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 ring ${{\mathbb K}}$ such as ${{\mathbb K}}=\mathbb{Q}$ or ${{\mathbb K}}=\mathbb{Z } _{2}$, with a variable $T$,

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

Now 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. Then 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

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

Each of the morphism spaces is a module over the Novikov ring $\Lambda =\Lambda _{{{\mathbb K}}}$. For most versions of Fukaya categories, these modules should also carry a grading induced by brane structures over ${{\mathbb Z}}$ or ${{\mathbb Z}}_{N}$. To simplify notation, we will in the following use the universal construction of these modules as

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

as freely generated by a critical set

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

Note that this involves a choice of Hamiltonian diffeomorphism $\phi :M\to M$ or Morse function $f:L_{0}\to \mathbb{R}$, respectively, for each pair of objects. Only for $L_{0}\pitchfork L_{1}$ there is a canonical choice of $\phi ={\text{id}}_{M}$.

Here the use of the Morse complex for the isomorphism spaces is based on input from the Mirror Symmetry community.

Further input is needed to decide how to track or deal with the choices of Hamiltonian symplectomorphisms algebraically, and what gradings to use - resulting from appropriate brane structures.

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,

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

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 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 $\overline {\mathcal {M}}^{0}(x_{0};x_{1},\ldots ,x_{d})$,

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

Here $\nu$ denotes a regularization of the moduli space (e.g. a perturbation), ${\text{w}}:\overline {\mathcal {M}}^{0}(x_{0};x_{1},\ldots ,x_{d};\nu )\to {\mathbb {K}}$ is a weight function determined by the regularization (e.g. ${\text{w}}(b)=1$ 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.

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. These are most effectively phrased in terms of 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 $|c_{d}\otimes \ldots \otimes c_{1}|=d$. 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

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

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 $|c'|=0$. The sign is determined by $\|c_{n}\otimes \ldots \otimes c_{1}\|=|c_{n}|+\ldots +|c_{1}|-n$, where $|c_{i}|\in \mathbb{Z }$ (or $|c_{i}|\in \mathbb{Z } _{N}$ with even $N$) denotes the grading induced by brane structures. 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

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

To prove these, we will identify the summands with the boundary facets of a moduli space of expected dimension 1 and appeal to the fact that the boundary of a sufficiently regular moduli space is null homologous. More precisely, using linearity it suffices to prove the $A_{\infty }$-relation for $x_{i}\in {\text{Crit}}(L_{i},L_{{i-1}})$

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

By construction of the $\mu ^{d}$ this is equivalent to the following identity for each fixed $x_{0}\in {\text{Crit}}(L_{0},L_{d})$, where we abbreviate $\underline x=(x_{1},\ldots ,x_{n}),\underline {x}'=(x_{{n+1}},\ldots ,x_{{n+m}}),\underline {x}''=(x_{{n+m+1}},\ldots ,x_{d}).$

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

This identity follows from the fact that the regularized moduli spaces are constructed such that the boundary of each 1-dimensional part is given by Cartesian products of 0-dimensional parts,

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

More precisely, 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 $y\in {\text{Crit}}(L_{n},L_{{m+n}})$. Their symplectic area is additive $\omega ((b,b'))=\omega (b)+\omega (b')$ and the weight functions on these moduli spaces are related by ${\text{w}}((b,b'))=(-1)^{{\|\underline x\|}}{\text{w}}(b){\text{w}}(b')$. Moreover, the symplectic area is constant on every connected component of the moduli space, so that the claimed identity can be partitioned 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,

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