Moduli spaces of pseudoholomorphic polygons

table of contents

To construct the moduli spaces from which the composition maps are defined we fix an auxiliary almost complex structure $J:TM\to TM$ which is compatible with the symplectic structure in the sense that $\omega (\cdot ,J\cdot )$ defines a metric on $M$. (Unless otherwise specified, we will use this metric in all following constructions.) Then given Lagrangians $L_{0},\ldots ,L_{d}\subset M$ and generators $x_{0}\in {\text{Crit}}(L_{0},L_{d}),x_{1}\in {\text{Crit}}(L_{0},L_{1}),\ldots ,x_{d}\in {\text{Crit}}(L_{{d-1}},L_{d})$ of their morphism spaces, we need to specify the (compactified) moduli space $\overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$. We will do this by combining two special cases which we discuss first.

Pseudoholomorphic polygons for pairwise transverse Lagrangians

If each consecutive pair of Lagrangians is transverse, i.e. $L_{0}\pitchfork L_{1},L_{1}\pitchfork L_{2},\ldots ,L_{{d-1}}\pitchfork L_{d},L_{d}\pitchfork L_{0}$, then our construction is based on pseudoholomorphic polygons

$u:\Sigma \to M,\qquad u((\partial \Sigma )_{i})\subset L_{i},\qquad \overline \partial _{J}u=0,$

where $\Sigma =\Sigma _{{\underline {z}}}:=D\setminus \{z_{0},\ldots ,z_{d}\}$ is a disk with $d+1$ boundary punctures in counter-clockwise order $z_{0},\ldots ,z_{d}\subset \partial D$, and $(\partial \Sigma )_{i}$ denotes the boundary component between $z_{i},z_{{i+1}}$ (resp. between $z_{{d}},z_{0}$ for i=d). More precisely, we construct the (uncompactified) moduli spaces of pseudoholomorphic polygons for any tuple $x_{i}\in L_{i}\cap L_{{i+1}}$ for $i=0,\ldots ,d$ as in [S]:

${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}):={\bigl \{}(\underline {z},u)\,{\big |}\,{\text{1. - 2.}}{\bigr \}}/\sim$

where

1. $\underline {z}=(z_{0},z_{1},\ldots ,z_{d})\subset \partial D$ is a tuple of pairwise disjoint marked points on the boundary of a disk, in counter-clockwise order.
2. $u:\Sigma _{{\underline {z}}}\to M$ is a smooth map satisfying
   • the Cauchy-Riemann equation $\overline \partial _{J}u=0$,
   • Lagrangian boundary conditions $u((\partial \Sigma )_{i})\subset L_{i}<\infty$,
   • the finite energy condition $\textstyle \int _{{\Sigma }}u^{*}\omega <\infty$,
   • the limit conditions $\lim _{{z\to z_{i}}}u(z)=x_{i}$ for $i=0,1,\ldots ,d$.

Here two pseudoholomorphic polygons are equivalent $(\underline {z},u)\sim (\underline {z}',u')$ if there is a disk automorphism $\psi :D\to D$ that preserves the complex structure on $D$, the marked points $\psi (\underline {z})=\underline {z}'$, and relates the pseudoholomorphic polygons by reparametrization, $u=u'\circ \psi$.

The case $d=0$ is not considered in this part of the moduli space setup since $L_{0},L_{d}=L_{0}$ are never transverse. TODO: However, it might appear in the construction of homotopy units?

For $d=1$, the twice punctured disks are all biholomorphic to the strip $\Sigma _{{\{z_{0},z_{1}\}}}\simeq \mathbb{R} \times [0,1]$, so that we could equivalently set up the moduli spaces ${\mathcal {M}}(x_{0};x_{1})$ by fixing the domain $\Sigma _{{d=1}}:=\mathbb{R} \times [0,1]$ and defining the equivalence relation $\sim$ only in terms of the shift action $u(s,t)\mapsto u(\tau +s,t)$ of $\tau \in \mathbb{R}$.

For $d\geq 2$, the moduli space of domains

${\mathcal {M}}_{{d+1}}:={\frac {{\bigl \{}\Sigma _{{\underline {z}}}\,{\big |}\,\underline {z}=\{z_{0},\ldots ,z_{d}\}\in \partial D\;{\text{pairwise disjoint}}{\bigr \}}}{\Sigma _{{\underline {z}}}\sim \Sigma _{{\underline {z}'}}\;{\text{iff}}\;\exists \psi :\Sigma _{{\underline {z}}}\to \Sigma _{{\underline {z}'}},\;\psi ^{*}i=i}}$

can be compactified to form the Deligne-Mumford space $\overline {\mathcal {M}}_{{d+1}}$, whose boundary and corner strata can be represented by trees of polygonal domains $(\Sigma _{v})_{{v\in V}}$ with each edge $e=(v,w)$ represented by two punctures $z_{e}^{-}\in \Sigma _{v}$ and $z_{e}^{+}\in \Sigma _{w}$. The thin neighbourhoods of these punctures are biholomorphic to half-strips, and a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.

To construct the compactified moduli spaces $\overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ we have to add various strata to the moduli space of pseudoholomorphic polygons without breaking or nodes ${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ defined above. This is done precisely in the general construction below, but roughly requires to

• include degenerate pseudoholomorphic polygons given by a tuple of pseudoholomorphic maps $u_{v}:\Sigma _{v}\to M$ whose domain is a nontrivial tree of domains $[(\Sigma _{v})_{{v\in V}},(z_{e}^{\pm })_{{e\in E}}]\in \overline {\mathcal {M}}_{{d+1}}$;
• allow for Floer breaking at each puncture of the domains $\Sigma _{v}$, i.e. a finite string of pseudoholomorphic strips in ${\mathcal {M}}(x;x'),{\mathcal {M}}(x';x''),\ldots ,{\mathcal {M}}(x^{{(k)}};x_{i})$;
• allow for disk bubbling at any boundary point of the above domains, i.e. a tree, each of whose vertices is represented by a pseudoholomorphic disk, with edges representing nodes - given by marked points on different disks at which the maps satisfy a matching condition;
• allow for sphere bubbling at any (boundary or interior) point of each of the above domains, i.e. a tree, each of whose vertices is represented by a pseudoholomorphic sphere, with edges representing nodes - given by marked points on different spheres at which the maps satisfy a matching condition.

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We will see that sphere bubbling does not contribute to the boundary stratification of these moduli spaces, so that the boundary stratification and thus the algebraic structure arising from these moduli spaces is induced by Floer breaking and disk bubbling. The boundary strata arising from Floer breaking are fiber products of other moduli spaces of pseudoholomorphic polygons over finite sets of Lagrangian intersection points, which indicates an algebraic composition in this finitely generated Floer chain complex. Disk bubbling, on the other hand, in the present setting yields boundary strata that are fiber products over the Lagrangian submanifold specified by the boundary condition. The corresponding algebraic composition requires a push-pull construction on some space of chains, currents, or differential forms on the Lagrangian. However, such constructions require transversality of the chains to the evaluation maps from the regularized moduli spaces, so that a rigorous construction of the $A_{\infty }$-structure in this setting - as in the approach by Fukaya et al - requires a complicated infinite iteration. We will resolve this issue as in [JL] by following another earlier proposal by Fukaya-Oh to allow disks to flow apart along a Morse trajectory, thus yielding disk trees which are constructed next - still ignoring sphere bubbling - before we put everything together to a general construction of the compactified moduli space.

Pseudoholomorphic disk trees for a fixed Lagrangian

If the Lagrangians are all the same, $L_{0}=L_{1}=\ldots =L_{d}=:L$, then our construction is based on pseudoholomorphic disks

$u:D\to M,\qquad u(\partial D)\subset L,\qquad \overline \partial _{J}u=0.$

Such disks (modulo reparametrization by automorphisms of the disk) also arise from compactifying other moduli spaces of pseudoholomorphic curves in which energy concentrates at a boundary point. To capture this bubbling algebraically, we work throughout with the Morse function $f:L\to \mathbb{R}$ chosen in the setup of the morphism space ${\text{Hom}}(L,L)=\textstyle \sum _{{x\in {\text{Crit}}(f)}}\Lambda x$. We also choose a metric on $L$ so that the gradient vector field $\nabla f$ satisfies the Morse-Smale conditions and an additional technical assumption in [1] which guarantees a natural smooth manifold-with-boundary-and-corners structure on the compactified Morse trajectory spaces $\overline {\mathcal {M}}(L,L),\overline {\mathcal {M}}(p^{-},L),\overline {\mathcal {M}}(L,p^{+}),\overline {\mathcal {M}}(p^{-},p^{+})$ for $p^{\pm }\in {\text{Crit}}(f)$. This smooth structure is essentially induced by the requirement that the evaluation maps at positive and negative ends ${{\rm {ev^{\pm }}}}:\overline {\mathcal {M}}(\ldots )\to L$ are smooth. With that data and the fixed almost complex structure $J$ we can construct the moduli spaces of pseudoholomorphic disk trees for any tuple $x_{0},x_{1},\ldots ,x_{d}\in {{\rm {Crit}}}(f)$ as in JL:

${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}):={\bigl \{}(T,\underline {\gamma },\underline {z},\underline {u})\,{\big |}\,{\text{1. - 5.}}{\bigr \}}/\sim$

where

1. $T$ is an ordered tree with the following structure on the sets of vertices $V$ and edges $E$:
   • The edges $E\subset V\times V\setminus \Delta _{V}$ are oriented towards the root vertex $v_{0}\in V$ of the tree, i.e. for $e=(v,w)\in E$ the outgoing vertex $w$ is still connected to the root after removing $e$. Thus each vertex $v\in V$ has a unique outgoing edge $e_{v}^{0}=(v,\;\cdot \;)\in E$ and a (possibly empty) set of incoming edges $E_{v}^{{{\rm {in}}}}=\{e=(\;\cdot \;,v)\in E\}$. Moreover, the set of incoming edges is ordered, $E_{v}^{{{\rm {in}}}}=\{e_{v}^{1},\ldots ,e_{v}^{{|v|-1}}\}$ with $|v|$ denoting the valence - number of attached edges - of $v$).
   • The set of vertices is partitioned $V=V^{m}\sqcup V^{c}$ into the sets of main vertices $V^{m}$ and the set of critical vertices $V^{c}=\{v_{0}^{c},v_{1}^{c},\ldots v_{d}^{c}\}$. The latter is ordered to start with the root $v_{0}^{c}=v_{0}$, which is required to have a single edge $\{e_{{v_{0}}}^{1}\}=E_{{v_{0}}}^{{{\rm {in}}}}$, and then contains d leaves $v_{i}^{c}$ of the tree (i.e. with $E_{{v_{i}^{c}}}^{{{\rm {in}}}}=\emptyset$), with order induced by the orientation and order of the edges (with the root being the minimal vertex).
2. $\underline {\gamma }=(\underline {\gamma }_{e})_{{e\in E}}$ is a tuple of generalized Morse trajectories in the following compactified Morse trajectory spaces:
   • $\underline {\gamma }_{e}\in \overline {\mathcal {M}}(x_{i},x_{j})$ for any edge $e=(v_{i}^{c},v_{j}^{c})$ between critical vertices;
   • $\underline {\gamma }_{e}\in \overline {\mathcal {M}}(x_{i},L)$ for any edge $e=(v_{i}^{c},w)$ from a critical vertex $v_{i}^{c}$ to a main vertex $w\in V^{m}$;
   • $\underline {\gamma }_{e}\in \overline {\mathcal {M}}(L,x_{j})$ for any edge $e=(v,v_{j}^{c})$ from a main vertex $v\in V^{m}$ to a critical vertex $v_{j}^{c}$;
   • $\underline {\gamma }_{e}\in \overline {\mathcal {M}}(L,L)$ for any edge $e=(v,w)$ between main vertices $v,w\in V^{m}$.
3. $\underline {z}=(\underline {z}_{v})_{{v\in V^{m}}}$ is a tuple of boundary marked points as follows:
   • For each main vertex $v$ there are $|v|$ pairwise disjoint marked points $\underline {z}_{v}=(z_{e}^{v})_{{e\in \{e_{v}^{0}\}\cup E_{v}^{{{\rm {in}}}}}}\subset \partial D$ on the boundary of a disk.
   • The order $\{e_{v}^{0}\}\cup E_{v}^{{{\rm {in}}}}=\{e_{v}^{0},e_{v}^{1},\ldots ,e_{v}^{{|v|-1}}\}$ of the edges corresponds to a counter-clockwise order of the marked points $z_{{e_{v}^{0}}}^{v},z_{{e_{v}^{1}}}^{v},\ldots ,z_{{e_{v}^{{|v|-1}}}}^{v}\in \partial D$.
   • The marked points can also be denoted as $z_{e}^{-}=z_{e}^{v}$ and $z_{e}^{+}=z_{e}^{w}$ by the edges $e=(v,w)\in E$ for which $v\in V^{m}$ or $w\in V^{m}$.
4. For each main vertex $v$ there is a pseudoholomorphic disk, that is a smooth map $u_{v}:D\to M$ satisfying
   • the Cauchy-Riemann equation $\overline \partial _{J}u_{v}=0$,
   • Lagrangian boundary conditions $u_{v}(\partial D)\subset L<\infty$,
   • the finite energy condition $\textstyle \int _{D}u_{v}^{*}\omega <\infty$.
   • The pseudholomorphic disks can also be indexed as $u_{e}^{-}=u_{v}$ and $u_{e}^{+}=u_{w}$ by the edges $e=(v,w)\in E$ for which $v\in V^{m}$ or $w\in V^{m}$. In that notation, they satisfy the matching conditions with the generalized Morse trajectories $u_{e}^{\pm }(z_{e}^{\pm })={{\rm {ev}}}^{\pm }(\underline {\gamma }_{e})$ whenever $v_{e}^{\pm }\in V^{m}$.
5. The disk tree is stable in the sense that any main vertex $v\in V^{m}$ whose disk has zero energy $\textstyle \int u_{v}^{*}\omega =0$ (which is equivalent to $u_{v}$ being constant) has valence $|v|\geq 3$.

Finally, two pseudoholomorphic disk trees are equivalent $(T,\underline {\gamma },\underline {z},\underline {u})\sim (T',\underline {\gamma }',\underline {z}',\underline {u}')$ if there is a tree isomorphism $\zeta :T\to T'$ and a tuple of disk automorphisms $(\psi _{v}:D\to D)_{{v\in V^{m}}}$ preserving the complex structure on $D$ such that

• $\zeta$ preserves the tree structure and order of edges;
• the Morse trajectories are the preserved $\underline {\gamma }_{e}=\underline {\gamma }'_{{\zeta (e)}}$ for every $e\in E$;
• the marked points are preserved $\psi _{v}(\underline {z}_{v})=\underline {z}'_{{\zeta (v)}}$ for every $v\in V^{m}$;
• the pseudoholomorphic disks are related by reparametrization, $u_{v}=u'_{{\zeta (v)}}\circ \psi _{v}$ for every $v\in V^{m}$.

Any equivalence class of disk trees $[(T,\underline {\gamma },\underline {z},\underline {u})]$ induces a domain tree $[(T',\underline {\ell },\underline {z})]$ as follows:

1. The tree $T'$ is obtained from $T$ by replacing critical vertices and their outgoing edges by incoming semi-infinite edges of the new tree $T'$. We also replace the critical root vertex and its incoming edge by an outgoing semi-infinite edge of the new tree $T'$. The new tree $T'$ retains the orientations of edges and inherits an order of the edges from $T$. Its root is the unique main vertex from which there was an edge to the critical root vertex in $T$.
2. Every vertex $v\in V'=V^{m}$ of $T'$ then represents a disk domain $D_{v}=D$.
3. Every edge $e\in E'=E$ is labeled with the length $\ell _{e}:=\ell (\underline {\gamma }_{e})\in [0,\infty ]$ of the associated generalized Morse trajectory. For the semi-infinite edges, this length is automatically $\ell _{e}=\infty$ since the associated Morse trajectories are semi-infinite.
4. The domain for each vertex $v\in V'$ is marked by $|v|$ boundary points $\underline {z}_{v}=(z_{e}^{v})_{{e\in \{e_{v}^{0}\}\cup E_{v}^{{{\rm {in}}}}}}\subset \partial D_{v}$, ordered counter-clockwise.
5. Two such trees are equivalent $[(T,\underline {\ell },\underline {z})]\sim [(T',\underline {\ell }',\underline {z}')]$ if there is a tree isomorphism $\zeta :T\to T'$ and a tuple of disk automorphisms $(\psi _{v}:D\to D)_{{v\in V^{m}}}$ preserving the complex structure on $D$ such that $\zeta$ preserves the ordered tree structure and lengths $\ell _{e}=\ell '_{{\zeta (e)}}$ for every $e\in E$, and the marked points are preserved $\psi _{v}(\underline {z}_{v})=\underline {z}'_{{\zeta (v)}}$ for every $v\in V^{m}$.

For $d\geq 2$, such a tree is called stable if every vertex has valence $|v|\geq 3$ - i.e. there are at least three marked points on each disk $D_{v}$. While the above trees are not necessarily stable, they induce unique stable rooted metric ribbon trees $(T,\underline {\ell })$ in the sense of [Def.2.7, MW], by forgetting the marked points, forgetting every leaf of valence 1 and its outgoing edge, and replacing every vertex $v$ of valence 2 and its incoming and outgoing edges $(v^{-},v),(v,v^{+})$ by a single edge $(v^{-},v^{+})$ of length $\ell _{{(v^{-},v^{+})}}=\ell _{{(v^{-},v)}}+\ell _{{(v,v^{+})}}$. The space of such stable rooted metric ribbon trees - where a tree containing an edge of length $\ell _{e}=0$ is identified with the tree in which this edge and its adjacent vertices are replaced by a single vertex - is another topological representation of the Deligne Mumford space $\overline {\mathcal {M}}_{{d+1}}$, as discussed in [BV]. Its boundary strata are given by trees with interior edges of length $\ell _{e}=\infty$.

moduli spaces of pseudoholomorphic disk trees for any tuple $x_{0},x_{1},\ldots ,x_{d}\in {{\rm {Crit}}}(f)$ as in JL:

${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}):={\bigl \{}(T,\underline {\gamma },\underline {z},\underline {u})\,{\big |}\,{\text{1. - 5.}}{\bigr \}}/\sim$

$todo$

TODO: if J with no spheres, then compact and trivial isotropy [JL,Prop.2.5]

... otherwise add spheres as below and gnerally nontrivial isotropy

General moduli space of pseudoholomorphic polygons

Make up for differences in Hamiltonian symplectomorphisms applied to each Lagrangian by a domain-dependent Hamiltonian perturbation to the Cauchy-Riemann equation

Finally, the symplectic area function $\omega :\overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})\to \mathbb{R}$ in each case is given by TODO

Fredholm index ${\mathcal {M}}^{k}(\ldots )=\{b\in {\mathcal {M}}(\ldots )\,|\,IND...=k\}$