# Polyfold constructions for Fukaya categories

Fukaya categories were first introduced by Fukaya, Oh, Ohta, Ono in ca.2000. They capture the chain level information contained in Lagrangian Floer theory and its product structures. For an introduction see e.g. Auroux' Beginner's introduction to Fukaya categories [A].

There are different constructions (and even more proposals) depending on the properties of the (fixed) ambient symplectic manifold.

For example, Seidel [S] considers exact symplectic manifolds and constructs an (uncurved) Fukaya $A_{\infty }$ category whose objects are exact Lagrangians (with a brane structure), whose morphism spaces are Floer complexes (depending on the choice of a Hamiltonian), and whose composition operations are given by counting pseudoholomorphic polygons with boundary on the Lagrangians. (Here the benefit of the exactness assumption is that bubbling is excluded, so that the moduli spaces can be regularized by geometric methods (choices of Hamiltonian perturbations and almost complex structures). On the other hand, exact symplectic manifolds (on which the symplectic 2-form is exact) are necessarily noncompact, so one needs to assume certain boundedness and convexity conditions to ensure that pseudoholomorphic curves do not escape to infinity.)

This wiki will focus on the main difficulty that is not addressed in Seidel's book: How to regularize the moduli spaces of pseudoholomorphic polygons when geometric methods fail (e.g. due to sphere bubbling), and how to capture disk bubbling algebraically. To limit the classical analytic challenges in studying the pseudoholomorphic curves involved, we restrict our constructions to a fixed compact symplectic manifold $(M,\omega )$. Then - depending on various open choices and algebraic packaging for which we seek input from the Mirror Symmetry community - the Fukaya category ${\text{Fuk}}(M)$ consists of the following data:

## Objects

An object $L\in {\text{Obj}}_{{\operatorname {Fuk}(M)}}$ of $\operatorname {Fuk}(M)$ is a compact Lagrangian submanifold $L\subset M$ equipped with a brane structure.

Here input from the Mirror Symmetry community is needed

to determine what specific brane structures should be used. For the time being, we will treat brane structures as abstract gadgets that induce gradings on Floer or Morse complexes (which will form the morphism spaces) and orientations on the moduli spaces of pseudoholomorphic curves (from which we will construct the composition maps).

For special symplectic manifolds (those equipped with an almost complex structure J for which all J-holomorphic spheres are constant) we could work with Lagrangians without additional brane structure. In this case of trivial brane structures we will not have gradings or orientations, and thus will have to (and can) work with ${{\mathbb K}}={{\mathbb Z}}_{2}$ coefficients in the following.

## Morphisms

The morphism spaces of an $A_{\infty }$-category form graded modules over a ring. For Fukaya categories this typically is the Novikov ring over a fixed field ${{\mathbb K}}$ such as ${{\mathbb K}}=\mathbb{Q}$ or ${{\mathbb K}}=\mathbb{Z } _{2}$, with a variable $T$,

$\textstyle \Lambda :=\Lambda _{{{\mathbb K}}}:=\left\{\sum _{{i=0}}^{\infty }a_{i}T^{{\lambda _{i}}}\,\left|\,0\leq \lambda _{i}{\underset {i\to \infty }{\to }}\infty ,a_{i}\in {\mathbb {K}}\right.\right\}$

The natural construction of the morphism spaces ${\text{Hom}}(L_{0},L_{1})$ arises from the geometry of the Lagrangian intersection $L_{0}\cap L_{1}$ as follows:

For two Lagrangians $L_{0},L_{1}\subset M$ that are transverse (i.e. $T_{x}L_{0}\oplus T_{x}L_{1}=T_{x}M\;{\text{for each}}\;x\in L_{0}\cap L_{1}$) and hence have a finite set of intersection points, the natural choice of morphism space is the Floer chain complex

$\textstyle {\text{Hom}}(L_{0},L_{1}):=\sum _{{x\in L_{0}\cap L_{1}}}\Lambda \,x$.

When $L_{0},L_{1}\subset M$ are not transverse, then the construction of the Floer chain complex usually proceeds by choosing a Hamiltonian symplectomorphism $\phi :(M,\omega )\to (M,\omega )$ such that $\phi (L_{0}),L_{1}$ are transverse. So we define the Floer complex resp. Fukaya category morphism space by ${\text{Hom}}(L_{0},L_{1}):={\text{Hom}}(\phi (L_{0}),L_{1})$.

When considering the isomorphism space ${\text{Hom}}(L,L)$ of a fixed Lagrangian $L_{0}=L_{1}=L$, then the Hamiltonian symplectomorphism $\phi$ can be obtained by lifting a Morse function $f:L\to \mathbb{R}$ to a Hamiltonian function in a Lagrangian neighborhood $T^{*}L\subset M$. After extending it suitably, the intersection points can be identified with the critical points, $\phi (L)\cap L={\text{Crit}}f$. Thus in this case we define the morphism space by the Morse chain complex

$\textstyle {\text{Hom}}(L,L):=\sum _{{x\in {\text{Crit}}f}}\Lambda \,x$.

The universal construction of the morphism spaces is

$\textstyle {\text{Hom}}(L_{0},L_{1}):=\sum _{{x\in {\text{Crit}}(L_{0},L_{1})}}\Lambda \,x$

as module over the Novikov ring $\Lambda =\Lambda _{{{\mathbb K}}}$ that is freely generated by a finite critical set

${\text{Crit}}(L_{0},L_{1}):={\begin{cases}{\text{Crit}}(f)&;L_{0}=L_{1},\\\phi _{{L_{0},L_{1}}}(L_{0})\cap L_{1}&;L_{0}\neq L_{1}.\end{cases}}$

This requires the choice of a Hamiltonian diffeomorphism $\phi _{{L_{0},L_{1}}}:M\to M$ or Morse function $f:L_{0}\to \mathbb{R}$, respectively, for each pair of objects.

For $L_{0}\pitchfork L_{1}$ we will prescribe the canonical choice of $\phi _{{L_{0},L_{1}}}:={\text{id}}_{M}$. For each nontransverse pair $L_{0}\neq L_{1}$ we fix an autonomous Hamiltonian function $H_{{L_{0},L_{1}}}:M\to \mathbb{R}$ such that the time-1 flow $\phi _{{L_{0},L_{1}}}:=\phi _{{X_{{L_{0},L_{1}}}}}^{1}:M\to M$ of the associated Hamiltonian vector field $X_{{L_{0},L_{1}}}:=-J\nabla H_{{L_{0},L_{1}}}$ yields the desired transversality $\phi _{{L_{0},L_{1}}}(L_{0})\pitchfork L_{1}$. Moreover, we can make these choices symmetric by setting $H_{{L_{1},L_{0}}}=-H_{{L_{0},L_{1}}}$ so that $\phi _{{L_{1},L_{0}}}=\phi _{{L_{0},L_{1}}}^{{-1}}$. This identifies the morphism spaces ${\text{Hom}}(L_{0},L_{1})\cong {\text{Hom}}(L_{1},L_{0})$ via the bijection of critical sets

$\phi _{{L_{0},L_{1}}}:{\text{Crit}}(L_{1},L_{0})=\phi _{{L_{0},L_{1}}}^{{-1}}(L_{1})\cap L_{0}\;\to \;\phi _{{L_{0},L_{1}}}(L_{0})\cap L_{1}={\text{Crit}}(L_{0},L_{1}).$

For most versions of Fukaya categories, these modules also carry a grading induced by brane structures $|x|\in \mathbb{Z }$ (or $|x|\in \mathbb{Z } _{N}$ with even $N$) for all $x\in {\text{Crit}}(L_{0},L_{1})$. (When working with trivial brane structures, we set $|x|=0$.)

Here input from the Mirror Symmetry community is needed

on how to deal with the choices of Hamiltonian symplectomorphisms algebraically, and what gradings to use - resulting from appropriate brane structures.

The use of the Novikov ring and the Morse complex for the isomorphism spaces is based on such input.

## Composition Operations

While a category has a single composition map ${\text{Hom}}(L_{0},L_{1})\times {\text{Hom}}(L_{1},L_{2})\to {\text{Hom}}(L_{0},L_{2})$, an $A_{\infty }$-category has composition maps of every order $d\geq 1$, which are $\Lambda$-linear maps from a tensor product of morphism spaces,

$\mu ^{d}:{\text{Hom}}(L_{{d-1}},L_{d})\otimes \ldots \otimes {\text{Hom}}(L_{1},L_{2})\otimes {\text{Hom}}(L_{0},L_{1})\to {\text{Hom}}(L_{0},L_{d}).$

When the morphism spaces carry a grading induced by brane structures, then the composition operation $\mu ^{d}$ has degree $2-d$ (i.e. shifts the grading down by this amount). In particular, the $d=1$ composition map $\mu ^{1}:{\text{Hom}}(L_{0},L_{1})\to {\text{Hom}}(L_{0},L_{1})$ is a differential (of degree 1) on the morphism space - namely the Floer differential in the case of the Fukaya category. For Fukaya categories of non-exact symplectic manifolds, disk bubbling will moreover result in curvature terms in the $A_{\infty }$-relations, which are encoded in terms of a $d=0$ composition for each Lagrangian brane,

$\mu ^{0}:\Lambda \to {\text{Hom}}(L,L),\qquad \mu ^{0}:\lambda \mapsto \lambda \,\mu ^{0}(1).$

By linearity it suffices to construct these composition maps for any pure tensor given by intersection points $x_{i}\in {\text{Crit}}(L_{i},L_{{i-1}})$. These constructions will result from appropriate ways of counting elements of moduli spaces of pseudoholomorphic polygons $\overline {\mathcal {M}}^{0}(x_{0};x_{1},\ldots ,x_{d})$,

$\mu ^{d}(x_{d}\otimes \ldots \otimes x_{2}\otimes x_{1})=\sum _{{x_{0}\in {\text{Crit}}(L_{d},L_{0})}}\;\sum _{{b\in \overline {\mathcal {M}}^{0}(x_{0};x_{1},\ldots ,x_{d};\nu )}}\nu (b)\,T^{{\omega (b)}}\,x_{0}.$

Here $\nu$ denotes a regularization of the moduli space (e.g. a perturbation), which in particular induces a weight function $\nu :\overline {\mathcal {M}}^{0}(x_{0};x_{1},\ldots ,x_{d};\nu )\to {\mathbb {K}}$ (e.g. $\nu (b)=1$ in case ${\mathbb {K}}=\mathbb{Z } _{2}$), and $\omega :\overline {\mathcal {M}}^{0}(x_{0};x_{1},\ldots ,x_{d};\nu )\to \mathbb{R}$ is a symplectic area function. Finally, the superscript in $\overline {\mathcal {M}}^{0}(\ldots )$ indicates the part of the moduli space of expected dimension 0.

An example of this regularization construction by means of polyfold theory can be found in [J.Li thesis], which constructs a curved $A_{\infty }$-algebra $(\mu ^{d})_{{d\geq 0}}$ on the Morse complex ${{\rm {Hom}}}(L,L)$ of a fixed Lagrangian $L$.

More precisely, [J.Li thesis] assumes that $M$ contains no $J$-holomorphic spheres, which allows to work with coefficients in the Novikov ring over $\mathbb{Z } _{2}$. Generalizing this work to all compact symplectic manifolds requires inclusion of sphere bubble trees, which cause nontrivial isotropy. In that case, the need for multivalued perturbations rules out working with $\mathbb{Z } _{2}$-coefficients, and requires the construction of coherent orientations on the regularized moduli spaces.

## Curved ${\mathbf {A}}_{{\mathbf {\infty }}}$-Relations

The $A_{\infty }$-relations generalize the associativity relation for classical composition of morphisms in categories. They also describe the failure of the Floer differential to square to zero, due to a curvature term. So, more precisely, we need to establish the curved ${\mathbf {A}}_{{\mathbf {\infty }}}$-relations.

The curved $A_{\infty }$-relations can be phrased as $\widehat \mu \circ \widehat \mu =0$, where $\widehat \mu$ is given by the composition maps $(\mu ^{d})_{{d\geq 0}}$ acting on the total complex

$\widehat C={\mathbb {K}}\;\oplus \;\bigoplus _{{d\geq 1}}\;\bigoplus _{{L_{0},\ldots ,L_{d}\in {\text{Obj}}_{{\operatorname {Fuk}(M)}}}}{\text{Hom}}(L_{{d-1}},L_{d})\otimes \ldots \otimes {\text{Hom}}(L_{1},L_{2})\otimes {\text{Hom}}(L_{0},L_{1}).$

We will denote the length of a pure tensor $c_{d}\otimes \ldots \otimes c_{1}\in {\text{Hom}}(L_{{d-1}},L_{d})\otimes \ldots \otimes {\text{Hom}}(L_{0},L_{1})$ by $\ell (c_{d}\otimes \ldots \otimes c_{1})=d$. Now the abstract sum of all composition maps $\widehat \mu :\widehat C\to \widehat C$ is a $\Lambda$-linear map on $\widehat C$ that is given on pure tensors by

$\widehat \mu \,:\;c_{d}\otimes \ldots \otimes c_{1}\;\mapsto \sum _{{c_{d}\otimes \ldots \otimes c_{1}=c''\otimes c'\otimes c}}(-1)^{{\|c\|}}\;c''\otimes \mu ^{{\ell (c')}}(c')\otimes c.$

Here we sum over all decompositions $c_{d}\otimes \ldots \otimes c_{1}=(c''=c_{d}\otimes \ldots \otimes c_{{m+n+1}})\otimes (c'=c_{{m+n}}\otimes \ldots \otimes c_{{n+1}})\otimes (c=c_{n}\otimes \ldots \otimes c_{1})$ into pure tensors of lengths $d-m-n,m,n\geq 0$, in particular allow $c'=1\;\;{\scriptstyle ({\text{or}}\;c=1\;{\text{or}}\;c''=1)}$ to have length $\ell (c')=0$. The sign $\|c_{n}\otimes \ldots \otimes c_{1}\|=|c_{n}|+\ldots +|c_{1}|-n$ is determined by the length \ell(c_n \otimes \ldots \otimes c_1) = n) and the grading induced by brane structures $|c_{i}|\in \mathbb{Z }$ (or $|c_{i}|\in \mathbb{Z } _{N}$ with even $N$). With this notation, the curved $A_{\infty }$-relations for $(\mu ^{d})_{{d\geq 0}}$ are $\widehat \mu \circ \widehat \mu =0$.

Spelling this out, the first two relations are

$\mu ^{1}(\mu ^{0}(1))=0;\qquad \mu ^{2}(c,\mu ^{0}(1))-(-1)^{{|c|}}\mu ^{2}(\mu ^{0}(1),c)+\mu ^{1}(\mu ^{1}(c))=0.$

Using linearity it suffices to prove the $A_{\infty }$-relations for $d\geq 1$ and $c_{i}=x_{i}\in {\text{Crit}}(L_{i},L_{{i-1}})$

$\sum _{{m,n\geq 0}}(-1)^{{\|x_{n}\otimes \ldots \otimes x_{1}\|}}\mu ^{{d-m}}(x_{d}\otimes \ldots \otimes x_{{n+m+1}}\otimes \mu ^{m}(x_{{n+m}}\otimes \ldots \otimes x_{{n+1}})\otimes x_{n}\otimes \ldots \otimes x_{1})=0,$

where $\|x_{n}\otimes \ldots \otimes x_{1}\|=|x_{n}|+\ldots +|x_{1}|-n$.

To prove these identities we will identify the summands with Cartesian products of 0-dimensional parts of regularized moduli spaces, identify these with the boundary facets of 1-dimensional regularized moduli space,

$\partial \overline {\mathcal {M}}^{1}(x_{0};x_{1},\ldots ,x_{d};\nu )=\bigsqcup _{{m,n\geq 0}}\bigsqcup _{{y\in {\text{Crit}}(L_{n},L_{{m+n}})}}\overline {\mathcal {M}}^{0}(x_{0};\underline {x}',y,\underline {x}''';\nu )\times \overline {\mathcal {M}}^{0}(y;\underline {x}'';\nu ),$

where we abbreviate $\underline {x}'=(x_{1},\ldots ,x_{n}),\underline {x}''=(x_{{n+1}},\ldots ,x_{{n+m}}),\underline {x}'''=(x_{{n+m+1}},\ldots ,x_{d}),$ and appeal to the fact that the boundary of a sufficiently regular moduli space is null homologous. In addition, the proof relies on additivity of the symplectic area $\omega ((b,b'))=\omega (b)+\omega (b')$ and requires the regularizations of these moduli spaces to be related by $\nu ((b,b'))=(-1)^{{\|\underline {x}'\|}}\nu (b)\nu (b')$.

More precisely, by construction of the $\mu ^{d}$ the $A_{\infty }$-relation above is equivalent to the identity for each fixed $x_{0}\in {\text{Crit}}(L_{0},L_{d})$

$\sum _{{m,n\geq 0}}\sum _{{(b,b')\in \overline {\mathcal {M}}^{0}(x_{0};\underline {x}',y,\underline {x}''';\nu )\times \overline {\mathcal {M}}^{0}(y;\underline {x}'';\nu )}}(-1)^{{\|\underline {x}'\|}}\nu (b)\nu (b')\,T^{{\omega (b)}}\,T^{{\omega (b')}}=0.$

Now the relationship between 1- and 0-dimensional regularized moduli spaces above says more precisely that the boundary of the 1-dimensional regularized moduli space $\overline {\mathcal {M}}^{1}(x_{0};x_{1},\ldots ,x_{d};\nu )$ consists of pairs $(b,b')\in \overline {\mathcal {M}}^{0}(x_{0};\underline {x}',y,\underline {x}''';\nu )\times \overline {\mathcal {M}}^{0}(y;\underline {x}'';\nu )$ for any $m,n\geq 0$ and $y\in {\text{Crit}}(L_{n},L_{{m+n}})$. Since the symplectic area is constant on every connected component of the moduli space, we can partition the claimed identity into sums of the weight function over the boundary of a union of components, $\overline {\mathcal {M}}_{{w_{0}}}^{1}:=\overline {\mathcal {M}}^{1}(x_{0};x_{1},\ldots ,x_{d};\nu )\cap \omega ^{{-1}}(w_{0})$ for some $w_{0}\in \mathbb{R}$. Now the $A_{\infty }$-relations finally follow from a version of Stokes' theorem for the boundary of a 1-dimensional regularized moduli space,

$\textstyle \sum _{{\underline {b}\in \partial \overline {\mathcal {M}}_{{w_{0}}}^{1}}}\nu (\underline {b})=0.$

## Invariance

There are many choices involved in the above construction of a Fukaya category. These can roughly be separated into geometric, local choices (the almost complex structure $J$, and Hamiltonian diffeomorphisms $\phi$ resp. Morse functions $f$) and abstract, global choices (the perturbation $\nu$ and setups of ambient space/bundle in which it gets constructed). Sometimes we can even obtain the required regularization by a geometric choice of the perturbations.

To prove independence of the Fukaya category (up to an appropriate notion of equivalence) from a geometric/local choice, we can construct continuation maps

as in [3.4 Salamon notes] or [(10c) Seidel book] to obtain direct $A_{\infty }$-functors between the Fukaya categories for two different choices. Such constructions use moduli spaces of solutions of a Cauchy-Riemann PDE in which the geometric data (almost complex structure and Hamiltonian perturbation term) interpolates between the two different choices.

For abstract/global choices, such a continuation map PDE is not available

since the abstract maps $u\mapsto \nu (u)$ are generally not given by a local differential operator - i.e. $\nu (u)(z)$ depends on the entire map $u:\Sigma \to M$, not just on the value $u(z)$ and some derivatives of $u$ at the point $z\in \Sigma$. Instead, one is led to considering a 1-parameter family of Cauchy-Riemann PDEs which interpolates between the two abstract choices. This homotopy method was originally introduced by Floer, and is outlined in the exact $A_{\infty }$-context in [(10e) Seidel book]. For general symplectic manifolds, curvature terms form a major obstacle which can be resolved in special cases by obstruction bundle gluing-like analysis [J.Li thesis].

A general invariance proof will likely require a setup along the following lines: To compare the Fukaya categories ${{\rm {Fuk}}}_{0},{{\rm {Fuk}}}_{1}$ resulting from two different sets of choices, use a homotopy between the perturbation data to construct an $A_{\infty }$-category ${{\rm {Fuk}}}_{{[0,1]}}$ together with two restriction functors ${{\rm {Fuk}}}_{{[0,1]}}\to {{\rm {Fuk}}}_{0}$ and ${{\rm {Fuk}}}_{{[0,1]}}\to {{\rm {Fuk}}}_{1}$.

Here input from the Mirror Symmetry community is needed

"as to what algebraic notion of curved equivalence is desired ... and possible given the context of abstract perturbations that cannot exclude bubbling, even if the two categories under comparison are 'unobstructed'.