\delta \mathbb{B}_{S_k}]}\\
\boldsymbol{\Omega}^{\mathbb{B}}_{0,1}&=\sum_{\substack{k\geq 2,S_1, \dots S_k\\ i_1,\dots, i_k, j_1, \dots, j_k}}\frac{(-i\hbar)^k}{(k+S_1+\dots+S_k)!}\varint_{\partial_2\Sigma_3}\Big(\hat{R}_{2k-1}\Big)^{i_1\dots i_k}_{\bar{j};j_1\dots j_k} dx^{\bar{j}}[\mathbb{B}_{i_1}\mathbb{B}_{S_1}]\dots\\
&\hspace{8cm}\times[\mathbb{B}_{i_k}\mathbb{B}_{S_k}] \frac{\delta^{|j_1+\dots+j_k|+|S_1|+\dots+|S_k|}}{\delta [\mathbb{B}_{j_1}\dots \mathbb{B}_{j_k}][\delta \mathbb{B}_{S_1}]\dots [\delta \mathbb{B}_{S_k}]}.
\end{split}
\end{equation}

For $n>1$, the situation gets more complicated. We solve equation (<ref>) numerically. Empirically, for an even number of bulk vertices, we witness the absence of solutions. This implies immediately $\boldsymbol{\Omega}^{\mathbb{B}}_{(2)}=0$.

In the case $n=3$, the number of Feynman diagrams for the vertices I–IX increases dramatically with respect to the $n=1$ case. This increment is tamed since the necessity of having a degree 1 operator will decrease their number. However, we are not able to provide an explicit as well as general form for the BFV operator along the same lines as in Proposition <ref>. We rely on examples which we show in Appendix <ref>.

Here we are assuming that the dimension of our target manifold $M$ is at least 4, if this would not be the case, then we would not have the 3 bulk vertices contribution to the BFV boundary operator. Hence, the number of bulk vertices allowed is bounded by the dimension of $M$. This was already noticed in <cit.>. The difference here is that this reflects the “odd Grassmanian nature" of the RW model with respect to CS theory (see Remark <ref>).

§.§ BFV boundary operator in the $\mathbb{A}$-representation

In the $\mathbb{A}$-representation, the arrows coming from the globalized vertices are allowed to leave the collapsing subgraph. Therefore, our arguments about the degree counting are not valid here. However since the coboundary operator has a total degree 1, while $\mathbb{A}^i$ has total degree 0, we can have at most 1 bulk vertex, i.e. $\boldsymbol{\Omega}^{\mathbb{A}}_{\mathrm{pert}}=\boldsymbol{\Omega}^{\mathbb{A}}_{1,0}+\boldsymbol{\Omega}^{\mathbb{A}}_{0,1}$ with

\begin{equation}
\begin{split}
\boldsymbol{\Omega}^{\mathbb{A}}_{1,0}&= \sum_{k\geq0}\,\,\varint_{\partial_1\Sigma_3}\sum_{J_1,\dots, J_r, I_1,\dots, I_s} \frac{(-i\hbar)^{|I_1|+\dots +|I_s|}}{(|I_1|+\dots +|I_s|)!}\Big(\hat{R}_k\Big)^{I_1\dots I_s}_{j;J_1\dots J_r}dx^j\prod^{r+s=k+1}_{r=1, s=1}[\mathbb{A}^{J_r}]\frac{\delta^{|I_s|}}{\delta [\mathbb{A}^{I_s}]},\\
\boldsymbol{\Omega}^{\mathbb{A}}_{0,1}&= \sum_{k\geq3}\,\,\varint_{\partial_1\Sigma_3}\sum_{J_1,\dots, J_r, I_1,\dots, I_s} \frac{(-i\hbar)^{|I_1|+\dots +|I_s|}}{(|I_1|+\dots +|I_s|)!}\Big(\hat{R}_k\Big)^{I_1\dots I_s}_{\bar{j};J_1\dots J_r}dx^{\bar{j}}\prod^{r+s=k+1}_{r=1, s=1}[\mathbb{A}^{J_r}]\frac{\delta^{|I_s|}}{\delta [\mathbb{A}^{I_s}]},
\end{split}
\end{equation}

where we label by the multiindex $J_r$ the arrows emanating from a boundary vertex towards the globalized vertex, by the multiindex $I_s$ the leaves emanating from the bulk vertex. The sum of $r$ and $s$ has to be $k+1$ since these are the total number of arrows leaving and arriving at a globalized vertex $(R_k)_jdx^j$ (or $(R_k)_{\bar{j}}dx^{\bar{j}}$).

§.§ Flatness of the qGBFV operator for the globalized split RW model

In this section, we prove that the qGBFV operator for the globalized split RW model squares to zero. The proof follows along the same lines as in <cit.>, we will remark where there are differences and refer to their work when the procedure is identical.  Before entering into the details of the proof, we should mention that their proof (and the proof of the mdQME) depends on two assumptions: unimodularity and absence of hidden faces (anomaly-free condition). The first means that tadpoles are not allowed. In the case of the globalized split RW model, we notice that this assumption is not needed since tadpoles vanish <cit.>.

We assume that the globalized split RW model is anomaly-free, i.e. for every graph $\Gamma$, we have that

\begin{equation}
\varint_{F_{\geq 3}}\omega_\Gamma=0,
\end{equation}

where by $F_{\geq 3}$, we denote the union of the faces where at least three bulk vertices collapse in the bulk (also called hidden faces <cit.>).

It is well known that Chern–Simons theory is not an anomaly-free theory <cit.>. The construction of the quantum theory there depends on the choice of gauge-fixing. The appearance of anomalies can be resolve by choosing a framing and framing-dependent counter terms for the gauge-fixing. A famous example of an anomaly-free theory is given by the Poisson sigma model <cit.> since by the result of Kontsevich <cit.> any 2-dimensional theory is actually anomaly-free.
A general method for dealing with theories that do have anomalies is to add counter terms to the action. If the differential form $\omega_\Gamma$, which is integrated over the hidden faces, is exact, one can use the primitive form to cancel the anomalies by the additional vertices that appear.

Since the integrals we will consider are fiber integrals, we will apply of Stokes' theorem for integration along a compact fiber with corners, i.e.

\begin{equation}
\end{equation}

where $\pi_*$ denotes the fiber integration.
In particular, the application of Stokes' theorem to a fiber integral yiels

\begin{equation}
\label{stokes}
(d_x+d_{\bar{x}})\varint_{\text{C}_{\Gamma}}\omega_\Gamma=\varint_{\text{C}_{\Gamma}}(d+d_{\bar{x}})\omega_\Gamma-\varint_{\partial \text{C}_{\Gamma}}\omega_\Gamma,
\end{equation}

where $d$ is the differential on $M\times C_\Gamma$.

The 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}

According to <cit.>, the flatness of $\nabla_{\mathrm{G}}$ is equivalent to the equation

\begin{equation}
\label{flatness_GBFV_3}
i\hbar d_M\boldsymbol{\Omega}_{\partial \Sigma_3}-\frac12\bigg[\boldsymbol{\Omega}_{\partial \Sigma_3},\boldsymbol{\Omega}_{\partial \Sigma_3}\bigg]=0.
\end{equation}

This equation was proven for a globalized split AKSZ theory in <cit.>, in which the $d_M$ is just the de Rham differential on the body of the target manifold. However, in our case, $d_M$ is the sum of the holomorphic and antiholomorphic Dolbeault differentials on $M$.

We prove Eq. (<ref>) for $\boldsymbol{\Omega}^{\mathbb{B}}$. For $\boldsymbol{\Omega}^{\mathbb{A}}$, the proof is analogous as discussed in <cit.>. Suppose we apply $d_M$ to a term of the form

\begin{equation}
\boldsymbol{\Omega}^{\mathbb{B}}_\Gamma=\varint_{\partial_2\Sigma_3} \sigma_\Gamma \bigg(\Big(\hat{R}_k\Big)_jdx^j; \Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg)^I_{J_1\dots J_s}[\mathbb{B}^{J_1}]\dots [\mathbb{B}^{J_s}]\frac{\delta}{\delta [\mathbb{B}^{I}]},
\end{equation}

where $k$ could be any number greater than 0. Here, we chose the easiest term to express with more clarity what is going on. As in <cit.>, we apply Stokes' theorem. However, this is different to the corresponding situation in <cit.> since in our theory we have also red vertices[For the sake of clarity, we stress again that in <cit.>, the vertices are only “black" since $d_M$ is the de Rham differential on the body of the target manifold.], which is portrayed by the fact that $\sigma_\Gamma$ depends also on $\Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}$. We obtain

\begin{equation}
\begin{split}
(d_x+d_{\bar{x}})\boldsymbol{\Omega}^{\mathbb{B}}_\Gamma=\varint_{\partial_2\Sigma_3}\bigg\{(d_x+d_{\bar{x}})\sigma_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_jdx^j; \Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg)\bigg\}^I_{J_1\dots J_s}[\mathbb{B}^{J_1}]\dots [\mathbb{B}^{J_s}]\frac{\delta}{\delta [\mathbb{B}^{I}]}  +[\Omega^\mathbb{B}_0,\boldsymbol{\Omega}^{\mathbb{B}}_\Gamma],
\end{split}
\end{equation}

where the second term is produced when $d_x$ acts on the $\mathbb{B}$ fields (we do not have a corresponding term for $d_{\bar{x}}$ since we do not have fields $\mathbb{B}^{\bar{i}}$ terms to act on). By applying again Stokes' theorem, we have:

\begin{equation}
\begin{split}
(d_x+d_{\bar{x}})\sigma_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_jdx^j; \Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg)&=(d_x+d_{\bar{x}})\varint_{\tilde{\text{C}}_{\Gamma}}\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_jdx^j; \Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg)\\
&=\varint_{\tilde{\text{C}}_{\Gamma}}(d+d_{\bar{x}})\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_jdx^j; \Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg)\\
&\quad\pm\varint_{\partial\tilde{\text{C}}_{\Gamma}}\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_jdx^j; \Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg).
\end{split}
\end{equation}

In principle, $d$ is the differential on $M\times \mathrm{C}_\Gamma$, hence it can be decomposed as $d=d_x+d_1+d_2$, where $d_1$ denotes the part of the differential acting on the propagator and $d_2$ the part acting on $\mathbb{B}$ fields (and, more generally, on $\mathbb{A}$ fields). We do not have a corresponding antiholomorphic differential on $M\times \mathrm{C}_\Gamma$ since the propagators and the fields are all holomorphic. This is different with respect to the case considered in <cit.>.

As in <cit.>, we have $d\omega_{\Gamma} =d_x\omega_{\Gamma}$ and in the boundary integral we have three classes of faces. The first two types of faces, where more than two bulk points collapse and where a subgraph $\Gamma$ collapse at the boundary, can be proved as in <cit.>. In particular, the former vanishes by our assumptions that the theory is anomaly-free (see Assumption <ref>), while the second produces exactly the term $\frac12\bigg[\boldsymbol{\Omega}^{\mathbb{B}}_{\mathrm{pert}},\boldsymbol{\Omega}^{\mathbb{B}}_{\mathrm{pert}}\bigg]$ by <cit.>. On the other hand, the third case, when two bulk vertices collapse, has some differences with respect to the analogous situation in <cit.> due to the already mentioned further presence of red vertices. Here we distinguish four cases:

* when a red vertex collapses with a black vertex, then these faces cancel out with
$d_x\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg)$ by the dCME (<ref>);

* when a black vertex collapses with a red vertex, then these faces cancel out with
$d_{\bar{x}}\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_jdx^j\bigg)$ by the dCME (<ref>);

* when two black vertices collapse, then these faces cancel out with $d_x\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_{j}dx^{j}\bigg)$ by the dCME (<ref>);

* when two red vertices collapse, then these faces cancel out with $d_{\bar{x}}\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg)$ by the dCME (<ref>).

By $\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_{\bar{j}}dx^{\bar{j}}\bigg)$ or $\omega_{\Gamma}\bigg(\Big(\hat{R}_k\Big)_{j}dx^{j}\bigg)$, we mean the part of the subgraph $\Gamma'$, which contains a red or black vertex.

This proves (<ref>), thus $(\nabla_{\textup{G}})^2\equiv 0$.

§.§ Proof of the mdQME for the globalized split RW model

In this section, we are going to prove the mdQME for the globalized split RW model. The proof follows similarly as in <cit.>. As before, we will refer to their work when the situation is identical and point out eventual differences.

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}

Let $\mathcal{G}$ denote the set of Feynman graphs of the theory. Then, we can write the full covariant quantum state for the globalized split RW model as

\begin{equation}
\label{proof_state}
\boldsymbol{\hat{\psi}}\surgR=T_{\Sigma_3}\sum_{\Gamma\in\mathcal{G}}\varint_{\text{C}_\Gamma}\omega_\Gamma \Big(\hat{R}_jdx^j; \hat{R}_{\bar{j}}dx^{\bar{j}}\Big),
\end{equation}

where the combinatorial prefactor $\frac{(-i\hbar)^{\loops(\Gamma)}}{\vert\Aut(\Gamma)\vert}$ is included in $\omega_\Gamma$ (by $\loops$ we denote the number of loops of a graph $\Gamma$) and we denote the configuration space $\text{C}_\Gamma(\Sigma_3)$ by $\text{C}_\Gamma$ for simplicity. We note that $\omega_\Gamma$ is a ($\mathcal{V}_{\Sigma_3,x}$-dependent) differential form on $\text{C}_\Gamma\times M$. Again, following <cit.>, we can apply Stokes' theorem (<ref>) and we get

\begin{equation}
d_M\varint_{\text{C}_\Gamma}\omega_\Gamma\Big(\hat{R}_jdx^j; \hat{R}_{\bar{j}}dx^{\bar{j}}\Big)=\varint_{\text{C}_\Gamma}(d+d_{\bar{x}})\omega_\Gamma\Big(\hat{R}_jdx^j; \hat{R}_{\bar{j}}dx^{\bar{j}}\Big)-\varint_{\partial\text{C}_\Gamma}\omega_\Gamma\Big(\hat{R}_jdx^j; \hat{R}_{\bar{j}}dx^{\bar{j}}\Big).
\end{equation}

As mentioned in Remark <ref>, the $d$ inside the integral is the total differential on $\text{C}_\Gamma(\Sigma_3)\times M$, and thus we can split it as

\begin{equation}
\end{equation}

where $d_1$ denotes the part of the differential acting on the propagators in $\omega_\Gamma$ and $d_2$ is the part acting on $\mathbb{B}$ and $\mathbb{A}$ fields.

With this setup, which is basically analogous to the one in <cit.>, except for the presence of the red vertices and $d_{\bar{x}}$ already extensively discussed, Eq. (<ref>) is verified by proving three relations

* a relation between the application of $d_1$ and of $\Delta_{\mathcal{V}_{\Sigma_3, x}}$ to the quantum state,

* a relation between the application of $d_2$ and of $\Omega_0$ to the quantum state,

* a relation between the application of $d_M$ and of the boundary contributions to the quantum state.

The proofs of these relations can be carried from <cit.> over to the globalized split RW model without any problem. The only difference is when they prove that the contributions in $\partial \text{C}_\Gamma$ consisting of diagrams with two bulk vertices collapsing vanish (which is needed for the third relation). In our case one should consider again three contributions: when two bulk black vertices collapse, when two bulk red vertices collapse, when a red vertex and a black one collapse. The vanishing of these terms follows from Eqs. (<ref>), (<ref>), (<ref>), (<ref>). The rest of the procedure is identical to <cit.>.

§ OUTLOOK AND FUTURE DIRECTION

Our globalization construction leads to an interesting extension of some aspects in the program presented in <cit.> for manifolds with boundary and cutting-gluing techniques. In particular, it would be of interest to understand some relations to the deformation quantization of Kähler manifolds in the guise of <cit.>, especially using the constructions of <cit.>, and Berezin–Toeplitz quantization as presented in <cit.> (possibly for the noncompact case).
It also leads to a more general globalization construction of an algebraic index theory formulation by using the BV formalism together with Fedosov's globalization approach as presented in <cit.>. Moreover, it might also be related to a case of twisted topological field theories, known as Chern–Simons–Rozansky–Witten TFTs, constructed by Kapustin and Saulina in <cit.>. In particular, they use the BRST formalism to produce interesting observables as Wilson loops and thus one might be able to combine it with ideas of <cit.>. Another direction would be the study of the RW invariants through our construction for hyperKähler manifolds. We guess that this would require studying observables of RW theory in the BV-BFV formulation, but the globalization procedure should tell something about these 3-manifold invariants. We hope that this might also be compatible with some generalizations of RW invariants in the non-hyperKähler case as discussed in <cit.>.

§ TOPOLOGICAL QUANTUM FIELD THEORIES

This appendix gives a brief introduction to perturbative and functorial constructions of topological (quantum) field theories, especially we recall Atiyah's TQFT axioms.

§.§ Brief introduction to perturbative quantum field theory

On a spacetime manifold $\Sigma$, consider a space of fields[The space of fields is usually given by sections of some vector bundle over $\Sigma$.] $F_{\Sigma}$ and an action functional $S_\Sigma$ which is required to be local. This means that the action is the integral of a density-valued Lagrangian $\mathscr{L}$, called Lagrangian density, depending on the fields and on a finite number of higher derivatives. In particular, $S_\Sigma: F_{\Sigma}\rightarrow \mathbb{C}$, with

\begin{equation}
S_{\Sigma}(\phi)=\varint_{\Sigma}\mathscr{L}(\phi, \partial\phi, \dots),
\end{equation}

where $\phi\in F_{\Sigma}$ is a field. The set of data consisting of $(\Sigma, F_{\Sigma}, S_{\Sigma})$ defines a classical Lagrangian field theory.

During the years, physicists have developed several approaches to quantum field theory. Roughly, we can split them into perturbative and non-perturbative methods. Here, we focus on the former. Note that by perturbative, we mean semiclassical: in physics jargon perturbation theory is the idea of expanding through a formal power series around the coupling constant of the action. In the perturbative setting, the protagonist of the story is the partition function $Z$, which encodes all the information about the quantum theory it portraits. In general, we can express it through a path integral as

\begin{equation}
\label{ov.tft:part_funct_1}
Z_{\Sigma}=\varint_{F_{\Sigma}} e^{\frac{i}{\hbar}S_{\Sigma}(\phi)}\mathscr{D}[\phi],
\end{equation}

where $\hbar$ is the reduced Planck constant.

In (<ref>), $\mathscr{D}$ denotes a formal measure on $F_{\Sigma}$. Depending on the space of fields $F_\Sigma$, this measure is often mathematically ill-defined. Nevertheless, one can define (<ref>) by considering the methods of perturbative expansion around critical points of $S_\Sigma$ in a formal power series in $\hbar$ with coefficients given by Feynman graphs (see e.g. <cit.>).

Let us make the above discussion more precise. Consider $\Sigma$ to be a manifold with boundary $\partial \Sigma$ and $B_{\partial \Sigma}$ to be the space of boundary values of the fields on $M$. Since the boundary manifold is the boundary of $\Sigma$, we have a restriction map $F_{\Sigma}\xrightarrow[]{\pi}B_{\partial \Sigma}$. The partition function is thus a complex-valued function on $B_{\partial \Sigma}$ which can be written as

\begin{equation}
\label{ov.tft:part_funct_2}
Z_{\Sigma}(\phi_{\partial\Sigma}; \hbar)=\varint_{\{ \phi\in F_{\Sigma}\mid\  \phi\vert_{\partial \Sigma}:=\phi_{\partial \Sigma}\}} e^{\frac{i}{\hbar}S_{\Sigma}(\phi)}\mathscr{D}[\phi],
\end{equation}

where $\phi_{\partial \Sigma}$ is a point in $B_{\partial \Sigma}$.

The manifold $\Sigma$ may be complicated and, as a result, the computation of $Z_{\Sigma}$ can become difficult. Therefore, it would be desirable to cut $\Sigma$ into smaller and, hopefully, easier pieces, compute the partition function there and then glue them together to get the overall state. Suppose, $\Sigma$ is closed and cut it in two disjoint manifolds $\Sigma_1$ and $\Sigma_2$ along a common boundary $\Sigma$, i.e. $\Sigma= \Sigma_1\sqcup_{\partial \Sigma}\Sigma_2$. If we paste them together, we expect the following condition to hold

\begin{equation}
Z_{\Sigma}=\varint_{\phi_{\partial\Sigma}\in B_{\partial \Sigma}}Z_{\Sigma_1}(\phi_{\partial \Sigma})Z_{\Sigma_2}(\phi_{\partial \Sigma})\mathscr{D}[\phi_{\partial \Sigma}].
\end{equation}

§.§ Brief introduction to functorial quantum field theory

The functorial approach to QFT was developed by Segal in the context of conformal field theory <cit.> and by Atiyah for TQFT <cit.>. However, this description is general and it allows us to describe any QFT.

According to Atiyah's axioms, an $n$-dimensional topological field quantum field theory consists of the following set of data:

* A Hilbert space $\mathcal{H}(\Sigma)$, called the space of states, associated to a closed oriented[The orientation endows the manifolds with symbols $\{in, out\}$ which denote incoming or outgoing orientation.] $(n-1)$-manifold $\Sigma$,

* A linear map of vector spaces $Z_M: \mathcal{H}_\mathrm{in}\rightarrow \mathcal{H}_\mathrm{out}$, called partition function, associated to an oriented $n$-cobordism[See Example <ref> for a definition.] $M$ from $\Sigma_\mathrm{in}$ to $\Sigma_\mathrm{out}$ (i.e. the boundary of $M$ is assumed to be given as $\partial M=\Sigma_\mathrm{in}\sqcup \Sigma_\mathrm{out}$).

* Orientation-preserving diffeomorphisms $\phi: \Sigma_1\rightarrow \Sigma_2$ which act on $\mathcal{H}_\Sigma$ through unitary maps $\rho(\phi): \mathcal{H}_{\Sigma_\mathrm{in}}\rightarrow \mathcal{H}_{\Sigma_\mathrm{out}}$, with $\rho$ a representation.

* Orientation-reversing identity diffeomorphisms $s_\Sigma: \Sigma\rightarrow \Bar{\Sigma}$, where we denote by $\Bar{\Sigma}$, the manifold with opposite orientation. These diffeomorphisms act by $\mathbb{C}$-anti-linear maps $\sigma_{\Sigma}\coloneqq\rho(s_\Sigma): \mathcal{H}_{\Sigma}\rightarrow \mathcal{H}_{\Bar{\Sigma}}$.

This set of data is required to satisfy the following axioms:

(i)(Multiplicativity) For two closed oriented $(n-1)$-manifolds $\Sigma$ and $\Sigma'$, the space of states is multiplicative, i.e.

\begin{equation}
\mathcal{H}_{\Sigma \sqcup \Sigma'}=\mathcal{H}_\Sigma \otimes \mathcal{H}_{\Sigma'}.
\end{equation}

For two $n$-cobordisms $M: \Sigma_\mathrm{in}\rightarrow\Sigma_\mathrm{out}$ and $M': \Sigma'_\mathrm{in}\rightarrow\Sigma'_\mathrm{out}$, the partition function is multiplicative

\begin{equation}
Z_{M\sqcup M'}=Z_M\otimes Z_{M'}:\quad  \mathcal{\Sigma}_\mathrm{in}\otimes \mathcal{\Sigma'}_\mathrm{in}\rightarrow \mathcal{\Sigma}_\mathrm{out}\otimes \mathcal{\Sigma'}_\mathrm{out}
\end{equation}

(ii)(Gluing) Let $M_1: \Sigma_1\rightarrow \Sigma_2$, $M_2: \Bar{\Sigma}_2\rightarrow \Sigma_3$ be two $n$-cobordisms, the glued cobordisms can be constructed by gluing along the common $\Sigma_2$-component as $M_1 \cup_{\Sigma_2}M_2: \Sigma_1 \rightarrow \Sigma_3$. The associated partition function is then obtained by composing the partition functions for $M_1$ and $M_2$ as linear maps:

\begin{equation}
Z_{M_1 \cup_{\Sigma_2}M_2}=Z_{M_2}\circ Z_{M_1}:\quad \mathcal{H}_{\Sigma_1}\rightarrow \mathcal{H}_{\Sigma_3}.
\end{equation}

(iii)(Involutivity) $Z(\Bar{\Sigma})=Z(\Sigma)^\vee$, where $Z(\Sigma)^\vee$ is the dual vector space.

(iv) $\mathcal{H}_{\emptyset}=\mathbb{C}$ and $Z_{\Sigma \times [0,1]}=\Id:\mathcal{\Sigma}\rightarrow\mathcal{\Sigma}$.

(v) For $\phi: M\rightarrow M'$ a diffeomorphism, the following diagram commutes:

[column sep=large, row sep=large]
ℋ_Σ_in[r, "Z_M"][d, "ρ(ϕ|_Σ_in)"']    ℋ_Σ_out[d, "ρ(ϕ|_Σ_out)"]

ℋ_Σ'_in[ r,"Z_M'"']     ℋ_Σ'_out

It follows that $Z_M$ is invariant under diffeomorphisms of $M$ relative to its boundary components.

(vi)(Symmetry) The natural diffeomorphism $\Sigma \sqcup \Sigma'\rightarrow \Sigma' \sqcup \Sigma$ is sent by $\rho$ to the natural isomorphism $\mathcal{H}_{\Sigma}\otimes \mathcal{H}_{\Sigma'}\rightarrow\mathcal{H}_{\Sigma'}\otimes \mathcal{H}_{\Sigma}$.

(vii) The partition function for the cylinder $\Sigma \times [0,1]$ viewed as a cobordism $\Sigma \times \Bar{\Sigma}\rightarrow\emptyset$ composed with the anti-linear map $(\sigma_\Sigma)^{-1}:\mathcal{H}_{\Sigma}\rightarrow \mathcal{H}_\Sigma$ yields the Hermitian inner product $\braket{-,-}: \mathcal{H}_\Sigma \times \mathcal{H}_\Sigma \rightarrow\mathbb{C}$.

Let $M$ be a closed $n$-manifold, which can be regarded as a cobordism $\emptyset \rightarrow \emptyset$. The associated partition function $Z_M\in \mathbb{C}$ is an invariant under orientation-preserving diffeomorphisms on $M$.
In general, for a mapping torus
$\frac{\Sigma \times [0,1]}{\Sigma \times \{0\}\stackrel{\phi}{\sim} \Sigma \times \{1\}}$ with $\phi:\Sigma\rightarrow\Sigma$ a gluing diffeomorphism, the axioms above imply that $Z=\Tr_{\mathcal{H}_\Sigma}\rho(\phi)$. In particular, for the product manifold $\Sigma \times S^1$ formed by identifying the opposite ends of the cylinder:

\begin{equation}
\label{ov:tft.partfunc_withS1}
Z_{\Sigma \times S^1}=\Tr_{\mathcal{H}_\Sigma}(\Id)=\dim \mathcal{H}_{\Sigma}\in \mathbb{Z}_{\geq 0}.
\end{equation}

This implies that the space of states is finite-dimensional.

§.§.§ Atiyah's axioms for TQFTs

Atiyah's axioms can be reformulated in the categorical language. We start with some prerequisites before arriving to the definition of a TQFT.

Let us consider a symmetric monoidal category $\mathbf{C}$: it is a category equipped with a bifunctor $\otimes: \mathbf{C}\times \mathbf{C}\rightarrow \mathbf{C}$, called monoidal product which allows to, roughly speaking, “multiply" objects. This product is well-defined because it is associative up to natural isomorphisms (in jargon it satisfies the pentagon equations <cit.>). Moreover, a monoidal category is symmetric when for all the objects $A, B\in \mathbf{C}$ there are natural isomorphisms

\begin{equation}
\beta_{A,B}: A\otimes B\rightarrow B\otimes A
\end{equation}

compatible with the associativity of the monoidal structure (they satisfy the hexagon equations <cit.>).

We consider two examples, which we need for later:

* The category $\mathbf{Vect}_{\mathbb{K}}$ whose objects are $\mathbb{K}$-vector spaces for some field $\mathbb{K}$ and morphisms are $\mathbb{K}$-linear maps. It is monoidal with the usual tensor product as monoidal product ($\otimes :=\otimes_{\mathbb{K}}$) and with unit $\boldsymbol{1}:=\mathbb{K}$. Moreover, one can show that it is symmetric.

* The category of oriented cobordisms$, \mathbf{Cob}^{\text{or}}_n$. The objects are oriented closed $(n-1)$- dimensional manifolds and morphisms are diffeomorphisms classes of bordisms. In a more down to Earth language, this means that the morphisms are given by the bulk of an oriented compact $n$-dimensional manifold with boundary, whose boundary components are the objects. We can compose a morphism with another morphism simply by gluing along the common boundaries. It has a monoidal structure where the monoidal product is given by the disjoint union and the unit object is the empty set $\emptyset$ viewed as an $(n-1)$-dimensional manifold. The objects are endowed with orientations labeled by symbols $\{\mathrm{in}, \mathrm{out}\}$.

Atiyah's axioms can be reformulated in a short way as:

Let $(\mathbf{C},\otimes)$ be a symmetric monoidal category. An $n$-dimensional oriented closed topological field theory (TFT) is a symmetric monoidal functor

\begin{equation}
Z: \mathbf{Cob}^{\text{or}}_n\rightarrow \mathbf{C}.
\end{equation}

An $n$-dimensional oriented topological quantum field theory (TQFT) is a symmetric monoidal functor

\begin{equation}
Z: \mathbf{Cob}^{\text{or}}_n\rightarrow \mathbf{Vect}_{\mathbb{C}}.
\end{equation}

Note that the target category contains also infinite-dimensional vector spaces. However, an analogue of Eq. (<ref>), implies that the state spaces are finite-dimensional.

As seen in Definition <ref>, the category of smooth oriented cobordisms is usually used to describe a TQFT. However, cobordisms may possess other geometric structures such as conformal structure, spin structure, framing, boundaries, etc. Consequently, the associated field theory will be conformal QFT, spin or framed TQFT, etc. For example, for Yang-Mills theories and sigma models, the source category is the category of smooth Riemannian manifolds with a collar at the boundary.

As first example, let us consider a cobordism represented by some pair of pants with genus 1 (see Fig. <ref>). The TQFT $F$ assigns to each boundary component a Hilbert space, i.e. $Z(\partial_k\Sigma)=\mathcal{H}_k$ for $k=1,2,3$. Since $Z$ is a symmetric monoidal functor, we have $Z(\partial_1\Sigma\sqcup \partial_2\Sigma\sqcup \partial_3\Sigma)=\mathcal{H}^{\vee}_1\otimes \mathcal{H}^{\vee}_2\otimes\mathcal{H}_3$. As said before, each cobordism comes with a certain orientation: $\partial_1\Sigma$ as well as $\partial_2\Sigma$ are incoming boundaries (which we denote in the figure by an incoming arrow), while $\partial_3\Sigma$ is an outgoing boundary (which we denote in the figure by an outgoing arrow). Associated to $\partial_1\Sigma$ and $\partial_2\Sigma$, we have an incoming Hilbert space $\mathcal{H}_{\text{in}}\coloneqq \mathcal{H}^{\vee}_1\otimes \mathcal{H}^{\vee}_2\cong \mathcal{H}_1\otimes \mathcal{H}_2$ and an outgoing Hilbert space $\mathcal{H}_{\text{out}}\coloneqq\mathcal{H}_3$ associated to $\partial_3\Sigma$. The state $\psi$ corresponding to this
cobordism and the given TQFT is given as the value of the morphism represented by the genus 1 pair of pants above (i.e. the bounding manifold $M$) under $F$.

[rotate=270,transform shape,tqft, view from=incoming,
every incoming boundary component/.style=fill=gray
tqft/pair of pants,
every lower boundary component/.style=draw,
every incoming lower boundary component/.style=solid,
every outgoing lower boundary component/.style=dashed,
shade, shading angle=270,

[label=below:$\uparrow$] at (A-outgoing boundary 1) ;
[label=below:$\uparrow$] at (A-outgoing boundary 2) ;
[label=above:$\uparrow$] at (A-incoming boundary 1) ;
[rotate=90] at (A-outgoing boundary 1) ;
[rotate=90] at (A-outgoing boundary 2) ;
[rotate=90] at (A-incoming boundary 1) ;
[rotate=90] at (-1.2,-0.9) $M$;

Cobordism $M$ represented by pair of pants of genus 1 with boundary components $\partial_1\Sigma$, $\partial_2\Sigma$, $\partial_3\Sigma$.

As already mentioned in Section <ref>, a closed manifold $\Sigma$ can be seen as a cobordism $\emptyset \rightarrow \emptyset$. We can cut it in two disjoint manifolds $\Sigma_1$ and $\Sigma_2$ along a common boundary $\partial\Sigma$, i.e. $\Sigma= \Sigma_1\sqcup_{\partial \Sigma}\Sigma_2$. Then we can assign an opposite orientation to $\partial_1\Sigma_1$ with respect to the orientation of $\partial_1\Sigma_2$. The same can be done to  $\partial_2\Sigma_1$ with respect to the orientation of $\partial_2\Sigma_2$. The two manifolds with boundary $\Sigma_1$ and $\Sigma_2$ can be glued back together to recover the partition function of the closed manifold $\Sigma$, see Fig. <ref>.

[rotate=-90,transform shape,tqft, view from=incoming,

tqft/reverse pair of pants,
every lower boundary component/.style=draw,
every incoming boundary component/.style=solid, fill=white, draw=white,
every outgoing boundary component/.style=solid, draw=white,
every outgoing lower boundary component/.style=dashed,
shade, shading angle=90,

every lower boundary component/.style=draw,
every incoming lower boundary component/.style=solid,
every outgoing lower boundary component/.style=dashed,

anchor=incoming boundary 1,name=B, at=(A-outgoing boundary 1)
[rotate=90] at (A-incoming boundary 1) ;
[rotate=90] at (A-incoming boundary 2) ;
[rotate=90] at (-0.6,-0.5) $\Sigma_1$;

[rotate=270,transform shape,tqft, view from=incoming,
every incoming boundary component/.style=fill=gray,
every lower boundary component/.style=draw=gray
tqft/pair of pants,

every incoming lower boundary component/.style=solid,draw=gray,
every outgoing lower boundary component/.style=dashed,
shade, shading angle=270,

every lower boundary component/.style=draw,

fill=gray, every outgoing boundary component/.style=draw=white,

anchor=outgoing boundary 1,name=B, at=(A-incoming boundary 1)

[rotate=90] at (A-outgoing boundary 1) ;
[rotate=90] at (A-outgoing boundary 2) ;

[rotate=90] at (-1.2,-0.9) $\Sigma_2$;

Gluing of two manifolds $\Sigma_1$ and $\Sigma_2$ along a common boundary $\partial \Sigma$.

It is important to highlight that the functorial approach to TQFT is not based on any perturbative framework, therefore, its nature is intrinsically non perturbative.

Successively, in <cit.>, Baez and Dolan suggested enhancing Atiyah's notion of TQFT to a functor from the $(\infty, n)$-extension of the cobordism category. Their idea is to allow gluing as well as cutting with higher codimension data. Moreover, they conjectured these TQFTs to be completely classifiable:  this conjecture is known as the Cobordism hypothesis. In <cit.>, Lurie provided a complete classification result for fully extended TQFTs formulated in the language of $(\infty,n)$-categories, a generalization of the notion of a category.

§ ELEMENTS OF FORMAL GEOMETRY

In this appendix, we want to explain some of the important notions of formal geometry developed in <cit.> which are used for the globalization procedure.

§.§ Formal power series on vector spaces

Let $V$ be a finite dimensional vector space. The polynomial algebra on $V$ is given by

\begin{equation}
\Sym^\bullet (V^\vee) = \bigoplus_{k=0}^\infty \Sym^k(V^\vee).
\end{equation}

If we choose $\{e_1, \dots, e_n\}$ to be a basis of $V$, with dual basis $\{y_1, \dots , y_n\}$, then elements $f \in \Sym^\bullet (V^\vee)$ are given by

\begin{equation}
f(y)=\sum_{i_1,\dots,i_n=1}^\infty f_{i_1,\dots, i_n}y_1^{i_1}\dots y_n^{i_n}=\sum_I f_Iy^{I},
\end{equation}

with only finitely many non-vanishing $f_I$. We have denoted by $I$ a multi-index and $y^I=y^{i_1}\dots y^{i_n}$, $y^{\emptyset} \coloneqq 1$.

This algebra can be completed to the algebra of formal power series $\reallywidehat{\Sym}^\bullet(V^\vee)$, with infinitely many nonzero coefficients $f_I$. Note that both algebras $\Sym^\bullet(V^\vee)$ and $\reallywidehat{\Sym}^\bullet (V^\vee)$ are commutative, with the multiplication of formal power series or polynomials, respectively, generated by $V^\vee$. One can specify derivations of these algebras by their value on these generators, therefore, the map

V ⊗^∙(V^∨)    →(^∙(V^∨))

v ⊗f    ↦(V^∨∋α↦α(v) ·f)

is an isomorphism with inverse

(^∙(V^∨))     →V ⊗^∙(V^∨)

D     ↦∑_i=1^n e_i ⊗D(y^i).

In coordinates this corresponds to sending $e_i \mapsto \frac{\partial}{\partial y^i}$.

§.§ Formal exponential maps

Let $M$ be a manifold and let  $\varphi: U \rightarrow M$, with $U \subset TM$ an open neighbourhood of the zero section. For $x \in M$, $y \in T_xM \cap U$, we can write $\varphi(x,y)=\varphi_x(y)$.

We call $\varphi$ a generalized exponential map if for all $x \in M$ we have that

* $\varphi_x(0)=x$,

* $d\varphi_x(0)=\Id_{T_xM}$.

In local coordinates

\begin{equation}
\varphi_x^i(y)=x^i+y^i+\frac{1}{2}\varphi_{x,jk}^iy^jy^k+\frac{1}{3!} \varphi_{x,jkl}^iy^jy^ky^l+\dots,
\end{equation}

where the $x_i$ and $y_i$ are  respectively the base and the fiber coordinates. Two generalized exponential maps are identified if their corresponding jets agree at all orders.

A formal exponential map is an equivalence class of generalized exponential maps. A formal exponential map is completely specified by the sequence of functions $\Big (\varphi_{x,i_1,\dots,i_k}^i \Big)_{k=0}^\infty$.

From now on, we will abuse notation and we will denote equivalence classes and their representatives by $\varphi$. One can produce a section $\sigma \in \Gamma(\reallywidehat{\Sym}^\bullet(T^\vee M))$ from a formal exponential map $\varphi$ and a function $f \in \mathcal{C}^\infty(M)$,  via $\sigma_x=\mathrm{T}\varphi_x^*f$, with $\mathrm{T}$ the Taylor expansion in the fiber coordinates around $y=0$ and the pullback defined by any representative of $\varphi$. We will denote this section by $\mathrm{T}\varphi^*f$, and note that it is independent of the choice of the representative since it only depends on the jets.

§.§ Grothendieck connection

On $\reallywidehat{\Sym}^\bullet(T^\vee M)$ we can define a flat connection $\Gr$, satisfying  the property that

\begin{equation}
\Gr \sigma=0 \iff \sigma=\mathrm{T}\varphi^*f,
\end{equation}

for some $f\in \mathcal{C}^\infty(M)$. Namely, $\Gr=d+R$ with $R\in \Gamma\Big(T^\vee M \otimes TM \otimes \reallywidehat{\Sym}^\bullet(T^\vee M)\Big)$, a 1-form with values in derivations[We can use the isomorphism (<ref>) to identify derivations of $\reallywidehat{\Sym}^\bullet(T^\vee M)$ with $\Gamma\Big(TM \otimes \reallywidehat{\Sym}^\bullet(T^\vee M)\Big)$.] of $\reallywidehat{\Sym}^\bullet(T^\vee M)$. In local coordinates, $R$ is defined as $R_idx^i$ and

\begin{equation}
R_i(x;y)\coloneqq-\bigg[\bigg(\frac{\partial \varphi}{\partial y}\bigg)^{-1}\bigg]^k_j\frac{\partial \varphi^j}{\partial x^i}\frac{\partial}{\partial y^k}= Y_i^k(x;y)\frac{\partial}{\partial y^k}.
\end{equation}

Hence, we have

\begin{equation}
R(x;y)=R_i(x;y)dx^i=Y_i^k \frac{\partial}{\partial y^k}dx^i.
\end{equation}

The connection $\Gr$ is called the Grothendieck connection[In the setting of field theory we have called this the classical Grothendieck connection in order to distinguish it from its quantum counterpart.].

For $\sigma \in \Gamma(\reallywidehat{\Sym}^\bullet(T^\vee M))$, $R(\sigma)$ is expressed via the Taylor expansion (in the $y$ coordinates) of

\begin{equation}
-d_y\sigma \circ (d_y\varphi)^{-1} \circ d_x\varphi: \Gamma(TM) \rightarrow \Gamma(\reallywidehat{\Sym}^\bullet(T^\vee M)),
\end{equation}

and therefore, $R$ does not depend on the coordinate choice. For a vector field $\xi=\xi^i \frac{\partial}{\partial x^i}$, we have

\begin{equation}
\Gr^\xi=\xi + \hat{\xi},
\end{equation}

\begin{equation}
\hat{\xi}(x;y)=\iota_\xi R(x;y)=\xi^i(x)Y_i^k(x;y) \frac{\partial}{\partial y^k}.
\end{equation}

The classical Grothendieck connection is flat (i.e. $D^2_G=0$). Moreover, the flatness condition translates into

\begin{equation}
\label{MC_R}
\end{equation}

which is a Maurer–Cartan (MC) equation for $R$.

It can be proved that its cohomology is concentrated in degree 0 and is given by

\begin{equation}
H^0_{\Gr}\Big(\Gamma(\reallywidehat{\Sym}^\bullet(T^\vee M))\Big)=\mathrm{T}\varphi^* \mathcal{C}^\infty(M) \cong \mathcal{C}^\infty(M).
\end{equation}

§.§ Formal vertical tensor fields

Let $E \rightarrow M$ be any tensorial bundle[A tensorial bundle is any bundle which is a  tensor product or antisymmetric or symmetric product of the tangent or cotangent bundle, or a direct sum thereof.], for example $E=\bigwedge^k TM$. Its sections are called tensor fields of type E.

The associated formal vertical bundle to $E$ is then $\hat{E} \coloneqq E \otimes \reallywidehat{\Sym}^\bullet(T^\vee M)$. Its sections are called formal vertical tensors of type $E$.

These bundles can be thought of as tensors of the same type on $TM$ where the dependence on fiber directions is formal.

The formal exponential map defines an injective map

\begin{equation}
\mathrm{T}\varphi^*:E \rightarrow \hat{E}
\end{equation}

via the Taylor expansion of a tensor field pulled back[Note that $\varphi$ is a local diffeomorphism and hence we can define the pullback of contravariant tensors as the pushforward of the inverse.] to $U$ by $\varphi$.

Furthermore, we can let $R$ act by formal derivatives and therefore, we get a Grothendieck connection $\Gr=d+R$ on any formal vertical tensor bundle. Similarly, as before, we have:

* $\Gr$ is flat;

* flat sections of $\Gr$ are precisely the ones in the image of $\mathrm{T}\varphi^*$;

* the cohomology of $\Gr$ is concentrated in degree 0 and given by the flat sections, i.e. $\hat{E}$-valued 0-forms.

§.§ Changing the formal exponential map

We will denote by $\varphi$ be a family of formal exponential maps depending on a parameter $t$ belonging to an open interval $I$. One can then associate to this family a formal exponential map $\psi$ for the manifold $M \times I$ by

\begin{equation}
\psi(x,t,y,\tau) \coloneqq (\varphi_x(y),t+\tau),
\end{equation}

with $\tau$ the tangent variable to $t$. The corresponding connection $\tilde{R}$ is defined as follows. Let $\tilde{\sigma}$ be a section of $\reallywidehat{\Sym}^\bullet\Big(T^\vee(M\times I)\Big)$, by definition we have:

\begin{equation}
\tilde{R}(\tilde{\sigma})=-(d_y\tilde{\sigma},d_\tau \tilde{\sigma}) \circ \begin{pmatrix}
(d_y\varphi)^{-1} & 0 \\
0 & 1\end{pmatrix} \circ \begin{pmatrix}
d_x\varphi & \Dot{\varphi} \\
0 & 1
\end{pmatrix}.
\end{equation}

Hence, $\tilde{R}=R+Cdt+T$, with $R$ defined as in Section <ref>, but with a $t$-dependence now, and $T=-dt\frac{\partial}{\partial \tau}$.

The MC equation (<ref>) can be reformulated for $\tilde{R}$ observing that :

* $d_x T=d_tT=0$,

* $T$ commutes with $R$ and $C$.

The $(2,0)$-form component of the MC equation over $M \times I$ yields again the MC equation for $R$, while the $(1,1)$- component reads

\begin{equation}
\Dot{R}=d_xC+[R,C].
\end{equation}

Under a change of formal exponential map, $R$ changes by a gauge transformation having as generator the section $C$ of $\hat{\mathfrak{X}}(TM) \coloneqq TM \otimes \reallywidehat{\Sym}^\bullet(T^\vee M)$. Finally, if $\sigma$ is a section in the image of $\mathrm{T}\varphi^*$, a simple computation yields

\begin{equation}
\Dot{\sigma}=-L_C\sigma.
\end{equation}

One can think of it as the associated gauge transformation for sections.

§.§ Extension to graded manifolds

The previous results can be generalized to the category of graded manifolds exploiting the algebraic reformulation of formal exponential maps developed in <cit.>.

More concretely, given a formal exponential map $\varphi$ on a smooth manifold $M$, one can construct a map

\begin{equation}
\pbw: \Gamma(\reallywidehat{\Sym}^\bullet(TM)) \rightarrow \mathcal{D}(M)
\end{equation}

from sections of the completed symmetric algebra of the tangent bundle to the algebra of differential operators $\mathcal{D}$ by defining

\begin{equation}
\pbw\Big(X_1 \odot \dots \odot X_n\Big)(f)=\frac{d}{dt_1}\Bigg|_{t_1=0} \dots \frac{d}{dt_n}\Bigg|_{t_n=0} f\Big(\varphi(t_1X_1+\dots+t_nX_n)\Big),
\end{equation}

where we denote by $\odot$ the symmetric product.
One can also define this map in the category of graded manifolds by choosing a torsion-free connection $\nabla$ on the tangent bundle of a graded manifold $M$ with Christoffel symbols $\Gamma^k_{ij}$. In particular, there still exists an element
$R^{\nabla} \in \Omega^1\Big(M,TM \otimes \reallywidehat{\Sym}^\bullet(T^\vee M)\Big)$ with the property that $\Gr=d_M+R^{\nabla}$ is a flat connection on $\reallywidehat{\Sym}^\bullet(T^\vee M)$, i.e.

\begin{equation}
\end{equation}

In local coordinates $\{x^i\}$ on $M$ and $\{y^i\}$ on $TM$, we have

\begin{equation}
\begin{split}
\delta &=dx^i\frac{\partial}{\partial y^i},\\
\Gamma &=-dx^i\Gamma^k_{ij}(x)y^j\frac{\partial}{\partial y^k}, \\
A^{\nabla}&=dx^i \sum_{|J| \geq 2}A^k_{i,J}(x)y^J \frac{\partial}{\partial y^k}.
\end{split}
\end{equation}

We define $R_i \in \Gamma(M, \reallywidehat{\Sym}^\bullet(T^\vee M) \otimes TM)$ and $Y_i^k \in \Gamma(M, \reallywidehat{\Sym}^\bullet(T^\vee M))$ via

\begin{equation}
R^{\nabla}=R_i(x;y)dx^i=Y_i^k(x;y)dx^i\frac{\partial}{\partial y^k}.
\end{equation}

In particular note that $\Gr$ extends to a differential on $\Omega^{\bullet}(M,\reallywidehat{\Sym}^\bullet(T^\vee M))$. T

The Taylor expansion of a function $f \in \mathcal{C}^\infty(M)$ can be defined as <cit.>

\begin{equation}
\label{1}
\mathrm{T}\varphi^* f \coloneqq \sum_I \frac{1}{I!}y^I\pbw \Big(\underset{\leftarrow}{\partial_x^I}\Big)(f),
\end{equation}

\begin{equation}
\Big(\underset{\leftarrow}{\partial_x^I}\Big)=\underbrace{\partial_{x_1} \odot \dots \odot \partial_{x_1}}_{i_1} \odot \dots \odot \underbrace{\partial_{x_n} \odot \dots \odot \partial_{x_n}}_{i_n}.
\end{equation}

One can prove that (<ref>) has still the same properties, i.e. the image of $\mathrm{T}\varphi^*$ consists precisely of the $\Gr$-closed sections of $\reallywidehat{\Sym}^\bullet(T^\vee M)$.

We can describe how the exponential map varies under the choice of a connection mimicking the construction for the smooth case described in Section <ref>. More concretely, assume we have a smooth family $\nabla^t$ of connections on $TM$, then we can associate to that family a connection $\tilde{\nabla}$ on $M \times I$. The associated $R^{\tilde{\nabla}}$ can be split as in Section <ref>

\begin{equation}
\end{equation}

where $C \in \Gamma(M,\reallywidehat{\Sym}^\bullet(T^\vee M))$. As previously, $\Gr^2=0$ means

\begin{equation}
\Dot{R}^{\nabla^t}=d_MR^{\nabla^t}+[C^{\nabla^t},R ^{\nabla^t}],
\end{equation}

and for any section $\sigma$ in the image of $\mathrm{T}\varphi^*$ we have

\begin{equation}
\Dot{\sigma}=-L_{C^{\nabla^t}} \sigma.
\end{equation}

§ ELEMENTS OF DERIVED GEOMETRY

In Section <ref>, we have introduced the BV formalism as a way to deal with non-isolated critical points for the action of a gauge theory. In other words, this means that the critical locus of the action functional (i.e. the set of points such that $\delta S=0$) is singular. The BV formalism instructs us to resolve the singularities homologically by taking the derived critical locus of the action functional, which is a smooth object in the category of derived spaces: this is done by the Koszul resolution of the critical locus. More generally, this procedure can be understood globally in the setting of derived algebraic geometry (DAG) <cit.>. However, for the present work, we do not require the whole DAG language. For us it is sufficient to work with a “tamed" version of DAG, namely the framework developed by Costello in <cit.> to deal with formal mapping stacks which capture the geometry of derived critical loci in nonlinear sigma models.

§.§ Category of derived manifolds

Here, we want to define the category of derived manifolds. Let us start with the objects.

Denote by $\Omega^\bullet(M)$ the de Rham algebra of a manifold $M$, which, in other words, is a sheaf of commutative differentially graded algebras.

A derived manifold (over $\mathbb{R}$) is a pair $(M, \mathcal{A})$, where $M$ is a smooth manifold and $\mathcal{A}$ is a sheaf of unital differentially graded $\Omega^\bullet(M)$, satisfying the conditions

* As a sheaf of $\mathcal{C}^\infty(M)$-algebras, $\mathcal{A}$ is locally free and of finite rank.

* There is a morphism $\A\rightarrow\mathcal{C}^\infty(M)$ of sheaves of $\Omega^\bullet(M)$-algebras and the kernel of this map is a sheaf of nilpotent ideals.

* The topology of $M$ has a basis such that the cohomology $\A(U)$ is concentrated in nonpositive degrees for each basis set of $U$.

Trivially, any manifold $M$ with $\A=\cinfty$ is a derived manifold.

Let $M$ be a manifold, take $\A=\derham$ equipped with a de Rham differential. The pair $(M,\derham)$ is derived manifold which we denote by $M_{\,\text{dR}}$.

Let $M$ be a complex manifold, take $\A=\Omega^{0,\bullet}(M)$ equipped with a Dolbeault differential $\Bar{\partial}$. The pair $(M,\Omega^{0,\bullet}(M))$ is complex derived manifold.

A morphism of derived manifolds $(M,\A)\rightarrow(N,\mathcal{B})$ is a smooth map $f:M\rightarrow N$ together with a morphism $\phi:f^{-1}\mathcal{B}\rightarrow\A$ of $f^{-1}\Omega^\bullet(N)$-algebras such that the diagram

[column sep=large, row sep=large]
f^-1ℬ[r, "ϕ"][d]    [d]

f^-1𝒞^∞(N)[ r]    𝒞^∞(M)

We denote by $\mathbf{DMan}$ the category with objects given by derived manifolds and morphisms given by the ones we have just defined.

The notion of morphisms between derived manifolds is further enriched by introducing weak equivalences between derived manifolds. For this purpose, we will use the nilpotent differential graded (dg) ideal $I$ of $(M,\A)$  defined as the kernel of the map $\A\rightarrow\mathcal{C}^\infty(M)$. Here, we have a filtration by powers of the nilpotent ideal. Let $\Grad\A$ denote the associated graded algebra with degree $k$ part $\Grad^k\A\coloneqq F^k\A/F^{k+1}\A$ and the induced differential.

A morphism $(f,\phi):(M,\A)\rightarrow(N,\mathcal{B})$ of derived manifolds is a weak equivalence if $f$ is a differomoprhisms and the induced map

\begin{equation}
\Grad\phi:f^{-1}\Grad\mathcal{B}\rightarrow\Grad\A
\end{equation}

is a quasi-isomorphism.

Having a filtration has also another aim: it should mirror the role of the tower of quotients of a local Artinian algebra in formal deformation theory. In that context, in many situations, it is useful to proceed by Artinian induction: let $(A, \mathfrak{m})$ be a local Artinian algebra over $\mathbb{R}$, there is a tower

\begin{equation}
A=A/\mathfrak{m}^{n+1}\rightarrow A/\mathfrak{m}^{n}\rightarrow \dots \rightarrow A/\mathfrak{m}\cong \mathbb{R}.
\end{equation}

This tower is then used to prove some properties of $A$. Following these ideas, in fact, derived manifolds can be used to study derived deformation theory as Artinian algebras are used to study formal deformation theory. Now, let us define Artinian dg algebras and make these ideas more precise. We will be concise so we refer to <cit.> for a more detailed exposition.

An Artinian dg algebra $R$ over a field $\mathbb{K}$ is a finite dimensional dg algebra over $\mathbb{K}$, concentrated in non positive degrees, with a unique nilpotent dg ideal $\mathfrak{m}$ such that $R/\mathfrak{m}\cong\mathbb{K}$.

The relation between Artinian dg algebra and derived manifolds is explained by the following Proposition

There is a fully faithful embedding

\begin{align}
\begin{split}
\Spec: \mathbf{dgArt}^{op}_{\mathbb{K}}&\rightarrow\mathbf{DMan},\\
R&\mapsto \Spec R\coloneqq(\pt, R).
\end{split}
\end{align}

The importance of Artinian dg algebras comes from being a sort of “test object" in formal derived deformation theory.

A formal derived moduli problem over $\mathbb{K}$ is a functor

\begin{equation}
\end{equation}

where $\mathbf{sSets}$ is the category of simplicial sets and $X$ is such that $X(\mathbb{K})$ is contractible and $X$ preserves certain homotopy limits.

Loosely speaking, Artinian dg algebras are points with nilpotent directions in derived manifolds. Hence, studying formal moduli problems corresponds to studying the formal neighbourhoods of such points.

We can now combine Definition <ref> and Definition <ref>. We generalize the formal derived moduli problems by extending the functor $X$ to a functor $\mathbf{DMan}^{op}\rightarrow \mathbf{sSets}$. In this way, we can study formal moduli problems parametrized by a smooth manifold $M$ (before they were parametrized by an Artinian algebra).

§.§ Derived stacks

In this section, we are going to introduce briefly the derived stacks. These are the spaces studied in derived algebraic geometry.

Recall the functor of points approach in algebraic geometry: a scheme can be defined as a functor from the category of commutative $\mathbb{K}$-algebras, i.e. $\mathbf{CAlg}_{\mathbb{K}}$, to the category of sets. Motivated by the study of moduli problems, since the focus was to classify objects with their isomorphisms, the target category was extended to the category of groupoids (a small category whose morphisms are invertible). These new functors were called stacks. A further generalization is called higher stacks, where the interest is to classify objects up to a higher notion of equivalence rather than isomorphisms (e.g. quasi-isomorphisms). The target category in this case is extended to the category of simplicial sets. Finally derived stacks (or derived higher stacks) arrive when we enlarge the source category to $\mathbf{DCAlg}_{\mathbb{K}}$, i.e. the category of simplicial commutative $\mathbb{K}$-algebras. This category has a natural model category structure, which allows to do homotopy theory. Hence, derived stacks are defined as functors $\mathbf{DCAlg}_{\mathbb{K}}\rightarrow \mathbf{sSets}$ which send equivalences in the source category to weak homotopy equivalences on the target and satisfy a descent condition <cit.> .

The related definition in Costello's approach <cit.> is similar, the only difference is for the source category which is the category of derived manifolds $\mathbf{DMan}^{op}$.

A derived stack or derived space is a functor:

\begin{equation}
\end{equation}

such that:

* $X$ takes weak equivalences of derived manifolds to weak equivalences of simplicial sets.

* $X$ satisfies Čech descent.

The notion of Čech descent is outside the scope of the present work, we refer to <cit.> for a definition.

In the following, we will study a particular type of derived stack with a geometric interpretation, i.e. the derived stack represented by $\Linf$-spaces.

§.§ L-spaces

The heart of the philosophy of deformation theory consists of the following statement: “every formal derived moduli problem is represented by an $\Linf$-algebra". For the explicit statement see <cit.>. We will see how this works in our setting, but before, we need some definitions.

Let $A$ be a commutative differential graded algebra (cdga) with a nilpotent dg ideal $I$ and $A^{\#}$ be the underlying graded algebra, with zero differential. A curved $\Linf$-algebra over  is a finitely generated projective $A^{\#}$-module $V$ together with a derivation of cohomological degree 1:

\begin{equation}
\end{equation}

such that:

* $(\reallywidehat{\Sym}^\bullet(V[1]^\vee),d)$ is a cdga over $A$.

* $d$ preserves the ideal $\Sym^{>0}(V[1]^\vee)$ modulo the nilpotent ideal $I$.

If we take the Taylor components of $d$, we obtain the maps

\begin{equation}
d_n: V^\vee\rightarrow\bigwedge\nolimits^n(V[n-2]^\vee),\quad n\geq 0
\end{equation}

which, upon dualization, become the $\Linf$-brackets

\begin{equation}
\ell_n: \bigwedge\nolimits^n(V[n-2])\rightarrow V
\end{equation}

The completed symmetric algebra over $A^{\#}$ of the 1-shifted dual of $V$, i.e.  $\reallywidehat{\Sym}^\bullet(V[1]^\vee)$, is known as the Chevalley–Eilenberg complex of $V$, and denoted in the following by $C^{\bullet}(V)$.

A morphism of curved $\Linf$-algebras $\phi:V\rightarrow W$ over $A$ is a map $\pi^*:C^\bullet(W)\rightarrow C^\bullet(V)$ of cdga's over $A$ which respects the filtration by the ideal $I$.

Let $\mathfrak{g}$ be the curved $\Linf$-algebra and $\alpha$ a degree 1 element of $\mathfrak{g}$. The Maurer–Cartan element associated to $\alpha$ is

\begin{equation}
\label{MC}
MC(\alpha)=\sum^{\infty}_{n=0}\frac{1}{n!}\ell_n(\alpha^{\otimes n}).
\end{equation}

The Maurer–Cartan equation for $\alpha$ is $MC(\alpha)=0$.

To render (<ref>) well-defined, since it involves an infinite sum, we will only consider Maurer–Cartan elements in nilpotent $\Linf$-algebras.

As we have preannounced before, formal moduli problems are described by $\Linf$-algebras. Explicitly, let $\mathfrak{g}$ be an $\Linf$-algebra and $R$ be an Artinian dg algebra over $\mathbb{R}$, with maximal ideal $\mathfrak{m}$. Then, we can associate a formal derived moduli problem to $\mathfrak{g}$ by sending $R$ to the simplicial set

\begin{equation}
\end{equation}

of solution of the Maurer–Cartan equation of the nilpotent $\Linf$-algebra $\mathfrak{g}\otimes_{\mathbb{R}}\mathfrak{m}$. More precisely, the Maurer–Cartan functor $MC_{\mathfrak{g}}$ is a formal derived moduli problem (see <ref>). Moreover, if we send an $\Linf$-algebra to its associated Maurer–Cartan functor, we obtain an equivalence of categories <cit.>.

However, we are interested in $\Linf$-algebras parametrized by smooth manifolds and derived stacks, the global counterparts to the notion of formal moduli problems, which has a local nature.

Let X be a smooth manifold.

* A curved $\Linf$-algebra over $\Omega^\bullet(X)$ consists of a $\mathbb{Z}$-graded topological[By topological, we mean that the fibers are topological vector spaces and the transition maps are continuous.] vector bundle $\pi: V\rightarrow X$ and the structure of an $\Linf$-algebra on its sheaf of smooth sections, denoted by $\mathfrak{g}$, where the base algebra is over $\Omega^\bullet(X)$ with nilpotent ideal $I=\Omega^{\geq 1}(X)$.

* An $\Linf$-space is a pair $(X,\mathfrak{g})$, where $\mathfrak{g}$ is a curved $\Linf$-algebra over $\Omega^\bullet(X)$.

Now, we will explain how every $\Linf$-space defines a derived stack. It works in the same manner as $\Linf$-algebras determine formal moduli problems. Let $B\mathfrak{g}\coloneqq(X,\mathfrak{g})$ denote an $\Linf$-space. For a smooth map $f:Y\rightarrow X$, a curved $\Linf$-algebra is formed over $\Omega^\bullet(Y)$ as

\begin{equation}
f^*\mathfrak{g}\coloneqq f^{-1}\mathfrak{g}\otimes_{f^{-1}\Omega^\bullet(X)}\Omega^\bullet(Y).
\end{equation}

Let $\Delta^n$ be the standard $n$-simplex in $\mathbb{R}^n.$ For $B\mathfrak{g}$ an $\Linf$-space, its functor of points is the functor

\begin{equation}
\end{equation}