Let $X$ be a derived stack endowed with an $\calO$-orientation of dimension $d$, and let $(F,\omega)$ be a derived Artin stack with an $n$-shifted symplectic structure $\omega$. Then the derived mapping stack $\mathbf{Map}(X,F)$ carries a natural $(n-d)$-shifted symplectic structure.

The BV-BFV formalism, as described in Section <ref>, can be easily extended to higher codimensions in the classical setting, but needs more sophisticated methods in the quantum setting <cit.>. As discussed in <cit.>, the quantum setting needs to couple the methods of deformation quantization to the methods of geometric quantization, both in the shifted setting. Fortunately, also the methods for shifted Poisson structures and shifted deformation quantization have been developed in <cit.>. The setting of geometric quantization in the shifted picture has been recently considered in <cit.>. The methods developed in <cit.> have been considered for the setting of codimension 1 structures (BV-BFV) in <cit.> for the setting of AKSZ topological field theories by using Lagrangian correspondences and have been recently (fully) extended by Calaque, Haugseng and Scheimbauer <cit.>.

[Chern–Simons theory]
As an example we want to discuss the 3-dimensional case of Chern–Simons theory (see Section <ref>). Recall that the phase space is given by the moduli space of flat $G$ connections for some compact Lie group $G$. In the codimension 0 case, we are working over a closed oriented 3-manifold $\Sigma$ (see Figure <ref>) which carries the induced BV symplectic structure, i.e. a $(-1)$-shifted structure. In the codimension 1 case we are working over a closed oriented 2-manifold $N_1\subset \Sigma$ (e.g. some Riemann surface $\Sigma_g$ of genus $g$), to which we can associate a phase space endowed with the Atiyah–Bott symplectic structure <cit.>, which is a 0-shifted structure. In the codimension 2 case we are working over a closed oriented 1-manifold $S^1\cong N_2\subset \Sigma$, for which the phase space is given by the stack of conjugacy classes $[G/G]$ and we can consider a $1$-shifted symplectic structure by the canonical 3-form on $G$. Finally, in the codimension 3 case we consider a point $\mathrm{pt}$ (0-manifold), for which the phase space is given by the classifying stack $\mathrm{B}G=[\mathrm{pt}/G]$ endowed with the 2-shifted symplectic form given by the invariant pairing (Killing form) on the Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)$.

[shading=ball, ball color=gray!10, draw] (0,0) circle (2);

[dashed, fill=blue!10, opacity=.3] (0,0) ellipse (2cm and .5cm);
[rotate=30,dashed,fill=green!10,opacity=.3] (0,0) ellipse (2cm and .5cm);
[rotate=150,dashed,fill=yellow!10,opacity=.3] (0,0) ellipse (2cm and .5cm);

Examples of 2-dimensional defects sitting inside the 3-ball.

§ AKSZ FORMULATION OF DW THEORY

In this section we want to describe a way of obtaining the DW action functional as in Section <ref> in terms of an AKSZ theory by choosing a suitable gauge-fixing Lagrangian submanifold $\calL$ <cit.>.

§.§ AKSZ data

Let $\mathfrak{h}$ be a Lie algebra and consider its Weil model[For a Lie algebra $\mathfrak{h}$, the Weil model is given by the algebra $\calW(\mathfrak{h}):=\bigwedge^\bullet(\mathfrak{h^*}\oplus\mathfrak{h}^*[1])$ together with a differential $\dd_\calW\colon \calW(\mathfrak{h})\to \calW(\mathfrak{h})$ such that $(\calW(\mathfrak{h}),\dd_\calW)$ is a differential graded algebra. The differential is defined to act on $\mathfrak{h}^*$ as the differential for the Chevalley–Eilenberg algebra of $\mathfrak{h}$ plus the degree shift differential. In particular $\dd_\calW=\dd_{\mathrm{CE}(\mathfrak{h})}+\dd_s$, where $\dd_s$ acts by degree shift $\mathfrak{h}^*\to \mathfrak{h}^*[1]$ for elements in $\mathfrak{h}$ and by $0$ for elements in $\mathfrak{h}^*[1]$.] $(\calW(\mathfrak{h}),\dd_\calW)$.
Note that $\calW(\mathfrak{h})$ can be endowed with a natural symplectic form of degree $+3$ if $\mathfrak{h}$ is endowed with an invariant, nondegenerate symmetric pairing $\langle\enspace,\enspace\rangle$. We want to consider this symplectic manifold to be the target for a 4-dimensional AKSZ theory. The graded vector space $\mathfrak{h}[1]\oplus\mathfrak{h}[2]$ is endowed with the symplectic structure

\[
\omega=\langle\delta X,\delta Y\rangle,
\]

where $X$ denotes the coordinate of degree $+1$ and $Y$ denotes the coordinate of degree $+2$. We can define a Hamiltonian of degree $+4$ by

\[
\Theta(X,Y)=\frac{1}{2}\langle Y,Y\rangle+\frac{1}{2}\langle Y,[X,X]\rangle.
\]

One can check that the Hamiltonian vector field of $\Theta$ with respect to $\omega$ is given by the Weil differential $\dd_\calW$. We have

\begin{equation}
\label{eq:Weil_differential_relation}
\dd_\calW X=Y+\frac{1}{2}[X,X],\qquad \dd_\calW Y=[X,Y].
\end{equation}

Let now $\Sigma$ be a 4-manifold and consider the BV space of fields

\[
\calF_\Sigma=\Map(T[1]\Sigma,\mathfrak{h}[1]\oplus \mathfrak{h}[2])\cong \Omega^\bullet(\Sigma)\otimes\mathfrak{h}[1]\oplus \Omega^\bullet(\Sigma)\otimes \mathfrak{h}[2].
\]

The superfields are given by

\begin{align}
\mathbf{X}(u,\theta)&=X+A\theta^{i_1}+\chi\theta^{i_1}\theta^{i_2}+\psi^\dagger\theta^{i_1}\theta^{i_2}\theta^{i_3}+Y^\dagger\theta^{i_1}\theta^{i_2}\theta^{i_3}\theta^{i_4},\\
\mathbf{Y}(u,\theta)&=Y+\psi\theta^{i_1}+\chi^\dagger\theta^{i_1}\theta^{i_2}+A^\dagger\theta^{i_1}\theta^{i_2}\theta^{i_3}+X^\dagger\theta^{i_1}\theta^{i_2}\theta^{i_3}\theta^{i_4}.
\end{align}

for local coordinates $(u_i,\theta^i)$ on $T[1]\Sigma$.
The BV symplectic form is given by

\[
\omega_\Sigma=\int_{T[1]\Sigma}\boldsymbol{\mu}_4\langle \delta \mathbf{X},\delta\mathbf{Y}\rangle,
\]

and the AKSZ-BV action is given by

\[
\calS_\Sigma=\int_{T[1]\Sigma}\boldsymbol{\mu}_4\left(\langle\mathbf{Y},\dd_\Sigma\mathbf{X}\rangle+\frac{1}{2}\langle\mathbf{Y},\mathbf{Y}\rangle+\frac{1}{2}\langle \mathbf{Y},[\mathbf{X},\mathbf{X}]\rangle\right).
\]

We have denoted by $\boldsymbol{\mu}_4:=\dd^4u\dd^4\theta$ the supermeasure on $T[1]\Sigma$.
If we expand things into components and perform Berezinian integration, we get

\begin{equation}
\omega_\Sigma=\int_\Sigma\dd^4u \left(\delta X \delta X^\dagger+\delta A\delta A^\dagger+\delta\chi\delta\chi^\dagger+\delta\psi^\dagger\delta\psi+\delta Y^\dagger\delta Y\right)
\end{equation}

\begin{multline}
\calS_\Sigma= \int_\Sigma\dd^4u\bigg(\langle \psi,\dd_A\chi\rangle+\frac{1}{2}\langle Y,[\chi,\chi]\rangle+\langle \psi^\dagger,(\dd_A Y+[X,\psi])\rangle\\+\langle \chi^\dagger,(F_A+[X,\chi])\rangle+\langle A^\dagger,(\psi+\dd_AX)\rangle+\langle Y^\dagger,[X,Y]\rangle\\+\left\langle X^\dagger,\left(\frac{1}{2}[X,X]\right)\right\rangle+\frac{1}{2}\langle \chi^\dagger,\chi^\dagger\rangle\bigg),
\end{multline}

where $\dd_A=\dd_\Sigma+[A,\enspace]$ denotes the covariant derivative of $A$ and $F_A=\dd_\Sigma A+\frac{1}{2}[A,A]$. The BV transformations on the superfields are then given by

\begin{align}
Q_\Sigma \mathbf{X}&=\dd_\Sigma\mathbf{X}+\mathbf{Y}+\frac{1}{2}[\mathbf{X},\mathbf{X}],\\
\end{align}

i.e., in superfield notation, the cohomological vector field is given by

\begin{equation}
\end{equation}

In component fields and after Berezinian integration, we get the cohomological vector field

\begin{multline}
Q_\Sigma=\int_\Sigma\dd^4u\Bigg(\left(Y+\frac{1}{2}[X,X]\right)\frac{\delta}{\delta X}+(\psi+\dd_AX)\frac{\delta}{\delta A}+(\chi^\dagger+F_A+[X,\chi])\frac{\delta}{\delta \chi}\\
+(\dd_A\chi+A^\dagger+[X,\psi^\dagger])\frac{\delta}{\delta \psi^\dagger}+(\dd_A\psi^\dagger+X^\dagger+[X,Y^\dagger])\frac{\delta}{\delta Y^\dagger}+[X,Y]\frac{\delta}{\delta Y}\\
+(\dd_AY+[X,\psi])\frac{\delta}{\delta \psi}+(\dd_A\psi+[X,\chi^\dagger]+[\chi,Y])\frac{\delta}{\delta \chi^\dagger}\\
+(\dd_A\chi^\dagger+[X,A^\dagger]+[\psi^\dagger,Y]+[\chi.\psi])\frac{\delta}{\delta A^\dagger}\\
+(\dd_AA^\dagger+[X,X^\dagger]+[Y^\dagger,Y]+[\psi^\dagger,\psi]+[\chi,\chi^\dagger])\frac{\delta}{\delta X^\dagger}\Bigg),
\end{multline}

where we have used the relation in (<ref>).

§.§ A suitable gauge-fixing

Consider a Riemannian metric on $\Sigma$ such that we can take a splitting of the fields $\chi,\chi^\dagger$ into self-dual and anti self-dual parts

\begin{align}
\chi&=\chi^++\chi^-,\\
\chi^\dagger&=\chi^{+\dagger}+\chi^{-\dagger}.
\end{align}

We can take the gauge-fixing where $\chi^-=\chi^{+\dagger}=0$ and all other forms are coexact when using Hodge decomposition. This will imply that $X^\dagger=Y^\dagger=0$. Introduce new fields $\bar X$ and $b$ with $\deg(\bar X)=\deg(b)=0$ and $\gh(\bar X)=-1$ and $\gh(b)=0$ together with corresponding anti-fields $\bar X^\dagger$ and $b^\dagger$ with $\deg(\bar X^\dagger)=\deg(b^\dagger)=4$, such that $\gh(\bar X^\dagger)=0$ and $\gh(b^\dagger)=-1$, respectively. Moreover, we consider the term

\[
\calS^{(1)}_\Sigma=\int_\Sigma\left(\langle \bar X^\dagger,(b+[X,\bar X])\rangle+\langle b^\dagger,([X,b]-[Y,\bar X])\rangle\right).
\]

Additionally, introduce more new fields $(\bar Y,\eta)$ with $\deg(Y)=\deg(\eta)=0$ and $\gh(\bar Y)=-2$ and $\gh(\eta)=-1$ together with corresponding anti-fields $\bar Y^\dagger$ and $\eta^\dagger$ with $\deg(\bar Y^\dagger)=\deg(\eta^\dagger)=4$, such that $\gh(\bar Y^\dagger)=+1$ and $\gh(\eta^\dagger)=0$. We consider the term

\[
\calS^{(2)}_\Sigma=\int_\Sigma\left(\langle \bar Y^\dagger,(\eta+[X,\bar Y]\rangle+\langle \eta^\dagger,([X,\eta]+[\bar Y,Y])\rangle\right).
\]

The gauge-fixed BV action is then obtained by

\[
\calS^\mathrm{gf}_\Sigma=\calS_\Sigma+\calS^{(1)}_\Sigma+\calS^{(2)}_\Sigma
\]

with gauge-fixing fermion

\begin{equation}
\label{eq:gauge-fixing_fermion}
\varPsi^\mathrm{gf}=\int_\Sigma\langle \bar X,\dd *A\rangle+\int_\Sigma\langle \bar Y,\dd*\psi\rangle.
\end{equation}

such that $\calS^\mathrm{gf}_\Sigma=\calS_\Sigma+Q_\Sigma\varPsi^\mathrm{gf}$.
The auxiliary fields $\chi^{-\dagger}$ have ghost number zero and can be actually integrated out. One can check that the gauge-transformations for each field squares to zero except the one for $\chi^+$. Namely, we have

\[
\delta^2\chi^+=(\dd_A\psi)^++[Y,\chi^+],
\]

which is the equation of motion for this setting. Hence, the tuple

\[
(A,Y,\psi,\chi^+,\bar Y,\eta)
\]

is the same as in the construction of Section <ref>.

§.§ BV-BFV formulation

As an AKSZ theory, there a nice way of treating DW theory as a gauge-theory on 4-manifolds with boundary. Let $\Sigma$ be a 4-manifold with 3-dimensional boundary $\de\Sigma$. As obtained in Section <ref>, the BV theory of the AKSZ construction of DW theory is given by

\[
\]

One can then easily formulate a BFV theory on the boundary $\de\Sigma$ by setting

\begin{align}
\calF^\de_{\de\Sigma}&:=\Map(T[1]\de\Sigma,\mathfrak{h}[1]\oplus\mathfrak{h}[2])\cong \Omega^\bullet(\de\Sigma)\otimes\mathfrak{h}[1]\oplus\Omega^\bullet(\de\Sigma)\otimes \mathfrak{h}[2],\\
\omega^\de_{\de\Sigma}&:=\int_{T[1]\de\Sigma}\bbmu_3\langle\delta\mathds{X},\delta\mathds{Y}\rangle,\\
\calS^\de_{\de\Sigma}&:=\int_{T[1]\de\Sigma}\bbmu_3\left(\langle\mathds{Y},\dd_{\de\Sigma}\mathds{X}\rangle+\frac{1}{2}\langle \mathds{Y},\mathds{Y}\rangle+\frac{1}{2}\langle\mathds{Y},[\mathds{X},\mathds{X}]\rangle\right),
\end{align}

where $\bbmu_3:=\dd^3\mathbb{u}\dd^3\bbtheta$ denotes the supermeasure on $T[1]\de\Sigma$ for local coordinates $(\mathbb{u}_i,\bbtheta^i)$ on $T[1]\de\Sigma$.
The boundary superfields are given by

\begin{align}
\mathds{X}(\mathbb{u},\bbtheta)&=\mathbb{X}+\mathbb{A}\bbtheta^{i_1}+\bbchi\bbtheta^{i_1}\bbtheta^{i_2}+\bbpsi^\dagger\bbtheta^{i_1}\bbtheta^{i_2}\bbtheta^{i_3}+\mathbb{Y}^\dagger\bbtheta^{i_1}\bbtheta^{i_2}\bbtheta^{i_3}\bbtheta^{i_4},\\
\mathds{Y}(\mathbb{u},\bbtheta)&=\mathbb{Y}+\bbpsi\bbtheta^{i_1}+\bbchi^\dagger\bbtheta^{i_1}\bbtheta^{i_2}+\mathbb{A}^\dagger\bbtheta^{i_1}\bbtheta^{i_2}\bbtheta^{i_3}+\mathbb{X}^\dagger\bbtheta^{i_1}\bbtheta^{i_2}\bbtheta^{i_3}\bbtheta^{i_4}.
\end{align}

The cohomological vector field $Q^\de_{\de\Sigma}$ is given by the Hamiltonian vector field of $\calS^\de_{\de\Sigma}$:

\[
\iota_{Q^\de_{\de\Sigma}}\omega^\de_{\de\Sigma}=\delta\calS^\de_{\de\Sigma}.
\]

It is easy to see that $\omega^\de_{\de\Sigma}$ is exact with primitive 1-form

\[
\alpha^\de_{\de\Sigma}=\int_{T[1]\de\Sigma}\bbmu_3\langle\mathds{Y},\delta\mathds{X}\rangle.
\]

If we expand things into components and perform Berezinian integration, we get

\begin{equation}
\omega^\de_{\de\Sigma}=\int_{\de\Sigma}\dd^3\mathbb{u} \left(\delta \mathbb{X} \delta \mathbb{X}^\dagger+\delta \mathbb{A}\delta \mathbb{A}^\dagger+\delta\bbchi\delta\bbchi^\dagger+\delta\bbpsi^\dagger\delta\bbpsi+\delta \mathbb{Y}^\dagger\delta \mathbb{Y}\right)
\end{equation}

\begin{multline}
\calS^\de_{\de\Sigma}= \int_{\de\Sigma}\dd^3\mathbb{u}\bigg(\langle \bbpsi,\dd_\mathbb{A}\bbchi\rangle+\frac{1}{2}\langle \mathbb{Y},[\bbchi,\bbchi]\rangle+\langle \bbpsi^\dagger,(\dd_\mathbb{A} \mathbb{Y}+[\mathbb{X},\bbpsi])\rangle\\+\langle \bbchi^\dagger,(\mathbb{F}_\mathbb{A}+[\mathbb{X},\bbchi])\rangle+\langle \mathbb{A}^\dagger,(\bbpsi+\dd_\mathbb{A}\mathbb{X})\rangle+\langle \mathbb{Y}^\dagger,[\mathbb{X},\mathbb{Y}]\rangle\\+\left\langle \mathbb{X}^\dagger,\left(\frac{1}{2}[\mathbb{X},\mathbb{X}]\right)\right\rangle+\frac{1}{2}\langle \bbchi^\dagger,\bbchi^\dagger\rangle\bigg),
\end{multline}

where $\dd_\mathbb{A}=\dd_{\de\Sigma}+[\mathbb{A},\enspace]$ denotes the covariant derivative of $\mathbb{A}$ and $\mathbb{F}_\mathbb{A}=\dd_{\de\Sigma} \mathbb{A}+\frac{1}{2}[\mathbb{A},\mathbb{A}]$. The BV transformations on the superfields are then given by

\begin{align}
Q^\de_{\de\Sigma} \mathds{X}&=\dd_{\de\Sigma}\mathds{X}+\mathds{Y}+\frac{1}{2}[\mathds{X},\mathds{X}],\\
\end{align}

i.e., in superfield notation, the cohomological vector field is given by

\begin{equation}
\end{equation}

In component fields and after Berezinian integration, we get the cohomological vector field

\begin{multline}
Q^\de_{\de\Sigma}=\int_{\de\Sigma}\dd^3\mathbb{u}\Bigg(\left(\mathbb{Y}+\frac{1}{2}[\mathbb{X},\mathbb{X}]\right)\frac{\delta}{\delta \mathbb{X}}+(\bbpsi+\dd_\mathbb{A}\mathbb{X})\frac{\delta}{\delta \mathbb{A}}+(\bbchi^\dagger+\mathbb{F}_\mathbb{A}+[\mathbb{X},\bbchi])\frac{\delta}{\delta \bbchi}\\
+(\dd_\mathbb{A}\bbchi+\mathbb{A}^\dagger+[\mathbb{X},\bbpsi^\dagger])\frac{\delta}{\delta \bbpsi^\dagger}+(\dd_\mathbb{A}\bbpsi^\dagger+\mathbb{X}^\dagger+[\mathbb{X},\mathbb{Y}^\dagger])\frac{\delta}{\delta \mathbb{Y}^\dagger}+[\mathbb{X},\mathbb{Y}]\frac{\delta}{\delta \mathbb{Y}}\\
+(\dd_\mathbb{A}\mathbb{Y}+[\mathbb{X},\bbpsi])\frac{\delta}{\delta \bbpsi}+(\dd_\mathbb{A}\bbpsi+[\mathbb{X},\bbchi^\dagger]+[\bbchi,\mathbb{Y}])\frac{\delta}{\delta \bbchi^\dagger}\\
+(\dd_\mathbb{A}\bbchi^\dagger+[\mathbb{X},\mathbb{A}^\dagger]+[\bbpsi^\dagger,\mathbb{Y}]+[\bbchi,\bbpsi])\frac{\delta}{\delta \mathbb{A}^\dagger}\\
+(\dd_\mathbb{A}\mathbb{A}^\dagger+[\mathbb{X},\mathbb{X}^\dagger]+[\mathbb{Y}^\dagger,\mathbb{Y}]+[\bbpsi^\dagger,\bbpsi]+[\bbchi,\bbchi^\dagger])\frac{\delta}{\delta \mathbb{X}^\dagger}\Bigg).
\end{multline}

By setting $\pi_\Sigma\colon \calF_\Sigma\to \calF^\de_{\de\Sigma}$ to be the restriction of the fields to the boundary, we get that the modified CME

\[
\]

is satisfied and that the cohomological vector field $Q_\Sigma$ is indeed projectable, i.e. we have

\[
\delta\pi_\Sigma Q_\Sigma=Q^\de_{\de\Sigma}.
\]

§ QUANTIZATION OF DW THEORY IN THE BV-BFV SETTING

§.§ The BFV boundary operator

Assume now that we can write the boundary as a disjoint union $\de\Sigma=\de_1\Sigma\sqcup\de_2\Sigma$ where $\de_1\Sigma$ and $\de_2\Sigma$ have opposite orientation. We choose the convenient polarization $\calP$ on $\de\Sigma$ consisting of choosing the $\frac{\delta}{\delta \mathds{Y}}$-polarization ($\mathds{X}$-representation) on $\de_1\Sigma$ and the $\frac{\delta}{\delta\mathds{X}}$-polarization ($\mathds{Y}$-representation) on $\de_2\Sigma$ (see Figure <ref>).
As we have seen in Section <ref>, we can compute the BFV boundary operator $\Omega^\calP_{\de\Sigma}$ as the ordered standard quantization of the boundary action $\calS^\de_{\de\Sigma}$ with respect to the chosen polarization.
We get

\[
\Omega^\calP_{\de\Sigma}=\Omega^\calP_0+\Omega^\calP_\mathrm{pert},
\]

\begin{equation}
\label{eq:boundary_operator_0}
\Omega^\calP_0:=\underbrace{\I\hbar\int_{\de_1\Sigma}\sum_i\dd_{\de_1\Sigma}\mathds{X}^i\frac{\delta}{\delta\mathds{X}^i}}_{=:\Omega_0^\mathds{X}}+\underbrace{\I\hbar\int_{\de_2\Sigma}\sum_i\dd_{\de_2\Sigma}\mathds{Y}_i\frac{\delta}{\delta\mathds{Y}_i}}_{=:\Omega^\mathds{Y}_0},
\end{equation}

\begin{equation}
\label{eq:boundary_operator_pert}
\Omega^\calP_\mathrm{pert}=\Omega^\mathds{X}_\mathrm{pert}+\Omega^\mathds{Y}_\mathrm{pert},
\end{equation}

\begin{equation}
\label{eq:boundary_operator_pert_X}
\Omega^\mathds{X}_\mathrm{pert}:=\int_{\de_1\Sigma}\left(-\frac{\hbar^2}{2}\sum_{i,j}\frac{\delta^2}{\delta\mathds{X}^i\delta\mathds{X}^j}+\frac{\I\hbar}{2}\sum_{i,j,k}c_{ij}^k\mathds{X}^i\mathds{X}^j\frac{\delta}{\delta\mathds{X}^k}\right),
\end{equation}

\begin{equation}
\label{eq:boundary_operator_pert_Y}
\Omega^\mathds{Y}_\mathrm{pert}:=\int_{\de_2\Sigma}\left(\frac{1}{2}\sum_{i,j}\delta^{i}_j\mathds{Y}_i\mathds{Y}_j-\frac{\hbar^2}{2}\sum_{i,j,k}c_{ij}^k \mathds{Y}_k\frac{\delta^2}{\delta\mathds{Y}_i\delta\mathds{Y}_j}\right),
\end{equation}

where $\delta^{i}_j$ denotes the Kronecker delta and $c_{ij}^k$ the structure constants of the Lie bracket $[\enspace,\enspace]$.

§.§ Composite fields

In order to regularize the higher functional derivatives, one needs to introduce the concept of composite fields as in <cit.>. In particular, we want to regard the higher functional derivates as a single first order derivative with respect to the composite field. In order to get it coherent to the naive interpretation where a higher functional derivative concentrates the fields on some diagonal, we should also understand the product of integrals as containing the diagonal contributions for the corresponding composite field. We consider the following bullet product of integrals

\begin{multline}
\label{eq:bullet_product}
\left(\int_{\de_k\Sigma}\sum_i\alpha_i\Phi^i\right)\bullet\left(\int_{\de_k\Sigma}\sum_j\beta_j\Phi^j\right):=\\
\end{multline}

where $\alpha$ and $\beta$ are smooth forms which depend on the bulk and residual fields and we denote by $[\Phi^i\Phi^j]$ the composite field. We can now interpret the operator $\int_{\de_k\Sigma}\phi^{ij}\frac{\delta^2}{\delta\Phi^i\delta\Phi^j}$ as $\int_{\de_k\Sigma}\phi^{ij}\frac{\delta}{\delta[\Phi^i\Phi^j]}$. Hence, we get

\[
\int_{\de_k\Sigma}\phi^{ij}\frac{\delta^2}{\delta\Phi^i\delta\Phi^j}\left(\left(\int_{\de_k\Sigma}\alpha_i\Phi^i\right)\bullet\left(\int_{\de_k\Sigma}\beta_j\Phi^j\right)\right)=\int_{\de_k\Sigma}\alpha_i\beta_j\phi^{ij}.
\]

More general, let $I:=(i_1,\ldots,i_n)$ be a multi-index and note that we can replace the higher derivative $\frac{\delta^n}{\delta\Phi^{i_1}\dotsm \delta\Phi^{i_n}}$ by the first order derivative $\frac{\delta}{\delta[\Phi^I]}:=\frac{\delta}{\delta[\Phi^{i_1}\dotsm \Phi^{i_n}]}$ with respect to the composite field $[\Phi^I]:=[\Phi^{i_1}\dotsm \Phi^{i_n}]$. The higher functional derivatives appearing in the BFV boundary operator are then regularized by using composite fields and it acts on the state space given by regular functionals, i.e. on the algebra which is generated by linear combinations of expressions of the form

\begin{multline}
\label{eq:condition_1}
\int_{\overline{\mathrm{Conf}_{m_1}(\de_1\Sigma)}\times\overline{\mathrm{Conf}_{m_2}(\de_2\Sigma)}}\mathsf{Z}^{J_1^1\dotsm J_1^{\ell_1}J_2^1\dotsm J^{\ell_2}\dotsm}_{I_1^1\dotsm I_1^{r_1}I_2^{1}\dotsm I_2^{r_2}\dotsm}\pi_1^*\prod_{j=1}^{r_1}\left[\mathds{X}^{I_1^j}\right]\dotsm \pi_{m_1}^*\prod_{j=1}^{r_{m_1}}\left[\mathds{X}^{I^j_{m_1}}\right]\times\\
\times \pi^*_1\prod_{j=1}^{\ell_1}\left[\mathds{Y}_{J^j_1}\right]\dotsm\pi^*_{m_2}\prod_{j=1}^{\ell_{m_2}}\left[\mathds{Y}_{J^j_{m_2}}\right],
\end{multline}

where $I^j_i$ and $J^j_i$ are multi-indices and $\mathsf{Z}^{J_1^1\dotsm J_1^{\ell_1}J_2^1\dotsm J^{\ell_2}\dotsm}_{I_1^1\dotsm I_1^{r_1}I_2^{1}\dotsm I_2^{r_2}\dotsm}$ is a smooth form on the product $\overline{\mathrm{Conf}_{m_1}(\de_1\Sigma)}\times\overline{\mathrm{Conf}_{m_2}(\de_2\Sigma)}$. If we denote by $\exp_\bullet$ the exponential defined through the bullet product $\bullet$ constructed as in (<ref>), we can see that

\[
\int_{\de_k\Sigma}\phi^I\frac{\delta}{\delta\Phi^I}\Big\langle\exp_\bullet\left(\calS^\mathrm{res}_\Sigma+\calS^\mathrm{source}_\Sigma\right)\Big\rangle=\left\langle\int_{\de_k\Sigma}\phi^I\frac{\delta}{\delta\Phi^I}\exp_\bullet\left(\calS^\mathrm{res}_\Sigma+\calS^\mathrm{source}_\Sigma\right)\right\rangle,
\]

\begin{align*}
\calS^\mathrm{res}_\Sigma&:=\int_{\de_1\Sigma}\langle \mathsf{y},\mathds{X}\rangle-\int_{\de_2\Sigma}\langle\mathds{Y},\mathsf{x}\rangle,\\
\calS^\mathrm{source}_\Sigma&:=\int_{\de_1\Sigma}\langle\mathscr{Y},\mathds{X}\rangle-\int_{\de_2\Sigma}\langle \mathds{Y},\mathscr{X}\rangle,
\end{align*}

The full boundary state is then defined through the regularization of composite fields by using (<ref>):

\begin{equation}
\label{eq:full_state}
\mathsf{Z}^{\bullet,\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}:=\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}\Big\langle\exp_\bullet\left(\calS^\mathrm{res}_\Sigma+\calS^\mathrm{source}_\Sigma\right)\Big\rangle,
\end{equation}

where $\big\langle\enspace\big\rangle$ denotes the expectation value with respect to $\calS_\Sigma$, i.e. for an observable $O\in\calO_\calF$, we have

\[
\big\langle O\big\rangle:=\int\exp(\I\calS_\Sigma(\mathbf{X},\mathbf{Y})/\hbar)O(\mathbf{X},\mathbf{Y})\mathscr{D}[\mathbf{X}]\mathscr{D}[\mathbf{Y}].
\]

Let $I_1,\ldots, I_{m_1}$ and $J_1,\ldots,J_{m_2}$ be multi-indices for $m_1,m_2\geq 1$.
Denote by $\Gamma_k'$ for $k=1,2$ the graphs with $m_1$ vertices on $\de_k\Sigma$, where vertex $s$ has valency $\vert I_s\vert\geq 1$, with adjacent half-edges oriented inwards and decorated with boundary fields $\left[\Phi^{I_1}_k\right],\ldots,\left[\Phi^{I_{m_1}}_k\right]$ all evaluated at the point of collapse $\mathbb{u}\in \de_k\Sigma$. We also want them to have $\vert J_1\vert+\dotsm +\vert J_{m_2}\vert$ outward leaves if $k=1$ and $\vert J_1\vert+\dotsm+\vert J_{m_2}\vert$ inward leaves if $k=2$, decorated with functional derivatives with respect to the boundary fields:

\[
\I\hbar\frac{\delta}{\delta\left[\Phi^{J_1}_k\right]},\ldots,\I\hbar\frac{\delta}{\delta\left[\Phi^{J_{m_2}}_k\right]}
\]

evaluated at the point of collapse $\mathbb{u}\in \de_k\Sigma$. Moreover, there are no outward leaves if $k=2$ and no inward leaves if $k=1$. Denote by $\sigma_{\Gamma_i'}$ the differential form given by

\[
\sigma_{\Gamma_k'}:=\int_{\overline{\mathrm{C}_{\Gamma_k'}(\mathbb{H}^4)}}\omega_{\Gamma_k'},
\]

where $\omega_{\Gamma_k'}$ denotes the product of limiting propagators at the point of collapse $\mathbb{u}\in\de_k\Sigma$ and vertex tensors.
Then we can construct the corresponding full BFV boundary operator through the regularization of composite fields as

\begin{equation}
\label{eq:full_boundary_operator}
\Omega^{\bullet,\calP}_{\de\Sigma}=\Omega^\calP_0+\Omega^{\bullet,\mathds{X}}_\mathrm{pert}+\Omega^{\bullet,\mathds{Y}}_\mathrm{pert},
\end{equation}

\begin{align}
\label{eq:principal_part_X}
\Omega^{\bullet,\mathds{X}}_\mathrm{pert}&:=\sum_{m_1,m_2\geq 0}\,\sum_{\Gamma_1'}\frac{(-\I\hbar)^{\ell(\Gamma_1')}}{\vert\mathrm{Aut}(\Gamma_1')\vert}\int_{\de_1\Sigma}\left(\sigma_{\Gamma_1'}\right)^{J_1\dotsm J_{m_2}}_{I_1\dotsm I_{m_1}}\prod_{j=1}^{m_1}\left[\mathds{X}^{I_j}\right]\left((-1)^{4m_2}(\I\hbar)^{m_2}\frac{\delta^{\vert J_1\vert+\dotsm+\vert J_{m_2}\vert}}{\delta\left[\mathds{X}^{J_1}\dotsm \mathds{X}^{J_{m_2}}\right]}\right)\\
\label{eq:principal_part_Y}
\Omega^{\bullet,\mathds{Y}}_\mathrm{pert}&:=\sum_{m_1,m_2\geq 0}\,\sum_{\Gamma_2'}\frac{(-\I\hbar)^{\ell(\Gamma_2')}}{\vert\mathrm{Aut}(\Gamma_2')\vert}\int_{\de_2\Sigma}\left(\sigma_{\Gamma_2'}\right)^{J_1\dotsm J_{m_2}}_{I_1\dotsm I_{m_1}}\prod_{j=1}^{m_1}\left[\mathds{Y}_{I_j}\right]\left((-1)^{4m_2}(\I\hbar)^{m_2}\frac{\delta^{\vert J_1\vert+\dotsm+\vert J_{m_2}\vert}}{\delta\left[\mathds{Y}_{J_1}\dotsm \mathds{Y}_{J_{m_2}}\right]}\right)
\end{align}

§.§ Feynman rules

In order to formulate a BV-BFV quantization, i.e. for the boundary state and the BFV boundary operator, we need to consider the Feynman graphs on the source manifold $\Sigma$ (see Figure <ref> for an example) for DW theory. The graphs are determined by the Feynman rules of the theory and the corresponding degree count (see Section <ref>).
The Feynman rules are determined by the AKSZ action functional $\calS_\Sigma$ of DW theory. In particular, the interaction vertices are given by the ones as in Figure <ref>. The boundary vertices are given by Figure <ref> and <ref>.

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5
[rounded corners=20pt,top color=gray!50, bottom color=gray!.5](5.5,1)–(8,0)–(10,-1)–(12,-1)–(12,-3)–(10,-3)–(8,-4)–(5.2,-5.2)–(5,-3)–(7,-2)–(5,-1)–(5.5,1);
[fill=white] (9.5,-2) arc (0:360:0.7cm and 0.3cm);
(8.8,-2.3) arc (-90:0:0.9cm and 0.5cm);
(8.8,-2.3) arc (-90:-180:0.9cm and 0.5cm);
[color=white, top color=gray!80, bottom color=gray!10.5] (6,0) arc (0:360:0.5cm and 1cm);
[thick, color=red] (6,0) arc (0:360:0.5cm and 1cm);
[color=white, bottom color=gray!80, top color=gray!10.5] (6,-4) arc (0:360:0.5cm and 1cm);
[thick, color=red] (6,-4) arc (0:360:0.5cm and 1cm);
[color=white, top color=gray!80, bottom color=gray!10.5] (12.1,-2) arc (0:360:0.5cm and 1cm);
[thick, color=blue] (12.1,-2) arc (0:360:0.5cm and 1cm);
[label=left:$\mathds{X}$] at (5,0);
[label=left:$\mathds{X}$] at (5,-4);
[label=right:$\mathds{Y}$] at (12.1,-2);
[label=above:$\de_1^{(1)}\Sigma$] at (5.5,1);
[label=below:$\de_1^{(2)}\Sigma$] at (5.5,-5);
[label=below:$\de_2\Sigma$] at (11.6,-3);
[Bullet=gray,label=above:] at (7,-1);
[Bullet=gray,label=left:] at (9,-0.8);
[Bullet=gray,label=right:] at (10,-1.3);
[Bullet=gray,label=right:] at (7.5,-3);
[Bullet=gray,label=right:] at (10.3,-2.5);
[Bullet=gray,label=right:] at (6,0);
[Bullet=gray,label=right:] at (6,-4);
[Bullet=gray,label=right:] at (11.1,-2);
[-latex[scale=3.0]] (6,0) to[bend left] (7,-1);
[-latex[scale=3.0]] (6,0) to[bend right] (7,-1);
[-latex[scale=3.0]] (7,-1) to[bend left] (9,-0.8);
[-latex[scale=3.0]] (9,-0.8) to[bend left] (10,-1.3);
[-latex[scale=3.0]] (10,-1.3) to[bend left] (10.3,-2.5);
[-latex[scale=3.0]] (10.3,-2.5) to[bend left] (7.5,-3);
[-latex[scale=3.0]] (7.5,-3) to[bend left] (7,-1);
[-latex[scale=3.0]] (6,-4) to[bend right] (7.5,-3);
[-latex[scale=3.0]] (6,-4) to[bend left] (7.5,-3);
[-latex[scale=3.0]] (10,-1.3) to[bend left] (11.1,-2);
[-latex[scale=3.0]] (10.3,-2.5) to[bend right] (11.1,-2);

Example of a Feynman graph on the source manifold.

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5

[Bullet=cyan] at (0,0);
at (0,.5)1;
[-latex[scale=3.0]] (0,0)–(-1,-1);
[-latex[scale=3.0]] (0,0)–(1,-1);
[Bullet=red] at (4,0);
at (3.3,0)$c^k(\mathbf{X})$;
[-latex[scale=3.0]] (4,0)–(4,-1);
[-latex[scale=3.0]] (3,1)–(4,0);
[-latex[scale=3.0]] (5,1)–(4,0);
at (8,1)$\mathsf{x}$;
[-latex[scale=3.0]] (8.25,1)–(10,1);
at (8,-1)$\mathsf{y}$;
[-latex[scale=3.0]] (10,-1)–(8.28,-1);

The Feynman rules for DW theory. The first type of vertex corresponds to the term $\frac{1}{2}\langle\mathbf{Y},\mathbf{Y}\rangle=\frac{1}{2}\sum_i\mathbf{Y}_i\mathbf{Y}_i$. There are two outgoing arrows and no incoming arrow for the first vertex type. The second type of vertex corresponds to the term $\frac{1}{2}\langle\mathbf{Y},[\mathbf{X},\mathbf{X}]\rangle=\frac{1}{2}\sum_{i,j,k}c^k_{ij}\mathbf{X}^i\mathbf{X}^j\mathbf{Y}_k$, where $c^k_{ij}$ are the structure constants of the Lie bracket and where we have denoted $c^k(\mathbf{X}):=c^k_{ij}\mathbf{X}^i\mathbf{X}^j$. There are at most two incoming arrows and exactly one outgoing arrow for the second vertex type. Finally, we also have leaves corresponding to the residual fields. Note that an arrow between vertices represents the propagator from the starting to the end point.

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5

[color=red,thick] (0,0)–(4,0);
[color=blue,thick] (6,-2)–(10,-2);
[Bullet=gray, label=above:$\mathbb{u}_1$] at (2,0);
[label=above:$\mathds{X}$] at (1,0);
[Bullet=gray, label=below:$\mathbb{u}_2$] at (8,-2);
[label=below:$\mathds{Y}$] at (9,-2);
[-latex[scale=3.0]] (2,0)–(2,-1);
[-latex[scale=3.0]] (8,-1)–(8,-2);

The Feynman rules for vertices on the boundary

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5

[color=red,thick] (0,0)–(4,0);
[color=blue,thick] (6,-2)–(10,-2);
[Bullet=gray, label=above:$[\mathds{X}^{i_1}\dotsm\mathds{X}^{i_n}]$] at (2,0);
[Bullet=gray, label=below:$[\mathds{Y}_{i_1}\dotsm\mathds{Y}_{i_n}]$] at (8,-2);
[-latex[scale=3.0]] (2,0)–(1,-1);
[-latex[scale=3.0]] (2,0)–(1.3,-1);
[-latex[scale=3.0]] (2,0)–(1.5,-1);
[-latex[scale=3.0]] (2,0)–(1.8,-1);
[-latex[scale=3.0]] (2,0)–(2,-1);
[-latex[scale=3.0]] (2,0)–(2.3,-1);
[] at (2.5,-0.8)...;
[-latex[scale=3.0]] (2,0)–(3,-1);
[-latex[scale=3.0]] (7,-1)–(8,-2);
[-latex[scale=3.0]] (7.3,-1)–(8,-2);
[-latex[scale=3.0]] (7.5,-1)–(8,-2);
[-latex[scale=3.0]] (7.8,-1)–(8,-2);
[-latex[scale=3.0]] (8,-1)–(8,-2);
[-latex[scale=3.0]] (8.3,-1)–(8,-2);
[] at (8.5,-1.2)...;
[-latex[scale=3.0]] (9,-1)–(8,-2);

The Feynman rules for composite field vertices on the boundary

§.§ The partition function

The partition function (or boundary state) is then formally given by the functional integral where we integrate out the fluctuations, i.e. by

\begin{equation}
\label{eq:partition_function_DW_theory}
\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}(\mathds{X},\mathds{Y},\mathsf{x},\mathsf{y};\hbar)=\int_{\calL\subset\calY'} \exp\left(\I\calS_\Sigma(\mathbf{X},\mathbf{Y})/\hbar\right)\mathscr{D}[\mathscr{X}]\mathscr{D}[\mathscr{Y}]
% &=\int_{\calL}\exp\left(\I\calS_\Sigma/\hbar\right)\mathscr{D}[\mathbb{X}]\mathscr{D}[\mathbb{X}^\dagger]\mathscr{D}[\mathbb{Y}]\mathscr{D}[\mathbb{Y}^\dagger]\mathscr{D}[\mathbb{A}]\mathscr{D}[\mathbb{A}^\dagger]\mathscr{D}[\bbchi]\mathscr{D}[\bbchi^\dagger]\mathscr{D}[\bbpsi]\mathscr{D}[\bbpsi^\dagger],
\end{equation}

where the Lagrangian submanifold is given by the gauge-fixing as in Section <ref>, i.e.

\[
\calL=\mathrm{graph}(\dd\varPsi^\mathrm{gf}),
\]

where $\varPsi^\mathrm{gf}$ is the gauge-fixing fermion constructed as in (<ref>). The integral (<ref>) is defined perturbatively as a formal power series in $\hbar$ in terms of the Feynman graphs given by the according Feynman rules as in Section <ref>, i.e. we can rewrite it as a perturbative expansion around critical points of $\calS_\Sigma$ as

\begin{equation}
\label{eq:perturbative_expansion}
\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}(\mathds{X},\mathds{Y},\mathsf{x},\mathsf{y};\hbar)=T_\Sigma\exp\left(\frac{\I}{\hbar}\sum_\Gamma \frac{(-\I\hbar)^{\ell(\Gamma)}}{\vert\mathrm{Aut}(\Gamma)\vert}\int_{\overline{\mathrm{Conf}_\Gamma(\Sigma)}}\omega_\Gamma(\mathds{X},\mathds{Y},\mathsf{x},\mathsf{y})\right),
\end{equation}

where we sum over connected Feynman graphs $\Gamma$ and where $\ell(\Gamma)$ denotes the number of loops of $\Gamma$. Moreover, the integral in (<ref>) is over the configuration space of the vertex set of $\Gamma$ regarded as points in $\Sigma$ where we integrate a differential form $\omega_\Gamma$ depending on the boundary fields and residual fields.

§.§ Degree count and Feynman graphs

Note that we want the dimension of the configuration space $\overline{\mathrm{Conf}_\Gamma(\Sigma)}$ to match the form degree of the differential form $\omega_\Gamma$ in order to perform the integrals in (<ref>). Note that the dimension of the configuration space is given by

\[
\dim \overline{\mathrm{Conf}_\Gamma(\Sigma)}=4n+3m,
\]

where $n$ denotes the amount of vertices in the bulk and $m$ the amount of vertices on the boundary. Using the fact that the propagator $\mathscr{P}\in \Omega^3(\overline{\mathrm{Conf}_2(\Sigma)})$ is a 3-form on the configuration space of two points, the form degree of the differential form $\omega_\Gamma$ is given by $6\cdot\#I+3\cdot\#II$, where $\#I$ denotes the amount of first type vertices and $\#II$ the amount of second type vertices as in Figure <ref>. Thus we get the system of equations

\begin{align}
\begin{split}
\label{eq:degree_count_1}
\end{split}
\end{align}

Now if $m=0$, we get $2\cdot\#I=\#II$. Hence, one can check that the diagrams which give a contribution are either wheels with an even amount of type $II$ vertices and $\frac{1}{2}\cdot\#I$ vertices attached to them as in Figure <ref>.

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5

[Bullet=red,label=left:$u_1$] at (1,-.5);
[Bullet=red,label=above:$u_2$] at (2,.5);
[Bullet=red,label=right:$u_3$] at (3,-.5);
[Bullet=red,label=below:$u_4$] at (2,-1.5);
[-latex[scale=3.0]] (1,-.5)–(2,.5);
[-latex[scale=3.0]] (2,.5)–(3,-.5);
[-latex[scale=3.0]] (3,-.5)–(2,-1.5);
[-latex[scale=3.0]] (2,-1.5)–(1,-.5);
[Bullet=cyan,label=above:$u_5$] at (1,.5);
[Bullet=cyan,label=below:$u_6$] at (3,-1.5);
[-latex[scale=3.0]] (1,.5)–(1,-.5);
[-latex[scale=3.0]] (1,.5)–(2,.5);
[-latex[scale=3.0]] (3,-1.5)–(3,-.5);
[-latex[scale=3.0]] (3,-1.5)–(2,-1.5);
[Bullet=red,label=left:$u_1$] at (5,0);
[Bullet=red,label=above:$u_2$] at (6,.5);
[Bullet=red,label=above:$u_3$] at (7,0);
[Bullet=red,label=below:$u_4$] at (7,-1);
[Bullet=red,label=below:$u_5$] at (6,-1.5);
[Bullet=red,label=left:$u_6$] at (5,-1);
[-latex[scale=3.0]] (5,0)–(6,0.5);
[-latex[scale=3.0]] (6,.5)–(7,0);
[-latex[scale=3.0]] (7,0)–(7,-1);
[-latex[scale=3.0]] (7,-1)–(6,-1.5);
[-latex[scale=3.0]] (6,-1.5)–(5,-1);
[-latex[scale=3.0]] (5,-1)–(5,0);
[Bullet=cyan,label=above:$u_7$] at (5,1);
[Bullet=cyan,label=right:$u_8$] at (8,-.5);
[Bullet=cyan,label=below:$u_9$] at (5,-2);
[-latex[scale=3.0]] (5,1)–(5,0);
[-latex[scale=3.0]] (5,1)–(6,.5);
[-latex[scale=3.0]] (8,-.5)–(7,0);
[-latex[scale=3.0]] (8,-.5)–(7,-1);
[-latex[scale=3.0]] (5,-2)–(6,-1.5);
[-latex[scale=3.0]] (5,-2)–(5,-1);
[Bullet=red,label=left:$u_1$] at (-3,-1);
[Bullet=red, label=right:$u_2$] at (-1,-1);
[-latex[scale=3.0]] (-3,-1) to[bend right] (-1,-1);
[-latex[scale=3.0]] (-1,-1) to[bend right] (-3,-1);
[Bullet=cyan,label=above:$u_3$] at (-2,0);
[-latex[scale=3.0]] (-2,0) to (-3,-1);
[-latex[scale=3.0]] (-2,0) to (-1,-1);

Example of wheel graphs appearing due to the degree count. In the first graph we have $\#II=2$ and $\#I=1$, in the second graph we have $\#II=4$ and $\#I=2$ and in the third graph we have $\#II=6$ and $\#I=3$. In fact, the first graph does not give any contribution, since by Kontsevich's lemma <cit.> all the graphs with double edges (i.e. graphs where there exist two vertices which have exactly two arrows (propagators) connecting them) vanish. This can be seen by using the angle form on $\mathbb{H}^4$.

every loop/.style=min distance=20mm,in=50,out=130,looseness=10
Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5

[Bullet=gray, label=below:$u$] at (4,0);
[-latex[scale=3.0]] (4,0) edge  [loop above] node  ();

Short loop graphs (tadpoles) are not allowed since the propagator is singular on the diagonal. By a unimodularity condition which needs to be satisfied on $\mathfrak{h}[1]\oplus \mathfrak{h}[2]$, one can actually exclude these graphs. The condition is that the structure constants $c^k_{ij}$ of the Lie bracket on $\mathfrak{h}[1]\oplus\mathfrak{h}[2]$ satisfy $\sum_ic^i_{ij}=0$.
In fact, the unimodularity condition can be dropped if the Euler characteristic of $\Sigma$ vanishes.

When $m\not=0$, the system of equations (<ref>) gives us $2\cdot\#I-\#II-3m=0$. Example of Feynman graphs in this setting are given in Figure <ref>.

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5

[thick,color=blue] (0,0) to (4,0);
[thick,color=blue] (6,0) to (10,0);
[Bullet=red,label=left:$u_1$] at (1,1);
[Bullet=cyan,label=above:$u_2$] at (2,2);
[Bullet=cyan,label=above:$u_3$] at (3,1.5);
[Bullet=gray, label=below:$\mathbb{u}$] at (2,0);
[-latex[scale=3.0]] (1,1) to (2,0);
[-latex[scale=3.0]] (2,2) to (1,1);
[-latex[scale=3.0]] (2,2) to (2,0);
[-latex[scale=3.0]] (3,1.5) to (2,0);
[-latex[scale=3.0]] (3,1.5) to (1,1);
[Bullet=red,label=above:$u_1$] at (7.5,1.5);
[Bullet=red,label=above:$u_2$] at (8.5,1.5);
[Bullet=cyan,label=above:$u_3$] at (6.5,2);
[Bullet=cyan,label=above:$u_4$] at (6.5,1);
[Bullet=cyan,label=above:$u_5$] at (9.5,1);
[Bullet=cyan,label=above:$u_6$] at (9.5,2);
[Bullet=gray, label=below:$\mathbb{u}_1$] at (7.5,0);
[Bullet=gray, label=below:$\mathbb{u}_2$] at (8.5,0);
[-latex[scale=3.0]] (7.5,1.5) to (8.5,1.5);
[-latex[scale=3.0]] (6.5,2) to (7.5,1.5);
[-latex[scale=3.0]] (6.5,2) to (7.5,0);
[-latex[scale=3.0]] (6.5,1) to (7.5,1.5);
[-latex[scale=3.0]] (6.5,1) to (7.5,0);
[-latex[scale=3.0]] (9.5,2) to (8.5,1.5);
[-latex[scale=3.0]] (9.5,2) to[bend right] (8.5,0);
[-latex[scale=3.0]] (9.5,1) to[bend right] (8.5,0);
[-latex[scale=3.0]] (9.5,1) to[bend left] (8.5,0);
[-latex[scale=3.0]] (8.5,1.5) to (7.5,0);

Example of Feynman graphs when $m\not=0$. In the left graph we have $m=1$, $\#I=2$ and $\#II=2$. In the right graph we have $m=2$, $\#I=4$ and $\#II=2$. Similarly as befor, the second graph does not give any contribution since by Kontsevich's lemma <cit.> all the graphs with double edges vanish.

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5

[Bullet=red,label=left:$u_1$] at (0,-.5);
[Bullet=red,label=left:$u_2$] at (1,.5);
[Bullet=red,label=above:$u_3$] at (2,-.5);
[Bullet=red,label=left:$u_4$] at (1,-1.5);
[-latex[scale=3.0]] (0,-.5)–(1,.5);
[-latex[scale=3.0]] (1,.5)–(2,-.5);
[-latex[scale=3.0]] (2,-.5)–(1,-1.5);
[-latex[scale=3.0]] (1,-1.5)–(0,-.5);
[Bullet=cyan,label=left:$u_5$] at (0,.5);
[Bullet=cyan,label=right:$u_6$] at (2,-1.5);
[-latex[scale=3.0]] (0,.5)–(0,-.5);

at (.5,1)$\mathsf{y}$;
at (1.5,1)$\mathsf{x}$;
[-latex[scale=3.0]] (0,.5)–(.35,.85);
[-latex[scale=3.0]] (1.365,.865)–(1,.5);
[-latex[scale=3.0]] (2,-1.5)–(2,-.5);

at (1.5,-2)$\mathsf{y}$;
at (.5,-2)$\mathsf{x}$;
[-latex[scale=3.0]] (2,-1.5)–(1.65,-1.85);
[-latex[scale=3.0]] (.62,-1.8556)–(1,-1.5);
[Bullet=red,label=left:$u_1$] at (5,0);
[Bullet=red,label=left:$u_2$] at (6,.5);
[Bullet=red,label=above:$u_3$] at (7,0);
[Bullet=red,label=right:$u_4$] at (7,-1);
[Bullet=red,label=below:$u_5$] at (6,-1.5);
[Bullet=red,label=below:$u_6$] at (5,-1);
[-latex[scale=3.0]] (5,0)–(6,0.5);
[-latex[scale=3.0]] (6,.5)–(7,0);
[-latex[scale=3.0]] (7,0)–(7,-1);
[-latex[scale=3.0]] (7,-1)–(6,-1.5);
[-latex[scale=3.0]] (6,-1.5)–(5,-1);
[-latex[scale=3.0]] (5,-1)–(5,0);
[Bullet=cyan,label=left:$u_7$] at (5,1);
[Bullet=cyan,label=right:$u_8$] at (8,-.5);
[Bullet=cyan,label=below:$u_9$] at (5,-2);
[-latex[scale=3.0]] (5,1)–(5,0);

at (5.5,1.5)$\mathsf{y}$;
at (6.5,1.5)$\mathsf{x}$;
[-latex[scale=3.0]] (5,1)–(5.35,1.35);
[-latex[scale=3.0]] (6.4,1.35)–(6,0.5);
[-latex[scale=3.0]] (8,-.5)–(7,0);

at (8.5,-1.5)$\mathsf{y}$;
at (7.5,-1.5)$\mathsf{x}$;
[-latex[scale=3.0]] (8,-.5)–(8.4,-1.3);
[-latex[scale=3.0]] (7.4,-1.35)–(7,-1);
[-latex[scale=3.0]] (5,-2)–(6,-1.5);

at (4,-1.5)$\mathsf{y}$;
at (4,-.5)$\mathsf{x}$;
[-latex[scale=3.0]] (5,-2)–(4.17,-1.6);
[-latex[scale=3.0]] (4.15,-.6)–(5,-1);

Example of Feynman graphs with leaves (residual fields). Note that the $\mathsf{y}$-leaves give a contribution of form degree 2, whereas the $\mathsf{x}$-leaves give a contribution of form degree $1$. Gluing residual fields will induce a propagator of form degree $3$.

§.§ The modified Quantum Master Equation

We can now prove the following theorem:

The BV-BFV partition function $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}$ for DW theory satisfies the modified Quantum Master Equation, i.e.

\[
\big(\hbar^2\Delta_{\calV_\Sigma}+\Omega^{\bullet,\calP}_{\de\Sigma}\big)\mathsf{Z}^{\bullet,\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}=0,
\]

where $\Omega^{\bullet,\calP}_{\de\Sigma}=\Omega_0^\calP+\Omega^{\bullet,\calP}_\mathrm{pert}$ with $\Omega_0^\calP$ being the unperturbed quantization part as defined in (<ref>) and $\Omega^{\bullet,\calP}_\mathrm{pert}:=\Omega^{\bullet,\mathds{X}}_\mathrm{pert}+\Omega^{\bullet,\mathds{Y}}_\mathrm{pert}$ is fully determined by the boundary configuration space integrals.

The proof is on the level of graphs. First, we want to describe the construction of $\Omega^{\bullet,\calP}_\mathrm{pert}$ in terms of boundary configuration space integrals. Consider the compactified configuration space $\overline{\mathrm{Conf}}_\Gamma$ and $\omega_\Gamma$ the corresponding differential form, for some Feynman graph $\Gamma$. Then, using Stokes' theorem, we get $\int_{\overline{\mathrm{Conf}}_\Gamma}\dd\omega_\Gamma=\int_{\de\overline{\mathrm{Conf}}_\Gamma}\omega_\Gamma$. When we apply the de Rham differential $\dd$ to $\omega_\Gamma$ on the left-hand-side, it can either act on an $\mathds{X}$-field, $\mathds{Y}$-field or on the propagator $\mathscr{P}$. Clearly, the part acting on the $\mathds{X}$- or $\mathds{Y}$-fields corresponds to the action of $\frac{1}{\I\hbar}\Omega_0^\calP$, whereas the part acting on the propagator will correspond to the action of $-\I\hbar\Delta_{\calV_\Sigma}$ on the partition function $\mathsf{Z}^{\bullet,\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}$. The integral over the boundary configuration space on the right-hand-side contains terms of different possible strata for the corresponding manifold with corners. It either contains integrals over boundary components where two vertices collapse in the bulk, which we denote as situation $(a)$, or integrals over boundary components where more than two vertices collapse in the bulk[Usually called hidden faces.], which we denote by situation $(b)$, or integrals over boundary components where two or more (bulk and/or boundary) vertices collapse at the boundary or one single bulk vertex collapses to the boundary, which we denoted by situation $(c)$. For situation $(a)$, we can note that since we assume the CME to hold which equivalently translates to the fact that

\[
\frac{1}{2}\sum_{i=1}^4\pm \frac{\delta}{\delta\mathbf{X}^i}\left(\langle\mathbf{Y},\mathbf{Y}+[\mathbf{X},\mathbf{X}]\rangle\right)\cdot \frac{\delta}{\delta\mathbf{Y}_i}\left(\langle\mathbf{Y},\mathbf{Y}+[\mathbf{X},\mathbf{X}]\rangle\right)=0,
\]

the combinatorics of the Feynman graphs in the perturbative expansion leads to the cancellation of such terms when summing over all graphs. For situation $(b)$, one can use the usual vanishing theorems <cit.> in order to get rid of faces where all vertices of a connected component of a Feynman graph collapse. For situation $(c)$, note that we can split such integrals into an integral over a subgraph $\Gamma'\subset\Gamma$ and an integral over the graph which can be obtained by identifying all the vertices of $\Gamma'$ and deleting all edges of $\Gamma'$ which we denote by $\Gamma/\Gamma'$. Hence, one can define the action of $\frac{\I}{\hbar}\Omega^{\bullet,\calP}_\mathrm{pert}$ as the sum of the boundary contributions coming from the subgraphs $\Gamma'\subset \Gamma$ and thus we have

\[
\Omega^{\bullet,\calP}_\mathrm{pert}\mathsf{Z}^{\bullet,\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}:=\sum_\Gamma\sum_{\Gamma'\subset\Gamma}\int_{\overline{\mathrm{C}_{\Gamma'}(\mathbb{H}^4)}\times \overline{\mathrm{Conf}_{\Gamma/\Gamma'}(\Sigma)}}\omega_{\Gamma'}.
\]

It is important to note that in the setting of DW theory, there are no higher corrections for the definition of the principal part of $\Omega^{\bullet,\calP}_{\de\Sigma}$, i.e. the term $\Omega^{\bullet,\calP}_\mathrm{pert}$ which is in general of the form (<ref>)+(<ref>). This is due to the fact that we are working with $4$-dimensional source manifolds and by the result of the following lemma.

Let $\Sigma$ be a $d$-dimensional source manifold. If $d$ is even, then the principal part of $\Omega^{\bullet,\calP}_{\de\Sigma}$ is directly given by the ordered standard quantization of the boundary action $\calS^\de_{\de\Sigma}$. If $d$ is odd, the principal part of $\Omega^{\bullet,\calP}_{\de\Sigma}$ is given by the ordered standard quantization of the modified boundary action

\[
\tilde\calS^\de_{\de\Sigma}:=\calS^\de_{\de\Sigma}-\I\hbar\sum_{j=0}^{\left[\frac{d-3}{4}\right]}\int_{\de\Sigma}\gamma_j\tr\ad_{\mathbf{X}}^{d-4j},
\]

where $\gamma_j$ is a closed $4j$-form on $\de\Sigma$ which is an invariant polynomial, with universal coefficients, of the curvature of the connection used in the construction of the propagator.

Moreover, we can prove the following theorem:

The full BFV boundary operator $\Omega^{\bullet,\calP}_{\de\Sigma}$ for DW theory squares to zero and hence
we have a well-defined BV-BFV cohomology for DW theory, i.e. the operator $\hbar^2\Delta_{\calV_\Sigma}+\Omega^{\bullet,\calP}_{\de\Sigma}$ squares to zero.

It is easy to see that if $\Omega^{\bullet,\calP}_{\de\Sigma}$ squares to zero, so does $\hbar^2\Delta_{\calV_\Sigma}+\Omega^{\bullet,\calP}_{\de\Sigma}$ since $\Delta_{\calV_\Sigma}$ squares to zero and $[\Delta_{\calV_\Sigma},\Omega^{\bullet,\calP}_{\de\Sigma}]=0$. Consider again a subgraph $\Gamma'\subset\Gamma$ as in the definition of $\Omega^{\bullet,\calP}_{\de\Sigma}$ in the proof of Theorem <ref> with corresponding differential form $\sigma_{\Gamma'}$ over the configuration space $\overline{\mathrm{C}_{\Gamma'}(\mathbb{H}^4)}$. Using Stokes' theorem, we get $\int_{\overline{\mathrm{C}_{\Gamma'}(\mathbb{H}^4)}}\dd\sigma_{\Gamma'}=\int_{\de\overline{\mathrm{C}_{\Gamma'}(\mathbb{H}^4)}}\sigma_{\Gamma'}$. Similarly as before, we can obtain that the part where the de Rham differential acts on the boundary fields in $\sigma_{\Gamma'}$ corresponds to the action of $\frac{1}{\I\hbar}\Omega_0^\calP$. Moreover, we can obtain the similar three situations as in the proof of Theorem <ref>. In particular, the terms of situation $(a)$ cancel out when summing over all graphs, the terms of situation $(b)$ are excluded by the vanishing theorems and the terms of situation $(c)$ lead to the action of $\Omega^{\bullet,\calP}_\mathrm{pert}$ when summing over all graphs. Hence, we have

\[
\Omega_0^\calP\Omega^{\bullet,\calP}_\mathrm{pert}+\Omega^{\bullet,\calP}_\mathrm{pert}\Omega_0^\calP+(\Omega^{\bullet,\calP}_\mathrm{pert})^2=0,
\]

since $(\Omega_0^\calP)^2=0$.

Instead of considering the boundary operator in terms of boundary configuration space integrals, one can also take the explicit form of $\Omega^{\bullet,\calP}_{\de\Sigma}$ as in (<ref>) in order to prove that it squares to zero. In particular, for the full perturbation part, one can take the sum of the corresponding full versions of (<ref>) and (<ref>).

By Witten's approach <cit.> (see also Section <ref>), the expectation value of the observable $O$ as in (<ref>) with respect the BV-BFV partition function for the DW AKSZ-BV-action $\calS_\Sigma$ should be given by a Floer cohomology class associated to the boundary after integrating out the residual fields (see <cit.> for a possible integration theory on $\calV_\Sigma$). The resulting boundary state would then be of the form

\[
\Psi_{\de\Sigma}(\mathds{X},\mathds{Y})=\int_{\calV_\Sigma}\underbrace{\int_{\calL}\exp(\I\calS_\Sigma(\mathbf{X},\mathbf{Y})/\hbar)O(\mathbf{X},\mathbf{Y})\mathscr{D}[\mathscr{X}]\mathscr{D}[\mathscr{Y}]}_{=\left\langle\prod_{j=1}^dO^{(\gamma_j)}\right\rangle=\left\langle \prod_{j=1}^d\int_{\gamma_j}W_{k_j}\right\rangle}\in HF^\bullet(\de\Sigma)\subset\calH^\calP_{\de\Sigma}.
\]

For a particular polarization adapted to the boundary condition given by the Floer cohomology class $\Psi_{\de\Sigma}$, the boundary state space $\calH^\calP_{\de\Sigma}$ in this case should include the Floer cohomology classes and thus the BFV boundary operator $\Omega^{\bullet,\calP}_{\de\Sigma}$ should be given in terms of the Floer differential. Note that here we want $\mathfrak{h}=\mathfrak{su}(2)$. The gauge-fixing Lagrangian $\calL$ is given by the one considered in Section <ref>.

§.§ The modified differential Quantum Master Equation

Following the construction in <cit.>, we can perform globalization on the target of the DW AKSZ theory by using methods of formal geometry as developed in <cit.>. For a manifold $M$ and an open neighborhood $U\subset TM$ of the zero section, we can consider a generalized exponential map $\varphi\colon U\to M$ such that $\varphi(x,y):=\varphi_x(y)$, i.e. $\varphi$ satisfies the properties:

* $\varphi_x(0)=x,\quad \forall x\in M$,

* $\dd\varphi_x(0)=\id_{T_xM},\quad \forall x\in M$.

Locally, we get

\begin{equation}
\label{eq:local_generalized_exponential_map}
\varphi_x^i(y)=x^i+y^i+\frac{1}{2}\varphi^i_{x,jk}y^jy^k+\frac{1}{3!}\varphi^i_{x,jk\ell}y^jy^ky^\ell+\dotsm
\end{equation}

where $(x^i)$ are coordinates on the base and $(y^i)$ are coordinates on the fiber. A formal exponential map is given by the equivalence class of generalized exponential maps with respect to the equivalence relation which identifies two generalized exponential maps when their jets agree to all orders. By abuse of notation, we will also denote formal exponential maps by $\varphi$.
We can define a flat connection $D$ on $\widehat{\Sym}(T^*M)$, where $\widehat{\Sym}$ denotes the completed symmetric algebra. The connection is called Grothendieck connection <cit.> and can be locally written as $D=\dd+R$, where $\dd$ denotes the de Rham differential on $M$ and $R$ denotes a 1-form with values in derivations of the completed symmetric algebra of the cotangent bundle[Note that flatness of $D$ is equivalent to the Maurer–Cartan equation $\dd R+\frac{1}{2}[R,R]=0$.]. For a section $\sigma\in\Gamma(\widehat{\Sym}(T^*M))$, we get that $R$ acts on $\sigma$ through the Lie derivative, i.e. we have  $R(\sigma)=L_R\sigma$. Locally, we can express $R=R_\ell\dd x^\ell$, where $R_\ell:=R^j_\ell(x,y)\frac{\de}{\de y^j}$ and

\[
R^j_\ell(x,y):=-\frac{\de\varphi^k}{\de x^\ell}\left(\left(\frac{\de\varphi}{\de y}\right)^{-1}\right)^j_k=-\delta_\ell^j+\mathrm{O}(y).
\]

Thus, for a section $\sigma\in \Gamma(\widehat{\Sym}(T^*M))$, we have

\[
R(\sigma)=L_R\sigma=R_\ell(\sigma)\dd x^\ell=-\frac{\de\sigma}{\de y^j}\frac{\de\varphi^k}{\de y^\ell}\left(\left(\frac{\de\varphi}{\de y}\right)^{-1}\right)^j_k\dd x^\ell.
\]

Moreover, one can extend the Grothendieck connection $D$ to the complex $\Gamma(\bigwedge^\bullet T^*M\otimes \widehat{\Sym}(T^*M))$ consisting of $\widehat{\Sym}(T^*M)$-valued forms.

A section $\sigma\in \Gamma(\widehat{\Sym}(T^*M))$ is $D$-closed if and only if $\sigma=\mathsf{T}\varphi^*f$ for some $f\in C^\infty(M)$, where $\mathsf{T}$ denotes the Taylor expansion around the fiber coordinates at zero. Moreover, the $D$-cohomology is concentrated in degree zero and is given by

\[
H_D^0(\widehat{\Sym}(T^*M))=\mathsf{T}\varphi^*C^\infty(M)\cong C^\infty(M).
\]

The main part of the proof of Proposition <ref> uses methods from cohomological perturbation theory. let us now consider the AKSZ construction of DW theory as in Section <ref>. In particular, the techniques of <cit.> can be used here since the target is of the form $\mathfrak{h}[1]\oplus\mathfrak{h}[2]$. Consider a 4-dimensional source manifold (possibly with boundary) and recall the BV space of fields of DW theory

\[
\calF_{\Sigma}=\Map(T[1]\Sigma,\mathfrak{h}[1]\oplus\mathfrak{h}[2])\cong\Map(T[1]\Sigma,(T[1]\mathfrak{h})[1]).
\]

Consider then the formal exponential map given by

\begin{align*}
\varphi\colon T\mathfrak{h}\cong\mathfrak{h}\oplus\mathfrak{h}&\to \mathfrak{h},\\
(x,y)&\mapsto \varphi_x(y):=x+y.
\end{align*}

Note also that a particularly easy class of solutions of AKSZ theories is given by constant ones of the type $x=(x,0)\colon T[1]\Sigma\to \mathfrak{h}[1]\oplus \mathfrak{h}[2]$, where $x=\mathrm{const}$.
Thus, we can consider the linearized space of fields at a constant solution $x$, given by

\[
\varphi_x^*\calF_\Sigma:=\Map(T[1]\Sigma,(T[1]T_x\mathfrak{h})[1]).
\]

Let $(\mathbf{X},\mathbf{Y})\in\calF_\Sigma$. Then the corresponding lifts by $\varphi_x$ are given by

\[
\widehat{\mathbf{X}}:=\varphi_x^{-1}(\mathbf{X}),\qquad \widehat{\mathbf{Y}}:=(\dd\varphi_x)^*\mathbf{Y}.
\]

The Grothendieck connection can be computed by noticing that $R^j_\ell(x,y)=-\delta_\ell^j$.
We can then define the formal global action by

\[
\widehat{\calS}_{\Sigma,x}:=\int_\Sigma\left(\widehat{\mathbf{Y}}_i\dd_\Sigma\widehat{\mathbf{X}}^i+\widehat{\Theta}_x(\widehat{\mathbf{X}},\widehat{\mathbf{Y}})-\widehat{\mathbf{Y}}_\ell\dd_{\mathcal{EL}} x^\ell\right).
\]

Here we have denoted $\widehat{\Theta}_x(\widehat{\mathbf{X}},\widehat{\mathbf{Y}}):=\mathsf{T}\widehat{\varphi}_x^*\Theta(\mathbf{X},\mathbf{Y})$, where $\widehat{\varphi}_x\colon \varphi_x^*\calF_\Sigma\to \calF_\Sigma$. Note also that $\dd_{\mathcal{EL}}$ denotes the de Rham differential on the moduli space $\mathcal{EL}$ of classical solutions, which can be identified with the target by considering constant solutions. One can then check that the differential CME (dCME) holds:

\begin{equation}
\label{eq:dCME}
\dd_x\widehat{\calS}_{\Sigma,x}+\frac{1}{2}(\widehat{\calS}_{\Sigma,x},\widehat{\calS}_{\Sigma,x})=0.
\end{equation}

If $\Sigma$ has boundary, one can show that a differential version of the mCME (<ref>) is also satisfied <cit.>. For the quantum case, we have the following theorem:

The formal global BV-BFV partition function for DW theory, given formally by the functional integral

\[
\widehat{\mathsf{Z}}^{\bullet,\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}=\int_{\calL}\exp(\I\widehat{\calS}_{\Sigma,x}/\hbar),
\]

satisfies the modified differential QME (mdQME)

\begin{equation}
\label{eq:mdQME}
\left(\dd_x-\I\hbar\Delta_{\calV_\Sigma}+\frac{\I}{\hbar}\Omega^{\bullet,\calP}_{\de\Sigma}\right)\widehat{\mathsf{Z}}^{\bullet,\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}=0.
\end{equation}

Moreover, using similar methods as in <cit.>, we can show the following theorem:

The quantum Grothendieck BFV (qGBFV) operator

\[
\nabla_\mathsf{G}:=\dd_x-\I\hbar\Delta_{\calV}+\frac{\I}{\hbar}\Omega^{\bullet,\calP}
\]

is flat and behaves well under change of data. Moreover, it defines a cohomology theory on the globally extended state space.

This proof is similar to the one in <cit.>. In particular, the flatness of $\nabla_\mathsf{G}$ is equivalent to the equation

\begin{equation}
\I\hbar\dd_x\Omega^{\bullet,\calP}_{\de\Sigma}-\frac{1}{2}\left[\Omega^{\bullet,\calP}_{\de\Sigma},\Omega^{\bullet,\calP}_{\de\Sigma}\right]=0.
\end{equation}

Using the explicit expression of $\Omega^{\bullet,\calP}_{\de\Sigma}$ through configuration space integrals by using the perturbed parts (<ref>) and (<ref>), we can apply Stokes' theorem

\begin{equation}
\label{eq:Stokes}
\dd_x\int_{\overline{\mathrm{Conf}_\Gamma(\Sigma)}}\omega_\Gamma=\int_{\overline{\mathrm{Conf}_\Gamma(\Sigma)}}\dd\omega_\Gamma\pm \int_{\de\overline{\mathrm{Conf}_\Gamma(\Sigma)}}\omega_\Gamma,
\end{equation}

where $\dd=\dd_x+\dd_1+\dd_2$ is the differential on $(\mathfrak{h}[1]\oplus\mathfrak{h}[2])\times \overline{\mathrm{Conf}}_\Gamma(\Sigma)$ with $\dd_1$ the differential acting on the propagator, i.e. on the residual fields, and $\dd_2$ the differential acting on the boundary fields. Then by <cit.>, we can observe that the boundary face where more than two bulk vertices in a subgraph $\Gamma'\subset \Gamma$ collapse to the boundary was shown to be $\frac{1}{2}\left[\Omega^{\bullet,\mathds{Y}}_\mathrm{pert},\Omega^{\bullet,\mathds{Y}}_\mathrm{pert}\right]$ (and similarly for the $\mathds{X}$-representation). For the case where exactly two bulk vertices collapse, we can observe that these faces cancel with $\dd_x\omega_{\Gamma'}$ by using the dCME (<ref>).

Again, using the configuration space integral formulation of the partition function as in (<ref>), we can apply Stokes' theorem (<ref>) in order to obtain similar relations of $\dd_1$ with $\Delta_{\calV_\Sigma}$, $\dd_2$ with $\Omega_0^\calP$ and $\dd_x$ with the boundary contribution when applied to the partition function $\widehat{\mathsf{Z}}^{\bullet,\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}$ as in <cit.>.

§ NEKRASOV'S PARTITION FUNCTION, EQUIVARIANT BV FORMALISM AND EQUIVARIANT FLOER (CO)HOMOLOGY

§.§ Seiberg–Witten theory

In <cit.>, Seiberg and Witten have formulated a way of describing low-energy behaviour for special supersymmetric gauge field theories. Moreover, they have formulated topological invariants of 4-manifolds which can be shown to be equivalent to the Donaldson polynomials but in general easier to compute.

The gauge group is fixed to be $\mathrm{SU}(2)$ in this setting.
Recall that an $\calN=2$ chiral multiplet (or vector multiplet) includes a gauge field $A_\mu$, two Weyl fermions[Recall that a Weyl fermion (or Weyl spinor) is a spinor which satisfies the Weyl equations $\sigma^\mu\de_\mu\psi=0$, where $\sigma^\mu$ denotes the Pauli matrices for $\mu=0,\ldots,3$.] $\lambda,\psi$ and a scalar field $\phi$ in the adjoint representation, i.e. we have something of the form

\[
\]

where the brackets $(\!(\enspace)\!)$ here indicate that we have singlets $A_\mu,\phi$ and a doublet $(\lambda,\psi)$. When $\calN=1$, we can express this in one vector multiplet $W_\alpha$ (containing $A_\mu$ and $\lambda$) and a chiral multiplet $\Phi$ (containing $\phi$ and $\psi$).
In $\calN=1$ superspace, one can express the fermionic low-energy effective action in terms of a holomorphic function $\mathsf{F}$ as

\begin{equation}
\label{eq:Seiberg-Witten_action}
S^\mathrm{SW,eff}=\frac{1}{4\pi}\mathrm{Im}\,\left(\int\dd^4\theta \frac{\de\mathsf{F}(A)}{\de A}\bar A+\int \dd^2\theta\frac{1}{2}\frac{\de^2\mathsf{F}(A)}{\de A^2}W_\alpha W^{\alpha}\right),
\end{equation}

where $\theta$ denotes the angle for the global $\mathrm{SU}(2)$-action, $A$ denotes the $\calN=1$ chiral multiplet in the $\calN=2$ vector multiplet whose scalar component is given by some complex parameter which labels the vacua. The holomorphic function $\mathsf{F}$ is in fact the free energy which can be expressed in terms of the Seiberg–Witten formula for periods[Recall that a period of a closed differential form $\omega$ over some $n$-cycle $C$ is given by $\int_C\omega$.] of some differential $\dd \mathsf{S}$ on some algebraic curve $C$ (Seiberg–Witten curve).

§.§ Nekrasov's partition function

In <cit.>, Nekrasov constructed a regularized partition function $\mathsf{Z}^\mathrm{Nek}$ for the supersymmetric gauge field theories appearing in the construction of Seiberg–Witten, i.e. based on $\calN=2$ supersymmetric Yang–Mills theory. The idea was to use equivariant integration with respect to a natural symmetry group for a long-distance cut-off regularization with parameter $\varepsilon$. Moreover, he proposed that for $\varepsilon\to 0$, asymptotically we get $\log\mathsf{Z}^\mathrm{Nek}\sim -\frac{1}{\varepsilon^2}\mathsf{F}$. The proof was given by Nekrasov and Okounkov for $\mathrm{U}(r)$-gauge theories with matter fields in fundamental and adjoint representations of the gauge group and for 5-dimensional theories compactified on the circle.
Consider a connection $A$ on a trivial rank $r>1$ bundle over $\Sigma=\R^4$ such that the Yang–Mills action functional $S^{\mathrm{YM}}_\Sigma(A)=\int_\Sigma\|F_A\|^2<\infty$. In this case, as we have seen in Section <ref>, it is bounded from below

\[
8\pi^2k_A\leq S^\mathrm{YM}_{\Sigma}(A)=\int_\Sigma\|F_A\|^2.
\]

Moreover, as we have also seen in Section <ref>, if $A$ is an instanton (anti self-dual connection), we get equality. Donaldson showed in <cit.> that the moduli space of anti self-dual connections $\calM_\mathrm{ASD}$ is equal to the moduli space of holomorphic bundles on $\Sigma=\C^2\cong \R^4$ which are trivial at infinity. One can describe Nekrasov's theory as a $(\Aut(\Sigma)\times \GL(r))$-equivariant integration over a certain partial compactification

\[
\calM_\mathrm{ASD}\subset \overline{\calM}_r=\{\text{framed torsion-free sheaves on $\Sigma$ of rank $r$}\}.
\]

Note that each element of $\mathrm{Aut}(\Sigma)\times \mathrm{GL}(r)$ can always be considered in the form

\[
\begin{pmatrix}\varepsilon_1&\\& \varepsilon_2\end{pmatrix}\times \begin{pmatrix}a_1& & \\&\ddots&\\&&a_r\end{pmatrix}.
\]

In particular, $\varepsilon_1$ and $\varepsilon_2$ can be considered as two rotations of $\C^2$ regarded as the generators of a torus $\mathbb{T}$.
Let $\calM$ be any smooth algebraic variety and consider a torus $\mathbb{T}$ acting on it and $\calE$ a $\mathbb{T}$-equivariant coherent sheaf on $\calM$. Furthermore, let $\overline{\Sigma}$ be a projective surface, e.g. $\mathbb{P}^2$, and choose an embedding $\Sigma\hookrightarrow \overline{\Sigma}$ with a framing[Note that $\mathrm{GL}(r)\cong\mathrm{Aut}(\calO^{\oplus r}_D)$ acts on $\phi$ which is the same as the action of constant gauge transformations on instantons.] $\phi\colon\calE\vert_D\to \calO_D^{\oplus r}$ of $\calE$ where $D=\overline{\Sigma}\setminus \Sigma$. Then one can consider the localization formula <cit.>, given by

\[
\chi(\calM,\mathcal{E})=\chi\big(\calM^\mathbb{T},\mathcal{E}\vert_{\calM^\mathbb{T}}\otimes \Sym\big(N^*\big(\calM/\calM^\mathbb{T}\big)\big)\big)\in \mathrm{K}_\mathbb{T}(\mathrm{pt})\left[\frac{1}{1-\varepsilon^\nu}\right]\subset \Q(\mathbb{T}),
\]

where $\varepsilon^\nu$ denote the weights of the normal bundle $N\big(\calM/\calM^\mathbb{T}\big)$ and where $\Sym$ denotes the symmetric algebra. Here we have denoted by $\mathrm{K}_\mathbb{T}(\mathrm{pt})$ the $\mathbb{T}$-equivariant $K$-theory over a point and $\Q(\mathbb{T})$ denotes the representation ring of $\mathbb{T}$.

If a point $p\in \calM^\mathbb{T}$ is isolated, and if the tangent space at that point is given by $T_p\calM=\sum_i \varepsilon^{\nu_i}$ as a $\mathbb{T}$-module, then

\[
\Sym\big(N^*\big(\calM/p\big)\big)=\prod_i \frac{1}{1-\varepsilon^{\nu_i}}.
\]

This product is given by the character of the $\mathbb{T}$-action on functions on the formal neighborhood of $p\in\calM$.

We will only consider the case where all fixed points are isolated, hence $\chi\big(\calM_r^\mathbb{T}\big)$ is a finite sum over $r$-tuples of partitions.
Consider the Hilbert scheme (see also <cit.> for the definition of a Hilbert scheme)

\[
\mathrm{Hilb}(\Sigma,k):=\{\calI\subset \C[u_1,u_2] \mid \text{$\calI$ ideal of codimension $k$},\,\, u_1,u_2\in\Sigma\}
\]

and let $\varepsilon:=(\varepsilon_1,\varepsilon_2)$ acting on it by $f(u)\mapsto f(\varepsilon^{-1}\cdot u)$. In particular, if $\calI$ is fixed, we get[A partition of a number $k$ is a monotone sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsm \lambda_{\ell(\lambda)}\geq 0)$ consisting of nonnegative integers whose sum is equal to $k$. Most of the times, we denote by $\vert\lambda\vert:=\sum_{j=1}^{\ell(\lambda)}\lambda_j=k$ the size and $\ell(\lambda)$ the length of a partition $\lambda$.]

\begin{align*}
\mathrm{Hilb}(\Sigma,k)^\varepsilon&=\{\calI\subset \C[u_1,u_2]\mid \text{$\calI$ monomial ideal of codimension $k$},\,\, u_1,u_2\in\Sigma\}\\
&\cong\{\text{partitions of $k$}\}.
\end{align*}

Table <ref> gives a good illustration for the relation between these ideals and the partitions.

1     $u_1$     $u_1^2$     $u_1^3$     $u_1^4$     $u_1^5$     $u_1^6$     $u_1^7$     gray!30$\boldsymbol{u_1^8}$     gray!15$u_1^9$
$u_2$     $u_1u_2$     $u_1^2u_2$     $u_1^3u_2$     $u_1^4u_2$     $u_1^5u_2$     gray!30$\boldsymbol{u_1^6u_2}$     gray!15$u_1^7u_2$     gray!15$u_1^8u_2$     gray!15$u_1^9u_2$
$u_2^2$     $u_1u_2^2$     $u_1^2u_2^2$     $u_1^3u_2^2$     gray!30$\boldsymbol{u_1^4u_2^2}$     gray!15$u_1^5u_2^2$     gray!15$u_1^6u_2^2$     gray!15$u_1^7u_2^2$     gray!15$u_1^8u_2^2$     gray!15$u_1^9u_2^2$
$u_2^3$     $u_1u_2^3$     $u_1^2u_2^3$     gray!30$\boldsymbol{u_1^3u_2^3}$     gray!15$u_1^4u_2^3$     gray!15$u_1^5u_2^3$     gray!15$u_1^6u_2^3$     gray!15$u_1^7u_2^3$     gray!15$u_1^8u_2^3$     gray!15$u_1^9u_2^3$
$u_2^4$     gray!30$\boldsymbol{u_1u_2^4}$     gray!15$u_1^2u_2^4$     gray!15$u_1^3u_2^4$     gray!15$u_1^4u_2^4$     gray!15$u_1^5u_2^4$     gray!15$u_1^6u_2^4$     gray!15$u_1^7u_2^4$     gray!15$u_1^8u_2^4$     gray!15$u_1^9u_2^4$
$u_2^5$     gray!15$u_1u_2^5$     gray!15$u_1^2u_2^5$     gray!15$u_1^3u_2^5$     gray!15$u_1^4u_2^5$     gray!15$u_1^5u_2^5$     gray!15$u_1^6u_2^5$     gray!15$u_1^7u_2^5$     gray!15$u_1^8u_2^5$     gray!15$u_1^9u_2^5$
gray!30$\boldsymbol{u_2^6}$     gray!15$u_1u_2^6$     gray!15$u_1^2u_2^6$     gray!15$u_1^3u_2^6$     gray!15$u_1^4u_2^6$     gray!15$u_1^5u_2^6$     gray!15$u_1^6u_2^6$     gray!15$u_1^7u_2^6$     gray!15$u_1^8u_2^6$     gray!15$u_1^9u_2^6$

Illustration for the ideal $\calI_\lambda\subset \mathcal{O}_\Sigma$ generated by monomials $u_1^8,u_1^6u_2,\ldots,u_2^6$ which corresponds to the partition $\lambda=(8,6,4,3,1,1)$ of $k=23$. The entries in the table for $\lambda$ correspond to a basis of $\mathcal{O}_\Sigma/\calI_\lambda$.

Consider the collection

\[
\mathcal{E}:=\calI_{\lambda^{(1)}}\oplus\dotsm \oplus \calI_{\lambda^{(r)}}\in \calM,
\]

where $\calI_{\lambda^{(j)}}$ denotes the ideal corresponding to the partition $\lambda^{(j)}$.
To compute the tangent space $T_\mathcal{E}\calM$, we can use the modular interpretation of $\calM$. Generally, we can compute the tangent space to the moduli space of (coherent) sheaves by using the $\mathrm{Ext}^1$-groups[Recall first that, for $D=\mathrm{Spec}(k[t]/t^2)$, a deformation over $D$ of a coherent sheaf $\calE$ over a scheme $X$ is defined to be a coherent sheaf $\calE'$ on $X':=X\times D$, flat over $D$, together with a homomorphism $\calE'\to\calE$ such that the induced map $\calE'\otimes_Dk\to \calE$ is an isomorphism. Then there is a theorem (see e.g. <cit.>) which says that deformations of $\calE$ over $D$ are in natural one-to-one correspondence with elements of $\mathrm{Ext}^1_X(\calE,\calE)$, with the zero element corresponding to the trivial deformation. Finally, by the universal property of the moduli space $\calM$, its tangent space at $\calE$ consists of the deformations of $\calE$ over $D$.]. So we get

\begin{align}
\begin{split}
&=\bigoplus_{1\leq i,j\leq r} a_j/a_i\otimes \mathrm{Ext}^1_{\overline{\Sigma}}\big(\calI_{\lambda^{(i)}},\calI_{\lambda^{(j)}}(-D)\big).
\end{split}
\end{align}

In fact, we get

\[
\Sym(T^*_\mathcal{E}\calM)=\prod_{i,j}\E\big(\lambda^{(i)},\lambda^{(j)},a_j/a_i\big)^{-1},
\]

\begin{equation}
\label{eq:character}
\E(\lambda,\mu,u):=\prod_{\text{$w$ weights of $u\otimes \mathrm{Ext}^1_{\overline{\Sigma}}\big(\calI_\lambda,\calI_\mu(-D)\big)$}}(1-w^{-1}).
\end{equation}

For a partition $\lambda$, consider the generating function

\[
\mathbb{G}_\lambda=\chi^{\C^2}(\mathcal{O}_\Sigma/\calI_\lambda)=\sum_{u_i^au_j^b\not\in \calI_\lambda}\varepsilon_1^{-a}\varepsilon_2^{-b},
\]

where the sum corresponds to the boxes $\Box=(a+1,b+1)$ in the table of $\lambda$. Moreover, define[Note that these numbers will be negative for $\Box\not\in \lambda$.] the arm-length for the partition $\lambda$ of a box $\Box=(j,i)$ by $a_\lambda(\Box):=\lambda_i-j$ and the leg-length by $\ell_\lambda(\Box):=\lambda_j'-i$, where $\lambda'$ denotes the transposed table of $\lambda$.
Then the character in (<ref>) is given by

\begin{align}
\label{eq:character_1}
\mathrm{Ext}^1_{\overline{\Sigma}}(\calI_\lambda,\calI_\mu(-D))&=\mathbb{G}_\mu+\varepsilon_1\varepsilon_2\overline{\mathbb{G}}_\lambda-(1-\varepsilon_1)(1-\varepsilon_2)\mathbb{G}_\mu\overline{\mathbb{G}}_\lambda\\
\label{eq:character_2}
&=\sum_{\Box\in\mu}\varepsilon_1^{-a_\mu(\Box)}\varepsilon_2^{\ell_\lambda(\Box)+1}+\sum_{\Box\in \lambda}\varepsilon_1^{a_\lambda(\Box)+1}\varepsilon_2^{-\ell_\mu(\Box)}.
\end{align}

The proof of (<ref>) is given in <cit.> and the proof of (<ref>) is given in <cit.>.

Note that $\overline{\varepsilon^d}=\varepsilon^{-d}$ denotes the usual duality for representations and characters. Moreover, in particular, there are $\vert\lambda\vert+\vert\mu\vert$ factors in (<ref>).

Define now

\begin{equation}
\overline{\calM}:=\prod_i \overline{\calM}_{r_i}\supset \calM_\mathrm{ASD}(\Sigma,\mathrm{U}(r_i)).
\end{equation}

Then we can define a preliminary partition function as

\begin{align}
\label{eq:preliminary_partition_function}
\begin{split}
\mathsf{Z}^{\mathrm{pre}}_\Sigma&:=\chi\left(\overline{\calM},\prod_i z_i^{c_2(\mathcal{E}_i)}\chi_\mathrm{top}(\mathbb{M})\right)\\
&=\sum_{\text{$r$-tuple of partitions}}z^{\#\Box}\prod_{\eta,\nu \text{ $r$-tuples}\atop\text{interactions with mass $u_k$}}\E(\eta,\nu,u_k)^{\pm 1},
\end{split}
\end{align}

where $r=\sum_i r_i$ denotes the total rank and

\[
z^{\#\Box}:=\prod_i z_i^{c_2(\mathcal{E}_i)}=z_1^{\sum_i\vert \lambda^{(i)}\vert}z_2^{\sum_j\vert\mu^{(j)}\vert}\dotsm
\]

We have denoted by $\mathbb{M}$ in (<ref>) the matter that is described by fermions which are defined through their representations of the gauge group of the form $(\C^{r_i})^*\otimes \C^{r_j}$.

Using the discussions before, we can define Nekrasov's partition function by

\begin{equation}
\label{eq:Nekrasov_partition_function}
\mathsf{Z}^{\mathrm{Nek}}_\Sigma:=\mathsf{Z}^\mathrm{pert}\mathsf{Z}^{\mathrm{pre}}_\Sigma\Big|_{\E\mapsto \Hat{\E},}
\end{equation}

where $\E$ is defined as in (<ref>),

\[
\Hat{\E}(\lambda,\mu,u):=\prod_{\text{$w$ weights of $u\otimes \mathrm{Ext}^1_{\overline{\Sigma}}\big(\calI_\lambda,\calI_\mu(-D)\big)$}\atop\text{same $w$}}(w^{1/2}-w^{-1/2})
\]

and $\mathsf{Z}^\mathrm{pert}$ denotes a perturbation factor given by

\begin{equation}
\label{eq:Nekrasov_perturbative}
\mathsf{Z}^\mathrm{pert}(\varepsilon,a_1,\ldots,a_r)\risingdotseq \prod_{1\leq i_1,i_2\leq r}\,\,\, \prod_{j_1,j_2\geq 1}\I(a_{i_1}-a_{i_2}+\varepsilon(j_1-j_2)),
\end{equation}

where $\risingdotseq$ denotes the equality up to regularization of the product on the right-hand-side.
A suitable regularization is given by (see e.g. <cit.>) using Barne's double $\Gamma$-function <cit.>. Define

\begin{equation}
\zeta_2(s,w\mid c_1,c_2):=\frac{1}{\Gamma(s)}\int_0^\infty\frac{\dd t}{t}t^s\frac{\exp(-wt)}{\prod_i(1-\exp(-c_it))},\quad c_1,c_2\in\R,\quad \mathrm{Re}\,(w)\gg0,
\end{equation}

which has a meromorphic extension in $s$ with poles at $s=1,2$. Define now

\[
\Gamma_2(w\mid c_1,c_2):=\exp\left(\frac{\dd}{\dd s}\zeta_2(s,w\mid c_1,c_2)\right)\Bigg|_{s=0}.
\]

Using the difference equation

\[
\]

we can see that it extends to a meromorphic function of $w$. Moreover, define

\begin{equation}
\label{eq:Lambda}
\Lambda:=\exp(-4\pi^2\beta/r)
\end{equation}

for some parameter $\beta>0$ and the scaled perturbation factor

\begin{equation}
\mathsf{Z}^\mathrm{pert}(\varepsilon,a_1,\ldots,a_r,\Lambda):=\prod_{1\leq i_1,i_2\leq r}\Gamma_2\left(\frac{\I(a_{i_1}-a_{i_2})}{\Lambda}\,\Bigg|\,\frac{\I\varepsilon}{\Lambda},\frac{-\I\varepsilon}{\Lambda}\right)^{-1},
\end{equation}

where $\Gamma_2$ is analytically continued to imaginary arguments by using

\[
\Gamma_2(xw\mid xc,-xc)=x^{\frac{w^2}{2c^2}-\frac{1}{12}}\Gamma_2(w\mid c,-c),\quad \forall x\not\in (-\infty,0].
\]

Note also that

\[
\Gamma_2(0\mid 1,1)=\exp(-\zeta'(-1)).
\]

The instanton equations (ASD equations) are conformally invariant and can be translated to a punctured 4-sphere $S^4=\R^4\cup\{\infty\}$ by stereographic projection. Recall that Uhlenbeck's theorem for removing singularities (Theorem <ref>) implies that any instanton on $\R^4$ extends to one on $S^4$. Hence, there is an interpretation for an instanton at $\infty$. Consider the group

\[
\mathcal{G}_0:=\{g\colon S^4\to \mathrm{U}(r)\mid g(\infty)=1\}
\]

and define $\calM(r,k):=\calM^k_\mathrm{ASD}(S^4,\mathrm{U}(r))/\mathcal{G}_0$. Note that $\dim_\R\calM(r,k)=4rk<\infty$ but nevertheless $\calM(r,k)$ is not compact and thus has infinite volume. This is basically due to two problems: An element of $\calM(r,k)$ is, roughly speaking, a nonlinear superposition of $k$ instantons with charge $+1$. Some can be point-like[This means that their curvature might be concentrated in a peak with respect to the Dirac delta function.], others can go to $\infty$. Uhlenbeck's construction will only solve the first problem by replacing point-like instantons by points in $\R^4$, but it does not solve the second problem, while Nekrasov's idea of using equivariant integration (see Appendix <ref>) gives a suitable regularization procedure for those instantons.
We can rewrite Nekrasov's partition function as

\begin{equation}
\label{eq:Nekrasov_partition_function_explicit}
\mathsf{Z}^\mathrm{Nek}_{\Sigma\to S^4}(\varepsilon,a_1,\ldots,a_r,\Lambda)=\mathsf{Z}^\mathrm{pert}(\varepsilon,a_1,\ldots,a_r,\Lambda)\sum_{k\geq 0}\Lambda^{2rk}\int_{\overline{\calM}(r,k)}1,
\end{equation}

where the integral can be computed via equivariant localization (see Appendix <ref>, Equation (<ref>)) by using the formula

\[
\int_\Sigma1=\sum_{u\in \Sigma^{\C^\times}}\frac{1}{\det \xi|_{T_u\Sigma}}.
\]

Here we consider the group $\C^\times$ acting on a complex manifold $\Sigma$ and $\Sigma^{\C^\times}$ denotes the isolated fixed points with respect to this action. Moreover, $\xi=(\mathrm{diag}(-\I\varepsilon,\I\varepsilon),\mathrm{diag}(\I a_1,\ldots, \I a_r))\in \mathrm{SU}(2)\times \mathrm{SU}(r)$ and $\mathsf{Z}^{\mathrm{pert}}$ is defined as in (<ref>).

A way of expressing the gluing of Nekrasov's partition function for 5-manifolds with boundary has been studied in <cit.>.

§.§ Equivariant BV formalism

The equivariant formulation of DW theory and Nekrasov's construction are a main motivation to formulate a gauge formalism which deals with field theories in the equivariant setting. The equivariant extension of the BV formalism, which has been considered in <cit.>, is a suitable method to deal with the AKSZ construction of DW theory as in Section <ref> by extending it naturally to equivariant solutions of the CME, as it was also shown in <cit.>. We want to give the main ideas of this approach.

§.§.§ Equivariant (co)homology

consider a Lie algebra $\mathfrak{g}$ and define a $\mathfrak{g}$-differential algebra $\calO$ to be a dg algebra $(\calO,\dd)$ with Lie derivative and contraction $L_\xi\in\mathrm{Der}^0\calO$ and $\iota_\xi\in\mathrm{Der}^{-1}\calO$, respectively for all $\xi\in\mathfrak{g}$ which is linear in $\xi$ and satisfies the usual equations in Cartan's calculus. Define the subalgebra $\calO_b:=\{O\in \calO\mid L_\xi O=\iota_{\xi}O=0,\,\, \forall \xi\in\mathfrak{g}\}$. Choosing a basis $(e_a)$ of $\mathfrak{g}$, we can define its Weil model $\calW(\mathfrak{g}):=\left(\bigwedge^\bullet\mathfrak{g}^*\otimes \Sym(\mathfrak{g}^*),\dd_\calW\right)$ where the differential is given by

\begin{align}
\dd_\calW\theta^i&:=u^i+\frac{1}{2}[\theta,\theta]^i,\\
\dd_\calW u^i&:= [\theta,u]^i.
\end{align}

Here, $\theta$ denote the odd coordinates of degree $+1$ on $\bigwedge^\bullet\mathfrak{g}^*$ and $u$ the even coordinates of degree $+2$ on $\Sym(\mathfrak{g}^*)$. Moreover, we have $\iota_{e_i}=\frac{\de}{\de\theta^i}$ and $L_{e_i}=\{\iota_{e_i},\dd_\calW\}$. Note that that the subcomplex $\calO_b$ is given by $\mathfrak{g}$-invariant polynomials in $u$ endowed with the differential given by $\dd_\calW$ restricted to $\calO_b$, i.e. we have $(\C[u]^\mathfrak{g}\cong \Sym(\mathfrak{g}^*)^\mathfrak{g},\dd_\calW\vert_{\calO_b})$. Let $H^\bullet_G(\calO)$ the cohomology of the subcomplex $\calO_G:=(\calO\otimes \calW(\mathfrak{g}))_b$. This is called the Weil model for $H^\bullet_G(\calO)$. Similarly, we can consider the Cartan model for $H^\bullet_G(\calO)$ by considering the graded algebra $\calO[u]:=\calO\otimes \Sym(\mathfrak{g}^*)$ together with the differential $\dd_G:=\dd-u^i\iota_{e_i}$ and the diagonal $\mathfrak{g}$-action. It is easy to check that $(\dd_G)^2=u^iL_{e_i}$. Thus, we have a dg algebra $(\calO[u]^\mathfrak{g},\dd_G)$ with $\calO[u]^\mathfrak{g}:=\{O_u\in\calO[u]\mid L_\xi O_u=0,\,\, \forall \xi\in\mathfrak{g}\}$. In fact, we have an isomorphism between the cohomology of $(\calO[u]^\mathfrak{g},\dd_G)$ and $H^\bullet_G(\calO)$. Indeed, one can check that the map

\[
\mathrm{Exp}^\mathfrak{\mathfrak{g}}:=\exp\left(-(\iota_{e_a}\otimes\theta^a)\right)\colon \calO\otimes \calW(\mathfrak{g})\to \calO\otimes \calW(\mathfrak{g})
\]

can be restricted to an isomorphism of dg-algebras $\mathrm{Exp}^{\mathfrak{g}}\colon \calO_G\to \calO[u]^\mathfrak{g}$.

§.§.§ Equivariant extension for AKSZ-BV theories

We can formulate an equivariant extension of the AKSZ construction as follows: Let $\Sigma$ be a $d$-manifold togeher with a Lie algebra $\mathfrak{g}$ acting on it by the vector fields $v_\xi$ for some $\xi\in\mathfrak{g}$. Let $\calM$ be a graded manifold together with a symplectic form $\omega$ of degree $d-1$ and a smooth Hamiltonian function $\Theta$ on $\calM$ of degree $d$. The Hamiltonian vector field of $\Theta$ is denoted by $D_\Theta$. The space of fields is given by $\calF_\Sigma=\Map(T[1]\Sigma,\calM)$ and the cohomological vector field by

\[
Q_\Sigma=\hat{\dd}_\Sigma+\hat D_\Theta=(\calS_\Sigma,\enspace),
\]

where $\calS_\Sigma=\calS_0+\calS_\Theta$ with $\calS_0$ and $\calS_\Theta$ being the Hamiltonian functions of $\hat{\dd}_\Sigma$ and $\hat D_\Theta$, respectively. The “hat” denotes the lift to a vector field on the mapping space $\Map(T[1]\Sigma,\calM)$ of the corresponding vector field either on the source or on the target. It is easy to check that since $(D_\Theta)^2=0$ and $(Q_\Sigma)^2=0$, the CME $(\calS_\Sigma,\calS_\Sigma)=0$ holds. Denote by $\calO_{\calF_\Sigma}$ the $\mathfrak{g}$-dg algebra of functionals on $\calF_\Sigma$ endowed with the differential $\hat{\dd}_\Sigma+\hat Q$. Consider then the contraction $\hat\iota_{v_\xi}$ and Lie derivative $\hat L_{v_\xi}$ on $\calO_{\calF_\Sigma}$ for $\xi\in\mathfrak{g}$ and their corresponding Hamiltonian functions $\calS_{\hat\iota_{v_\xi}}$ and $\calS_{\hat L_{v_\xi}}$, respectively. Define also $\calO_{\calF_\Sigma}[u]:=\calO_{\calF_\Sigma}\otimes \Sym(\mathfrak{g}^*)$. If we choose a basis $(e_i)$ of $\mathfrak{g}$, we can consider the equivariant Cartan model of the BV action as

\[
\calS_\Sigma^\mathrm{Cartan}=\calS_\Sigma-u^i\calS_{\hat\iota_{v_i}},
\]

and therefore we get an equivariant model for the cohomological vector field as

\[
\]

Define now $L_\xi:=-\xi^ic^k_{ij}u^j\frac{\de}{\de u^k}$ and $\calL_\xi:=L_\xi+\hat{L}_{v_\xi}$ and note that then $\calL_\xi\calS_\Sigma^\mathrm{Cartan}=0$. This implies that $\calS^\mathrm{Cartan}_\Sigma\in\calO[u]^\mathfrak{g}$ and it satisfies an equivariant version of the CME

\begin{equation}
\label{eq:equivariant_master_equation}
\frac{1}{2}(\calS^\mathrm{Cartan}_\Sigma,\calS^\mathrm{Cartan}_\Sigma)+u^i\calS_{\hat{L}_{v_i}}=0.
\end{equation}

We further define

\[
\calT_{\Sigma}:=\frac{1}{2}(\calS^\mathrm{Cartan}_\Sigma,\calS^\mathrm{Cartan}_\Sigma)-\I\hbar\Delta \calS^\mathrm{Cartan}_\Sigma=-u^i\calS_{\hat{L}_{v_i}}-\I\hbar \Delta \calS^\mathrm{Cartan}_\Sigma,
\]

such that $\calT_\Sigma=-u^i(\calS_{\hat{L}_{v_i}}+\I\hbar\Delta\calS_{\hat{\iota}_{v_i}})-\I\hbar\Delta\calS^\mathrm{Cartan}_\Sigma$. If we apply $\Delta$ to $\calS_0$ and $\calS_{\hat{\iota}_{v_i}}$, we can use the fact that they are quadratic in the fields and thus we get

\begin{equation}
\label{eq:equivariant_BV_1}
(\Delta\calS_0,\enspace)=(\Delta\calS_{\hat{\iota}_{v_\xi}},\enspace)=0,\quad\forall X\in\mathfrak{g}.
\end{equation}

This gives us that

\[
\Delta\calS_{\hat{L}_{v_\xi}}=\Delta(\calS_0,\calS_{\hat{\iota}_{v_\xi}})=(\Delta\calS_0,\calS_{\hat{\iota}_{v_\xi}})\pm (\calS_0,\Delta\calS_{\hat{\iota}_{v_\xi}})=0,
\]

which needs to hold for consistency with the definition of a BV algebra as in Section <ref>. Hence, $\Delta\calT_\Sigma=0$ and thus $[\Delta,\hat{L}_{v_\xi}]=0$. We can then check that $Q^\mathrm{Cartan}_\Sigma\calT_\Sigma=0$.

Let us briefly consider a more general case. For an observable $O$, note that we want the equation $\Delta O+\frac{\I}{\hbar}Q_\Sigma O=0$, with $Q_\Sigma=(\calS_\Sigma,\enspace)$, to be satisfied in order for point (2) of Theorem <ref> to hold. This equation can be rephrased by using the twisted BV Laplacian $\Delta_{\calS_\Sigma}:=\exp(-\I\calS_\Sigma/\hbar)\Delta\exp(\I\calS_\Sigma/\hbar)$ according to the definition of the BV Laplacian on functions on the space of fields by Equation (<ref>).
In this case, $O$ is called a BV observable. However, without assuming the Quantum Master Equation, we can define

\[
\calT_\Sigma:=\left(\frac{\I}{\hbar}\right)^2\exp(-\I\calS_\Sigma/\hbar)\Delta \exp(\I\calS_\Sigma/\hbar)=\frac{1}{2}(\calS_\Sigma,\calS_\Sigma)-\I\hbar\Delta \calS_\Sigma
\]

and note that the twisted operator is given by

\[
\Delta_{\calS_\Sigma,\calT_\Sigma}:=\exp(-\I\calS_\Sigma/\hbar)\Delta(\exp(\I\calS_\Sigma/\hbar)O)=\Delta O+\frac{\I}{\hbar}Q_\Sigma O+\left(\frac{\I}{\hbar}\right)^2\calT_\Sigma O.
\]

The observation is that then

\[
\Delta\calT_\Sigma+\frac{\I}{\hbar}Q_\Sigma\calT_\Sigma=0,
\]

since $\Delta_{\calS_\Sigma,\calT_\Sigma}\calT_\Sigma=0$ and $(\calT_\Sigma)^2=0$.

Moreover, if we apply $\Delta$ to $(\calS_{\hat{L}_{v_i}},\calS_{\hat{\iota}_{v_j}})$ we get $c^k_{ij}\Delta\calS_{\hat{\iota}_{v_k}}=0$. Using (<ref>), we also get $(\calT_\Sigma,O)=(\calT_\Sigma',O)$ with $\calT_\Sigma':=-u^i\calS_{\hat{L}_{v_i}}-\I\hbar\Delta\calS_\Theta$. Define now

\[
\calN_{\calT_\Sigma}:=\{O\in \calO_{\calF_\Sigma}[u]\mid (\calT_\Sigma',O)\in\calI_{\calT_\Sigma}\},
\]

where $\calI_{\calT_\Sigma}$ denotes the ideal generated by $\calT_\Sigma$ in $\calO_{\calF_\Sigma}[u]$ and consider the subalgebra

\[
\calN_{\calT_\Sigma}':=\{O\in\calO_{\calF_\Sigma}[u]\mid \calL_\xi O=0,\, \forall \xi\in\mathfrak{g}\}\subset\calN_{\calT_\Sigma}.
\]

Then, one can show that if (<ref>) holds, $\calN_{\calT_\Sigma}'$ is a Poisson subalgebra which is invariant under both $Q^\mathrm{Cartan}_\Sigma$ and $\Delta_{\calS_{\Sigma},\calT_\Sigma}$ and $\calT_\Sigma\in \calN_{\calT_\Sigma}'$. Moreover, it is easy to see that the ideal $\calI_{\calT_\Sigma}'\subset \calN_{\calT_\Sigma}'$ generated by $\calT_\Sigma$ is a $\Delta_{\calS_\Sigma,\calT_\Sigma}$-invariant Poisson ideal. Now one can define the algebra of quantum equivariant preobservables as $\calO_{\calF_\Sigma}':=\calN_{\calT_\Sigma}'/\calI_{\calT_\Sigma}'$ together with the induced differential

\begin{equation}
\label{eq:equivariant_BV_2}
\Delta_{\calS_\Sigma,\calT_\Sigma}[O]:=[\Delta_{\calS_\Sigma,\calT_\Sigma}O]=\left[\left(\Delta+\frac{\I}{\hbar}Q^\mathrm{Cartan}_\Sigma\right)O\right].
\end{equation}

A quantum equivariant observable can then be defined as a quantum equivariant preobservable which is additionally $\Delta_{\calS_\Sigma,\calT_\Sigma}$-closed.

If the target $\calM$ is a graded vector space $V$, we can write the space of fields as $\calF_\Sigma=\Omega^\bullet(\Sigma)\otimes V$. Consider an invariant metric on $\Sigma$ and define a the submanifold $\calL:=\Omega^{\bullet}_\mathrm{coex}(\Sigma)\otimes V$, where $\Omega^\bullet_\mathrm{coex}(\Sigma)$ denotes the subspace of coexact forms. Note that, in general, $\calL$ is only isotropic due to harmonic forms. However, by invariance of the metric, we have $[L_{v_\xi},\dd^*]=0$, and thus $i_\calL^*\calS_{\hat{L}_{v_\xi}}=0$, where $i_\calL\colon\calL\hookrightarrow \calF_\Sigma$. The foliation then defined by $(\calT_\Sigma,\enspace)$ is the same as the infinitesimal $\mathfrak{g}$-action and hence we have to require that the Lie group $G$ is compact.

§.§ Equivariant DW theory

Consider the set-up of Section <ref> and let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$ acting on the 4-manifold $\Sigma$ with vector fields $v_\xi$ for $\xi\in\mathfrak{g}$. If we replace now $\dd_\Sigma$ with the equivariant differential $\dd_G:=\dd_\Sigma-u^i\iota_{v_i}$, we get the equivariant extension of the action as

\begin{align}
\begin{split}
\calS^\mathrm{Cartan}_\Sigma&=\int_{T[1]\Sigma}\boldsymbol{\mu}_4\left(\langle \mathbf{Y},\dd_G\mathbf{X}\rangle+\frac{1}{2}\langle\mathbf{Y},\mathbf{Y}\rangle+\frac{1}{2}\langle\mathbf{Y},[\mathbf{X},\mathbf{X}]\rangle\right)\\
\end{split}
\end{align}

The BV transformations of the superfields are then given by

\begin{align}
\end{align}

i.e. in superfield notation, the equivariant cohomological vector field is given by

\begin{equation}
\label{eq:equivariant_cohomological_vector_field}
\end{equation}

In component fields and after Berezinian integration, we get the equivariant cohomological vector field

\begin{multline}
Q^\mathrm{Cartan}_\Sigma=\int_\Sigma\Bigg(\left(Y+\frac{1}{2}[X,X]-u^i\iota_{v_i}A\right)\frac{\delta}{\delta X}+(\psi+\dd_AX-u^i\iota_{v_i}\chi)\frac{\delta}{\delta A}\\+(\chi^\dagger+F_A+[X,\chi]-u^i\iota_{v_a}\psi^\dagger)\frac{\delta}{\delta\chi}+([X,Y]-u^i\iota_{v_i}\psi)\frac{\delta}{\delta Y}\\+(\dd_AY+[X,\psi]-u^i_{\iota_{v_i}}\chi^\dagger)\frac{\delta}{\delta \psi}\Bigg).
\end{multline}

§.§.§ Relation to Nekrasov's partition function

We can construct then a gauge-fixed solution for the equivariant CME (<ref>) similarly as before whenever the chosen metric on $\Sigma$ is invariant. In particular, we have

\begin{multline}
\label{eq:equivariant_gauge-fixing}
\calS^\mathrm{Cartan, gf}_\Sigma=\calS^\mathrm{Cartan}_\Sigma+\int_\Sigma\left(\langle \bar X^\dagger,(b+[X,\bar X])\rangle+\langle b^\dagger,(L_v\bar X+[X,b]-[Y,\bar X])\rangle\right)\\
+\int_\Sigma\left(\langle \bar Y^\dagger,(\eta+[X,\bar Y])\rangle+\langle \eta^\dagger,(L_v\bar Y+[X,\eta]+[\bar Y,Y])\rangle\right).
\end{multline}

If we redefine the fields, we can see that the transformations are the same as in <cit.>. In particular, one can take $v^i$ to be the two rotations of $\C^2$ and consider instead of $u^i$ the parameters[This is true whenever the evaluation map which evaluates $u^i$ at $\varepsilon^i$ commutes with the path integral with the additional assumption that $\varepsilon^i$ are not on the support of the equivariant cohomology class induced by the path integral.] $\varepsilon^i$. In fact, the (regularized) path integral quantization of this equivariant theory leads to Nekrasov's partition function (<ref>), i.e. we have

\[
\mathsf{Z}^\mathrm{BV}_{\Sigma}=\int_\calL\exp(\I\calS^\mathrm{Cartan}_\Sigma(\mathbf{X},\mathbf{Y})/\hbar)\mathscr{D}[\mathscr{X}]\mathscr{D}[\mathscr{Y}]=\mathsf{Z}^\mathrm{Nek}_{\Sigma},
\]

where $\calL$ denotes the gauge-fixing Lagrangian as in (<ref>).
It is important to note that in the mentioned construction, we are considering the perturbation around constant solutions rather than general instantons. Moreover, the methods presented in this section are only considered for the finite-dimensional case.

Similar equivariant extensions have been studied e.g. in <cit.> for a (global) $S^1$-equivariant setting of the Poisson sigma model on the disk, and in <cit.> where a classical BV-equivariance under source diffeomorphisms for 1-dimensional systems is considered. Generalizations of this construction would lead to equivariant cohomological methods similarly as for the methods which appear in <cit.>.

§.§ Equivariant Floer (co)homology and boundary structure

Since the perturbative expectation value for DW theory of an observable as in <cit.> reproduces the Floer (co)homology groups as the boundary states on the corresponding boundary 3-manifold (see also Section <ref>), we expect to produce similarly in a perturbative way an equivariant version of Floer (co)homology as the boundary state when considering equivariant DW theory on a 4-manifold with boundary. An equivariant extension of Floer (co)homology groups in the setting of 4-manifold topology was considered in <cit.>. The convenient model chosen there is the Cartan model similarly as discussed in Section <ref>.
In particular, for $G=\mathrm{SU}(2)$ and $\mathfrak{g}=\mathrm{Lie}(G)=\mathfrak{su}(2)$ we can construct equivariant Floer homology and cohomology as follows: For any $G$-manifold $N$, define complexes

\begin{align*}
\Omega_{G,\bullet}(N)&:=(\Sym(\mathfrak{g})\otimes \Omega^{\dim N-\bullet}(N))^G,\\
\Omega_G^\bullet(N)&:=(\Sym(\mathfrak{g^*})\otimes \Omega^\bullet(N))^G,
\end{align*}

endowed with the boundary and coboundary operators $\de_G$ and $\dd_G$, respectively. They define equivariant homology and cohomology $H_{G,\bullet}(N)$ and $H^\bullet_G(N)$, respectively. Consider the $\mathrm{SU}(2)$-invariant Chern–Simons action functional $S^\mathrm{CS}_N\colon \calA/\tilde\calG_0\to S^1$, where $\tilde\calG_0$ denotes the group of based gauge transformations of degree 0 and denote by $A$ and $B$ critical orbits of the perturbed action $S^\mathrm{CS}_N+f$ for which the moduli space of gradient lines between $A$ and $B$ then becomes a smooth $\mathrm{SU}(2)$-manifold $\widetilde{\calM}_0(A,B)$ (here $f$ denotes an element of a suitable class of perturbations). For $0<p<\infty$ and $\delta\in\R$, denote by $L^p_\delta(E)$, for some bundle $E\to \Sigma$ over a 4-manifold $\Sigma$, the Banach space completion of smooth and compactly supported sections of $E$ by using the norm

\[
\|\sigma\|_{L^p_\delta}:=\left(\int_\Sigma\dd\mathrm{vol}_\Sigma(\exp(\tau\delta)\vert \sigma\vert^p)\right)^{1/p},
\]

where $\tau\colon \End(\Sigma)\to (0,\infty)$ denotes a distance function on $\Sigma$ defined as in <cit.>. Furthermore, for $1\leq p<\infty$, $0\leq \ell<\infty$ and $\delta\in\R$, define the weighted Sobolev space $L^p_{\ell,\delta}(E)$ to be the completion of smooth and compactly supported sections of $E$ with respect to the norm

\[
\|\sigma\|_{L^p_{\ell,\delta}}:=\left(\int_\Sigma\dd\mathrm{vol}_\Sigma\left(\exp(\tau\delta)\sum_{0\leq k\leq \ell}\vert\nabla^k\sigma\vert^p\right)\right)^{1/p},
\]

where $\nabla^k:=\nabla\nabla\dotsm \nabla$ $k$-times and $\nabla\colon \Gamma^\infty_0(E\otimes_q T^*\Sigma)\to \Gamma^\infty_{0}(E\otimes_{q+1}T^*\Sigma)$ is the covariant derivative from the end-periodic[The definition of end-periodic connections is rather technical and we refer to <cit.> for a definition of end-periodic connections and manifolds. In fact, these objects are only defined for end-periodic 4-manifolds.] connections on $E$ and $T^*\Sigma$. We have denoted by $\Gamma^\infty_0$ the space of smooth sections with compact support.

We can then consider the Atiyah–Hitchin–Singer deformation theory <cit.> to the moduli space $\calM_\varepsilon(A,B)$ for $\varepsilon<0$, which is locally a smooth $(\mathcal{G}(N)^A\times\calG(N)^B)$-manifold whenever the cokernel of the following deformation operator vanishes.
Define the deformation and gauge-fixing operator at some instanton $A_t$ as

\[
a\mapsto \left(\frac{\dd}{\dd t}a+*\dd_{A_t}a+\de^2f_{A_t}a,\,\,\frac{\dd}{\dd t}a+\dd^*_{A_t}a\right)
\]

and note that this indeed defines a Fredholm operator

\[
L^p_{\ell,\varepsilon}(\Omega^1(N\times \R,\ad P))\to L^p_{\ell-1,\varepsilon}((\Omega^0\oplus\Omega^1)(N\times \R,\ad P)).
\]

Denote by $\mathrm{sf}(A,B)$ the index[This is again the spectral flow as defined in Section <ref>.] of this operator and note that we then get

\[
\dim\calM_0(A,B)=\mathrm{sf}(A,B)+3-\dim \mathcal{G}(N)^A-\dim\calG(N)^B.
\]

Define then the index of a critical orbit $A\in \calA/\tilde\calG_0$ of the perturbed Chern–Simons action functional $S^\mathrm{CS}_N+f$ as

\[
\]

where $\theta$ denotes a preferred trivial connection in $\calA/\tilde\calG_0$ and $\varepsilon$ is chosen to be negative and sufficiently small.
Fix a homotopy of the corresponding principal $\mathrm{SU}(2)$-bundle $P$ over $N$ and note that

\[
\dim\widetilde{\calM}_0(A,B)=i_N(A)-i_N(B)+\dim A-1.
\]

For the 3-manifold $N$, we can then define graded vector spaces

\begin{align*}
\end{align*}

with boundary and coboundary operators given by

\begin{equation}
(\delta_\calG)_{A,B}\Psi:=\begin{cases}\de_G\Psi,& \text{if $A=B$},\\ (-1)^{\deg\Psi}(e^-_{B})_*(e_A^+)^*\Psi,&\text{if $i_N(A)>i_N(B)$},\\ 0,&\text{otherwise}.\end{cases}
\end{equation}

\begin{equation}
(D_\calG)_{A,B}\Psi:=\begin{cases}\dd_G\Psi,& \text{if $A=B$},\\ (-1)^{\deg\Psi}(e^+_{A})_*(e_B^-)^*\Psi,&\text{if $i_N(A)>i_N(B)$},\\ 0,&\text{otherwise}.\end{cases}
\end{equation}

We have used the endpoint maps

\begin{align*}
e_A^+\colon\widetilde{\calM}_0(A,B)\to A,\\
e_B^-\colon\widetilde{\calM}_0(A,B)\to B.
\end{align*}

It is easy to see that the equivariant Floer complexes are actually filtered by index

\begin{align}
CF^{(m)}_{\calG,k}&:=\bigoplus_{i_N(A)+j=k\atop i_N(A)<m}\Omega_{G,j}(A),\\
CF^{k,(m)}_\calG&:=\bigoplus_{i_N(A)+j=k\atop i_N(A)\geq m}\Omega^j_G(A),
\end{align}

with the corresponding relative equivariant (co)homology groups

\begin{align}
\end{align}

The operators $\delta_\calG$ and $D_\calG$ square to zero and define equivariant Floer homology (resp. cohomology) $HF_{\calG,\bullet}(N)$ (resp. $HF^\bullet_\calG(N)$) of the 3-manifold $N$. Moreover, there is a pairing $HF_{\calG,\bullet}(N)\times HF^\bullet_\calG(N)\to \R$ which is defined on the level of chains by $\bigoplus_A\langle\enspace,\enspace\rangle_A$. Furthermore, these groups are independent of the metric on $N$ and the choice of perturbation withing a class of perturbations, up to natural isomorphisms. An orientation preserving diffeomorphism of $N$ induces a natural map on equivariant Floer (co)homology.

Equivariant Floer homology and cohomology are both modules over equivariant polynomials $\Sym(\mathfrak{g}^*)^G\cong\R[u]$ and the action of $\Sym(\mathfrak{g}^*)^G$ is symmetric with respect to the pairing. Moreover, the complexes defining equivariant homology and cohomology are filtered by index and there is an associated spectral sequence, whose $E^1$ and $E_1$ terms are the equivariant (co)homology of the critical locus.

The equivariant Floer groups for $S^3$ are simple since there is only one flat connection, the trivial one. Hence, we get

\begin{align*}
\end{align*}

[Poincaré $3$-sphere]
In the case of the Poincaré 3-sphere $S^3_P$, there are two irreducible flat connections $A$ and $B$. These are indexed by $i_{S^3_P}(A)=0$ and $i_{S^3_P}(B)=4$ by using the standard orientation induced from $S^3$. The first term in the spectral sequence of the equivariant Floer cohomology complex has then generators endowed with index 0 and 4 coming from the connections $A$ and $B$, together with a tower of $H^\bullet_G(\mathrm{pt})$ at index 0 which represents the trivial connection. Then the dimension of the nonzero terms will give lead to the fact that all higher differentials vanish and thus, after taking the quotient with respect to the $\Z$-translations, we get

\begin{align*}
HF^{\bullet}_\calG(S^3_P)&=H^\bullet_G(\mathrm{pt})\oplus H^\bullet(\mathrm{pt})\oplus H^{\bullet-4}(\mathrm{pt}),\\
HF_{\calG,\bullet}(S^3_P)&=H_{G,\bullet}(\mathrm{pt})\oplus H_\bullet(\mathrm{pt})\oplus H_{\bullet-4}(\mathrm{pt}).
\end{align*}

Consider now a $\mathrm{U}(2)$-bundle $P\to \Sigma=\Sigma_1\cup_N\Sigma_2$ over an oriented $4$-manifold $\Sigma$ with $\de\Sigma_1=N$ and $\de\Sigma_2=N^\mathrm{opp}$. Moreover, consider the moduli space $\calM_0(\Sigma,\mathrm{U}(2))$ of projectively anti self-dual connections on $P$ with fixed central parts. Moreover, assume that $\calM_0(\Sigma,\mathrm{U}(2))$ is compact. Define now $\calM_0(P_1,A)$ and $\calM_0(P_2,A)$ to be the moduli spaces of anti self-dual connections modulo gauge transformations asymptotic to the flat connection defined by $A$. Then we have
\begin{align}
\dim \calM_0(P_1,A)&=C_{\Sigma_1}-i_N(A),\\
\dim \calM_0(P_2,A)&=C_{\Sigma_2}+3-i_N(A),
\end{align}

for some constants $C_{\Sigma_1}$ and $C_{\Sigma_2}$. For the framed moduli space, we get the $\mathrm{SU}(2)$-invariant endpoint maps

\begin{align*}
e_A^1&\colon \calM_0(P_1,A)\to A,\\
e_A^2&\colon \calM_0(P_2,A)\to A.
\end{align*}

If $P_1$ and $P_2$ are trivial bundles such that $c_2(P_1)+c_2(P_2)=c_2(P)$, we get a gluing map with open image

\[
\calM_0(P_1,A)\times \calM_0(P_2,A)\to \calM_0(\Sigma,\mathrm{U}(2)).
\]

Let $\calM_0(\Sigma,\mathrm{U}(2))$ be compact and let $P_1,P_2$ run over bundles such that $c_2(P_1)+c_2(P_2)=c_2(P)$. Assume that for a generic one parameter family of metrics on $\Sigma_1$ no reducible connections exist on $P_1$ for any $A$. Assume further that $b_+^2(\Sigma_2)=0$ or $b^2_+(\Sigma_2)>1$.

* Let $a\in H^\bullet_G(\calA(P_1)/\calG_0(P_1))$ and denote by $\gamma:=\left[\calM_0(P_1,A)\right]_G\in H_{G,\bullet}(\calM_0(P_1,A))$ the fundamental class. Then

\[
\Psi_{\Sigma_1}(a):=\bigoplus_{i_N(A)\geq m}(e_A^1)_*(\gamma\cap a)\in CF_{\calG,\bullet}(N)
\]

defines an element of $HF^{(m)}_{\calG,\bullet}(N)$ whenever $m>C_{\Sigma_1}-8$; that is, it is closed and, up to boundaries, independent of the choice of metric on $\Sigma_1$.

* Let $b\in H^\bullet_G(\calA(P_2)/\calG_0(P_2))$. The integration over the fiber gives an element

\[
\calD(\Sigma_2)(b):=\bigoplus_{A}(e^2_A)_*b\in CF^\bullet_\calG(N)
\]

which is coclosed and up to coboundaries independent of the metric on $\Sigma_2$. Moreover, it also defines an element of $HF^{\bullet,(m)}_\calG(N)$ for all $m<-3-C_{\Sigma_2}$:

\[
\int_{\calM_0(\Sigma,\mathrm{U}(2))}a\cup b=\langle \Psi_{\Sigma_1}(a),\calD(\Sigma_2)(b)\rangle_{HF_{\calG}^{(m)}}.
\]

§.§.§ Irreducible connection on one side

Assume now that the $2d$-dimensional moduli space of anti self-dual connections on $P$ splits uniquely as

\[
\calM_0(\Sigma,\mathrm{U}(2))=\calM_0(P_1,A)\times_A \calM_0(P_2,A),
\]

along a $\Z/2$ connection $A$. Then, we get that $H_G^\bullet(A,\R)=\R[u]$ such that $u$ is of degree 4. Moreover, suppose that $\calM_0(P_2,A)\simeq S^2$ only consists of a single reducible connection which defines a line bundle reduction $\mathbb{L}\oplus \mathbb{L}^\mathrm{opp}\to \Sigma_2$. Recall from Section <ref> that the Donaldson polynomials are given by

\[
\calD(\Sigma)([C],\ldots,[C])=\int_{\calM_0(\Sigma,\mathrm{U}(2))}\mu([C])^d,
\]

for a $2$-dimensional homology class $[C]\in H_2(\Sigma,\Z)$. Let us first look at the $\Sigma_2$ side. There, we get that $\calM_0(P_2,A)\simeq S^2$ corresponds to a single reducible connection with endpoint map $e_A^2\colon \calM_0(P_2,A)\to A$ which sends $S^2$ to one single point. We can extend the map $\mu$ to equivariant cohomology $\mu\colon H_2(\Sigma_2,\Z)\to H^2_G(\calA(P_2)/\calG_0(P_2))$ and hence

\[
\mu([C])=-2\int_{C}c_1(\mathbb{L})v\in H^2_G(\calM_0(P_2,A))=\R[u],
\]

where $v$ is of degree 2 and such that $v^2=u$. Note that we have a push-forward $(e^2_A)_*\colon \R[v]\to \R[u]$ such that

\[
\begin{cases}v^k\mapsto u^{(k-1)/2},&\text{$k$ odd},\\ 0, &\text{otherwise}.\end{cases}
\]

Finally, the relative Donaldson polynomial in the equivariant Floer cohomology of $N$ defined by $\Sigma_2$ is given by

\begin{equation}
\label{eq:Donaldson_equivariant_Sigma_2}
\calD(\Sigma_2)(\mu([C])^d)=\begin{cases}(-2)^d\left({\displaystyle\int_Cc_1(\mathbb{L})}\right)^du^{(d-1)/2},&\text{$d$ odd},\\ 0,&\text{otherwise}.\end{cases}
\end{equation}

Now we look at the $\Sigma_1$ side. There we need $\Psi_{\Sigma_1}(\boldsymbol{1}):=(e^1_A)_*(\gamma)\in CF_{\calG,\bullet}(N)$ in the equivariant Floer homology of $N$. In particular, we get

\[
\Psi_{\Sigma_1}(\boldsymbol{1})=\int_{\calM_0(P_1,A)}\gamma=\frac{2^{(d-1)/2}}{(d-1)!}p_1^{(d-1)/2}\hat{u}^{(d-1)/2}\in H_{G,\bullet}(A)=\R[\hat{u}],
\]

where $H_{G,\bullet}(\mathrm{pt})=\R[\hat u]$. In fact, the framed moduli space of anti self-dual connections on $\Sigma_1$ is an $\mathrm{SO}(3)$-bundle and thus has a Pontryagin class $p_1$.

Note that the latter construction only holds if $d<4$, otherwise the compactness of the moduli spaces can be violated. Moreover, as it was mentioned in <cit.>, this construction also coincides with the Friedman–Morgan construction <cit.> if $\Sigma$ is the blow-up of an algebraic surface, in particular, $\Sigma_2=\C \mathbb{P}^2$ and $N=S^3$, thus the moduli space on $\Sigma$ has to split along the trivial connection.

Similarly as mentioned in Remark <ref>, this construction is expected to be consistent with a possible equivariant extension of the BV-BFV formalism in parallel to the BV case for closed manifolds. In the same way as the BV partition function of equivariant DW theory on closed manifolds yields Nekrasov's partition function, one should be able to produce the equivariant Floer groups as a BV-BFV partition function for equivariant DW theory in the presence of boundary. This yields a boundary state of the form

\[
\Psi^\mathrm{eq.}_{\de\Sigma}(\mathds{X},\mathds{Y})=\int_{\calV_\Sigma}\underbrace{\int_{\calL}\exp(\I\calS^\mathrm{Cartan}_\Sigma(\mathbf{X},\mathbf{Y})/\hbar)\widetilde{O}(\mathbf{X},\mathbf{Y})\mathscr{D}[\mathscr{X}]\mathscr{D}[\mathscr{Y}]}_{=\left\langle\prod_{j=1}^d\widetilde{O}^{(\gamma_j)}\right\rangle_\mathrm{eq.}=\left\langle \prod_{j=1}^d\int_{\widetilde{\gamma}_j}\widetilde{W}_{k_j}\right\rangle_\mathrm{eq.}}\in HF_\calG^\bullet(\de\Sigma)\subset\calH^\calP_{\de\Sigma},
\]

where $\widetilde{O},\widetilde{\gamma}_j$ and $\widetilde{W}_{k_j}$ are equivariant extensions of the respective objects as in Section <ref> and $\langle\enspace\rangle_\mathrm{eq.}$ denotes the expectation for the BV-BFV partition function with respect to $\calS^\mathrm{Cartan}_\Sigma$.
In particular, the boundary state space $\calH^\calP_{\de\Sigma}$ in this case should include the equivariant Floer cohomology and thus a possible equivariant extension of the BFV boundary operator $\Omega^{\bullet,\calP}_{\de\Sigma}$ should be given in terms of the equivariant Floer differential. Moreover, the polarization needs to be adapted to the boundary condition defined by the state $\Psi^\mathrm{eq.}_{\de\Sigma}$. Note that here we want $\mathfrak{h}=\mathfrak{su}(2)$.

§ RELATION TO DONALDSON–THOMAS THEORY

Donaldson–Thomas theory, starting first in <cit.>, was motivated by formulating Donaldson's construction of invariants of 4-manifolds in higher dimensions.
In <cit.>, Thomas constructed a holomorphic Casson invariant (see Definition<ref>) counting bundles on a Calabi-Yau 3-fold, by using the holomorphic Chern–Simons action functional (see Section <ref>). He developed the deformation theory necessary to obtain certain virtual moduli cycles in moduli spaces of stable sheaves whose higher obstruction groups vanish, which gives Donaldson- and Gromov–Witten- (see Section <ref>) like invariants of Fano 3-folds. Moreover, he defined the holomorphic Casson invariant of a Calabi–Yau 3-fold $X$ and proved that it is deformation invariant.

In particular, when considering real Chern–Simons theory as in Section <ref>, one can show that the Hessian of the action functional is symmetric and hence the deformation complex is defined by its critical locus, given by flat connections, is self-dual. Thus, by Poincaré duality, we get

\[
H^i(\ad E, A)\cong (H^{3-i}(\ad E,A))^*
\]

and hence the virtual dimension of the moduli space of flat connections $\calM_\mathrm{flat}$ is given by

\[
\mathrm{vdim}\,\calM_\mathrm{flat}=\sum_{i=0}^3(-1)^{i+1}\dim H^i(\ad E,A)=0.
\]

A similar observation is also true when considering holomorphic Chern–Simons theory (see Section <ref>). In particular, instead of Poincaré duality one can use Serre duality to obtain that the virtual dimension of the moduli space of holomorphic bundles is zero. This observation in fact leads to the countig of bundles over Calabi–Yau 3-folds and more generally curves in algebraic 3-folds.

§.§ DT invariants

Let $X$ be a Calabi–Yau 3-fold. Define

\[
\mathrm{Hilb}_\beta(X,k):=\{X_0\subset X\mid [X_0]=\beta,\, \chi_\mathrm{hol}(\mathcal{O}_{X_0})=k\}
\]

to be the Hilbert scheme depending on the 1-dimensional subschemes in the curve class $\beta\in H_2(X,\mathbb{Z})$ with holomorphic[Recall that this is defined through the alternating sum of the dimension of sheaf cohomology $\chi_\mathrm{hol}(\mathcal{O}_X):=\sum_{i=0}^{\dim X}(-1)^i\dim H^i(X,\mathcal{O}_X)$.] Euler characteristic $k\in\mathbb{Z}$. By Behrend's construction <cit.>, one can define the Donaldson–Thomas (DT) invariants as a weighted Euler characteristic of the underlying moduli space. In particular, it is defined as

\begin{equation}
\label{eq:DT_invariants}
\mathrm{DT}_{\beta,k}^X:=\chi_\mathrm{top}\left(\mathrm{Hilb}_{\beta}(X,k),\nu^X_{\beta,k}\right):=\sum_{l\in\Z}l\chi_\mathrm{top}\big(\big(\nu_{\beta,k}^X\big)^{-1}(l)\big),
\end{equation}

where $\chi_\mathrm{top}$ denotes the topological Euler characteristic and

\begin{equation}
\label{eq:Behrend_function}
\nu_{\beta,k}^X\colon \mathrm{Hilb}_{\beta}(X,k)\to \Z
\end{equation}

denotes Behrend's constructible[That is to say for a quasi-projective proper moduli scheme with a symmetric obstruction theory, in our case $\mathrm{Hilb}_{\beta}(X,k)$, the weighted Euler characteristic $\chi_\mathrm{top}(\mathrm{Hilb}_{\beta}(X,k),\nu_{\beta,k}^X)$ is the degree of the virtual fundamental class $\left[\mathrm{Hilb}_{\beta}(X,k)\right]^{\mathrm{vir}}$ (see Footnote <ref> for a definition of the virtual fundamental class).] function <cit.>.

A result of Brav, Bussi, Dupont and Joyce was to prove that the coarse moduli space of simple perfect complexes of coherent sheaves, with fixed determinant, on a Calabi–Yau 3-fold admits, locally, for the analytic topology, a potential, i.e. it is isomorphic to the critical locus of a function. As it was shown in <cit.>, such a result is not true for general symmetric obstruction theories. Thus, the existence of such local potentials will crucially depend on the existence of a $(-1)$-shifted symplectic structure (in the setting of <cit.>, see also Section <ref>) on the derived moduli stack of simple perfect complexes on a Calabi–Yau 3-fold $X$. As it was shown in <cit.>, there is an étale local version of Darboux's theorem for $k$-shifted symplectic structures with $k<0$. However, extending such a local structure theorem to general $n$-shifted symplectic structures, especially for the case of interest where $n\geq0$, might lead to the existence of DT invariants for higher dimensional Calabi–Yau manifolds. Moreover, the results of <cit.> might lead to a categoryfied version of DT theory and a motivic version of DT invariants similarly as in <cit.>.

§.§.§ Relation to instanton Floer homology

DT invariants can be interpreted as a complex version of a certain invariant of homology 3-spheres, called Casson invariant, and then related to the instanton Floer homology construction as it was shown in <cit.>. Let us first recall the definition of a Casson invariant.

Let $\mathscr{S}$ be the class of oriented integral homology $3$-spheres. A Casson invariant is a map $\lambda\colon \mathscr{S}\to \Z$ such that:

* $\lambda(S^3)=0$, and $\lambda(\mathscr{S})$ is not contained in any proper subgroup of $\Z$.

* For any homology sphere $N$ and knot $k\subset N$, the difference

\begin{equation}
\lambda'(k):=\lambda\left(N+\frac{1}{m+1}\cdot k\right)-\lambda\left(N+\frac{1}{m}\cdot k\right),\quad m\in\Z,
\end{equation}

is independent of $m$.

* Let $k\cup \ell$ be a link in a homology sphere $N$ with linking number $\mathrm{lk}(k,\ell)=0$ and note that then for any integers $m,n$ the manifold

\[
N+\frac{1}{m}\cdot k+\frac{1}{n}\cdot \ell
\]

is a homology 3-sphere. We have

\begin{equation}
\lambda''(k,\ell):=\lambda'\left(\ell\subset N+\frac{1}{m+1}\cdot k\right)-\lambda'\left(\ell\subset N+\frac{1}{m}\cdot k\right)=0,
\end{equation}

for any boundary link $k\cup \ell$ in a homology sphere $N$.

There is a Casson invariant $\lambda$ which is unique up to sign. Moreover, it has the following properties:

* $\lambda'(k_{3_1})=\pm 1$, where $k_{3_1}$ denotes the trefoil knot.

* $\lambda'(k\subset N)=\frac{1}{2}\Delta_{k\subset N}''(1)\cdot \lambda'(k_{3_1})$ for any knot $k\subset N$,

* $\lambda(-N)=-\lambda(N)$ where $-N$ stands for $N$ with reversed orientation,

* $\lambda(N_1\#N_2)=\lambda(N_1)+\lambda(N_2)$,

* $\lambda(N)=\mu(N)$ mod 2, where $\mu$ denotes the Rohlin invariant[For an oriented integral homology 3-sphere $N$, there exists a smooth simply-connected oriented 4-manifold $\Sigma$ with even intersection form such that $\de\Sigma=N$. Then the signature of $\Sigma$ is divisible by 8 and $\mu(N):=\frac{1}{8}\mathrm{sign}\,\Sigma$ mod 2 is independent of the choice of $\Sigma$ <cit.>. The number $\mu(N)$ is called Rohlin invariant of $N$.].

In <cit.>, Taubes has shown that the Casson invariant as in Theorem <ref> can be defined by using an infinite-dimensional generalization of the topological Euler characteristic using instanton Floer homology for an oriented homology 3-sphere. Let $G$ be a finitely presented group and $N$ a homology 3-sphere. Denote by $R(G)=\Hom(G,\mathrm{SU}(2))$ where $G$ is assumed to have the discrete topology. Note that one can turn $R(G)$ into a real algebraic variety by regarding $\mathrm{SU}(2)$ as $S^4$. The quotient $\calR(G):=R(G)/\mathrm{SU}(2)$ under the $\mathrm{SU}(2)$-action given by conjugation, is the $\mathrm{SU}(2)$-character variety of $G$. Denote the open set of irreducible representations by $R^*(G)$. Then we can consider the quotient $\calR^*(G):=R^*(G)/\mathrm{SO}(3)\subset \calR(G)$. Now let $\mathcal{R}^*(N):=\calR^*(\pi_1(N))$, where $\pi_1(N)$ denotes the fundamental group of $N$. Moreover, for some admissible perturbation function $f\colon \calA^*/\mathcal{G}\to \R/8\pi^2\Z$, define the perturbed moduli space

\[
\calR_f(N):=\{A \in \calA\mid *F_A-4\pi^2(\nabla f)(A)=0\}/\mathcal{G}.
\]

Finally, define $\calR^*_f(N)$ to be the subset of $\calR_f(N)$ consisting of the orbits of irreducible perturbed flat connections.
Then, we have the following theorem;

The Euler characteristic

\[
\chi_\mathrm{top}(N)=\frac{1}{2}\sum_{A\in \mathcal{R}^*_f(N)}(-1)^{\mu(A)}
\]

with Floer index $\mu(A):=\mathrm{sf}(\theta,A)\,\,\mathrm{mod}\,8$, where $\theta$ denotes the trivial connection, is independent of the holonomy perturbation $f$ and the metric on $N$ and equals up to an overall sign the Casson invariant of $N$.

However, working with a homology 3-sphere does simplify things tremendously since it prevents the existence of non-trivial reducible flat connections before and after small perturbations. In fact, this allows to use the standard cobordism argument to show that the alternating sum is indeed independent of the choice of metric. We refer also to <cit.> for an excellet overview.

§.§ DT partition function

Let $X$ be a Calabi–Yau 3-fold. Using the DT invariants (<ref>), we can define the DT partition function as a generating function by

\begin{equation}
\label{eq:DT_partition)_function_1}
\mathsf{Z}^\mathrm{DT}_X(q):=\sum_{(\beta,k)\in H_2(X,\mathbb{Z})\oplus\Z}\mathrm{DT}_{\beta,k}^Xq^\beta(-z)^k,
\end{equation}

where $q^\beta:=q_1^{\beta_1}\dotsm q_n^{\beta_n}$ with $\beta=\beta_1C_1+\dotsm +\beta_nC_n$ where $\{C_1,\ldots,C_n\}$ denotes a basis of $H_2(X,\Z)$ which is chosen such that $\beta_i\geq 0$ for any effective curve class. Note that $\mathsf{Z}^\mathrm{DT}_X(q)$ is a formal power series in $q_1,\ldots,q_n$ with coefficients in formal Laurent series in the complex variable $z$. In particular, since $X$ is a 3-fold, we can split the Hilbert scheme into components according to the degree and the arithmetic genus of $C$ when $\beta=[C]$. We get

\[
([C],\chi_\mathrm{hol}(\mathcal{O}_C))\in H_2(X,\Z)\oplus H_0(X,\Z).
\]

Consider a smooth divisor $D\subset X$ which algebraically takes the place of the boundary when $X$ is a smooth manifold. We can define the relative DT partition function corresponding to the pair $(X,D)$ as

\begin{equation}
\label{eq:DT_partition_function_2}
\mathsf{Z}^\mathrm{DT}_{X/D}(q):=\sum_{(\beta,k)\in H_2(X,\mathbb{Z})\oplus H_0(X,\Z)}q^\beta(-z)^k\int_{\mathrm{Hilb}_{\beta}(X,k)}\Xi(D),
\end{equation}

where $\Xi(D)$ denotes a collection of forms constructed from the boundary conditions. We need to make sense of the integral in (<ref>).
The integral localizes to a virtual fundamental class[Following <cit.>, for a scheme (or Deligne–Mumford stack) $X$ with cotangent complex $\mathbb{L}_X$, the virtual fundamental class is defined through a perfect obstruction theory of $E\to \mathbb{L}_X$, where $E\in D(\calO_X)$ (here $D(\calO_X)$ denotes the derived category of sheaves of $\calO_X$-modules). In particular, $E$ defines a vector bundle stack $\mathfrak{C}$ over $X$. The virtual fundamental class $[X]^\mathrm{vir}\in A_{\mathrm{rk}\,E}(X)$ is then defined as the intersection of the fundamental class of the intrinsic normal cone $[\mathfrak{C}_X]$ (the perfect obstruction theory $E\to\mathbb{L}_X$ induces a closed immersion of cone stacks $\mathfrak{C}_X\hookrightarrow\mathfrak{C}$) with the zero section of $\mathfrak{C}$, i.e. $[X]^\mathrm{vir}:=0^!_{\mathfrak{C}}[\mathfrak{C}_X]$. Note that we have denoted by $A_r(X)$ the Chow group of $r$-cycles modulo rational equivalence on $X$ with values in $\Z$.] $[\calM_\mathrm{DT}]^\mathrm{vir}\in H_{2\mathrm{vdim}}(X)$, where $\calM_\mathrm{DT}$ denotes the moduli space of supersymmetric configurations. In particular, for the Hilbert scheme, we get the virtual dimension[Typically computed by using some version of the Riemann–Roch theorem.]

\[
\mathrm{vdim}\,\mathrm{Hilb}_{\beta}(X,k)=\int_\beta c_1(X).
\]

Using this construction, we can formulate a more precise version of (<ref>) by

\begin{equation}
\label{eq:DT_partition_function_3}
\mathsf{Z}^\mathrm{DT}_{X/D}(q)=\sum_{(\beta,k)\in H_2(X,\mathbb{Z})\oplus H_0(X,\Z)}q^\beta(-z)^k\int_{[\mathrm{Hilb}_{\beta}(X,k)]^\mathrm{vir}}[\Xi](D),
\end{equation}

where $[\Xi](D)$ denotes the collection of cohomology classes constructed from the boundary conditions.

Note that in fact, using the virtual fundamental class, we can equivalently express the DT invariants (<ref>) as a virtual count by

\[
\mathrm{DT}^X_{\beta,k}=\int_{\left[\mathrm{Hilb}_\beta(X,k)\right]^\mathrm{vir}}1.
\]

§.§ Gluing of DT partition functions

We want to describe the gluing procedure for partition functions in DT theory. Let $X$ be given by two parts $X_1$ and $X_2$. The Hilbert scheme of the singular variety

\[
\]

is not nice (although well-defined) since if $C$ intersects $D$ in a nontransverse way, it will lead to different problems including the failure for the obstruction theory. However, one can resolve this by using expanded degenerations. There we allow $X_0$ to insert bubbles $\mathbb{B}$ such that

\[
X_0[\ell]=X_1\cup_{D_1}\underbrace{\mathbb{B}\cup_{D_2}\dotsm \cup_{D_{\ell}}\mathbb{B}}_{\ell}\cup_{D_{\ell+1}}X_2,
\]

where $\mathbb{B}:=\mathbb{P}(N(X_1/D)\oplus\mathcal{O}_D)$ is a $\mathbb{P}^1$-bundle over $D$ associated to the rank 2 vector bundle $N(X_1/D)\oplus\mathcal{O}_D$, where $N(X_1/D)$ denotes the normal bundle of $D$ in $X_1$. The divisors $D_1\cong D_2\cong\dotsm \cong D_{\ell+1}\cong D$ are all copies of $D$ which appear together with the bubbles $\mathbb{B}$. We consider now subschemes $C\subset X_0[\ell]$ of the form $C=C_0\cup C_1\cup \dotsm \cup C_\ell\cup C_{\ell+1}$ with components $\{C_i\}$ all being transverse to the divisors $\{D_i\}$. We can glue them together by the setting

\[
C_i\cap D_{i+1}=D_i\cap C_{i+1}.
\]

This produces a new bubble each time the intersection does not appear to be transversal. The moduli spaces for different $\ell$ can be captured into one single orbifold

\[
\mathrm{Hilb}(X_0[\bullet])=\bigcup_{\ell\geq 0}\mathrm{Hilb}(X_0[\ell])_{\mathrm{semistable}}\Big/(\mathbb{C}^\times)^\ell,
\]

where $\C^\times$ acts on the $\mathbb{P}^1$-bundle. Note that when talking about the smooth setting for a manifold $\Sigma$, we consider tubular neighborhoods of the boundary $\de\Sigma\times [0,1]$ in order to glue the two boundary pieces $\de\Sigma\times\{0\}$ and $\de\Sigma\times\{1\}$, which in the algebraic setting corresponds to the gluing of two distinguished divisors, namely $D_0=\mathbb{P}(N(X/D))\subset\mathbb{B}$ and $D_\infty=\mathbb{P}(\mathcal{O}_X)\subset \mathbb{B}$. In fact, the $\mathbb{C}^\times$-action on $\mathbb{P}^1$ preserves the divisors $D_0$ and $D_\infty$. The DT gluing formula is then given by setting

\begin{equation}
\label{eq:DT_gluing_1}
\int_{[\mathrm{Hilb}(X)]^\mathrm{vir}}[\Xi](D)=\int_{[\mathrm{Hilb}(X_0[\bullet])]^\mathrm{vir}}[\Xi](D),
\end{equation}

where $[\Xi](D)$ denotes a cohomology class defined by using boundary conditions away from $D$ or in any other way such that it is well-defined as a cohomology class on the whole family of the Hilbert schemes. Additionally, we need to require that the integral on the right-hand-side of (<ref>) can be computed in terms of DT counts in $X_1$ and $X_2$ relative to the divisor $D$. In particular, this means

\begin{equation}
\label{eq:DT_gluing_2}
\mathsf{Z}^\mathrm{DT}_X=\left\langle\mathsf{Z}^\mathrm{DT}_{X_1/D},(-z)^{\vert\,\cdot\,\vert}\mathsf{Z}^\mathrm{DT}_{X_2/D}\right\rangle_{H_\bullet(\mathrm{Hilb}(D))},
\end{equation}

where $\mathsf{Z}^\mathrm{DT}_{X_i/D}$ for $i=1,2$ is defined as in (<ref>) and $\vert\,\cdot\,\vert$ denotes the grading on $H_\bullet(\mathrm{Hilb}(D))=\bigoplus_kH_\bullet(\mathrm{Hilb}(D,k))$. We put the extra weight $(-z)^{\vert\,\cdot\,\vert}$ since

\[
\chi_\mathrm{hol}(\mathcal{O}_C)=\chi_\mathrm{hol}(\mathcal{O}_{C_1})+\chi_\mathrm{hol}(\mathcal{O}_{C_2})-\chi_\mathrm{hol}(\mathcal{O}_{C_1\cap D})
\]

whenever $C=C_1\cup_{C_1\cap D}C_2$ is a transverse union along the common intersection with $D$.

§ RELATION TO GROMOV–WITTEN THEORY

§.§ The moduli space of stable maps

Gromov–Witten (GW) theory <cit.> of a non-singular projective variety $X$ deals with the moduli spaces of stable maps[Recall that a stable map on a non-singular projective variety $X$ is a morphism $f$ from a pointed nodal curve $C$ to $X$ such that if $f$ is constant on any component of $C$, then that component is required to have at least three distinguished points. The distinguished points are either marked points, or points lying over nodes in the normalization of $C$.] constructed by Kontsevich (first appeared in <cit.>), which is a generalization for the moduli space $\overline{\calM}_{g,n}$ of $n$-pointed stable curves[This is a complete algebraic curve whose automorphism group, as an $n$-pointed curve, is finite.] of genus $g$ (see also <cit.>). As a set, $\overline{\calM}_{g,n}$ is the set of isomorphism classes of $n$-pointed stable curves of genus $g$. It actually turns out that $\overline{\calM}_{g,n}$ is a quasi-projective variety of dimension $3g-3+n$ and is given as a projective compacification[One can show that $\calM_{g,n}$ is not compact by noticing that in general a family of smooth curves over a noncomplete base is not extendable to a family of smooth curves over a completion of it. The completion can be realized by allowing fibers that have nodes at worst and using the stable reduction theorem, hence compactness of $\overline{\calM}_{g,n}$.] of the moduli space given by the isomorphism classes of smooth $n$-pointed stable curves of genus $g$, which is usually denoted by $\calM_{g,n}$.

In particular, the moduli space of stable maps from $n$-pointed nodal curves[Recall that a nodal curve is a complete algebraic curve such that each of its points is either smooth or locally complex-analytically isomorphic to a neighborhood of the origin within the locus of the equation $xy=0$ in $\C^2$ <cit.>.] of genus $g$ to a non-singular projective variety $X$ representing[Any stable map $f\colon C\to X$ represents a homology class $\beta\in H_2(X,\mathbb{Z})$ whenever $f_*[C]=\beta$.] the class $\beta\in H_2(X,\mathbb{Z})$ is denoted by $\overline{\calM}_{g,n}(X,\beta)$. In fact, $\overline{\calM}_{g,n}(X,\beta)$ is a Deligne–Mumford stack. It is easy to see that if $X$ is a point, we get that

\[
\overline{\calM}_{g,n}(X,0)=\overline{\calM}_{g,n}.
\]

A natural cohomology class on the moduli space of stable maps is given by the pullback of $X$ (see below), i.e. $\mathrm{ev}^*_i(\gamma)$ where $\gamma\in H^\bullet(X,\Z)$ for $i=1,\ldots, n$. At each point $[C,p_1,\ldots,p_n,f]\in\overline{\calM}_{g,n}(X,\beta)$, the cotangent space to $C$ at each point $p_i$ is a 1-dimensional vector space. Gluing all these spaces together, we get a line bundle $\mathbb{L}_i$ called $i$-th tautological line bundle. Denote by $\psi_i:=c_1(\mathbb{L}_i)$ its first Chern class. Then we have the string equation for $\overline{\calM}_{g,n}$

\begin{equation}
\label{eq:string_eq}
\int_{\overline{\calM}_{g,n+1}}\prod_{i=1}^n\psi_i^{\beta_i}=\sum_{i=1}^n\int_{\overline{\calM}_{g,n}}\psi_1^{\beta_1}\dotsm \psi_i^{\beta_i-1}\dotsm \psi_n^{\beta_n}.
\end{equation}

One can then easily prove the dilaton equation for $\overline{\calM}_{g,n}$

\begin{equation}
\label{eq:dilaton_eq}
\int_{\overline{\calM}_{g,n+1}}\psi_{n+1}\prod_{i=1}^n\psi_i^{\beta_i}=(2g-2+n)\int_{\overline{\calM}_{g,n}}\prod_{i=1}^n\psi_i^{\beta_i}
\end{equation}

for $2g-2+n>0$.

Moreover, define

\[
\langle \tau_{\beta_1},\ldots,\tau_{\beta_n}\rangle_g:=\int_{\overline{\calM}_{g,n}}\prod_{i=1}^n\psi_i^{\beta_i}.
\]

The virtual dimension of the moduli space of stable maps <cit.> is given by

\begin{equation}
\label{eq:virtual_dim_moduli_of_stable_maps}
\mathrm{vdim}\, \overline{\calM}_{g,n}(X,\beta)=\int_\beta c_1(X)+(\dim X-3)(1-g)+n.
\end{equation}

Similarly as for the DT construction, one can also consider the virtual fundamental class

\[
\Big[\overline{\calM}_{g,n}(X,\beta)\Big]^\mathrm{vir}\in H_{2\mathrm{vdim}}(X,\mathbb{Q}).
\]

In particular, if there is no obstruction for all stable maps, we get that the virtual fundamental class is equal to the ordinary fundamental class. As already mentioned, for $i=1,...,n$ there are evaluation maps $\mathrm{ev}_i\colon \overline{\calM}_{g,n}(X,\beta)\to X$ such that $\mathrm{ev}_i(f)=f(x_i)$ for $x_i\in C$. Thus, the classes $\gamma\in H^\bullet(X,\Z)$ can be pulled back to classes in $H^\bullet(\overline{\calM}_{g,n}(X,\beta),\Q)$ by the map $\mathrm{ev}_i^*\colon H^\bullet(X,\Z)\to H^\bullet(\overline{\calM}_{g,n}(X,\beta),\Q)$.

§.§ GW invariants

Given cohomology classes $\gamma_1,\ldots, \gamma_n\in H^\bullet(X,\Z)$, we can define the GW invariants by

\begin{equation}
\label{eq:GW_invariants}
\mathrm{GW}_{g,\beta}^X(\gamma_1,\ldots,\gamma_n):=\int_{\big[\overline{\calM}_{g,n}(X,\beta)\big]^\mathrm{vir}}\prod_{i=1}^n\mathrm{ev}^*_i(\gamma_i).
\end{equation}

According to Witten's notation, we may sometimes also write $\left\langle \prod_{i=1}^n\gamma_i\right\rangle^X_{g,\beta}=\langle\gamma_1\dotsm\gamma_n\rangle_{g,\beta}^X$ instead of $\mathrm{GW}_{g,\beta}^X(\gamma_1,\ldots,\gamma_n)$.

One can also define a generalized version of the GW invariants, called the gravitational descendant invariants or just descendant invariants, defined by

\begin{equation}
\label{eq:gd_invariants}
\left\langle\prod_{i=1}^n\tau_{a_i}(\gamma_i)\right\rangle_{g,\beta}^X:=\int_{\big[\overline{\calM}_{g,n}(X,\beta)\big]^\mathrm{vir}}\prod_{i=1}^n\mathrm{ev}^*_i(\gamma_i) \psi_{i}^{a_i},
\end{equation}

where $\gamma_i\in H^\bullet(X,\Z)$ and $a_i\in\mathbb{Z}_{>0}$.
We can extend the string (<ref>) and dilaton (<ref>) equations to the moduli space of stable maps $\overline{\calM}_{g,n}(X,\beta)$. The string equation is given by

\begin{equation}
\label{eq:string_eq2}
\left\langle\prod_{i=1}^n\tau_{a_i}(\gamma_i)\boldsymbol{1}\right\rangle_{g,\beta}^X
\end{equation}

where $\boldsymbol{1}\in H^\bullet(X,\Z)$ denotes the unit. The dilaton equation is given by

\begin{equation}
\label{eq:dilaton_eq2}
\left\langle\prod_{i=1}^n\tau_{a_i}(\gamma_i)\tau_1(\boldsymbol{1})\right\rangle_{g,\beta}^X=(2g-2+n)\left\langle \prod_{i=1}^n\tau_{a_i}(\gamma_i)\right\rangle_{g,\beta}^X.
\end{equation}

It is easy to see that (<ref>) reduces to (<ref>) and (<ref>) to (<ref>) if $X$ is a point.

§.§ GW partition function

One can then define the GW partition function as

\begin{align}
\label{eq:GW_partition_function}
\begin{split}
\mathsf{Z}^\mathrm{GW}_X(q,u)&:=\sum_{(\beta,g)\in H_2(X,\mathbb{Z})\oplus \Z}\mathrm{GW}^X_{\beta,g}(\gamma_1,\ldots,\gamma_n)q^\beta u^{2g-2}\\
&=\sum_{(\beta,g)\in H_2(X,\mathbb{Z})\oplus \Z}q^\beta u^{2g-2}\int_{\big[\overline{\calM}_{g,n}(X,\beta)\big]^\mathrm{vir}}\prod_{i=1}^n\mathrm{ev}^*_i(\gamma_i)
\end{split}
\end{align}

GW theory can be regarded as some $A$-mirror since the invariants can be thought of an $A$-model with path integral over the space of fields given through pseudo-holomorphic curves as for the construction of Lagrangian Floer homology (see Section <ref>). We refer to the excellent reference <cit.> for more details on GW theory and its connection to mirror symmetry.

Consider the generating series

\begin{equation}
\label{eq:generating_series_KdV}
\mathsf{F}(t_0,t_1,\ldots):=\sum_{g\geq 0,n\geq0\atop 2g-2+n\geq 0}\frac{1}{n!}\sum_{\beta_1,\ldots,\beta_n\geq 0}\left(\int_{\overline{\calM}_{g,n}}\prod_{i=1}^n\psi_i^{\beta_i}\right)t_0^{\beta_1}t_1^{\beta_2}\dotsm t_n^{\beta_n}.
\end{equation}

In <cit.>, Witten conjectured that the function $u:=\frac{\de^2\mathsf{F}}{\de t_0^2}$ satisfies the KdV hirarchy, i.e.

\begin{align}
\begin{split}
\label{eq:KdV_hirarchy}
\end{split}
\end{align}

where $x:=t_0$. Using the string equation (<ref>), we get

\begin{equation}
\label{eq:KdV_string_equation}
\frac{\de\mathsf{F}}{\de t_0}=\sum_{i\geq 0}t_{i+1}\frac{\de\mathsf{F}}{\de t_i}+\frac{t_0^2}{2}.
\end{equation}

If $\mathsf{F}$ satisfies condition (<ref>) together with (<ref>), one can uniquely determine the generating series $\mathsf{F}$. Witten's conjecture was proven by Kontsevich in <cit.>.

§.§ Topological recursion

Topological recursion started with the work of Eynard–Orantin in <cit.> which uses the recursive methods of symplectic invariants to solve matrix model loop equations. In particular, as an application they show how one can obtain e.g. a proof of Witten's conjecture (see Remark <ref>), Mirzakhani's recursive methods of deriving Weil–Petersson volumes <cit.> and certain constructions of topological string theories.
In particular, one is interested in the data given by a spectral curve $C=(\Sigma_g,x,y,B)$, where $\Sigma_g$ denotes a Riemann surface of genus $g\geq 0$, $x,y$ are two meromorphic functions on $\Sigma_g$ and $B$ is some 2-form on $\Sigma_g\times\Sigma_g$. Assume moreover that the zeros of $\dd x$ are all simple and do not coincide with the zeros of $\dd y$. Then, as a result of topological recusrsion, one obtains symmetric multidifferentials $\omega_{g,n}\in H^0(K_{\Sigma_g}(*D)^{\otimes n},(\Sigma_g)^n)^{S_n}$ for $g\geq 0$ and $n\geq 1$ such that $2g-2+n> 0$. These differentials are usually called correlators. We have denoted by $D$ the divisor of zeros of $\dd x=0$ and by $S_n$ the symmetric group of order $n$. Note that then the multidifferentials $\omega_{g,n}$ will be holomorphic outside of $\dd x=0$ and can have poles of any order when the given variables approach the divisor $D$. The key is to define the correlators recursively. In particular, we define first the exceptional cases

\begin{align}
\omega_{0,1}(p_1)&=y(p_1)\dd x(p_1),\\
\omega_{0,2}(p_1,p_2)&=B(p_1,p_2).
\end{align}

Then, the recursive relation for the correlators in general is given by

\begin{multline}
\label{eq:multidifferentials}
\omega_{g,n}(p_1,\boldsymbol{p}_I)=\sum_{\dd x(\alpha)=0}\mathrm{Res}_{p=\alpha} K(p_1,p)\times\\
\times\left\{\omega_{g-1,n+1}(p,\sigma_\alpha(p),\boldsymbol{p}_I)+\widehat{\sum_{h+h'=g\atop J\sqcup J'=I}^{\omega_{0,1}}}\omega_{h,1+\vert J\vert}(p,\boldsymbol{p}_J)\omega_{h',1+\vert J'\vert}(\sigma_\alpha(p),\boldsymbol{p}_{J'})\right\}.
\end{multline}

We have used the notation $I=\{2,3,\ldots,n\}$ and $\boldsymbol{p}_J=\{p_{j_1},\ldots,p_{j_k}\}$ for $J=\{j_1,j_2,\ldots,j_k\}\subseteq I$. Note that the holomorphic function $p\mapsto \sigma_\alpha(p)$ is the non-trivial involution which is locally defined at the ramification point $\alpha$ and satisfies $x(\sigma_\alpha(p))=x(p)$. Moreover, we have denoted by $\widehat{\sum^{\omega_{0,1}}}$ in the bracket the sum which leaves out the terms involving $\omega_{0,1}$. The recursion kernel is given by