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 [Seidel book]:

${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}):={\bigl \{}(\underline {z},u)\,{\big |}\,{\text{1. - 3.}}{\bigr \}}/\sim$

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 $i=0,1,\ldots ,d$.
3. The pseudoholomorphic polygon $(\underline {z},u)$ is stable in the sense that the map $u:\Sigma _{{\underline {z}}}\to M$ is nonconstant if the number of marked points is $d+1<3$.

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 $\psi (z_{i})=z'_{i}$, 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. However, it might appear in the construction of homotopy units?

The domains of the pseudoholomorphic polygons are strips for $d=1$ and represent elements in a Deligne-Mumford space for $d\geq 2$.

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 $\tau \in \mathbb{R}$. This is the only case in which the stability condition is nontrivial: It requires the maps $u:\mathbb{R} \times [0,1]\to M$ to be nonconstant.

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.

So far, all isotropy groups of this uncompactified moduli space ${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ are trivial; that is any disk automorphism $\psi :D\to D$ that fixes $d+1\geq 1$ marked points $\psi (z_{i})=z_{i}$, and preserves a pseudoholomorphic map $u=u\circ \psi$ must be the identity $\psi ={{\rm {id}}}_{D}$.

Proof:

In case $d=2$ this follows directly from the marked points, since any Mobius transformation that fixes three points is the identity.

In case $d=1$ this requires both the stability and finite energy conditions: The group of automorphisms that fix two marked points - i.e. the automorphisms of the strip - are shifts by $\mathbb{R}$. On the other hand, any J-holomorphic map $u:\mathbb{R} \times [0,1]\to M$ has nonnegative energy density $u^{*}\omega ={\bigl (}|\partial _{s}u|^{2}+|\partial _{t}u|^{2}{\bigr )}ds\wedge dt$ with $|\partial _{s}u|=|\partial _{t}u|$. If we now had nontrivial isotropy, i.e. $u(\tau +s,t)=u(s,t)$ for some $\tau >0$ and a nonconstant map $u$, then there would exist $s_{0},t_{0}\in \mathbb{R} \times [0,1]$ with $|\partial _{s}u(s_{0},t_{0})|=|\partial _{t}u(s_{0},t_{0})|>0$ and thus $\textstyle \int _{{[s_{0}-{\frac {1}{2}}\tau ]\times [0,1]}}^{{[s_{0}+{\frac {1}{2}}\tau ]\times [0,1]}}u^{*}\omega >0$. However, this is in contradiction to $u$ having finite energy,

$\infty >\textstyle \int _{{\mathbb{R} \times [0,1]}}u^{*}\omega =\sum _{{k\in \mathbb{Z } }}\int _{{[s_{0}+(k-{\frac {1}{2}})\tau ]\times [0,1]}}^{{[s_{0}+(k+{\frac {1}{2}})\tau ]\times [0,1]}}u^{*}\omega =\sum _{{k\in \mathbb{Z } }}\int _{{[s_{0}-{\frac {1}{2}}\tau ]\times [0,1]}}^{{[s_{0}+{\frac {1}{2}}\tau ]\times [0,1]}}u^{*}\omega .$

Next, 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 breaking and bubbling, in particular

• 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.

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. (On the other hand, sphere bubbling will be the only source of nontrivial isotropy.) 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, which is problematic for a combination of algebra and regularity reasons.

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 [J.Li thesis] 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 $e_{v}^{0}=(v,\;\cdot \;)\in E$ (except for the root vertex which has no outgoing edge) 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).

$\underline {\gamma }=(\underline {\gamma }_{e})_{{e\in E}}$ 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 $v,w\in V^{m}$.