List of Todos

We study the behavior of Donaldson's invariants of 4-manifolds based on the moduli space of anti self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang–Mills theory, known today as Donaldson–Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable $n$-dimensional field theories (in the sense of Baez–Dolan and Lurie) in the perturbative setting, especially when $n\leq 4$.
Additionally, we relate these constructions to Nekrasov's partition function by treating an equivariant version of Donaldson–Witten theory in the BV formalism.
Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov–Witten and Donaldson–Thomas theory and recall their correspondence conjecture for general Calabi–Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena.

§ INTRODUCTION

§.§ Overview and motivation

There is no doubt that the study of gauge theories had a big influence on modern mathematics and physics. A particularly influential and important one is the study of the topology of 4-manifolds by using Yang–Mills theory and the notion of anti self-dual connections, also called instantons, which can be considered as a class of critical points of the Yang–Mills action functional. An early attempt, maybe even one of the starting points, was the construction of instantons given in <cit.>. Shortly thereafter, Donaldson introduced in <cit.> his famous polynomial invariants which are a type of topological invariants based on the theory of characteristic classes on vector bundles (topological $K$-theory) and the construction of the moduli space of anti self-dual connections. In particular, these invariants are described as integrals of a product of certain cohomology classes over the moduli space. Thus, defining these invariants relies very much on the behaviour of the moduli space as a sufficiently “nice” manifold. Many different people, including Uhlenbeck <cit.>, Freed <cit.>, Freedman <cit.> and Taubes <cit.>, provided results which contributed to the fact that these moduli spaces are indeed “nice” enough. However, at first, these invariants have been only defined for the case when the 4-manifold is closed.

A new drive came into play when Floer introduced in <cit.> a type of topological invariant of closed 3-manifolds by using methods of gauge theories based on ideas of Witten regarding the constructions of Morse theory in the setting of perturbative quantum field theory <cit.>. In particular, he constructed an infinite-dimensional version of Morse theory based on the moduli space of anti self-dual connections together with the Chern–Simons action functional playing the role of the Morse function. The main invariants are then given by the homology groups, called instanton Floer homology groups, similarly as the Morse homology groups in the finite setting.
Braam and Donaldson then realized in <cit.> that the polynomial invariants defined by Donaldson can be extended to 4-manifolds with boundary by imposing that the invariants are then exactly valued in these Floer groups. Moreover, they proved a gluing formula which endows the invariant with the structure of a functorial TQFT according to Atiyah's axioms <cit.>.

Another type of a similar approach to a homology theory for Floer's construction was considered in <cit.> by using the symplectic manifolds and corresponding transversal Lagrangian submanifolds thereof. Besides important insights regarding the Arnold conjecture (see e.g. <cit.>), it was soon realized that this type of homology theory plays a fundamental role in order to formulate and understand the mirror symmetry appearing in string theory <cit.> from a homological point of view. In particular, it was Fukaya who constructed an $A_\infty$-category in <cit.> (see also <cit.>) from out of this notion and Kontsevich who used these categories to formulate a conjecture which, for two mirror Calabi–Yau manifolds, relates this category of one side of the mirror to the derived category of coherent sheaves of the other side of the mirror through an equivalence of triangulated categories. This is known famously today as the homological mirror symmetry conjecture <cit.>.

On the other hand, based on ideas of Atiyah <cit.>, Witten gave a way of obtaining Donaldson's polynomials by considering the perturbative expansion of the expectation value (path integral quantization) for a certain observable with respect to a local action functional. He also gave an argument involving the case of manifolds with boundary by interpreting the boundary states as the Floer groups in agreement with Donaldson's observation.
The perturbative methods of treating quantum gauge theories have developed through time by different approaches.

The Batalin–Vilkovisky (BV) formalism <cit.> provides a nice way of dealing with quantum gauge theories in a cohomological symplectic formalism <cit.> by using methods of functional integrals. In fact, the gauge-fixing there is equivalent to the choice of a Lagrangian submanifold which, by similar methods as the BRST formalism <cit.> and Faddeev–Popov ghosts <cit.>, gives a way of computing the partition function by the perturbative expansion into Feynman graphs. The methods described by Batalin and Vilkovisky are considered from a Lagrangian point of view, whereas the Hamiltonian counterpart was described in the work of Batalin, Fradkin, Fradkina and Vilkovisky <cit.>, usually called the BFV formalism. Note that these constructions can be considered separately for closed spacetime manifolds. However, one is often interested in the quantization picture for manifolds with boundary in order to use the concept of locality for the simplification through cutting and gluing properties. Additionally, everything should be consistent with Atiyah's TQFT axioms and Segal's axioms regarding conformal field theories <cit.>. Such an extension has been recently provided by Cattaneo, Mnev and Reshetikhin to deal with the classical and quantum formalism of local gauge theories on manifolds with boundary in the cohomological symplectic setting by coupling the BV construction in the bulk to the BFV construction on the boundary <cit.>. These constructions can be easily extended to higher codimensions in the classical setting, but need more sophisticated techniques in the quantum setting. Nevertheless, this formalism is expected to give a reasonable candidate for a perturbative formulation of (fully) extended TQFTs (in the sense of Baez–Dolan <cit.> or Lurie <cit.>) for the case of interest.

As it was shown in <cit.>, it turns out that the field theory constructed by Witten can be naturally formulated by using a type of BV theory developed by Alexandrov, Kontsevich, Schwarz and Zaboronsky (AKSZ) in <cit.> for a special gauge-fixing. This formulation extends in a nice way to the BV-BFV formalism since AKSZ theories often appear as suitable deformations of abelian $BF$ theory <cit.>. This is also the case for Donaldson–Witten theory, formulated as a 4-dimensional AKSZ theory with target $\mathfrak{h}[1]\oplus\mathfrak{h}[2]$ for some Lie algebra $\mathfrak{h}$.
An approach to globalize a special type of AKSZ theories has been given in <cit.>. There one starts with an AKSZ theory of any dimension which is of split-type. Choosing a background field as an element of the moduli space of classical solutions, one can vary the local theories over the target manifold by considering the Grothendieck connection as in the setting of formal geometry developed by Gelfand–Fuks <cit.>, Gelfand–Kazhdan <cit.> and Bott <cit.>. Within the BV-BFV formalism, this leads to more general gauge conditions, such as the modified differential Quantum Master Equation.

Another approach, which is related to Donaldson's construction, to obtain topological invariants of 4-manifolds is due to Seiberg and Witten <cit.> (see also <cit.> for a more mathematical introduction). They considered $\calN=2$ supersymmetric Yang–Mills theory[Recall that $\calN$ denotes the number of irreducible real spin representations in a supersymmetry (SUSY) algebra.] and formulated a set of equations (Seiberg–Witten equations) which contains the same information of the 4-manifold as the Yang–Mills equations but has the advantage that it is much easier to deal with. Solutions of these equations are usually called monopoles[According to the equations for magnetic monopoles in the theory of electromagnetism.].

The field-theoretic approach of Seiberg–Witten theory is in its nature given by an $A$-model as in <cit.>. The low energy effective action can actually be written in terms of a holomorphic function $\mathsf{F}$, called the Seiberg–Witten prepotential. In <cit.>, Nekrasov showed how one can compute this prepotential $\mathsf{F}$ as a certain limit by using techniques of equivariant localization for a given torus action to define a partition function proportional to the volume of the moduli space of instantons. In particular, this partition function is given as a generating function with coefficients given by the volumes of the connected components of the moduli space of instantons controlled through the instanton numbers. This is achieved by formulating a 6-dimensional theory (i.e. a given 4-manifold, or locally $\R^4$, times a torus $\mathbb{T}$) and then perform dimensional reduction.
This is also called the $\calN=2$ supersymmetric Yang–Mills theory in the $\Omega$-background, where $\Omega$ is a $4\times 4$ matrix with entries given by one of the two generators of the torus $\mathbb{T}$ with a sign or zero in a certain way. This background in fact allows one to integrate out all the fluctuations (high energy modes) appearing in the functional integral quantization, and hence get rid of any divergencies.

Since the underlying $\calN=2$ supersymmetric Yang–Mills theory in this setting is based on the techniques of equivariant localization, one can study an equivariant version of Donaldson–Witten theory in the AKSZ-BV setting and consider a (regularized) perturbative quantization in order to obtain the mentioned partition function. In order to deal with with the quantization of such an equivariant field theory, it is natural to formulate an equivariant version of the BV formalism as it was developed in <cit.>. However, the extension to manifolds with boundary is still open but following the constructions of Atiyah, Donaldson and Witten for the non-equivariant setting, the boundary states, hence the geometric quantization procedure for the state space in the BV-BFV formalism, should recover an equivariant version of instanton Floer homology as discussed in <cit.>.

An approach to the construction of Donaldson for higher dimensional gauge theories was initiated through the work of Donaldson and Thomas in <cit.> and developed further by Thomas in <cit.>. In particular, using a holomorphic version of Chern–Simons theory <cit.>, Thomas constructed topological invariants for Calabi–Yau 3-folds by constructing a holomorphic version of the Casson invariant <cit.> which counts the bundles over the given Calabi–Yau 3-fold and extends to the counting of curves in algebraic 3-folds. The formulation of these invariants in terms of a weighted Euler characteristic was given by Behrend in <cit.>. One can consider the Donaldson–Thomas partition function, similarly as for Gromov–Witten theory <cit.>, as a generating function for the corresponding invariants on any Calabi–Yau 3-fold $X$. Moreover, one can prove gluing formulae for these partition functions, first considered in <cit.> by considering an associated divisor and the properties of the Hilbert scheme of 1-dimensional subschemes of $X$. By the nature of its construction, Donaldson–Thomas theory is related to the construction of Nekrasov theory through its way of instanton counting (the modified moduli space of instantons in Nekrasov's theory is replaced by the Hilbert scheme of 1-dimensional subschemes of $X$ in Donaldson–Thomas theory).
Moreover, it seems plausible that the Donaldson–Thomas partition function is related to the Gromov–Witten partition function by a certain correspondence. Such a correspondence for general Calabi–Yau 3-folds was conjectures by Maulik, Nekrasov, Okounkov and Pandharipande in <cit.> and proven for toric 3-folds in <cit.> by Maulik, Oblomkov, Okounkov and Pandharipande. The correspondence seems to formulate a version of the homological mirror symmetry conjecture, which was already known to be true for toric 3-folds.

eq. AKSZ-DW  [dr,"VI"][d,"V"]      AKSZ-DW[r,shift right][dr,dashed] [l,swap,"IV"]      DW[l,shift right,swap,"II"] [r,"I"] [d,"III"]          Donaldson [d,"XI"]

Nekrasov [rrr,swap,bend right=60,"IX"]     eq. Floer[r,shift right]      Floer[dl,shift right][r,"VIII"] [l,swap,shift right,"VII"]     DT[dr,dashed,shift right] [r,shift left,"X"]            GW[l,shift left][d,dashed,shift right]

Lagr. Floer[rrr,dashed,crossing over,shift right][ur,swap,shift right,"XIII"][urrr, bend right=30,crossing over,swap,"XII"]            MS[lll,dashed,crossing over,swap, shift right,"XVI" xshift=-20pt][u,dashed,swap,shift right,"XIV"][ul,dashed,swap,shift right,crossing over,"XV"]

Diagrammatic illustration of the relations discussed in this paper.

Legend for Figure <ref>:

* Witten's construction to obtain Donaldson polynomials via field theory,

* AKSZ-BV formulation of Donaldson–Witten theory,

* When the source manifold has boundary. Boundary states are given by Floer groups,

* Equivariant formulation of the AKSZ-BV formulation of Donaldson–Witten theory through an equivariant BV construction,

* BV partition function of equivariant Donaldson–Witten theory cooresponds to Nekrasov's partition function,

* BV-BFV quantization of equivariant Donaldson–Witten theory produces equivariant Floer groups on the boundary,

* Equivariant formulation of Floer (co)homology,

* Through Taubes' construction: the Euler characteristic with respect to Floer homology for any homology 3-sphere coincides with its Casson invariant. In particular, Donaldson–Thomas invariants are a complex version of such Euler characteristic when using the Floer groups with respect to the holomorphic Chern–Simons action functional for certain Calabi–Yau manifolds,

* Through enumerative counting methods and the nature of its formulation,

* Conjectured correspondence by Maulik, Nekrasov, Okounkov and Pandharipande,

* Higher gauge theory approach when passing to complex geometry,

* Formulation of Gromov–Witten invariants through the moduli space of pseudo-holomorphic curves as an $A$-model path integral,

* The Atiyah–Floer conjecture,

* Ingredients for mirror symmetry,

* Ingredients for mirror symmetry,

* Ingredients for mirror symmetry.

§.§ Main purpose of the paper

The purpose of this paper is to provide the reader with:

* a concise overview of the field of 4-manifold topology, both from the pure mathematical and field-theoretic point of view. Especially, to explain some of the relations of the literature as well as to new developements such as the BV-BFV techniques developed recently.

* a concise overview of (instanton and Lagrangian) Floer (co)homology and how it fits into the 4D-3D bulk-boundary correspondence through Chern–Simons theory.

* a detailed formulation of a (global) perturbative quantization of the AKSZ formulation of Donaldson–Witten theory on source manifolds with boundary which fits into the gauge theory setting of the BV-BFV formalism by using methods of configuration space integrals.

* a short discussion of the quantization of higher defect theories through the recently developed methods of shifted (symplectic or Poisson) structures.

* a concise overview of the relation to Seiberg–Witten theory through Nekrasov's construction by using equivariant methods of localization and an equivariant version of the BV formalism in combination with Donaldson–Witten theory and a description of the relation with Donaldson's invariants through an equivariant version of instanton Floer (co)homology.

* a description of the relation to constructions of Donaldson–Thomas theory.

* a concise overview of the conjectured correspondence of Donaldson–Thomas theory and Gromov–Witten theory and connections to other fields, such as e.g. topological recursion or supergeometry.

§.§ Structure of the paper

The paper is structured as follows:

* In Section <ref> we recall the definition of the moduli space of anti self-dual connections (instantons) together with some properties and define Donaldson's invariants. Moreover, we briefly discuss Witten's construction by using methods of field theory in order to obtain the aforementioned invariants.

* In Section <ref> we recall the construction of Morse theory and Morse homology in order to construct instanton Floer homology by using the Chern–Simons action functional as a Morse function. We describe the relation of Donaldson's invariants with this homology theory through the setting of 4-manifolds with boundary.
Moreover, we recall the construction of Lagrangian Floer homology, the Atiyah–Floer conjecture and give a remark on an approach for proving it.

* In Section <ref> we recall the classical and quantum BV-BFV formalism, the notion of BV algebras and give the important examples of AKSZ theories and abelian $BF$ theories. Moreover, we briefly discuss higher codimension (defects, branes) extensions and give some ideas for the quantization approach.

* In Section <ref> we give an AKSZ construction of Donaldson–Witten theory and show how it fits into the classical BV-BFV setting.

* In Section <ref> we consider the quantization of the classical setting obtained in Section <ref> by describing the Feynman rules and the partition function. In particular, we prove that the Donaldson–Witten partition function satisfies a modified version of the Quantum Master Equation by using configuration space integrals and show that we get a well-defined cohomology theory on the boundary state space. Moreover, we construct a perturbative globalization approach by using methods of formal geometry and argue that the partition function constructed through the globalization approach lies in the kernel of a certain operator that squares to zero which also leads to a cohomology theory in this setting, similarly as for the nonglobal case.

* In Section <ref> we introduce Nekrasov's partition function in terms of polynomial counts through the Hilbert scheme of monomial ideals in $\C[u_1,u_2]$ and the $\mathrm{Ext}^1$-groups by using the modular interpretation of the tangent bundle. We also consider an equivariant version of the BV gauge formalism and hence consider an equivariant version of Donaldson–Witten theory. On closed 4-manifolds the quantum picture corresponds to Nekrasov's partition function, whereas for 4-manifolds with boundary it induces an equivariant version of instanton Floer (co)homology as the boundary state. We describe the notion of equivariant Floer (co)homology following the construction of Austin–Braam in the Cartan model. Moreover, we discuss its relation to Donaldson's invariants. In particular, we consider the fact that these equivariant (co)homology groups appear as the (dual) image of the invariants.

* In Section <ref> we recall Donaldson–Thomas invariants through their description as the relative topological Euler characteristic of the Hilbert scheme of 1-dimensional subschemes of a Calabi–Yau 3-fold by using Behrend's construction. We also recall Taubes' construction for their relation to Floer homology by considering  them as a holomorphic version of the Casson invariant. Moreover, we construct the Donaldson–Thomas partition function as a generating function with respect to the Donaldson–Thomas invariants, extend it to a relative version when considering appropriate divisors of the underlying Calabi–Yau 3-fold and describe their gluing properties.

* In Section <ref> we recall Kontsevich's moduli space of stable maps for a Calabi–Yau 3-fold and define the corresponding Gromov–Witten invariants. We also recall the partition function for Gromov–Witten invariants, similarly as before, as a generating series for these invariants and extend also this to a relative version for appropriate divisors. We briefly explain some ideas of topological recursion and its connection to Gromov–Witten invariants. To close the circle, we recall the Gromov–Witten/Donaldson–Thomas correspondence conjecture for Calabi–Yau 3-folds. Finally, we give some ideas for Gromov–Witten invariants on (graded) supermanifolds by considering ideas of Keßler–Sheshmani–Yau in the case of super Riemann surfaces.

§.§ Notation

Throughout the paper, $\Sigma$ will denote a 4-manifold and $N$ a 3-manifold. It might happen that $N$ appears as the boundary 3-manifold of a 4-manifold $\Sigma$, otherwise the boundary of $\Sigma$ is denoted by $\de\Sigma$. Riemann surfaces of genus $g$ will be denoted by $\Sigma_g$. Moduli spaces, of different flavor however, will be denoted by the calligraphic letter $\calM$ and specified through different decorations. We will denote real numbers by $\R$, complex numbers by $\C$ and integers by $\Z$.
If a (co)homology group takes values in $\R$, we will not further emphasize it, i.e. we write $H^\bullet(\Sigma)$ instead of $H^\bullet(\Sigma,\R)$, whereas for integer-valued groups we will try to indicate it, i.e. we write $H^\bullet(\Sigma,\Z)$. The exterior product between differential forms is sometimes explicitly written and sometimes left out to avoid any cumbersome notation. The Einstein summation convention is assumed, i.e. we sum over repeating indices, whenever the summation sign is not written explicitly. Algebro-geometrical objects (e.g. (projective) algebraic varieties, schemes, etc.) are usually denoted by $X$, which should be distinguished from the classical base field in the AKSZ formulation of Donaldson–Witten theory, which will be also denoted by $X$. Classical action functionals are usually denoted by $S$ (with additional decorations, depending on the given theory), whereas the BV action is denoted by the calligraphic version $\calS$. The same holds for the space of fields, usually denoted by $F$, and its BV counterpart $\calF$. The space of connections on some principal bundle $P$ is usually denoted by $\calA$ (sometimes also $\calA(P)$ to emphasize the bundle). Smooth maps on a manifold $M$ will be denoted either by $C^\infty(M)$ or by $\calO_M$ (as the structure sheaf) depending on the context. Sections of a bundle $E$ are denoted by $\Gamma(E)$.
The space of smooth differential forms on a manifold $M$ will be denoted by $\Omega^\bullet(M)$. Differential forms on $M$ with values in some bundle $E$ will be denoted by $\Omega^\bullet(M,E)$.
Finally, we will denote by $\I:=\sqrt{-1}$ the imaginary unit.

§.§ Acknowledgements
I would like to thank A. S. Cattaneo, A. Sheshmani, P. Safronov and N. Reshetikhin for discussions and comments.
This research was supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, and by the SNF grant No. 200020_192080.

§ DONALDSON–WITTEN THEORY

§.§ Moduli space of anti self-dual connections

Let $(\Sigma,g)$ be a 4-dimensional Riemannian manifold and let $\mathfrak{g}=\mathrm{Lie}(G)$ be the Lie algebra of a Lie group $G$. Moreover, let $P\to \Sigma$ be a principal $G$-bundle and consider its adjoint bundle $\ad P:=P\times_G\mathfrak{g}$.
Using the Hodge star operator $*$, we can define for a 2-form $\chi\in \Omega^2(\Sigma)$ its self-dual and anti self-dual parts by

\begin{equation}
\label{eq:SD_ASD}
\chi^\pm=\frac{1}{2}(\chi\pm *\chi).
\end{equation}

Note that in four dimensions we have $*\colon H^2(\Sigma)\to H^2(\Sigma)$. We can extend the splitting of (<ref>) naturally to differential forms with values in $\ad P$. In particular we can extend this to the curvature 2-form $F_A$ of a connection 1-form $A\in \Omega^1(\Sigma,\ad P)$. A connection $A\in \Omega^1(\Sigma,\ad P)$ is called anti self-dual or instanton if $F^+_A=0$. We define the instanton number of $A$ to be

\begin{equation}
\label{eq:instanton_number}
k_A:=\frac{1}{8\pi^2}\int_\Sigma\tr(F_A\land F_A).
\end{equation}

This number is an integer only if $G=\mathrm{SU}(2)$. Moreover, if the corresponding $\mathrm{SU}(2)$-bundle lifts to a 2-dimensional complex vector bundle $E$, we have $k_A=\int_\Sigma c_2(E)\in\Z$, i.e. the instanton number is determined by the second Chern class since $c_1(E)=0$ (because in this case $\tr(F_A\land F_A)=0$). Note that if $A$ is an anti self-dual connection, the instanton number is positive. Consider the Yang–Mills action functional[We will not always indicate the measure whenever it is clear. When we write $\int \|F_A\|^2$, we actually mean $\int \|F_A\|^2\dd\mu$.]

\begin{equation}
\end{equation}

where we have considered the curvature locally as $F_A=\frac{1}{2}F_{ij}\dd x^i\land \dd x^j$. Hence, we get that

\begin{equation}
S^\mathrm{YM}_\Sigma(A)=\int_\Sigma\|F_A\|^2=\int_\Sigma\|F_A^-\|^2+\int_\Sigma \|F_A^+\|^2\geq 8\pi^2k_A.
\end{equation}

This shows that the critical points of $S^\mathrm{YM}_\Sigma$ are bounded by $8\pi^2 k_A$, and in fact the minimum is attained if $A$ is anti self-dual, i.e. $F_A^+=0$. Note that $\tr(F_A\land F_A)=-\|F_A\|^2$ and thus $\tr(F_A\land F_A)=-\left(\|F_A^+\|^2-\|F_A^-\|^2\right)$.
Let us denote by $\mathcal{A}:=\Omega^1(\Sigma,\ad P)$ the space of connections on $P$. One can check that this is an infinite-dimensional space. Define the map

\begin{align}
\begin{split}
s\colon \mathcal{A}&\to \Omega^2_{\mathrm{ASD}}(\Sigma,\ad P),\\
A &\mapsto F_A^+,
\end{split}
\end{align}

where we have denoted by $\Omega^2_{\mathrm{ASD}}(\Sigma,\ad P)$ the anti self-dual 2-forms on $\Sigma$ with values in $\ad P$.
Moreover, denote by $\mathcal{G}:=\Gamma(\mathrm{Aut}(\ad P))$ the infinite-dimensional Lie group of gauge transformations of $\ad P$. Then we can define the moduli space of anti self-dual connections[Sometimes it is also called moduli space of instantons. Moreover, we drop the dependence on the instanton number $k$, the 4-manifold $\Sigma$ and the group $G$ from the notation and just write $\calM_\mathrm{ASD}$ whenever it is clear.] by

\begin{equation}
\label{eq:moduli_ASD}
\calM^k_{\mathrm{ASD}}(\Sigma,G):=s^{-1}(0)=\{[A]\in \mathcal{A}/\mathcal{G}\mid s(A)=0,\,k_A=k\}.
\end{equation}

§.§ Donaldson polynomials

In <cit.>, Donaldson defined topological invariants for 4-manifolds by using methods of gauge theory. If $\Sigma$ is an oriented, compact and simply connected 4-manifold, we can split $H^2(\Sigma)$ into eigenspaces $H^2_+(\Sigma)\oplus H^2_-(\Sigma)$ by using the Hodge star (dual and anti self-dual part). Let $b^2_+:=\dim H^2_+(\Sigma)$, $P\to \Sigma$ a principal $\mathrm{SU}(2)$-bundle with second Chern class $c_2(P)$ and $k:=\int_\Sigma c_2(P)\in \mathbb{Z}$, the instanton number (<ref>). The moduli space of anti self-dual connections $\calM_{\mathrm{ASD}}$ in this setting[For $k>0$, there is a more complicated notion of dimension called the virtual dimension (see Footnote <ref> for more details on the virtual fundamental class) for the moduli space $\calM_\mathrm{ASD}$. The virtual dimension and the dimension might be different for certain metrics. However, it was argued in <cit.> that for the subspace of Riemannian metrics on $\Sigma$ consisting of generic metrics, the virtual dimension and the actual dimension of $\calM_\mathrm{ASD}$ coincide, i.e.
\[\mathrm{vdim}\,\calM_\mathrm{ASD}=\dim\calM_\mathrm{ASD}=8k-3(1+b_+^2).\]
], as defined in (<ref>), is of dimension[A general formula for the dimension is given by $\dim \calM_\mathrm{ASD}=4a(G)k-\dim G(1+b_+^2)$, where $a(G)\in\Z$ is an integer depending on the group $G$ (in our case $a(\mathrm{SU}(2))=2$). Note that we have $\dim \mathrm{SU}(2)=3$. A first approach to the dimension was considered in <cit.> for $S^4$ which is given by $8k-3$ since then $b_+=0$.]

\begin{equation}
\label{eq:dimension_moduli}
\dim \calM_{\mathrm{ASD}}=8k-3(1+b_+^2),
\end{equation}

which is even if $b_+^2$ is odd. Let us assume that $b_+^2$ is indeed odd and let $\dim \calM_\mathrm{ASD}=2d$. We also need the notion of an irreducible connection. A connection $A\in \mathcal{A}$ is called irreducible, if the image of the holonomy representation is not contained[This is equivalent to saying that the holonomy group of the connection is precisely $\mathrm{SU}(2)$ and not a proper subgroup.] in a one parameter subgroup of $\mathrm{SU}(2)$. Denote the space of irreducible connections by $\mathcal{A}^*\subset \mathcal{A}$. Consider the quotient $\mathcal{A}^*/\mathcal{G}$ by the gauge transformations and note that $\calM_{\mathrm{ASD}}\subset \mathcal{A}^*/\mathcal{G}$ is a subspace of the quotient. Moreover, it is quite compact and almost defines an element of $H_{2d}(\mathcal{A}^*/\mathcal{G},\mathbb{Z})$.

Let $\mathcal{R}$ denote the space of all Riemannian metrics on $\Sigma$ of class $C^r$ for some $r\geq0$. Freed and Uhlenbeck have shown in <cit.> that for all $k>0$, the moduli space

\[
\calM_\mathcal{R}:=(\mathcal{A}^*/\mathcal{G})\times \mathcal{R}
\]

is in fact a manifold. Moreover, for $k>0$, there exists a subset of $\mathcal{R}'\subset\mathcal{R}$ (elements of $\mathcal{R}'$ are called generic metrics) such that $\calM_{\mathrm{ASD}}$ is a smooth submanifold of $\mathcal{A}^*/\mathcal{G}$ for all $g\in\mathcal{R}'$ and with virtual dimension equal to the expected dimension.

§.§.§ Uhlenbeck compactification

By a combination of two theorems in <cit.>, Uhlenbeck showed that there exists a natural compactification $\overline{\calM}_{\mathrm{ASD}}$ of the moduli space $\calM_{\mathrm{ASD}}$. If we denote the moduli space of anti self-dual connections by $\calM_\mathrm{ASD}^k$ to indicate its dependence of the instanton number $k$, we get that

\begin{equation}
\label{eq:Uhlenbeck_compactification}
\overline{\calM}_\mathrm{ASD}\subset \bigcup_{j=0}^k \calM_{\mathrm{ASD}}^{k-j}\cup\Sym^j(\Sigma).
\end{equation}

This holds for any generic metric $g\in\mathcal{R}'$. The following theorem due to Uhlenbeck will be very important also for later constructions:

Let $A$ be an anti self-dual connection in a principal bundle $P$ over the punctured $4$-ball $B^4\setminus\{0\}$. If the $L^2$-norm of the curvature $F_A$ of $A$ is finite, i.e. if

\[
\int_{B^4\setminus\{0\}}\|F_A\|^2<\infty,
\]

then there exists a gauge in which the bundle $P$ extends to a smooth bundle $\widetilde{P}$ over $B^4$ and the connection $A$ extends to a smooth anti self-dual connection $\widetilde{A}$ in $B^4$.

Note that Theorem <ref> implies in particular that every anti self-dual connection over $\R^4$ with bounded Yang–Mills action functional with respect to the $L^2$-norm can be obtained from an anti self-dual connection over $S^4=\R^4\cup\{\infty\}$.

[bottom color=gray!50, top color=gray!.5] (-1,0) circle (2);

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(1,0) arc (0:-180:2cm and 0.3cm);
[dashed] (1,0) arc (0:180:2cm and 0.3cm);
(a) at (-1,2.5) $\infty$;
(a) at (-1,2) $\bullet$;
(-1,2) arc (90:270:1cm and 2cm);
[dashed] (-1,2) arc (90:-90:1cm and 2cm);
[fill=gray!20,opacity=.3] (-8,-.5)–(-4,.5)–(7,.5)–(3,-.5)–cycle;

Compactification of $\R^4$ to $S^4$.

§.§.§ Construction of invariants

On $(\mathcal{A}^*/\mathcal{G})\times \Sigma$ one can define a principal $\mathrm{SO}(3)$-bundle $\widetilde{P}$ as $(\pi^*P\to \mathcal{A}^*\times \Sigma)/\mathcal{G}$ with $\pi\colon \mathcal{A}\times \Sigma\to \Sigma$ being the projection onto the second factor. Note that an $\mathrm{SO}(3)$-bundle $P\to \Sigma$ is classified by the second Stiefel–Whitney class $w_2(P)\in H^2(\Sigma,\mathbb{Z}/2)$ and the first Pontryagin class $p_1(P)\in H^4(\Sigma,\mathbb{Z})$. When $w_2(P)=0$, then the bundle lifts to an $\mathrm{SU}(2)$-bundle with second Chern class $c_2(P)=-\frac{1}{4}p_1(P)$. However, also if $w_2(P)\not=0$, everything can be done more or less similarly as before (although the orientability of the moduli space will be more complicated[In fact, if $w$ is a class in $H^2(\Sigma,\Z/2)$ and devide the integral lifts $c$ of $w$ into equivalence classes through the relation $c\sim c$ if and only if $\frac{1}{2}(c-c')$ is even. Note that when $\Sigma$ is spin, then there is only one equivalence class and there are two otherwise. The orientation on the moduli space is then induced by the orientation of $H_+^2$ and a choice of equivalence class of the integral lifts of the second Stiefel–Whitney class $w_2$ of the given $\mathrm{SO}(3)$-bundle.]).
Using the first Pontryagin class $p_1(\widetilde{P})\in H^4((\mathcal{A}^*/\mathcal{G})\times \Sigma,\mathbb{Z})$, we can define a map

\begin{align}
\begin{split}
\mu\colon H_2(\Sigma,\mathbb{Z})&\to H^2(\mathcal{A}^*/\mathcal{G},\mathbb{Z}),\\
[C]&\mapsto \int_{C}p_1(\widetilde{P}).
\end{split}
\end{align}

Thus, one can define polynomials $\calD(\Sigma)$ of degree $d$ on $H_2(\Sigma,\mathbb{Z})$ as a map

\begin{align}
\label{eq:Donaldson_polynomials}
\begin{split}
\calD(\Sigma)\colon H_2(\Sigma,\Z)\times\dotsm \times H_2(\Sigma,\Z)&\to \Z,\\
\end{split}
\end{align}

Suppose that $b_+^2>1$ and $k>\frac{3}{2}\left(\frac{b_+^2+1}{2}\right)$. Then the polynomials $\calD(\Sigma)$ are independent of the metric and indeed only depend on the homology classes of $C_1,\ldots, C_d$, hence $\calD(\Sigma)$ define topological invariants of $\Sigma$.

Suppose $\Sigma$ is a simply connected, oriented 4-manifold with $b_+^2$ odd and there is an orientation preserving diffeomorphism between $\Sigma$ and an oriented connected sum of manifolds $\Sigma_1$, $\Sigma_2$ both having $b_+^2>0$. Then $\calD(\Sigma)=0$ for all $k$.

§.§ Field theory formulation

In <cit.>, Witten provided a way of obtaining the Donaldson polynomials by formulating a topological quantum field theory and using methods of functional integrals. In particular, he computed the expectation value of a certain observable by a perturbative expansion for a suitable Lagrangian density.
The action that was used is given in components by

\begin{multline}
\label{eq:Witten_action}
S^\mathrm{DW}_\Sigma=\int_\Sigma\dd^4u\sqrt{g}\tr\bigg(\frac{3}{8}F_{ij}F^{ij}+\frac{1}{2}\phi D_iD^i\lambda-\I\eta D_i\psi^i+\I D_i\psi_j\chi^{ij}-\\-\frac{\I}{8}\phi[\chi_{ij},\chi^{ij}]-\frac{\I}{2}\lambda[\psi_i,\psi^i]-\frac{\I}{2}\phi[\eta,\eta]-\frac{1}{8}[\phi,\lambda]^2\bigg),
\end{multline}

where the fields are defined as in <cit.>. We will refer to this action as the classical Donaldson–Witten (DW) action functional.
Denote by $\Phi$ the collection of all fields of the theory. Then, Donaldson's polynomials (<ref>) can be obtained by using the correlation function with respect to (<ref>) as

\[
\left\langle O_{a_1}\dotsm O_{a_d}\right\rangle:=\int \exp(\I S^\mathrm{DW}_\Sigma(\Phi)/\hbar)\prod_{j=1}^dO_{a_j}(\Phi)\mathscr{D}[\Phi],
\]

for some observables $O_{a_1},\ldots, O_{a_d}$. The observables of interest are given by

\begin{equation}
\label{eq:Witten_observables}
O^{(\gamma)}:=\int_\gamma W_{k_\gamma},
\end{equation}

where $\gamma\in H^{k_\gamma}(\Sigma,\Z)$ and, for $k_\gamma=0,\ldots, 4$, we have $W_0:=\frac{1}{2}\tr(\phi\land \phi)$, $W_1:=\tr(\phi\land \psi)$, $W_2:=\tr(\frac{1}{2}\psi\land\psi+\I\phi\land F)$, $W_3:=\I\tr(\psi\land F)$ and $W_4:=-\frac{1}{2}\tr(F\land F)$. To each $O^{(\gamma)}$, one actually associates a $(4-k_\gamma)$-form $\mu(\gamma)$ on $\overline{\calM}_\mathrm{ASD}$ as in Section <ref>. Explicitly, we get

\begin{align}
\label{eq:correlation_function}
\begin{split}
\left\langle O^{(\gamma_1)}\dotsm O^{(\gamma_d)}\right\rangle&=\int \exp(\I S^\mathrm{DW}_\Sigma(\Phi)/\hbar)\prod_{j=1}^dO^{(\gamma_j)}(\Phi)\mathscr{D}[\Phi]\\
&=\int \exp(\I S^\mathrm{DW}_\Sigma(\Phi)/\hbar)\prod_{j=1}^d\int_{\gamma_j}W_{k_j}(\Phi)\mathscr{D}[\Phi]\\
\end{split}
\end{align}

The integral in (<ref>) is rigorously defined since the differential forms $\mu(\gamma)$ can be obtained from each observable $O^{(\gamma)}$ by integrating out the high energy modes.

§ INSTANTON AND LAGRANGIAN FLOER HOMOLOGY

§.§ Morse homology

Let us recall the construction of Morse homology. Fix a compact, closed manifold $\Sigma$. A function $f\colon \Sigma\to \R$ is called Morse function, if all its critical points are non-degenerate, i.e. for all critical points, the Hessian of $f$ is invertible. In other words, the section $\dd f$ is transverse to the zero section of $T^*\Sigma$.
Since the Hessian is self-adjoint, it has real spectrum and hence we define $\mathrm{ind}_pf$ to be the dimension of the sum of all negative eigenspaces. Moreover, we assume that for any two critical points $p,q$ of $f$ we have that $\mathrm{ind}_pf>\mathrm{ind}_qf$ implies $f(p)>f(q)$. The function $f$ is then said to be self-indexing. If we choose a Riemannian metric on $\Sigma$, we can look at the gradient $\nabla f$ and the corresponding flow equations

\begin{equation}
\label{eq:morse_flow}
\dot{\gamma}(t)=-\nabla_{\gamma(t)}f.
\end{equation}

For two critical points $p,q$ of $f$, we define the moduli space $\calM(p,q)$ of solutions of (<ref>) such that

\begin{equation}
\lim_{t\to -\infty}\gamma(t)=p,\qquad \lim_{t\to+\infty}\gamma(t)=q.
\end{equation}

Note that $\calM(p,q)$ is empty unless $f(p)>f(q)$. In fact, we have the following theorem:

Let $p,q$ be two distinct critical points of a Morse function $f$.
Then $\calM(p,q)$ is a manifold of dimension $\mathrm{ind}_pf-\mathrm{ind}_qf$.

There is a free and proper $\R$-action on $\calM(p,q)$ given by reparametrization $\gamma(t)\mapsto \gamma(t-a)$ for some $a\in \R$. We consider then the quotient

\[
\overline{\calM}(p,q):=\calM(p,q)/\R.
\]

One can check that if $\mathrm{ind}_pf=\mathrm{ind}_qf+1$, then $\overline{\calM}(p,q)$ is a compact, oriented 0-dimensional manifold. Hence, by counting its points by signs, we can deduce

\begin{equation}
\#\overline{\calM}(p,q)\in \Z.
\end{equation}

We can now construct a chain complex as follows. Define the chain groups to be given by

\begin{equation}
CM_k(\Sigma,f):=\bigoplus_{\substack{\text{$p$ critical point of $f$}\\ \mathrm{ind}_pf=k}}\Z\langle p\rangle.
\end{equation}

The boundary operator $\de$ will be constructed by counting flow lines. Namely, for critical points $p$ of $f$ with $\mathrm{ind}_pf=k$ we define the boundary operator by

\begin{equation}
\de p=\sum_{\substack{\text{$q$ critical point of $f$}\\ \mathrm{ind}_qf=k-1}}\#\overline{\calM}(p,q)\cdot q
\end{equation}

One can show (although it is non-trivial) that $\de^2=0$ and thus indeed defines a differential.
Finally, Morse homology is given by the homology of this complex, and we denote it by

\[
HM_\bullet(\Sigma,f):=\ker \de/\mathrm{im}\,\de.
\]

§.§ The (holomorphic) Chern–Simons action functional

An important action functional for further discussions is given by Chern–Simons theory <cit.>, which is a topological field theory on a 3-manifold with many connections to other mathematical theories <cit.>. Let us briefly recall its construction and extension to a holomorphic version which will be important later (see Section <ref>). Let $N$ be a real 3-manifold and consider a vector bundle $E\to N$ with structure group $G$. The curvature $F_A$ of a connection $A\in\calA$ defines a closed 1-form

\begin{equation}
\label{eq:1-form}
a\mapsto \frac{1}{4\pi^2}\int_N\tr(a\land F_A),\qquad a\in\Omega^1(N,\ad E)
\end{equation}

on $\calA$. In fact, we can extend (<ref>) to gauge equivalence classes since this expression is gauge invariant. Fix a base point $A_0$ in the space of gauge equivalence classes. One can show that (<ref>) actually appears as the exterior derivative of a local action functional given by

\begin{equation}
\label{eq:Chern-Simons_action}
S^\mathrm{CS}_N(A):=\int_N\tr\left(\dd_{A_0}a\land a+\frac{2}{3}a\land a\land a\right),\qquad A=A_0+a.
\end{equation}

This is the Chern–Simons action functional. One can check that this is gauge invariant under transformations connected to the identity. Moreover, on the gauge equivalence classes it is well-defined modulo $\Z$. Note also that critical points are given by flat connections. One can extend this picture to Calabi–Yau 3-folds, i.e. smooth compact Kähler 3-folds $X$ with trivial canonical bundle $K_X\cong \calO_X$. Consider now the space $\calA_\mathrm{hol}:=\{\bar\de\text{-operators on a fixed smooth bundle $E\to X$}\}$ and the closed 1-form

\begin{equation}
\label{eq:holomorphic_1-form}
a\mapsto \frac{1}{4\pi^2}\int_X\tr\left(a\land F^{0,2}_A\right)\land \dd\text{vol}_X,\qquad a\in\Omega^{0,1}(X,\ad E),
\end{equation}

where $F^{0,2}_A$ denotes the antiholomorphic curvature (i.e. with respect to $\bar\de$) of a holomorphic connection $A\in\calA_\mathrm{hol}$ and $\dd\text{vol}_X$ denotes the complex volume form on $X$. similarly as before, the 1-form (<ref>) is gauge invariant and thus descends to the space of gauge equivalence classes. Again, fixing a basepoint $A_0\in\calA_\mathrm{hol}$, (<ref>) appears as the exterior derivative of a local holomorphic action functional given by

\begin{equation}
\label{eq:holomorphic_Chern-Simons_action}
S^{\mathrm{hol},\mathrm{CS}}_X(A):=\frac{1}{4\pi^2}\int_X\tr\left(\bar\de_{A_0}a\land a+\frac{2}{3}a\land a\land a\right)\land \dd\text{vol}_X,\qquad A=A_0+a.
\end{equation}

This the holomorphic Chern–Simons action functional. Again, one can check that (<ref>) is gauge invariant with respect to transformations connected to the identity. The critical points of (<ref>) are given by integrable holomorphic structures on the bundle $E$.

§.§ A 4D-3D bulk-boundary correspondence on (infinite) cylinders

Let $\Sigma$ be a 4-manifold. Moreover, let $G=\mathrm{SU}(2)$ and consider a principal $G$-bundle $P\to \Sigma$. Then $P$ has one characteristic class, the second Chern class $c_2(P)\in H^2(\Sigma,\Z)$. Using the Chern–Weil formalism <cit.>, we can identify it with its de Rham cohomology representative in the image of $H^2(\Sigma,\R)$, which is given by

\begin{equation}
c_2(P)=\frac{1}{8\pi^2}\tr(F_A\land F_A),
\end{equation}

for some connection 1-form $A\in \mathcal{A}$ on $P$ and its corresponding curvature 2-form $F_A=\dd A+\frac{1}{2}[A,A]$. Indeed, this cohomology class is independent of the choice of connection and it is a closed 2-form. Note also that its integral over any 4-manifold $\Sigma$ is an integer, since $c_2(P)$ is an integral class.
If $G=\mathrm{SO}(3)$, we get that the corresponding class is given by the first Pontryagin class of the associated bundle $P^{\mathrm{SO}(3)}:=P\times_{\mathrm{SO}(3)}\R^3$, which is given by

\begin{equation}
p_1(P^{\mathrm{SO}(3)})=-\frac{1}{2\pi^2}\tr(F_A\land F_A).
\end{equation}

Note that these classes vanish on 3-manifolds. However, there is a way how we can construct invariants on 3-manifolds by using Chern–Simons theory (see Section <ref>). One can actually check that

\begin{equation}
\label{eq:Chern-Simons_inv}
\left(\int_\Sigma\tr(F_A\land F_A)\right)\Big/8\pi^2\Z
\end{equation}

depends only on the gauge equivalence class of $A\big|_{\de\Sigma}$ and not on $\Sigma$ nor on $A$ in the bulk. If $A$ extends[In fact, one can always extend an $\mathrm{SU}(2)$- or $\mathrm{SO}(3)$-bundle on a closed oriented 3-manifold over some compact oriented 4-manifold. In particular, this follows from the fact that for these groups the cobordism group is given by the third homology group of their classifying spaces.] to a connection $A'$ on an extended bundle $P'\to \Sigma'$, then we can glue $\Sigma$ to $\Sigma'$ along their common boundary $N$ and obtain a new bundle $P''\to \Sigma'':=\Sigma\cup_{N}\Sigma'$. Hence

\begin{equation}
\int_\Sigma\tr(F_{A}\land F_A)-\int_{\Sigma'}\tr(F_{A'}\land F_{A'})=\int_{\Sigma''}\tr(F_{A''}\land F_{A''})\in 8\pi^2\Z.
\end{equation}

Let $B$ be a connection on $P\to N$. Then the Chern–Simons action functional $S^\mathrm{CS}_{N}(B)$ at $B$ is given by (<ref>) where $F_A$ is the curvature of a connection $A$ that is given as an extension of $B$ to some bundle over $\Sigma$.
Let $B_0$ be a fixed connection on $P$ and consider a family of connections $(B_t)_{t\in[0,1]}$ such that $B_1=B$, which we can regard as a connection $A$ on $I\times P\to I\times N$. Then we have

\begin{equation}
S^\mathrm{YM}_{I\times N,B_0}(B)=\int_{I\times N}\|F_A\|^2=-\int_{I\times N}\tr(F_A\land F_A).
\end{equation}

Consider the path $B_t=B_0+tb$ for some $b$ such that $B=B_0+b$, and assume that $B_0$ is a trivial connection by choosing a trivialization for $G=\mathrm{SU}(2)$. Then

\begin{equation}
F_A=\dd(tb)+\frac{t^2}{2}[b\land b]=\dd t\land b+t\dd b+\frac{t^2}{2}[b\land b],
\end{equation}

which gives

\begin{equation}
\tr(F_A\land F_A)=\dd t\land \tr(b\land (2t\dd b+t^2[b\land b])).
\end{equation}

Hence, we get the following chain of equality:

\begin{align}
\begin{split}
S^\mathrm{YM}_{I\times N}(B)&=-\int_{I\times N}\tr(F_A\land F_A)\\
&=-\int_{I\times N}\dd t\land \tr(b\land(2t\dd b+t^2[b\land b]))\\
&=-\int_{N}\tr\left(b\land \dd b+\frac{2}{3}b\land b\land b\right)=-S^\mathrm{CS}_{N,B_0}(b).
\end{split}
\end{align}

Moreover, if $B_0$ does not arise from a trivialization, we get

\begin{equation}
S^\mathrm{YM}_{I\times N}(B)=-\int_{N}\tr\left(2b\land F_{B_0}+b\land \dd b+\frac{2}{3}b\land b\land b\right),
\end{equation}

where $F_{B_0}$ is the curvature of the connection $B_0$.

§.§ Instanton Floer homology

We will restrict ourselves to the case where $G=\mathrm{SU}(2)$ and $N$ is an oriented integral homology 3-sphere[This is a 3-manifold $N$ whose homology groups are the same as for $S^3$. Namely, $H_0(N,\Z)=H_3(N,\Z)\cong \Z$ and $H_1(N,\Z)=H_2(N,\Z)=0$.] endowed with a Riemannian metric. Denote by $\mathcal{A}$ the set of all connections on $P\to N$, which is an affine space over $\Omega^1(N,\ad P)$. Recall that a gauge transformation $g$ is a bundle automorphism of $P$ covering the identity map of $\Sigma$. For such a transformation $g\colon P\to P$, we can construct a new map $\hat g\colon P\to G$ by $g(p)=p\hat g(p)$, with the property $\hat g(ph)=h^{-1}\hat g(p)h$ for all $p\in P$ and $h\in G$. We call the set of all gauge transformations $\mathcal{G}$. The elements of $\mathcal{G}$ are identified with sections of $\Ad P=P\times_{\Ad} G$. We would like to construct a Morse chain complex (as in Section <ref>) over $\mathcal{A}/\mathcal{G}$. Indeed, this can be done since by completing $\mathcal{A}$ and $\mathcal{G}$ with respect to the Sobolev norm topology[For the analytic sides of the construction, one needs the notion of a Sobolev space $L^p_\ell$ of sections of bundles associated to $P$, i.e. sections locally represented by functions with first $\ell$ derivatives in $L^p$. There one can define a norm $\|\enspace\|_{L^p_\ell(A)}$ for any smooth connection $A$, for example, if $p=2$ and $\ell=1$, we have

\[
\|\sigma\|^2_{L^2_1(A)}=\|\dd_A\sigma\|^2_{L^2}+\|\sigma\|^2_{L^2}=\int_\Sigma\dd\mu\left(\| \dd_A\sigma\|^2+\|\sigma\|^2\right).
\]
], the spaces $\mathcal{A}$ and $\mathcal{G}$ have the structure of an infinite-dimensional manifold. Passing to the quotient will produce a manifold by a local slice theorem for the natural action of $\mathcal{G}$ on $\mathcal{A}$, which is given by pulling back 1-forms on $P$, i.e. $A\mapsto g^*A$. Note that the curvature $F_A$ of $A$ would transform to the curvature of $g^*A$ by $F_{g^*A}=\Ad_{\hat g}F_A$.

Denote by $\mathcal{A}^*\subset \mathcal{A}$ the subspace of irreducible connections and consider the tangent space of $\calA^*$ at a reference connection $B$, which is given by

\begin{equation}
T_{B}\mathcal{A}^*\cong\Omega^1(N,\ad P).
\end{equation}

Next, we would like to consider the tangent space at some equivalence class $[B]\in \mathcal{A}^*/\mathcal{G}$. For this purpose, we use the Hodge star $*\colon \Omega^j(N)\to \Omega^{3-j}(N)$ induced by the metric on $N$. Moreover, recall that on $j$-forms we have $*^2=(-1)^{j(3-j)}$. Denoting by $\dd_B\colon \Omega^j(N,\ad P)\to \Omega^{j+1}(N,\ad P)$ the covariant derivative with respect to $B$, we can define its formal adjoint by $\dd^*_B:=-*\dd_{B}*$. Then we can obtain

\begin{equation}
T_{[B]}(\mathcal{A}^*/\mathcal{G})\cong \ker(\dd_B^*).
\end{equation}

The key point for our Morse complex is that we want to consider the Chern–Simons action functional

\[
S^\mathrm{CS}_{N}\colon \mathcal{A}^*/\mathcal{G}\to \R/8\pi^2\Z
\]

to play the role of a Morse function[Of course, this requires the Chern–Simons action functional to actually be Morse, i.e. with critical points being nondegenerate. In particular, one can consider small perturbations $\varepsilon$ (holonomy perturbations) such that $S^\mathrm{CS}+\varepsilon$ is indeed sufficiently nice. Here, $\varepsilon\colon \calA^*/\calG\to \R/8\pi^2\Z$ is some admissible perturbation function.
In particular, the existence of regular values for the perturbed action is connected to the properties of being smooth and its differential to be a Fredholm operator.]. For convenience, we actually want to consider $-S^\mathrm{CS}_{N}$.
Recall from Section <ref> that critical points of the Chern–Simons action functional on $\mathcal{A}^*$ are given by flat connections and on $\mathcal{A}^*/\mathcal{G}$ by gauge equivalence classes of flat connections.

The Floer chains $CF_\bullet(N)$ are given by the $\Z$-module with generators $[B]$ being gauge equivalence classes of flat connections of the trivial $\mathrm{SU}(2)$-bundle over $N$. For two flat connections $B_0,B_1$ consider a path $t\mapsto B_t$ connecting them. The corresponding path of operators has a spectral flow which is defined by pursuing the net number of eigenvalues crossing zero. Denote by $\mathrm{Eig}^{\mp}(B_t)$ the set of eigenvalues that pass along the path of operators $B_t$ from negative to positive and by $\mathrm{Eig}^{\pm}(B_t)$ the set of eigenvalues that pass along the path of operators $B_t$ from positive to negative.
Following <cit.>, we define the spectral flow to be the number

\begin{equation}
\mathrm{sf}(B_0,B_1):=\#\mathrm{Eig}^\mp(B_t)-\#\mathrm{Eig}^\pm(B_t).
\end{equation}

Note that in finite dimensions, this corresponds to the difference of the index of critical points. This will replace the grading of the Floer chains for the infinite-dimensional setting.
Consider the space of connections $A$ on $\R\times P$, which is a bundle over $\R\times N$, satisfying:

* $A$ is anti-self dual, i.e. $F_A=-*F_A$,

* $\lim_{t\to -\infty}[A\vert_{\{t\}\times N}]=:A_-$,

* $\lim_{t\to +\infty}[A\vert_{\{t\}\times N}]=:A_+$,

* $A$ has finite energy, i.e. the curvature $F_A$ has finite $L_2$-norm:

\[
\|F_A\|_2^2=\int_{\R\times N} \|F_A\|^2<\infty.
\]

If we denote the space of such connections by $\widehat{\calM}(A_-,A_+)$, we can define a moduli space

\[
\calM(A_-,A_+):=\widehat{\calM}(A_-,A_+)/\mathcal{G},
\]

where $\mathcal{G}:=\Aut(\R\times P)=\{f\colon \R\times N\to \mathrm{SU}(2)\}$ denotes the gauge transformations. We can define an $\R$-action on $\calM(A_-,A_+)$ by shifting the $t$ variable and define

\[
\overline{\calM}(A_-,A_+):=\calM(A_-,A_+)/\R.
\]

In fact, there exists (see <cit.>) a small (holonomy) perturbation $\varepsilon>0$ of $S^\mathrm{CS}_{N}$ which leads to a moduli space $\calM_\varepsilon(A_-,A_+)$ such that $\overline{\calM}_\varepsilon(A_-,A_+)$ is a smooth oriented manifold with

\[
\dim \overline{\calM}_\varepsilon(A_-,A_+)=\mathrm{sf}(A_-,A_+)-1.
\]

Moreover, if $\dim\overline{\calM}_\varepsilon(A_-,A_+)=0$, then $\overline{\calM}_\varepsilon(A_-,A_+)$ is compact.
Finally, the boundary operator is defined through the counting of instantons as

\[
\de A_-:=\sum_{A_+\in \mathrm{\mathcal{A}_\mathrm{flat}}\atop\mathrm{sf}(A_-,A_+)=1}\,\,\#\overline{\calM}_\varepsilon(A_-,A_+)\cdot A_+,
\]

where $\mathcal{A}_\mathrm{flat}$ denotes the space of flat connections.
One can show that the resulting complex $CF_\bullet(N,\de)$ is actually independent of the metric and the perturbation and that $\de^2=0$. Therefore, we get a well-defined homology theory <cit.>.
The corresponding homology, denoted by $HF_\bullet(N):=\ker \de/\mathrm{im}\, \de$, is called instanton Floer homology[We will sometimes also just call it Floer homology, i.e. dropping the word instanton, whenever it is clear. We mainly consider this type of Floer homology in this paper. There are different types of Floer homology constructions such as e.g. Lagrangian Floer homology (see Section <ref>) which, e.g., plays an important role in the formulation of Kontsevich's homological mirror symmetry conjecture (Conjecture <ref>)].

§.§ Relation to Donaldson polynomials

After Floer defined his homology groups, Donaldson soon realized how they were related to the polynomials he has constructed. Good expositions can be also found in <cit.>. Assume that $\Sigma=\Sigma_1\cup_{N} \Sigma_2$, where $N$ is an oriented homology 3-sphere and $\Sigma_1$ (resp. $\Sigma_2$) is a simply connected 4-manifold with boundary $N$ (resp. $N^\mathrm{opp}$)[Here we denote $N^\mathrm{opp}$ to be $N$ with opposite orientation.]. By the assumption that $b^2_+>0$ for both $\Sigma_1$ and $\Sigma_2$, Donaldson defined polynomials

\begin{align}
\calD(\Sigma_1)\colon H_2(\Sigma_1,\mathbb{Z})\times \dotsm\times H_2(\Sigma_1,\mathbb{Z})&\to (HF_\bullet(N))^*,\\
\calD(\Sigma_2)\colon H_2(\Sigma_2,\mathbb{Z})\times \dotsm \times H_2(\Sigma_2,\mathbb{Z})&\to (HF_\bullet(N^\mathrm{opp}))^*,
\end{align}

that is that the polynomials $\calD$ are valued in the dual of the Floer homology on the boundary. In fact, one can define a pairing $\langle\enspace,\enspace\rangle_{HF}$ between elements of $(HF(N))^*$ and $(HF(N^\mathrm{opp}))^*$

\begin{equation}
\label{eq:HF_pairing}
\langle\enspace,\enspace\rangle_{HF}\colon (HF_j(N))^*\times (HF_{3-j}(N^\mathrm{opp}))^*\to \mathbb{Z}
\end{equation}

by using the fact that $CF_j(N)=CF_{3-j}(N^\mathrm{opp})$ and for irreducible flat connections $A\in CF_j(N)$ and $B\in CF_{j-1}(N)$, that both, $\langle \dd_N A,B\rangle$ and $\langle A,\dd_{N^\mathrm{opp}}B\rangle$, are the number of flow lines from $A$ to $B$ counted with sign. Here we have denoted by $\dd_N$ the de Rham differential on $N$ and by $\dd_{N^\mathrm{opp}}$ the de Rham differential on $N^\mathrm{opp}$.
In fact, the signs are the same and hence these numbers do agree. Note that by defining a cochain complex $CF^j:=\Hom(CF_j,\mathbb{Z})$, we get

\[
HF_j(N)\cong HF^{3-j}(N^\mathrm{opp}).
\]

We have

\[
\calD(\Sigma_1\cup_N\Sigma_2)=\langle \calD(\Sigma_1),\calD(\Sigma_2)\rangle_{HF},
\]

where $\langle\enspace,\enspace\rangle_{HF}$ denotes the pairing as in (<ref>).

Theorem <ref> tells us how Donaldson polynomials glue along boundaries of 4-manifolds. This will be interesting in connection to the perturbative field-theoretic approach of quantum gauge theories on manifolds with boundary which in the cohomological symplectic setting is compatible with cutting and gluing <cit.>. We will see how this result fits into this framework.

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(2.8,-0.3) 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=purple] (6,0) arc (0:360:0.5cm and 1cm);
[color=white, top color=gray!80, bottom color=gray!10.5] (8,0) arc (0:360:0.5cm and 1cm);
[thick, color=purple] (8,0) arc (0:360:0.5cm and 1cm);
[fill=white] (10,-0.28) arc (0:360:0.7cm and 0.3cm);
(9.3,-0.58) arc (-90:0:0.9cm and 0.5cm);
(9.3,-0.58) arc (-90:-180:0.9cm and 0.5cm);
(a) at (5.7,-1.4) $\de\Sigma_1=N$;
(a) at (7.5,1.3) $\de\Sigma_2=N^\mathrm{opp}$;
(a) at (26:2.5) $\Sigma_1$;
(a) at (10.5,1) $\Sigma_2$;

Gluing of two manifolds along a common boundary with opposite orientations. Think of $\Sigma_1$ and $\Sigma_2$ as two 4-manifolds with boundary 3-manifolds $\de\Sigma_1$ and $\de\Sigma_2$, respectively.

§.§ Field theory approach to instanton Floer homology

Based on ideas of Atiyah <cit.>, Witten gave an approach for a quantum field theoretic construction regarding the appearance of Floer homology in Donaldson theory for 4-manifolds with boundary in <cit.>. His construction uses a supersymmetric field theory approach to Morse theory developed in <cit.>. Moreover, he gave a quantum field theoretic interpretation for the gluing of 4-manifolds along a common boundary 3-manifold with pairing ground states on the boundary contained in the instanton Floer homology groups which recovers the result of Theorem <ref>.
Recall that, in general, we are considering expectation values to be given by an integral of the form

\begin{equation}\label{eq:expectation_path_integral}
\langle O\rangle=\int \exp(\I S(\Phi)/\hbar)O(\Phi)\mathscr{D}[\Phi],
\end{equation}

where $S$ is the action of the theory and $\Phi$ denotes the collection of all integration variables. Let $\Sigma$ be the underlying 4-manifold with boundary $\de\Sigma$ and consider the field theory as in Section <ref>. Choosing boundary conditions on $\de\Sigma$ is usually required for the computation of a path integral as in (<ref>). For the 3-manifold $\de\Sigma$ (more precisely, the infinite cylinder $\de\Sigma\times\R$), we can consider the associated state space $\calH_{\de\Sigma}$. Let $\Phi\vert_{\de\Sigma}$ be the restriction of all integration variables to $\de\Sigma$ and note that then $\calH_{\de\Sigma}$ denotes the space of functionals depending on $\Phi\vert_{\de\Sigma}$ and a state corresponds to a functional $\Psi(\Phi\vert_{\de\Sigma})$.
If we consider the state $\Psi(\Phi\vert_{\de\Sigma})$ to define a boundary condition, we can consider the path integral (<ref>) in terms of this condition to be

\begin{equation}\label{eq:expectation_boundary_condition}
\langle O\Psi(\Phi\vert_{\de\Sigma})\rangle=\int\exp(\I S^\mathrm{DW}_\Sigma(\Phi)/\hbar)O(\Phi)\Psi(\Phi\vert_{\de\Sigma})\mathscr{D}[\Phi],
\end{equation}

where $S^\mathrm{DW}_\Sigma$ is defined as in (<ref>) and

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

with $\gamma_j\in H_{k_j}(\Sigma,\Z)$ and $W_{k_j}$ defined as in Section <ref>.
In fact, (<ref>) turns out to be a topological invariant if $\Psi$ represents an instanton Floer cohomology class and it depends only on the cohomology class represented by $\Psi$. The observables are chosen similarly as for the Donaldson polynomials (see Section <ref>). Hence, one obtains the Donaldson polynomials with values in the (dual of the) instanton Floer cohomology groups as in Section <ref>.

§.§ Lagrangian Floer homology

Besides the instanton construction <cit.>, there is another type of Floer homology theory which uses the data of a symplectic manifold <cit.>. We will mainly use the excellent introductory paper <cit.> for this section and we refer to it and the references within for more details and further constructions using Lagrangian Floer homology. We should remark that there the construction is dual to Floer's original construction and thus the grading convention will be reversed (so we should rather speak of cohomology instead of homology, but we decide to keep the original term).
Let $(\Sigma,\omega)$ be a compact symplectic manifold and consider two compact Lagrangian submanifolds $\calL_0,\calL_1\subset \Sigma$. Then we can associate to the pair $(\calL_0,\calL_1)$ of Lagrangians a chain complex $CF(\calL_0,\calL_1)$ which is freely generated by the intersection points of $\calL_0$ and $\calL_1$ together with a differential $\de\colon CF(\calL_0,\calL_1)\to CF(\calL_0,\calL_1)$ such that $\de^2=0$ and thus we can consider its corresponding homology $HF(\calL_0,\calL_1):=\ker \de/\mathrm{im}\, \de$.
Moreover, if there is a Hamiltonian isotopy between two compact Lagrangian submanifolds $\calL_1$ and $\calL_1'$, we get an isomorphism $HF(\calL_0,\calL_1)\cong HF(\calL_0,\calL_1')$ and if there is a Hamiltonian isotopy between $\calL_0$ and $\calL_1$, then $HF(\calL_0,\calL_1)\cong H^\bullet(\calL_0)$. Similarly as before, Lagrangian Floer homology can be formally viewed as an infinite version of Morse homology with respect to the Morse function given by the functional defined on the universal cover of the path space $\mathrm{Path}(\calL_0,\calL_1):=\{\gamma\colon[0,1]\to \Sigma\mid \gamma(0)\in \calL_0,\, \gamma(1)\in \calL_1\}$ which is given by

\[
S(\gamma,[\Gamma])=-\int_\Gamma \omega,
\]

with $\gamma\in \mathrm{Path}(\calL_0,\calL_1)$ and $[\Gamma]$ the equivalence class of a homotopy $\Gamma\colon [0,1]\times[0,1]\to \Sigma$ between $\gamma$ and a fixed base point in the connected component of $\mathrm{Path}(\calL_0,\calL_1)$ containing $\gamma$. Usually, we want that the two Lagrangian submanifolds intersect transversally, such that there is a finite set of intersection points. The Lagrangian Floer chain complex is then given by

\[
CF(\calL_0,\calL_1):=\bigoplus_{p\in \calL_0\cap \calL_1}\Lambda\cdot p,
\]

where $\Lambda:=\{\sum_{i\geq 0}a_iT^{\lambda_i}\mid a_i\in \mathbb{K},\, \lambda_i\in\R,\, \lim_{i\to \infty}=+\infty\}$ denotes the Novikov field for some field $\mathbb{K}$.
Let $J$ be a $\omega$-compatible almost-complex structure on $\Sigma$. We can define the differential $\de$ by counting pseudo-holomorphic strips in $\Sigma$ with boundary contained in $\calL_0$ and $\calL_1$. This is parametrized through the following construction: Let $p,q\in \calL_0\cap \calL_1$. Then the coefficient of $q$ in $\de p$ is given by the moduli space of maps $u\colon \R\times[0,1]\to \Sigma$
such that:

* it solves the Cauchy–Riemann equation

\[
\bar \de_J u:=\frac{\de u}{\de s}+J(u)\frac{\de u}{\de t}=0,
\]

with respect to the boundary conditions

\[
\begin{cases}u(s,0)\in \calL_0\text{ and }u(s,1)\in \calL_1,\,\, \forall s\in \R,\\
\lim_{s\to +\infty}u(s,t)=p,\,\, \lim_{s\to -\infty}u(s,t)=q\end{cases}
\]

* it has finite energy (symplectic area of the strip):

\[
\int u^*\omega=\iint \left\vert \frac{\de u}{\de s}\right\vert^2\, \dd s\dd t<\infty.
\]

Let $\widehat{\calM}(p,q;[u],J)$ denote the moduli space defined through the conditions (1) and (2) above and $\calM(p,q;[u],J)$ the moduli space after taking the quotient by the $\R$-action given by reparametrization (i.e. $u(s,t)\mapsto u(s-a,t)$ for some $a\in\R$). We have denoted by $[u]$ the homotopy class of $u$ in $\pi_2(\Sigma,\calL_0\cap\calL_1)$. Moreover, define the Maslov index of the homotopy class $[u]$ as

\[
\mathrm{ind}([u]):=\mathrm{ind}_\R(D_{\bar\de_J,u})=\dim\ker D_{\bar\de_J,u}-\dim \mathrm{coker}\, D_{\bar\de_J,u},
\]

where $D_{\bar\de_J,u}$ denotes the linearization of $\bar\de_J$ at a given solution $u$. One can show that $D_{\bar \de_J,u}$ is indeed a Fredholm operator and thus one can compute its Fredholm index. In particular, it can be shown that $\widehat{\calM}(p,q;[u],J)$ is a smooth manifold of dimension $\mathrm{ind}([u])$ if all solutions of (1) and (2) are regular, i.e. the operator $D_{\bar\de_J,u}$ is surjective at each point $u\in\widehat{\calM}(p,q;[u],J)$. Thus, $\calM(p,q;[u],J)$ is an oriented 0-dimensional manifold whenever $\mathrm{ind}([u])=1$.
Compactness of the moduli space is given through Gromov's compactness theorem <cit.>. Indeed, Gromov showed that any sequence of $J$-holomorphic curves with uniformly bounded energy admits a subsequence which converges, up to reparametrization, to a nodal tree of $J$-holomorphic curves.
Denote by $\mathrm{LGr}(n)$ the Grassmannian of Lagrangian $n$-planes in the standard symlectic space $(\R^{2n},\omega_0)$ and by $\mathrm{LGr}(T\Sigma)$ the Grassmannian of Lagrangian planes in $T\Sigma$ as an $\mathrm{LGr}(n)$-bundle over $\Sigma$.
The $\Z$-grading of the complex is obtained by ensuring that the difference of the index of a strip depends only on the difference between the degrees of the two generators that it connects and not on its homotopy class[This is actually obstructed by the following two conditions: The first Chern class of $\Sigma$ has to be 2-torsion and the Maslov class $\mu_\calL\in \Hom(\pi_1(\calL),\Z)=H^1(\calL,\Z)$ of $\calL$ vanishes, i.e. $2c_1(T\Sigma)=0$, which allows to lift the Grassmannian $\mathrm{LGr}(T\Sigma)$ to a fiberwise universal cover $\widetilde{\mathrm{LGr}}(T\Sigma)$ given as the Grassmannian of graded Lagrangian planes in $T\Sigma$. In particular, if we have a nowhere vanishing section $\sigma\in \Gamma(\bigwedge^n_\C T^*\Sigma\otimes \bigwedge^n_\C T^*\Sigma)$, the argument of $\sigma$ assigns to each Lagrangian plane $\ell$ a phase $\phi(\ell):=\arg(\sigma\vert_\ell)\in S^1=\R/2\pi\Z$. Thus, one defines a graded lift of $\ell$ to be the choice of a real lift $\tilde{\varphi}(\ell)\in\R$ of $\varphi(\ell)$.
The Maslov class is defined as the obstruction for choosing graded lifts of the tangent spaces to $\calL$, i.e. lifting the section of $\mathrm{LGr}(T\Sigma)$ over $\calL$ given by $p\mapsto T_p\calL$ to a section of the infinite cyclic cover $\widetilde{\mathrm{LGr}}(T\Sigma)$. The Lagrangian $\calL$ together with such a choice of lift is called graded Lagrangian submanifold of $\Sigma$.]. Then we can construct the grading as follows: For all $p\in\calL_0\cap\calL_1$ we can obtain a homotopy class of a path connecting $T_p\calL_0$ and $T_p\calL_1$ in $\mathrm{LGr}(T_p\Sigma)$ by connecting the chosen graded lifts of the tangent spaces at $p$ through a path in $\widetilde{\mathrm{LGr}}(T_p\Sigma)$. Composing this path with the canonical short path from $T_p\calL_0$ and $T_p\calL_1$, denoted by $-\lambda_p$, we get a closed loop in $\mathrm{LGr}(T_p\Sigma)$. We can then define the degree of $p$ to be the Maslov index of this loop. One can check that for any strip $u$ connecting $p$ and $q$, we get

\[
\mathrm{ind}(u)=\deg(q)-\deg(p),
\]

The differential can be then defined as

\[
\de p=\sum_{q\in\calL_0\cap\calL_1\atop \mathrm{ind}([u])=1}\#\calM(p,q;[u],J) T^{\omega([u])}q,
\]

where $\#\calM(p,q;[u],J)\in\Z$ (or $\Z_2$) is the signed (or unsigned) count of pseudo-holomorphic strips connecting $p$ to $q$ in the class $[u]$, and $\omega([u])=\int u^*\omega<\infty$ is the symplectic area of these strips. One can observe that $\de$ is indeed of degree $+1$. The fact that it squares to zero is a non-trivial observation which requires some assumptions. Let us go back to the compactness argument. In our case, we consider $J$-holomorphic strips $u\colon\R\times[0,1]\to \Sigma$ with boundary on Lagrangian submanifolds $\calL_0$ and $\calL_1$. Then we have three situations that can appear: the first situation is called strip breaking (see Figure <ref>) and occurs if energy concentrates at either end $s\to \pm\infty$, i.e. there exists a sequence $(a_n)$ with $a_n\to \pm\infty$ such that the sequence of strips $u_n(s-a_n,t)$ converges to a non-constant limit strip. The second situation is called disk bubbling (see Figure <ref>) and occurs if energy concentrates at a point on the boundary of the strip, i.e. when $t\in\{0,1\}$, where suitable rescalings of $u_n$ converge to a $J$-holomorphic disk in $\Sigma$ with boundary completely contained in either $\calL_0$ or $\calL_1$. the third situation occurs if energy concentrates at an interior point of the strip, where suitable rescalings of $u_n$ converge to a $J$-holomorphic sphere in $\Sigma$.

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5
[rounded corners=12pt,fill=gray!20](0,0)–(1,1)–(3,1)–(4,0);
[rounded corners=12pt, fill=gray!20](4,0)–(3,-1)–(1,-1)–(0,0);
[rounded corners=12pt,fill=gray!20](4,0)–(5,1)–(7,1)–(8,0);
[rounded corners=12pt, fill=gray!20](8,0)–(7,-1)–(5,-1)–(4,0);
(0,0) to (-1,1);
(0,0) to (-1,-1);
(8,0) to (9,1);
(8,0) to (9,-1);
[label=above:$\calL_1$] at (2,1);
[label=above:$\calL_1$] at (6,1);
[label=below:$\calL_0$] at (2,-1);
[label=below:$\calL_0$] at (6,-1);
[Bullet=black, label=above:$q$] at (0,0);
[Bullet=black, label=above:$p$] at (8,0);
[Bullet=black, label=above:$r$] at (4,0);

Example of a broken strip situation.

Bullet/.style=fill=blue,draw,color=#1,circle,minimum size=3pt,scale=0.5
[rounded corners=12pt,fill=gray!20](0,0)–(1,1)–(3,1)–(4,0);
[rounded corners=12pt, fill=gray!20](4,0)–(3,-1)–(1,-1)–(0,0);
(0,0) to (-1,1);
(0,0) to (-1,-1);
(4,0) to (5,1);
(4,0) to (5,-1);
[label=above:$\calL_1$] at (2,1);
[label=below:$\calL_0$] at (2,-1);
[Bullet=black, label=above:$q$] at (0,0);
[Bullet=black, label=above:$p$] at (4,0);
[fill=gray!20] (3.5,1.5) circle (.75);
[Bullet=black] at (3.2,.8);

Example of a disk bubbling situation.

Strip breaking is in fact the ingredient needed for the differential $\de$ to square to zero whenever disk bubbling can be excluded. We can make sure that disk and sphere bubbles do not appear by imposing the condition $[\omega]\cdot \pi_2(\Sigma,\calL_j)=0$ for $j\in\{0,1\}$. There are also other more general ways to avoid disk and sphere bubbling, e.g. to impose a lower bound on the Maslov index by considering the case when the symplectic area of disks and their Maslov index are proportional to each other. One then speaks of monotone Lagrangian submanifolds in monotone symplectic manifolds.

§.§ The Atiyah–Floer conjecture

In <cit.>, Atiyah conjectured that instanton Floer homology should be related to Lagrangian Floer homology in the following way (see also <cit.> for a slightly different approach to the same conjecture using Lagrangian boundary conditions):

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 1$. Then the space of flat $\mathrm{SU}(3)$-connections on $\Sigma_g$, denoted by $\calA_\mathrm{flat}^{\mathrm{SU}(2)}(\Sigma_g)$, up to isomorphism has a symplectic structure. Suppose $N$ is an integral homology sphere with Heegaard splitting along the surface $\Sigma_g$, given by

\[
N=H^1_g\cup_{\Sigma_g} H_g^2,
\]

where $H^i_g$ denotes a handle-body of genus $g$. Then the space $\calA_\mathrm{flat}^{\mathrm{SU}(2)}(H^i_g)$ of flat connections on $\Sigma_g$ which extend to flat connections on $H^i_g$ determines a Lagrangian subspace of $\calA_\mathrm{flat}^{\mathrm{SU}(2)}(\Sigma_g)$. In particular, we have

\[
H_\mathrm{inst}F_\bullet(N)\cong H_\mathrm{Lagr}F_\bullet(\calA_\mathrm{flat}^{\mathrm{SU}(2)}(H^1_g),\calA_\mathrm{flat}^{\mathrm{SU}(2)}(H^2_g)),
\]

where $H_\mathrm{inst}F_\bullet$ denotes the instanton Floer homology and $H_\mathrm{Lagr}F_\bullet$ denotes the Lagrangian Floer homology. Here, $\calA_\mathrm{flat}^{\mathrm{SU}(2)}(H^1_g)$ and $\calA_\mathrm{flat}^{\mathrm{SU}(2)}(H^2_g)$ are considered as Lagrangian submanifolds of $\calA_\mathrm{flat}^{\mathrm{SU}(2)}(\Sigma_g)$. Usually, $H_\mathrm{Lagr}F_\bullet(\calA_\mathrm{flat}^{\mathrm{SU}(2)}(H^1_g),\calA_\mathrm{flat}^{\mathrm{SU}(2)}(H^2_g))$ is called the symplectic instanton Floer homology of $N$.

The Atiyah–Floer conjecture leads to a better understanding of the instanton Floer homology for 3-manifolds with boundary. In particular, for a 4-manifold together with a principal $\mathbb{P}\mathrm{U}(2)$-bundle, where $\mathbb{P}\mathrm{U}$ denotes the projective unitary group, one can define numerical invariants as in (<ref>) due to Donaldson. For a 3-manifold together with a principal $\mathbb{P}\mathrm{U}(2)$-bundle, one can define the instanton Floer homology group as in Section <ref>. For a 2-manifold together with a principal $\mathbb{P}\mathrm{U}(2)$-bundle, we can construct an $A_\infty$-category[An $A_\infty$-category is, roughly speaking, a category whose associativity condition for the composition of morphisms is relaxed in a higher unbounded homotopical way (see e.g. <cit.>). Usually, the morphisms are given by chain complexes as for the meaning of a linear category.]. Moreover, as we have seen in Section <ref>, the Donaldson invariants of a 4-manifold $\Sigma$ with boundary is valued in the instanton Floer homology of $\de\Sigma$. Finally, it is expected that the Floer homology of a 3-manifold $N$ with boundary gives an object in the $A_\infty$-category associated to $\de N$.

In <cit.>, Daemi and Fukaya have proposed a proof for the Atiyah–Floer conjecture. In particular, they construct a different version of Lagrangian Floer homology and translate the Atiyah–Floer conjecture to the equivalent conjecture which states that this new version is actually isomorphic to $H_\mathrm{inst}F_\bullet$. They proposed a solution to this equivalent conjecture by considering a mixture of the moduli space of anti self-dual connections and the moduli space of pseudo-holomorphic curves.
Moreover, they gave a formulation in terms of $A_\infty$-categories in order to state conjectures for stronger versions of some properties for instanton Floer homology of 2- and 3-manifolds.

The $A_\infty$-category whose objects are given by certain Lagrangian submanifolds of a given symplectic manifold $(\Sigma,\omega)$ and morphisms between objects are given by the Lagrangian Floer chain complexes is called the Fukaya category <cit.> and is denoted by $\mathrm{Fuk}(\Sigma,\omega)$ (see also <cit.>). In particular, one could also construct a more general version of this category by using Lagrangian foliations as the objects. Such a construction would be interesting in order to understand the boundary structure when combining with the methods of geometric quantization as for the bounary state space construction in the BV-BFV setting.

Based on the mirror symmetry conjecture ($A$- and $B$-model mirror construction) considered in string theory for Calabi–Yau 3-folds (see e.g. <cit.>), Kontsevich formulated a homological version in terms of equivalences of triangulated categories. For some projective variety $X$, let $D^b\mathrm{Coh}(X)$ denote the derived category of coherent sheaves on $X$. Kontsevich's conjecture is then formulated as follows:
[Homological mirror symmetry<cit.>]

Let $X$ and $Y$ be mirror dual Calabi–Yau varieties. Then there is an equivalence of triangulated categories

\begin{equation}
\label{eq:HMS}
\mathrm{Fuk}(X)\cong D^b\mathrm{Coh}(Y),
\end{equation}

Conjecture <ref> has been proven by various people for different types of mirror varieties (e.g. toric 3-folds, quintic 3-folds). However, a general direct proof still remains open.

§ THE BV-BFV FORMALISM

In this section, we want to recall the most important concepts of <cit.>. Good introductory references for the BV and BV-BFV formalisms are <cit.>. In order to avoid large formulae, we will not always write out the exterior product between differential forms, hence we remark that from now on this is automatically understood.

§.§ BV formalism

Let us start with the BV construction on a closed $d$-dimensional source manifold $\Sigma$.

A BV manifold is a triple $(\calF,\omega,\calS)$, where  $\calF$ is a $\Z$-graded supermanifold[Recall that a supermanifold is a locally ringed space $(\calF,\calO_\calF)$ such that the structure sheaf is locally, for some open set $U\subset \R^d$, given by $C^\infty(U)\otimes \bigwedge^\bullet V^*$, where $V$ is some finite-dimensional real vector space.], $\omega$ an odd symplectic form on $\calF$ of degree $-1$ and $\calS$ an even function on $\calF$ of degree 0 such that

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

where $(\enspace,\enspace)$ denotes the odd Poisson bracket induced by the odd symplectic form $\omega$.

Equation (<ref>) is called Classical Master Equation (CME).

We call $\calF$ the BV space of fields[Usually, the BV space of fields is given by the $(-1)$-shifted cotangent bundle of the BRST space of fields, i.e. $\calF_{\mathrm{BV}}=T^*[-1]\calF_{\mathrm{BRST}}$. The fiber fields are usually called anti-fields. For each field $\Phi$, there is a corresponding notion of anti-field through duality, denoted by $\Phi^\dagger$. Recall that $\gh(\Phi)+\gh(\Phi^\dagger)=-1$. Moreover, if $\calF_\Sigma=\Omega^\bullet(\Sigma)$, then $\deg(\Phi)+\deg(\Phi^\dagger)=\dim\Sigma$.], $\omega$ the BV symplectic form and $\calS$ the BV action (functional). Moreover, $(\enspace,\enspace)$ is often called BV bracket or anti-bracket. In the physics literature, the $\Z$-grading is called ghost number. We will denote the ghost number by $\mathrm{gh}$. When considering differential forms, we will denote the form degree by $\deg$.

The assignment of a source manifold $\Sigma$ to a BV manifold

\[
\]

is called a BV theory.

§.§.§ Quantization
For the quantum picture one considers a canonical second-order differential operator $\Delta$ on half-densities on $\calF$ with the properties

\begin{align}
\Delta^2&=0,\\
\Delta(fg)&=\Delta f g\pm f\Delta g\pm (f,g),\quad \forall f,g\in \mathrm{Dens}^{\frac{1}{2}}(\calF),
\end{align}

where $\mathrm{Dens}^{\frac{1}{2}}(\calF)$ denotes the space of half-densities on $\calF$. The operator $\Delta$ is called BV Laplacian. If $\Phi^i$ and $\Phi^\dagger_i$ denote field and anti-field respectively, we have

\[
\Delta f=\sum_{i}(-1)^{\mathrm{gh}(\Phi^i)+1}f\left\langle \frac{\overleftarrow{\delta}}{\delta \Phi^i},\frac{\overleftarrow{\delta}}{\delta \Phi^\dagger_i}\right\rangle, \quad f\in \mathrm{Dens}^{\frac{1}{2}}(\calF).
\]

We denote by $\frac{\overleftarrow{\delta}}{\delta\Phi}$ and $\frac{\overrightarrow{\delta}}{\delta\Phi}$ the left- and right-derivatives. Namely, we have

\begin{align*}
\frac{\overrightarrow{\delta}}{\delta\Phi^i}f&=(-1)^{\gh(\Phi^i)(\gh(f)+1)}f\frac{\overleftarrow{\delta}}{\delta\Phi^i},\\
\frac{\overrightarrow{\delta}}{\delta\Phi^\dagger_i}f&=(-1)^{(\gh(\Phi^i)+1)(\gh(f)+1)}f\frac{\overleftarrow{\delta}}{\delta\Phi^\dagger_i}.
\end{align*}

In particular, one can show that for any odd symplectic supermanifold $\calF$, there is a supermanifold $\calN$ such that $\calF\cong T^*[1]\calN$. Hence, functions on $\calF$ are given by multivector fields on $\calN$. Moreover, the Berezinian bundle on $\calF$ is given by

\[
\mathrm{Ber}(\calF)\cong \bigwedge^\mathrm{top}T^*\calN\otimes \bigwedge^\mathrm{top}T^*\calN.
\]

Thus, half-densities on $\calF$ are defined by

\[
\mathrm{Dens}^\frac{1}{2}(\calF):=\Gamma\left(\mathrm{Ber}(\calF)^\frac{1}{2}\right).
\]

One can show that there exists a canonical operator $\Delta^\frac{1}{2}_\calF$ on $\mathrm{Dens}^\frac{1}{2}(\calF)$ which squares to zero <cit.>. Then, fixing a non-vanishing, $\Delta^\frac{1}{2}_\calF$-closed reference half-density $\sigma\in\mathrm{Dens}^\frac{1}{2}(\calF)$, one can define a Laplacian on functions on $\calF$ by

\begin{equation}
\label{eq:twisting_BV_Laplacian}
\Delta_\sigma f:=\frac{1}{\sigma}\Delta^\frac{1}{2}_\calF(f\sigma).
\end{equation}

It is easy to check that $\Delta_\sigma$ squares to zero. For convenience, we write $\Delta$ when we actually mean $\Delta_\sigma$ in order to not emphasize on the choice of reference half-density. It is important to mention that this construction needs a suitable regularization in the case where $\calF$ is infinite-dimensional. See also <cit.> for describing the BV Laplacian naturally through a spectral sequence approach of a double complex using the odd symplectic structure.

Let $f,g\in \mathrm{Dens}^{\frac{1}{2}}(\calF)$ be two half-densities on $\calF$. Then

* if $f=\Delta g$ (BV exact), we get that

\[
\int_\calL f=0,
\]

for any Lagrangian submanifold $\calL\subset \calF$.

* if $\Delta f=0$ (BV closed), we get that

\[
\frac{\dd}{\dd t}\int_{\calL_t}f=0,
\]

for any smooth family $(\calL_t)$ of Lagrangian submanifolds of $\calF$.

For the application to quantum field theory, we want to consider[More precisely, we want to consider $f=\exp(\I\calS/\hbar)\rho$, where $\rho$ is some $\Delta$-closed reference half-density on $\calF$.]

\[
\]

The choice of Lagrangian submanifold is in fact equivalent to the choice of gauge-fixing. Thus, the second point of Theorem <ref> tells us that the the gauge-independence condition is encoded in the equation

\[
\Delta\exp(\I\calS/\hbar)=0,
\]

which is equivalent to

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

Equation (<ref>) is called Quantum Master Equation (QME). Note that the QME reduces to the CME for $\hbar\to 0$.

[AKSZ theory<cit.>]

A natural example of BV theories was formulated by Alexandrov, Kontsevich, Schwarz and Zaboronsky in <cit.>. A differential graded symplectic manifold of degree $d$ consists of a graded manifold $\calM$ endowed with an exact symplectic form $\omega=\dd_\calM \alpha$ of degree $d$ and a smooth Hamiltonian function $\Theta$ on $\calM$ of degree $d+1$ satisfying

\[
\{\Theta,\Theta\}=0,
\]

where $\{\enspace,\enspace\}$ is the Poisson bracket induced by $\omega$. Such a triple $(\calM,\omega=\dd_\calM\alpha,\Theta)$ is sometimes also called a Hamiltonian manifold.
A $d$-dimensional AKSZ sigma model with target a Hamiltonian manifold $(\calM,\omega=\dd_\calM\alpha,\Theta)$ of degree $d-1$ is the BV theory, which associates to a $d$-manifold $\Sigma$ the BV manifold $(\calF_\Sigma,\omega_\Sigma,\calS_\Sigma)$, where[This is the infinite-dimensional graded manifold adjoint to the Cartesian product (internal morphisms).]

\begin{equation}
\calF_\Sigma=\Map(T[1]\Sigma,\calM),
\end{equation}

the BV symplectic form $\omega_\Sigma$ is of the form

\begin{equation}
\omega_\Sigma=\int_{T[1]\Sigma}\boldsymbol{\mu}_d\omega_{\mu\nu}\delta \Phi^\mu\delta\Phi^\nu,
\end{equation}

and the BV action is given by

\begin{equation}
\calS_\Sigma=\int_{T[1]\Sigma}\boldsymbol{\mu}_d\left(\alpha_\mu(\Phi)\dd_\Sigma \Phi^\mu+\Theta(\Phi)\right).
\end{equation}

we have denoted by $\Phi\in\calF_\Sigma$ the superfields, by $\omega_{\mu\nu}$ the components of the symplectic form $\omega$, by $\alpha_\mu$ the components of $\alpha$ and $\Phi^\mu$ are the components of $\Phi$ in local coordinates $(u,\theta)$. Moreover, $\boldsymbol{\mu}_d:=\dd^du\dd^d\theta$ denotes the measure on $T[1]\Sigma$ and $\delta$ is the de Rham differential on $\calF_\Sigma$. Indeed, one can check that $\omega_\Sigma$ is symplectic of degree $-1$ and that $\calS_\Sigma$ is of degree 0 satisfying the CME

\[
\]

This is obtained by considering the transgression map

\[
\mathscr{T}\colon \Omega^\bullet(\calM)\to\Omega^\bullet(\Map(T[1]\Sigma,\calM)),
\]

given by the following diagram:

\[
\begin{tikzcd}
\Map(T[1]\Sigma,\calM)\times T[1]\Sigma \arrow[r,"\mathrm{ev}"] \arrow[d,swap,"\pi_1"]
& \calM\arrow[dl,dashed]\\
\Map(T[1]\Sigma,\calM) & \end{tikzcd}
\]

where $\pi_1$ denotes the projection to the first factor and $\mathrm{ev}$ denotes the evaluation map. The transgression map is then defined as $\mathscr{T}:=(\pi_1)_*\mathrm{ev}^*$. Note that the map $(\pi_1)_*$ denotes fiber integration. The $(-1)$-shifted symplectic form on $\Map(T[1]\Sigma,\calM)$ is then given by $\omega_\Sigma=\mathscr{T}(\omega)=(\pi_1)_*\mathrm{ev}^*\omega$.
Many theories of interest are in fact of AKSZ-type such as e.g. Chern–Simons theory <cit.>, the Poisson sigma model <cit.>, Witten's $A$- and $B$-model <cit.> and $2$-dimensional Yang–Mills theory <cit.>.

§.§ BV algebras

We want to recall some notions on BV algebras as in <cit.>, and how it is related to the original BV formalism. A braid algebra $\mathcal{B}r$ is a commutative DG algebra endowed with a Lie bracket $[\enspace,\enspace]$ of degree $+1$ satisfying the Poisson relations

\begin{equation}
[a,bc]=[a,b]c+(-1)^{\vert a\vert(\vert b\vert -1)}b[a,c],\qquad \forall a,b,c\in \mathcal{B}r
\end{equation}

An identity element in $\mathcal{B}r$ is an element $\boldsymbol{1}$ of degree 0 such that it is an identity for the product and $[\boldsymbol{1},\enspace]=0$.
A BV algebra $\mathcal{BV}$ is a commutative DG algebra endowed with an operator $\Delta\colon \mathcal{BV}_\bullet\to \mathcal{BV}_{\bullet+1}$ such that $\Delta^2=0$ and

\begin{align}
\label{app:eq:BV_Laplacian_2}
\begin{split}
\Delta(abc)&=\Delta(ab)c+(-1)^{\vert a\vert}a\Delta(bc)+(-1)^{(\vert a\vert-1)\vert b\vert}b\Delta(ac)\\
&-\Delta abc-(-1)^{\vert a\vert}a\Delta bc-(-1)^{\vert a\vert+\vert b\vert}ab\Delta c,\qquad \forall a,b,c\in \mathcal{BV}.
\end{split}
\end{align}

An identity in $\mathcal{BV}$ is an element $\boldsymbol{1}$ of degree 0 such that it is an identity for the product and $\Delta\boldsymbol{1}=0$. One can show that a BV algebra is in fact a special type of a braid algebra. More precisely, a BV algebra is a braid algebra endowed with an operator $\Delta\colon \mathcal{BV}_\bullet\to \mathcal{BV}_{\bullet+1}$ such that $\Delta^2=0$ and such that the bracket and $\Delta$ are related by

\begin{equation}
[a,b]=(-1)^{\vert a\vert}\Delta(ab)-(-1)^{\vert a\vert}\Delta ab-a\Delta b,\qquad\forall a,b\in \mathcal{BV}.
\end{equation}

Moreover, in a BV algebra we have

\begin{equation}
\Delta([a,b])=[\Delta a,b]+(-1)^{\vert a\vert-1}[a,\Delta b],\qquad \forall a,b\in \mathcal{BV}.
\end{equation}

The motivation for such an algebra structure comes exactly from the approach of the BV formalism in quantum field theory. Let $(\calF,\omega)$ be an odd symplectic (super)manifold.
Let $f\in \calO_\calF$, where $\calO_\calF$ denotes the space of functions on $\calF$, and consider its Hamiltonian vector field $X_f$. One can check that $\calO_\calF$ endowed with the BV anti-bracket

\begin{equation}
\label{eq:odd_Poisson_bracket}
(f,g):=(-1)^{\vert f\vert -1}X_f(g)
\end{equation}

is a braid algebra. Let $\mu\in \Gamma(\mathrm{Ber}(\calF))$ be a nowhere-vanishing section of the Berezinian bundle of $\calF$. As we have seen before, this represents a density which is characterized by the integration map $\int\colon \Gamma_0(\calF,\mathrm{Ber}(\calF))\to \R$, where $\Gamma_0$ denotes sections with compact support. Hence, $\mu$ induces an integration map on functions with compact support

\begin{equation}
\calO_\calF\ni f\mapsto \int_{\calL\subset\calF} f\mu^{1/2},
\end{equation}

for some Lagrangian submanifold $\calL\subset \calF$, where the integral exists.
Then one can define a divergence operator $\Div_\mu X$ by

\begin{equation}
\int_\calF (\Div_\mu X)f\mu=-\int_\calF X(f)\mu.
\end{equation}

For a vector field $X$, define $X^*:=-X-\Div_\mu X$. Then

\begin{equation}
\int_\calF fX(g)\mu=(-1)^{\vert f\vert\vert X\vert}\int_\calF X^*(f)g\mu.
\end{equation}

Moreover, $\Div_\mu(fX)=f\Div_\mu X-(-1)^{\vert f\vert \vert X\vert}X(f)$ and if $\calS\in \calO_\calF$ is an even function, then

\[
\Div_{\exp(\calS)\mu}X=\Div_\mu X+X(\calS).
\]

One can then define the BV Laplacian $\Delta$ to be the odd operator on $\calO_\calF$ given by

\begin{equation}
\label{eq:Delta_div}
\Delta f=\Div_\mu X_f.
\end{equation}

A BV (super)manifold $(\calF,\omega,\mu)$ is then an odd symplectic (super)manifold with Berezinian $\mu$ such that $\Delta^2=0$.

Let $(\calF,\omega,\mu)$ be a BV (super)manifold.

* The algebra $(\calO_\calF,(\enspace,\enspace),\Delta)$ is a BV algebra, where $\Delta$ is given as in (<ref>) and $(\enspace,\enspace)$ is the odd Poisson bracket coming from the odd symplectic form $\omega$ as in (<ref>).

* The Hamiltonian vector field associated to some $f\in \calO_\calF$ is given by the formula $X_f=-[\Delta,f]+\Delta f$, where $[\enspace,\enspace]$ denotes the commutator of operators.

* If $\calS\in \calO_\calF$ and $\Delta_\calS$ is the operator associated to the Berezinian $\exp(\calS)\mu$, then $\Delta_\calS=\Delta-X_\calS$ and $\Delta_\calS^2=X_{\Delta \calS+\frac{1}{2}(\calS,\calS)}$.

Note that point (3) of Proposition <ref> is exactly the case that we have in quantum field theory. Moreover, if

\begin{equation}
\label{eq:QME_BV_alg}
\Delta \calS+\frac{1}{2}(\calS,\calS)=0,
\end{equation}

we get that $\Delta_\calS^2=0$, which ensures a BV algebra structure. Note that here we have set $\I\hbar=1$.

§.§ BFV formalism

We want to continue with the Hamiltonian approach of the BFV formalism for closed source manifolds.

A BFV manifold is a triple $(\calF^\de,\omega^\de,Q^\de)$ such that $\calF^\de$ is a $\Z$-graded supermanifold, $\omega^\de$ is an even symplectic form of degree 0 and $Q^\de$ is a cohomological ($Q^2=0$) and symplectic ($L_Q\omega=0$) vector field on $\calF$ of degree $+1$. Moreover, the Hamiltonian function $\calS^\de$ of $Q^\de$ defined by the equation $\iota_{Q^\de}\omega^\de=\delta\calS^\de$, is required to satisfy the CME

\begin{equation}
\label{eq:CME_2}
\{\calS^\de,\calS^\de\}=0,
\end{equation}

where $\{\enspace,\enspace\}$ denotes the Poisson bracket of degree 0 induced by the even symplectic form $\omega^\de$.

Note that we have denoted by $\delta$ the de Rham differential on $\calF^\de$. Moreover, we can, equivalently, express Equation (<ref>) as $Q^\de\calS^\de=0$. Similarly, for a BV manifold $(\calF,\omega,\calS)$ we can consider the Hamiltonian vector field $Q$ associated to $\calS$ through the equation $\iota_Q\omega=\delta\calS$ (by abuse of notation, we also denote the de Rham differential on $\calF$ by $\delta$). Hence, we can express Equation (<ref>) as $Q\calS=0$.

We call a BFV manifold $(\calF^\de,\omega^\de,Q^\de)$ exact, if $\omega^\de$ is exact, i.e. there is a 1-form $\alpha$ on $\calF^\de$ such that $\omega^\de=\delta\alpha^\de$.

The assignment of a manifold $\Sigma$ to a BFV manifold

\[
\]

is called a BFV theory.

Given an AKSZ theory $(\calF_\Sigma=\Map(T[1]\Sigma,\calM),\omega_\Sigma,\calS_\Sigma)$ as in Example <ref> associated to a manifold with boundary $\Sigma$, we can easily construct its BFV data by restriction to the boundary. The BFV action thus given by

\[
\calS^\de_{\de\Sigma}=\calS_\Sigma\big|_{\de\Sigma}.
\]

Similarly we can obtain the BFV space of boundary fields

\[
\calF^\de_{\de\Sigma}=\Map(T[1]\de\Sigma,\calM)=\calF_{\Sigma}\big|_{\de\Sigma}
\]

and the BFV symplectic form

\[
\omega^\de_{\de\Sigma}=\omega_\Sigma\big|_{\de\Sigma}.
\]

§.§ BV-BFV formalism

Consider now a source manifold $\Sigma$ with boundary $\de\Sigma$.

A BV-BFV manifold over an exact BFV manifold $(\calF^\de,\omega^\de=\delta\alpha^\de,Q^\de)$ is a quintuple $(\calF,\omega,\calS,Q,\pi)$ such that $\calF$ is a $\Z$-graded supermanifold, $\omega$ is an odd symplectic form of degree $-1$ on $\calF$, $\calS$ is an even functional of degree 0, $Q$ is a cohomological, symplectic vector field on $\calF$ and $\pi\colon \calF\to \calF^\de$ a surjective submersion such that

* $\delta\pi Q=Q^\de$,

* $\iota_Q\omega=\delta\calS+\pi^*\alpha^\de$,

where $\delta\pi$ denotes the tangent map of $\pi$. In fact, condition (1) and (2) together imply the modified Classical Master Equation (mCME)

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

The assignment of a source manifold with boundary $\Sigma$ to a BV-BFV manifold $(\calF_\Sigma,\omega_\Sigma,\calS_\Sigma,Q_\Sigma,\pi_\Sigma)$ over an exact BFV manifold $(\calF^\de_{\de\Sigma},\omega^\de_{\de\Sigma},Q^\de_{\de\Sigma})$ is called a BV-BFV theory.

[rounded corners=35pt,top color=gray!50, bottom color=gray!.5](5.5,-1)–(4.2,-1)–(2,-2)–(0,0) – (2,2)–(4.2,1)–(5.5,1);
[fill=white] (3.5,0) arc (0:360:0.7cm and 0.3cm);
(2.8,-0.3) arc (-90:0:0.9cm and 0.5cm);
(2.8,-0.3) 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=purple] (6,0) arc (0:360:0.5cm and 1cm);
(a) at (20:2.5) $\Sigma$;
(a) at (6:6.5) $\de\Sigma$;
[->, bend angle=45, bend left] (8,-2) to (5.5,-1);
(a) at (8.52,-1.8) BFV;
[->, bend angle=45, bend left] (-2,1) to (1,0);
(a) at (-2.4,1.1) BV;

The BV theory is associated to the bulk and the BFV theory to the boundary such that we have a coherent coupling.

§.§.§ Quantization
For the quantization, one chooses a polarization[This is an involutive $\omega^\de$-Lagrangian subbundle of $T\calF^\de$.] $\calP$ on the boundary and assume a symplectic splitting[Actually, this is only needed locally, so one can also allow $\calF$ to be a BV bundle over $\calB^\calP$. A fiber bundle $\calF$ over a base $\calB$ with odd-symplectic fiber $(\calY,\omega)$ is called a BV bundle if the transition maps of $\calF$ are given by locally constant fiberwise symplectomorphisms. An even more general setting is given by a hedgehog fibration where the fibers of the fiber bundle are given by hedgehogs <cit.>.] of the BV space of fields with respect to it

\begin{equation}
\label{eq:splitting1}
\calF=\calB^\calP\times \calY.
\end{equation}

Here $\calB^\calP$ denotes the leaf space for the polarization $\calP$ which we assume to be smooth and $\calY$ is a complement. Note that we can always split the BV space of fields in such a way but it is important to assume that it is symplectic, i.e. that the BV symplectic form $\omega$ is constant on $\calB^\calP$. In fact, we consider $\omega$ to be a weakly nondegenerate[In the finite-dimensional case the BV-BFV formalism is not consistent with the nondegeneracy of $\omega$ on the whole space and thus one has to exactly assume nondegeneracy along the fibers.] 2-form on $\calY$ which extends to the product $\calB^\calP\times\calY$.
For $BF$-like theories[These are perturbations of $BF$ theories.], which includes AKSZ-type theories, one can think of $\calB^\calP$ to be the fields restricted to the boundary and $\calY$ to be the bulk part. Using the fact that the BFV space of boundary fields $(\calF^\de,\omega^\de)$ is a symplectic manifold, we consider a geometric quantization (see e.g. <cit.>) on the boundary to obtain a vector space $\calH^\calP$. In order to take care of regularization, we consider another splitting of the bulk part

\begin{equation}
\label{eq:splitting2}
\calY=\calV\times \calY',
\end{equation}

where $\calV$ denotes the space of residual fields[$\calV$ is often also called the space of low energy fields or zero modes which we assume to be finite-dimensional. The complement $\calY'$ is accordingly often called space of high energy fields or fluctuation fields]. Moreover, let $\mathcal{Z}:=\calB^\calP\times \calV$ denote the bundle of residual fields over $\calB^\calP$. Note that we also assume a splitting of the symplectic manifold $(\calY,\omega)$ into two symplectic manifolds

\begin{align*}
\calY&=\calV\times \calY',\\
\omega&=\omega_\calV+\omega_{\calY'}.
\end{align*}

Gauge-fixing is then equivalent to the choice of a Lagrangian submanifold $\calL$ of $(\calY',\omega_{\calY'})$. Define $\Hat{\calH}^\calP:=\mathrm{Dens}^\frac{1}{2}(\mathcal{Z})=\mathrm{Dens}^\frac{1}{2}(\calB^\calP)\widehat{\otimes}\mathrm{Dens}^{\frac{1}{2}}(\calV)$ and the BV Laplacian $\Hat{\Delta}^\calP:=\id\otimes \Delta_{\calV}$ on $\Hat{\calH}^\calP$. We have denoted by $\Delta_\calV$ the BV Laplacian on $\calV$.
Note that the space $\mathrm{Dens}^\frac{1}{2}(\calB^\calP)$ coincides with the vector space $\calH^\calP$ constructed by geometric quantization on the boundary. Thus we have

\[
\Hat{\calH}^\calP=\calH^\calP\widehat{\otimes}\mathrm{Dens}^\frac{1}{2}(\calV).
\]

Another important assumption is that for any $\Phi\in \mathcal{Z}$, the restriction of the action to $\{\Phi\}\times \calL$ has isolated critical points on $\{\Phi\}\times\calL$.

The state is then defined through the perturbative expansion into Feynman graphs of the BV integral
\begin{equation}
\label{eq:state}
\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}(\Phi)=\int_\calL \exp\left(\I\calS(\Phi)/\hbar\right)\mathscr{D}[\Phi]\in\widehat{\calH}^\calP,\quad \Phi\in\mathcal{Z}.
\end{equation}

If one uses a different choice of residual fields $\calV'$ such that $\calV$ fibers over $\calV'$ as a hedgehog (see <cit.> for the definition of a hedgehog fibration), then the corresponding quantum theories are BV equivalent in the sense of <cit.>. however, there is a minimal choice for the residual fields for which the assumption in Remark <ref> is satisfied by a good choice of Lagrangian submanifold $\calL$. For the case of abelian $BF$ theory, the minimal space of residual fields is given by the de Rham cohomology of the underlying source manifold.

If $\Delta\calS=0$, which we usually want to assume, the QME is replaced by the modified QME (mQME)

\begin{equation}
\label{eq:mQME_1}
\big(\hbar^2\Delta+\Omega^\calP\big)\exp\left(\I\calS/\hbar\right)=0,
\end{equation}

where, for local coordinates $(b_i)\in\calB^\calP$, we have that

\[
\Omega^\calP:=\calS^\de\left(b_i,-\I\hbar\frac{\delta}{\delta b_i}\right)
\]

is the standard ordering quantization of $\calS^\de$.
If $\calS$ depends on $\hbar$ and (or) $\Delta\calS\not=0$, we get the mQME from the assumption of the QME in the bulk by defining $\calS^\de_\hbar:=\calS^\de+O(\hbar)$ via the equation

\[
\pi^*\calS^\de_\hbar=\frac{1}{2}(\calS,\calS)-\I\hbar\Delta\calS
\]

and setting $\Omega^\calP$ to be the standard ordering quantization of $\calS^\de_\hbar$.
Note that the operator $\hbar^2\Delta+\Omega^\calP$ is of order $+1$.
Moreover, we assume that the operator $\hbar^2\Delta+\Omega^\calP$ squares to zero in order to have a well-defined BV cohomology. In the finite-dimensional setting we automatically get that $\Hat{\mathsf{Z}}$ solves the mQME:

\begin{equation}
\label{eq:mQME_2}
\big(\hbar^2\Delta_\calV+\Omega^\calP\big)\Hat{\mathsf{Z}}=0.
\end{equation}

In the infinite-dimensional setting, where we have to compute the Feynman graphs instead of an integral, the mQME is only expected to hold and requires a separate checking.

§.§ Gluing of BV-BFV partition functions

Let $\Sigma$ be a $d$-manifold and consider two $d$-manifolds $\Sigma_1,\Sigma_2$ such that $\Sigma=\Sigma_1\cup_{N}\Sigma_2$ where $N$ is the $(d-1)$-manifold which is identified with the common boundaries $\de\Sigma_1$ and $\de\Sigma_2$ (see Figure <ref>). Then in general, following Atiyah's TQFT axioms <cit.>, we can define the glued partition function by the pairing

\begin{equation}
\label{eq:gluing_general}
\mathsf{Z}_{\Sigma}=\int_N \mathsf{Z}_{\Sigma_1,\de\Sigma_1}\mathsf{Z}_{\Sigma_2,\de\Sigma_2}.
\end{equation}

[rounded corners=35pt,top color=gray!50, bottom color=gray!.5](6,-1)–(4.2,-1)–(2,-2)–(0,0)–(2,2)–(4.2,1)–(7,1)–(9.2,2)–(11,0)
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(7.3,-0.58) arc (-90:-180:0.9cm and 0.5cm);
(a) at (-13:5.8) $N$;
(a) at (26:2.5) $\Sigma_1$;
(a) at (8.5,1) $\Sigma_2$;

Gluing of $\Sigma_1$ and $\Sigma_2$ along $N$.

We want to see how this is adapted to the BV-BFV partition function.
Let $(\calY,\omega)$ be a direct product of two odd symplectic manifolds $(\calY',\omega')$ and $(\calY'',\omega'')$, i.e. $\calY=\calY'\times\calY''$, $\omega=\omega'+\omega''$. Then the space of half-densities on $\calY$ factorizes as

\[
\mathrm{Dens}^\frac{1}{2}(\calY)=\mathrm{Dens}^\frac{1}{2}(\calY')\widehat{\otimes}\mathrm{Dens}^\frac{1}{2}(\calY'').
\]

If we consider BV integration on the second factor for some Lagrangian submanifold $\calL\subset \calY''$, we can define a pushforward map on half-densities

\[
\int_\calL\colon \mathrm{Dens}^\frac{1}{2}(\calY)\xrightarrow{\id\otimes \int_\calL}\mathrm{Dens}^\frac{1}{2}(\calY').
\]

If we consider the space of fields $\calF=\calB\times\calY$ given by a product as in Section <ref>, and assuming that $\calY$ splits into a product of two odd symplectic manifolds $(\calV,\omega_\calV)$ and $(\calY',\omega_{\calY'})$, we can consider a version of the BV pushforward in terms of families over $\calB$ by

\[
\int_\calL\colon \mathrm{Dens}^\frac{1}{2}(\calF)\xrightarrow{\id\otimes \int_\calL}\mathrm{Dens}^\frac{1}{2}(\mathcal{Z}),
\]

where $\mathcal{Z}=\calB^\calP\times\calV$ for some chosen polarization $\calP$ and $\calL\subset \calY'$ is a Lagrangian submanifold. In particular, we have the coboundary operator $\hbar^2\Delta_\calV+\Omega^\calP$ on $\mathcal{Z}$, where $\Delta_\calV$ is the canonical BV Laplacian on $\calV$. In fact, the BV-BFV partition function is then defined through the family version of the BV pushforward.

The gluing of BV-BFV partition functions as in (<ref>) is given by

\begin{equation}
\label{eq:BV-BFV_gluing}
\mathsf{Z}^{\scriptscriptstyle\mathrm{BV}}_{\Sigma}=\int_{\mathcal{L}} \left\langle\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_1,\de\Sigma_1},\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_2,\de\Sigma_2}\right\rangle_N\in\mathrm{Dens}^\frac{1}{2}(\calV_\Sigma),
\end{equation}

where $\int_\mathcal{L}$ denotes the BV pushforward with respect to the map

\[
\calV_{\Sigma_1}\times\calV_{\Sigma_2}\to \calV_\Sigma,
\]

for some Lagrangian submanifold $\calL$ in the bulk complement of $\calV_\Sigma$ and $\langle\enspace,\enspace\rangle_N$ is the pairing in $\calH^\calP_N$.
Note that the product of the space of residual fields $\calV_{\Sigma_1}\times\calV_{\Sigma_2}$ is a hedgehog fibration as in <cit.>. The BV-BFV partition function depends only on residual fields when $\Sigma$ is closed. If we assume that the space of residual fields $\calV_\Sigma$ is finite-dimensional, which is the case for many theories of interest including AKSZ theories, we can compute the number valued partition function by integrating out the residual fields $\int_{\calV_\Sigma}\mathsf{Z}^{\scriptscriptstyle\mathrm{BV}}_{\Sigma}\in\C$.
If the glued manifold $\Sigma$ has boundary itself (see Figure <ref>), the BV-BFV partition function depends on the boundary fields $\calB^\calP_{\de\Sigma}$. In particular, we get

\[
\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}=\int_{\mathcal{L}} \left\langle\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_1,\de\Sigma_1},\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_2,\de\Sigma_2}\right\rangle_N\in \calH^\calP_{\de\Sigma}\widehat{\otimes}\mathrm{Dens}^\frac{1}{2}(\calV_\Sigma).
\]

[rounded corners=35pt,top color=gray!50, bottom color=gray!.5](9.5,-1)–(7,-2)–(4.2,-1)–(2,-2)–(0,0)–(2,2)–(4.2,1)–(7,2)–(9.5,1);
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(a) at (20:2.5) $\Sigma_1$;
(a) at (6:6.5) $\Sigma_2$;
(a) at (-8:10) $\partial \Sigma$;

Example of the case when $\Sigma=\Sigma_1\cup\Sigma_2$ has boundary.

[rounded corners=35pt,top color=gray!50, bottom color=gray!.5](9.5,-1)–(7,-2)–(4.2,-1)–(2,-2)–(0,0)–(2,2)–(4.2,1)–(7,2)–(9.5,1);
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[thick, color=purple] (11.5,0) arc (0:360:0.5cm and 1cm);
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(7.3,-0.58) arc (-90:0:0.9cm and 0.5cm);
(7.3,-0.58) arc (-90:-180:0.9cm and 0.5cm);
(a) at (20:2.5) $\Sigma_1$;
(a) at (6:6.5) $\Sigma_2$;
(a) at (-8:10) $\partial \Sigma$;

Closing the manifold $\Sigma$.

§.§ Example: abelian $BF$ theory

Let $\Sigma$ be a $d$-manifold with compact boundary $\de\Sigma$. The BV space of fields is then given by $\calF_\Sigma=\Omega^\bullet(\Sigma)[1]\oplus\Omega^\bullet(\Sigma)[d-2]$. Denote a field in $\calF_\Sigma$ by $\mathbf{X}\oplus\mathbf{Y}$, where $\mathbf{X}\in \Omega^\bullet(\Sigma)[1]$ and $\mathbf{Y}\in \Omega^\bullet(\Sigma)[d-2]$. The BV symplectic form is then given by

\[
\omega_\Sigma=\int_\Sigma\sum_i\delta\mathbf{Y}_i\delta\mathbf{X}^i,
\]

the BV action by

\[
\calS_\Sigma=\int_\Sigma\sum_i\mathbf{Y}_i\dd_\Sigma\mathbf{X}^i,
\]

and the cohomological vector field by

\[
\]

where $\delta$ denotes the de Rham differential on $\calF_\Sigma$ and $\dd_\Sigma$ the one on $\Sigma$.
The exact BFV manifold $(\calF^\de_{\de\Sigma},\omega^\de_{\de\Sigma}=\delta\alpha^\de_{\de\Sigma},Q^\de_{\de\Sigma})$ assigned to the boundary $\de\Sigma$ is given by the BFV space of fields $\calF^\de_{\de\Sigma}=\Omega^\bullet(\de\Sigma)[1]\oplus\Omega^\bullet(\de\Sigma)[d-2]$, the primitive 1-form

\[
\alpha^\de_{\de\Sigma}=(-1)^d\int_{\de\Sigma}\sum_i\mathds{Y}_i\delta\mathds{X}^i,
\]

the BFV boundary action

\[
\calS^\de_{\de\Sigma}=\int_{\de\Sigma}\sum_i\mathds{Y}_i\dd_{\de\Sigma}\mathds{X}^i,
\]

and cohomological vector field

\[
\]

We have denoted by $\mathds{X}\oplus\mathds{Y}=i_\Sigma^*(\mathbf{X}\oplus\mathbf{Y})\in \calF^\de_{\de\Sigma}$ for $\mathbf{X}\oplus\mathbf{Y}\in \calF_\Sigma$, where $i_\Sigma\colon \de\Sigma\hookrightarrow \Sigma$ denotes the inclusion. The surjective submersion $\pi_\Sigma\colon \calF_\Sigma\to \calF^\de_{\de\Sigma}$ is given by restricting to the boundary, i.e. $\pi_\Sigma=i_\Sigma^*$.

Consider the case where $\de\Sigma$ is given by the disjoint union of two compact manifolds $\de_1\Sigma$ and $\de_2\Sigma$ such that we can consider a splitting $\calF^\de_{\de\Sigma}=\calF^{\de}_{\de_1\Sigma}\times \calF^\de_{\de_2\Sigma}$. Then we can consider a polarization $\calP$ on $\calF^\de_{\de\Sigma}$ which is given by a direct product of polarizations on each factor. One can choose the convenient $\frac{\delta}{\delta\mathds{Y}}$-polarization on $\de_1\Sigma$ and identify the quotient leaf space with $\calB_{\de_1\Sigma}\cong\Omega^\bullet(\de_1\Sigma)[1]$ whose coordinates are given by the $\mathds{X}$ fields. On $\de_2\Sigma$ one can choose the convenient $\frac{\delta}{\delta\mathds{X}}$-polarization and similarly as before identify the quotient leaf space with $\calB_{\de_2\Sigma}\cong\Omega^\bullet(\de_2\Sigma)[d-2]$ whose coordinates are given by the $\mathds{Y}$ fields. The leaf space associated to the boundary polarization is then given by $\calB^\calP_{\de\Sigma}=\calB_{\de_1\Sigma}\times \calB_{\de_2\Sigma}$.

[rounded corners=35pt,top color=gray!50, bottom color=gray!.5](0,0)–(2,1)–(5,-1)–(2,-3)–(0,-2);
[fill=white] (3,-1) arc (0:360:0.7cm and 0.3cm);
(2.3,-1.3) arc (-90:0:0.9cm and 0.5cm);
(2.3,-1.3) arc (-90:-180:0.9cm and 0.5cm);
[thick, color=red, top color=gray!80, bottom color=gray!10.5] (0,-1) ellipse (.5cm and 1cm);
[thick, color=blue, top color=gray!80, bottom color=gray!10.5] (4.2,-1) ellipse (.25cm and .5cm);
(Y) at (-1,-1) $\frac{\delta}{\delta\mathds{Y}}$;
(X) at (5,-1) $\frac{\delta}{\delta\mathds{X}}$;
(1) at (-1,0) $\de_1\Sigma$;
(2) at (5,0) $\de_2\Sigma$;
(2) at (2,0) $\Sigma$;

Example of $\Sigma$ with two boundaries $\de_1\Sigma$ and $\de_2\Sigma$. On $\de_1\Sigma$ we choose the $\frac{\delta}{\delta\mathds{Y}}$-polarization (i.e. the $\mathds{X}$-representation) and on $\de_2\Sigma$ we choose the $\frac{\delta}{\delta\mathds{X}}$-polarization (i.e. the $\mathds{Y}$-representation).

The case when we have more than two boundary components is illustrated and explained in Figure <ref>. The case for boundary components with mixed polarization is explained in Figure <ref>. However, we will not consider this case in this paper[This case was studied for the Poisson sigma model (2-dimensional AKSZ theory) in <cit.> for the quantization of the relational symplectic groupoid. Another (a bit different) approach was considered for 2-dimensional Yang–Mills theory in <cit.>.].

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[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);
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[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);
[label=above:$\Sigma$] at (7,-1);

Example of a manifold with three boundary components where two components have the $\mathds{X}$-representation and one component has the $\mathds{Y}$-representation. Note that in this case we have $\de_1\Sigma=\de_1^{(1)}\Sigma\cup \de_1^{(2)}\Sigma$. In general we can have $r_1$ boundary components in the $\mathds{X}$-representation and $r_2$ boundary components in the $\mathds{Y}$-representation. In this case we have $\de_1\Sigma=\bigcup_{k=1}^{r_1}\de_1^{(k)}\Sigma$ and $\de_2\Sigma=\bigcup_{k=1}^{r_2}\de_2^{(k)}\Sigma$.

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[Bullet=black] at (0,0);
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[label=right:$\mathds{Y}$] at (-0.5,1.5);
[label=left:$\mathds{X}$] at (-3.5,1.5);
[label=left:$\mathds{Y}$] at (-3.5,-1.5);
[label=right:$\mathds{X}$] at (-0.5,-1.5);

Example of a boundary component with mixed boundary polarization.

In particular, we want to assume for the leaf space $\calB^\calP_{\de\Sigma}$ the property that for a suitably chosen local functional $f^\calP_{\de\Sigma}$, the restriction of the adapted BFV 1-form $\alpha^{\de,\calP}_{\de\Sigma}:=\alpha^\de_{\de\Sigma}-\delta f^\calP_{\de\Sigma}$ to the fibers of the polarization $\calP$ vanishes. In this case we can identify the space of boundary states $\calH^\calP_{\de\Sigma}$ with $\mathrm{Dens}^\frac{1}{2}(\calB^\calP_{\de\Sigma})$ by multiplying with $\exp(\I f^\calP_{\de\Sigma}/\hbar)$. Moreover, we can then modify the BV action $\calS_\Sigma$ to $\calS^\calP_\Sigma:=\calS_\Sigma+\pi^*_\Sigma f^\calP_{\de\Sigma}$. In our case, we consider the functional

\[
\]

the adapted BFV 1-form

\[
\alpha^{\de,\calP}_{\de\Sigma}=(-1)^d\int_{\de_1\Sigma}\sum_i\mathds{Y}_i\delta\mathds{X}^i-\int_{\de_2\Sigma}\sum_i\delta\mathds{Y}_i\mathds{X}^i
\]

and the modified BV action

\[
\calS^\calP_\Sigma=\int_\Sigma \sum_i\mathds{Y}_i\dd_{\de\Sigma}\mathds{X}+(-1)^{d-1}\int_{\de_2\Sigma}\sum_i\mathds{Y}_i\mathds{X}^i.
\]

Denote by $\widetilde{\mathds{X}}\oplus\widetilde{\mathds{Y}}:=\pi^*_\Sigma(\mathds{X}\oplus\mathds{Y})$ an extension of $\mathds{X}\oplus\mathds{Y}$ to $\calF_\Sigma$ and denote by $\widehat{\mathbf{X}}\oplus\widehat{\mathbf{Y}}$ the complement such that

\begin{align*}
\mathbf{X}&=\widetilde{\mathds{X}}+\widehat{\mathbf{X}},\\
\mathbf{Y}&=\widetilde{\mathds{Y}}+\widehat{\mathbf{Y}}.
\end{align*}

Note that we require that $i_1^*\widehat{\mathbf{X}}=0$ and $i_2^*\widehat{\mathbf{Y}}=0$, where $i_j\colon \de_j\Sigma\hookrightarrow \Sigma$ denotes the inclusion of the corresponding boundary components for $j=1,2$. Let us choose this section of the bundle $\calF_\Sigma\to \calB^\calP_{\de\Sigma}$. The modified BV action is then given by

\begin{multline*}
\calS^\calP_\Sigma=\int_\Sigma\sum_i\left(\widetilde{\mathds{Y}}_i\dd_{\de\Sigma}\widetilde{\mathds{X}}^i+\widetilde{\mathds{Y}}_i\dd_{\Sigma}\widehat{\mathbf{X}}^i+\widehat{\mathbf{Y}}_i\dd_{\de\Sigma}\widetilde{\mathds{X}}^i+\widehat{\mathbf{Y}}_i\dd_\Sigma\widehat{\mathbf{X}}^i\right)\\
\end{multline*}

Consider the last bulk term which we denote by $\widehat{\calS}_\Sigma:=\int_\Sigma\sum_i\widehat{\mathbf{Y}}_i\dd_\Sigma\widehat{\mathbf{X}}^i$. Note that, by the vanishing boundary conditions, we get the critical points defined by the equations $\dd_\Sigma\widehat{\mathbf{X}}=\dd_\Sigma\widehat{\mathbf{Y}}=0$. Moreover, consider the subcomplex of the de Rham complex $\Omega^\bullet(\Sigma)$ given by

\[
\Omega^\bullet_{\mathrm{D}j}(\Sigma):=\{\gamma\in\Omega^\bullet(\Sigma)\mid i^*_j\gamma=0\},
\]

where D stands for Dirichlet. The residual fields are then given by

\[
\calV_\Sigma=H^\bullet_{\mathrm{D}1}(\Sigma)[1]\oplus H^\bullet_{\mathrm{D}2}(\Sigma)[d-2],
\]

which is a finite-dimensional BV manifold. Note that we have a natural identification with relative cohomology $H^\bullet_{\mathrm{D}j}(\Sigma)\cong H^\bullet(\Sigma,\de_j\Sigma)$ for $j=1,2$.
Hence, we can define a canonical BV Laplacian by choosing coordinates $(z^j,z^\dagger_j)$ on $\calV_\Sigma$. Choosing a basis $([\kappa_j])$ of $H^\bullet_{\mathrm{D}1}(\Sigma)$ with representative $\kappa_j\in \Omega^\bullet_{\mathrm{D}1}(\Sigma)$ and the corresponding dual basis $([\kappa^j])$ of $H^\bullet_{\mathrm{D}2}(\Sigma)$ with representative $\kappa^j\in \Omega^\bullet_{\mathrm{D}2}(\Sigma)$ satisfying the relation $\int_\Sigma\kappa^i\kappa_j=\delta^i_j$, we can write the residual fields as

\begin{equation}
\mathsf{x}=\sum_jz^j\kappa_j,\qquad\mathsf{y}=\sum_jz^\dagger_j\kappa^j.
\end{equation}

The BV Laplacian on $\calV_\Sigma$ is given by

\[
\Delta_{\calV_\Sigma}=\sum_j(-1)^{(d-1)\gh(z^j)+1}\frac{\de^2}{\de z^i\de z^\dagger_j}
\]

and the BV symplectic form on $\calV_\Sigma$ is given by

\[
\omega_{\calV_\Sigma}=\sum_j(-1)^{(d-1)\gh(z^j)+1}\delta z^\dagger_j\delta z^j.
\]

Note that $\gh(z^j)=1-\gh(\kappa_j)$ and $\gh(z^\dagger_j)=\gh(\kappa_j)-2$. Consider another splitting of the fields

\begin{align*}
\widehat{\mathbf{X}}&=\mathsf{x}+\mathscr{X},\\
\widehat{\mathbf{Y}}&=\mathsf{y}+\mathscr{Y},
\end{align*}

where $\mathscr{X}\oplus\mathscr{Y}$ denotes the corresponding complement of the field $\mathsf{x}\oplus \mathsf{y}\in \calV_\Sigma$ as elements of a symplectic complement $\calY'_\Sigma$ of $\calV_\Sigma$. Note that $i_1^*\mathscr{X}=i_2^*\mathscr{Y}=0$. Using this, we can see that $\widehat{\calS}_\Sigma=\int_\Sigma\mathscr{Y}\dd_\Sigma\mathscr{X}$ which can be regarded as a quadratic function on $\Omega^\bullet_{\mathrm{D}1}(\Sigma)[1]\oplus\Omega^\bullet_{\mathrm{D}2}(\Sigma)[d-2]$ and has critical points given by closed forms. One can now choose a Lagrangian submanifold $\calL$ of $\calY'_\Sigma$ where $\widehat{\calS}_\Sigma$ has isolated critical points at the origin, which is to say that the de Rham differential $\dd$ has trivial kernel. This can be done by Hodge theory for manifolds with boundary. Consider the Hodge star operator $*$ for a chosen metric on $\Sigma$ with product structure near the boundary, i.e. there is a diffeomorphism $\varphi\colon \de\Sigma\supset U\to \de\Sigma\times [0,\varepsilon)$ for some $\varepsilon>0$ such that the restriction of $\varphi$ to $\de\Sigma$ is the identity on $\de\Sigma$ and the metric on $\Sigma$ restricted to the neighborhood $U$ has the form $\varphi^*(g_{\de\Sigma}+\dd t^2)$, where $g_{\de\Sigma}$ denotes the metric on the boundary and $t\in[0,\varepsilon)$ the coordinate on $\de\Sigma\times [0,\varepsilon)$. One can define the Lagrangian submanifold

\[
\calL=(\dd^*\Omega^{\bullet+1}_{\mathrm{N}2}(\Sigma)\cap\Omega^\bullet_{\mathrm{D}1}(\Sigma))[1]\oplus (\dd^*\Omega^{\bullet+1}_{\mathrm{N}1}(\Sigma)\cap\Omega^\bullet_{\mathrm{D}2}(\Sigma))[d-2],
\]

where $\dd^*:=*\dd*$ and

\[
\Omega^\bullet_{\mathrm{N}j}(\Sigma):=\{\gamma\in\Omega^\bullet(\Sigma)\mid i^*_j*\gamma=0\}
\]

is the space of Neumann forms relative to $\de_j\Sigma$. One can check that the restriction of $\widehat{\calS}_\Sigma$ to $\calL$ is nondegenerate. Moreover, one can show (see <cit.> for more details) that $\calL$ is is indeed Lagrangian in the symplectic complement $\calY'_\Sigma$ of $\calV_\Sigma$. One can then construct a propagator $\mathscr{P}$ as the integral of the chain contraction $K$ of $\Omega^\bullet_{\mathrm{D}1}(\Sigma)$ onto $H^\bullet_{\mathrm{D}1}(\Sigma)$. The gauge-fixing Lagrangian is then given as

\[
\calL=\mathrm{im}(K)[1]\oplus\mathrm{im}(K^*)[d-2].
\]

This can be done by choosing a Hodge chain contraction $K\colon \Omega^\bullet_{\mathrm{D}1}(\Sigma)\to \Omega^{\bullet-1}_{\mathrm{D}1}(\Sigma)$. We call a differential form $\beta\in \Omega^\bullet(\Sigma)$ ultra-Dirichlet relative to $\de_j\Sigma$ if the pullbacks to $\de_j\Sigma$ of all even normal derivatives of $\beta$ and pullbacks of all odd normal derivatives of $*\beta$ vanish. We call a differential form $\beta\in\Omega^\bullet(\Sigma)$ ultra-Neumann relative to $\de_j\Sigma$ if pullbacks to $\de_j\Sigma$ of all even normal derivatives of $*\beta$ and pullbacks of all odd normal derivatives of $\beta$ vanish. Denote by $\Omega^\bullet_{\widehat{\mathrm{D}}j}(\Sigma)$ and $\Omega^\bullet_{\widehat{\mathrm{N}}j}(\Sigma)$ the space of ultra-Dirichlet and ultra-Neumann forms, respectively. We call a differential form $\beta\in \Omega^\bullet(\Sigma)$ ultra-harmonic if it is closed with respect to $\dd$ and $\dd^*$. Denote by $\widehat{\mathrm{Harm}}^\bullet(\Sigma)$ the space of ultra-harmonic forms on $\Sigma$. Note that $\Omega^\bullet_{\widehat{\mathrm{D}}j}(\Sigma)$ and $\Omega^\bullet_{\widehat{\mathrm{N}}j}(\Sigma)$ are both subcomplexes of $\Omega^\bullet(\Sigma)$ with respect to $\dd$ and $\dd^*$, respectively. One can easily see that ultra-harmonic Dirichlet forms are ultra-Dirichlet and ultra-harmonic Neumann forms are ultra-Neumann.
Let $\Delta_\mathrm{Hodge}:=\dd\dd^*+\dd^*\dd$ denote the Hodge Laplacian.
The Hodge chain contraction is then given by $K=\dd^*/(\Delta_\mathrm{Hodge}+\pi_{\mathrm{Harm}})$, where $\pi_\mathrm{Harm}$ denotes the projection to (ultra-)harmonic forms. The propagator $\mathscr{P}$ is then a smooth form on the compactified configuration space $\overline{\mathrm{Conf}_2(\Sigma)}:=\overline{\{(u_1,u_2)\in\Sigma\times\Sigma\mid u_1\not=u_2\}}$. Moreover, consider the space $\mathfrak{D}:=\{(u_1,u_2)\in (\de_1\Sigma\times \Sigma)\cup(\de_2\Sigma\times \Sigma)\mid u_1\not=u_2\}$ (see Appendix <ref> for more on configuration spaces on manifolds with boundary). Then we get that $\mathscr{P}\in\Omega^{d-1}(\overline{\mathrm{Conf}_{2}(\Sigma)},\mathfrak{D})=\{\gamma\in \Omega^{d-1}(\overline{\mathrm{Conf}_2(\Sigma)})\mid i^*_\mathfrak{D}\gamma=0\}$, where $i_\mathfrak{D}\colon \mathfrak{D}\hookrightarrow \overline{\mathrm{Conf}_2(\Sigma)}$ denotes the inclusion. We can write the propagator as a path integral

\[
\mathscr{P}=\frac{1}{T_\Sigma}\frac{(-1)^d}{\I\hbar}\int_\calL \exp\left(\I\widehat{\calS}_\Sigma/\hbar\right)\pi_1^*\mathscr{X}\pi_2^*\mathscr{Y},
\]

where $\pi_j\colon \Sigma\times\Sigma\to \Sigma$ for $j=1,2$ denotes the projection onto the first and second factor, respectively and $T_\Sigma=\int_\calL\exp(\I\widehat{\calS}_\Sigma/\hbar)\in\C\otimes \mathrm{Dens}^\frac{1}{2}(\calV_\Sigma)/\{\pm 1\}$ is given in terms of the Ray–Singer torsion <cit.> which is defined through zeta regularization which does not depend on the choice of $\calL$ since the Ray–Singer torsion does not depend on the choice of metric.

Let $N$ be a $(d-1)$-manifold and let $\calH^n_{N,\ell}$ be the vector space of $n$-linear functionals on $\Omega^\bullet(N)[\ell]$ of the form

\[
\Omega^\bullet(N)[\ell]\ni \mathds{D}\mapsto \int_{N^n}\gamma\pi^*_1\mathds{D}\dotsm \pi^*_n\mathds{D}
\]

up to multiplication with a term given by the Reidemeister torsion <cit.> of $N$. We have denoted by $\gamma$ a distributional form on $N^n$.
The boundary state space $\calH^\calP_{\de\Sigma}$ is then given by

\[
\calH^\calP_{\de\Sigma}=\prod_{n_1,n_2=0}^\infty \calH^{n_1}_{\de_1\Sigma,1}\widehat{\otimes}\calH^{n_2}_{\de_2\Sigma,d-2}
\]

and $\widehat{\calH}^\calP_{\Sigma}=\calH^\calP_{\de\Sigma}\widehat{\otimes}\mathrm{Dens}^\frac{1}{2}(\calV_\Sigma)$. Perturbatively, the BV-BFV partition function is asymptotically given by

\begin{multline*}
T_\Sigma\exp(\I\calS^\mathrm{eff}_\Sigma/\hbar)\times\\\times\sum_{j\geq 0}\hbar^j\sum_{n_1,n_2\geq 0}\,\int_{(\de_1\Sigma)^{n_1}\times(\de_2\Sigma)^{n_2}}R^j_{n_1n_2}(\mathsf{x},\mathsf{y})\pi_{1,1}^*\mathds{X}^{i_1}\dotsm \pi^*_{1,n_1}\mathds{X}^{i_n}\pi^*_{2,1}\mathds{Y}_{i_1}\dotsm \pi^*_{2,n_2}\mathds{Y}_{i_n},
\end{multline*}

where $R^j_{n_1n_2}(\mathsf{x},\mathsf{y})$ denotes distributional forms on $(\de_1\Sigma)^{n_1}\times(\de_2\Sigma)^{n_2}$ with values in $\mathrm{Dens}^\frac{1}{2}(\calV_\Sigma)$. Note that we sum over $i_1,\ldots,i_n$. Moreover, we have denoted by $\calS^\mathrm{eff}_\Sigma$ the effective action[One can easily check that the partition function $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}=T_\Sigma\exp(\I\calS^\mathrm{eff}_\Sigma/\hbar)$, defined through the effective action $\calS^\mathrm{eff}_\Sigma$, satisfies the mQME (<ref>)] given by

\[
\calS^\mathrm{eff}_\Sigma=(-1)^{d-1}\left(\int_{\de_2\Sigma}\sum_i\mathds{Y}_i\mathsf{x}^i-\int_{\de_1\Sigma}\sum_i\mathsf{y}_i\mathds{X}^i\right)-(-1)^{2d}\int_{\de_2\Sigma\times\de_1\Sigma}\sum_i\pi^*_1\mathds{Y}_i\mathscr{P}\pi^*_{2}\mathds{X}^i.
\]

The BFV boundary operator acting on $\calH^\calP_{\de\Sigma}$ is then given by the ordered standard quantization of the BFV boundary action $\calS^\de_{\de\Sigma}$:

\[
\Omega^\calP_{\de\Sigma}=(-1)^d\I\hbar\left(\int_{\de_2\Sigma}\sum_i\dd_{\de\Sigma}\mathds{Y}_i\frac{\delta}{\delta\mathds{Y}_i}+\int_{\de_1\Sigma}\sum_i\dd_{\de\Sigma}\mathds{X}^i\frac{\delta}{\delta\mathds{X}^i}\right).
\]

Suppose now that we have two smooth $d$-manifolds $\Sigma_1$ and $\Sigma_2$ with common boundary component $(d-1)$-manifold $N$. We can then compute the glued partition function $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV}}_\Sigma$ for the glued  manifold $\Sigma=\Sigma_1\cup_N\Sigma_2$ (see Figure <ref>) out of the BV-BFV partition functions $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_1,\de\Sigma_1}$ on $\Sigma_1$ and $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_2,\de\Sigma_2}$ on $\Sigma_2$. We consider the transversal polarization on $\calF^\de_{N}$. In particular, we decompose the boundaries $\de\Sigma_1=\de_1\Sigma_1\sqcup\de_2\Sigma_1$ and $\de\Sigma_2=\de_1\Sigma_2\sqcup\de_2\Sigma_2$ such that $N\subset \de_1\Sigma_1$ and $N^\mathrm{opp}\subset\de_2\Sigma_2$. Let $\mathds{X}^N_1$ and $\mathds{Y}^N_2$ be the coordinates on $\Omega^\bullet(N)[1]$ and $\Omega^\bullet(N)[d-2]$, respectively. The glued state is then given by

\[
\mathsf{Z}^{\scriptscriptstyle\mathrm{BV}}_\Sigma=\int_{\{\mathds{X}^N_1,\mathds{Y}^N_2\}}\exp\left(\frac{\I}{\hbar}(-1)^{d-1}\int_N\sum_i(\mathds{Y}^N_2)_i(\mathds{X}^N_1)^i\right)\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_1,\de\Sigma_1}\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_2,\de\Sigma_2}.
\]

The glued partition function can be computed by considering the glued effective action and then considering $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV}}_\Sigma=T_{\Sigma_1}T_{\Sigma_2}\exp(\I\calS^\mathrm{eff}_\Sigma/\hbar)$.

This construction can be generalized to the case of $BF$-like theories (perturbations of abelian $BF$ theory), see <cit.> for more details. In particular, this is important since AKSZ theories often appear as $BF$-like theories.
This will be relevant especially for treating DW theory in the BV-BFV setting by regarding it as an AKSZ theory, which will be the content of Section <ref>. An important result regarding the gluing for $BF$-like theories is given by the following proposition.

Let $\Sigma$ be cut along a codimension-one submanifold $N$ into $\Sigma_1$ and $\Sigma_2$. Let $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_1,N}$ and $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_2,N^\mathrm{opp}}$ be the boundary states for $\Sigma_1$ and $\Sigma_2$ with a choice of residual fields and propagators and transverse ($\mathds{X}$ vs. $\mathds{Y}$) polarizations on $N$. Then the gluing of $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_1,N}$ and $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma_2,N^\mathrm{opp}}$ is the state $\mathsf{Z}^{\scriptscriptstyle\mathrm{BV-BFV}}_{\Sigma,\de\Sigma}$ for $\Sigma$ with the consequent choice of residual fields and propagators.

§.§ BV-BF$^k$V extension and shifted symplectic structures

If we move to higher defects, the symplectic form will be shifted by degree $+1$. In particular, if we consider a codimension $k$ submanifold $N_k\subset\Sigma$, then the symplectic gauge formalism associated to $N_k$ will be $(k-1)$-shifted. We will call the theory associated to a codimension $k$ submanifold a BF$^k$V theory and the coupling for each contiguous codimension (fully extended) a BV-BF$^k$V theory associated to the $d$-manifold $\Sigma$. The underlying mathematical theory for shifted symplectic structures was developed first in <cit.> by using methods and the language of derived algebraic geometry and studied further by various people. In <cit.>, they define first a symplectic form on a smooth scheme over some base ring $k$ of characteristic zero to be a $2$-form $\omega\in H^0(X,\Omega^{2,cl}_{X/k})$ which moreover is required to be nondegenerate, i.e. it induces an isomorphism $\Theta_\omega$ between the tangent and cotangent bundles of $X$.
Then one can define an $n$-shifted symplectic form on a derived Artin stack $X$ (in particular, they consider $X$ to be the solution of a derived moduli problem (see also <cit.>)) to be a closed $2$-form $\omega\in H^n(X,\bigwedge^2\mathbb{L}_{X/k})$ of degree $n$ on $X$ such that the corresponding morphism $\Theta_\omega\colon \mathbb{T}_{X/k}\to \mathbb{L}_{X/k}[n]$ is an isomorphism in the derived category of quasi-coherent sheaves $D^b\mathrm{QCoh}(X)$ on $X$. Here we have denoted by $\mathbb{L}_{X/k}$ the cotangent complex of $X$ and by $\mathbb{T}_{X/k}$ its dual, the tangent complex of $X$. Of course, it is important to mention what closed in this setting actually means. They define closedness of general $p$-forms by interpreting sections of $\bigwedge^2\mathbb{L}_{X/k}$ as functions on the derived loop stack $\calL X$ and consider it as some type of $S^1$-equivariance property. An important result of <cit.> is an existence result concerning a derived algebraic version of the AKSZ formulation in this setting. In particular, they prove the following theorem: