Ambient space

table of contents

This section will combine the techniques of [HWZ-GW] and [Li-thesis] to construct topological ambient spaces ${\mathcal {X}}(\underline {x})$ of the moduli spaces of pseudoholomorphic polygons $\overline {\mathcal {M}}(\underline {x})=\overline \partial _{{J,Y}}^{{-1}}(0)$, within which polyfold theory will construct the regularized moduli spaces $\overline {\mathcal {M}}(\underline {x};\nu )$. These will arise naturally from the data that was fixed when constructing the (unperturbed) moduli spaces of pseudoholomorphic polygons as part of the polyfold constructions for Fukaya categories.

Fixed data

We are given Lagrangians $L_{i}\subset M$ indexed cyclically by $i\in \mathbb{Z } _{{d+1}}$ and a tuple of generators $\underline {x}=(x_{0},\ldots ,x_{d})\in {\text{Crit}}(L_{0},L_{d})\times \ldots \times {\text{Crit}}(L_{{d-1}},L_{d})$ in their morphism spaces. These sets of generators depend on the choices of Morse functions $f_{i}:L_{i}\to \mathbb{R}$ on each Lagrangian and of Hamiltonian vector fields $X_{{L_{{i-1}},L_{i}}}$ whose time-1-flow produces transverse intersections $\phi _{{L_{{i-1}},L_{i}}}(L_{{i-1}})\pitchfork L_{i}$ whenever $L_{{i-1}}\neq L_{i}$. We moreover fix metrics on each $L_{i}$ so that the gradient vector fields $\nabla f_{i}$ are Morse-Bott and have standard Euclidean form near the critical points, such that the compactified Morse trajectory spaces $\overline {\mathcal {M}}(x_{i},L_{i})$ inherit a natural smooth structure (see [1]). On the other hand, the construction of the ambient space ${\mathcal {X}}(\underline {x})$ will be independent of the choice of almost complex structure on the symplectic manifold $M$, and the gluing construction for Hamiltonians that determines the PDEs $\overline \partial _{{J,Y}}u_{v}=0$ in [7. of general moduli space of pseudoholomorphic polygons].

The only auxiliary choice that we need to make is a (sufficiently small - as will be discussed elsewhere) Sobolev decay constant $\delta >0$.

The ambient set

We can now define the ambient space as set

${\mathcal {X}}(x_{0};x_{1},\ldots ,x_{d}):={\bigl \{}(T,\underline {\gamma },\underline {z},\underline {w},\underline {\beta },\underline {u})\,{\big |}\,{\text{1. - 8.}}{\bigr \}}/\sim$

which consists of the same data as the moduli spaces of pseudoholomorphic polygons $\overline {\mathcal {M}}(\underline {x})$, except for maps not necessarily being pseudoholomorphic but just of Sobolev $H^{{3,\delta }}$-regularity. (We indicate these differences in boldface.)

1. $T$ is an ordered tree with sets of vertices $V=V^{m}\cup V^{c}$ and edges $E$,

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equipped with orientations towards the root, orderings of incoming edges, and a partition into main and critical (leaf and root) vertices as follows:

• The edges $E\subset V\times V\setminus \Delta _{V}$ are oriented towards the root vertex $v_{0}\in V$ of the tree, so that 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 $E_{v}^{{{\rm {in}}}}=\{e=(\;\cdot \;,v)\in E\}$.
• Each set of incoming edges is ordered, $E_{v}^{{{\rm {in}}}}=\{e_{v}^{1},\ldots ,e_{v}^{{|v|-1}}\}$. This induces a cyclic order on the set of all edges $E_{v}:=\{e_{v}^{0},e_{v}^{1},\ldots ,e_{v}^{{|v|-1}}\}$ adjacent to $v$, by setting $e_{v}^{{|v|}}=e_{v}^{0}$, and we will denote consecutive edges in this order by $e=e_{v}^{i},e+1=e_{v}^{{i+1}}$. In particular this yields $e_{v}^{0}+i=e_{v}^{i}$.
• The set of vertices is partitioned $V=V^{m}\sqcup V^{c}$ into the sets of main vertices $V^{m}$ and critical vertices $V^{c}=\{v_{0}^{c},v_{1}^{c},\ldots v_{d}^{c}\}$. The latter is ordered to start with the root $v_{0}^{c}=v_{0}$ and then contains d leaves $v_{i}^{c}$ of the tree, with order induced by the orientation and order of the edges.
• The root vertex $v_{0}^{c}\in V^{c}$ has a single edge $\{e_{{v_{0}}}^{1}=(v,v_{0}^{c})\}=E_{{v_{0}}}^{{{\rm {in}}}}=E_{{v_{0}}}$, and this attaches to a main vertex $v\in V^{m}$ except for one special case: For $d=1$ and $L_{d}=L_{0}$ we allow the tree with a single edge $e=(v_{1}^{c},v_{0}^{c})$ between its two critical vertices $V=\{v_{0}^{c},v_{1}^{c}\}$.

2. The tree structure induces tuples of Lagrangians $\underline {L}=(\underline {L}^{v})_{{v\in V^{m}}}$

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that label the boundary components of domains in overall counter-clockwise order $L_{0},\ldots ,L_{d}$ as follows:

• For each main vertex $v\in V^{m}$ the Lagrangian label $\underline {L}^{v}=(L_{e}^{v})_{{e\in E_{v}}}$ is a cyclic sequence of Lagrangians $L_{e}^{v}\in \{L_{0},\ldots ,L_{d}\}$ indexed by the adjacent edges $E_{v}$ (which will become the boundary condition on $(\partial \Sigma ^{v})_{e}$).
• For each edge $e=(v^{-},v^{+})\in E$ the Lagrangian labels satisfy a matching condition as follows:
  • The edge from a critical leaf $v^{-}=v_{i}^{c}\in V^{c}$ requires $L_{e}^{{v^{+}}}=L_{i},L_{{e-1}}^{{v^{+}}}=L_{{i-1}}$.
  • The edge to the critical root $v^{+}=v_{0}^{c}\in V^{c}$ requires $L_{e}^{{v^{-}}}=L_{0},L_{{e-1}}^{{v^{-}}}=L_{d}$.
  • Any edge between main vertices $v^{-},v^{+}\in V^{m}$ requires $L_{{e}}^{{v^{-}}}=L_{{e-1}}^{{v^{+}}}$ and $L_{{e-1}}^{{v^{-}}}=L_{{e}}^{{v^{+}}}$.
  • Since $T$ has no further leaves, this determines the Lagrangian labels uniquely.

3. $\underline {\gamma }=(\underline {\gamma }_{e})_{{e\in E}}$ is a tuple of generalized Morse trajectories

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in the following compactified Morse trajectory spaces:

• Any edge $e=(v_{i}^{c},w)$ from a critical leaf $v_{i}^{c}$ to a main vertex $w\in V^{m}$ is labeled by a half-infinite Morse trajectory $\underline {\gamma }_{e}\in \overline {\mathcal {M}}(x_{i},L_{i})$ if $L_{{i-1}}=L_{i}$, resp. by the constant $\underline {\gamma }_{e}\equiv x_{i}\in {{\rm {Crit}}}(L_{{i-1}},L_{i})$ in the discrete space $\phi _{i}(L_{{i-1}})\cap L_{i}$ if $L_{{i-1}}\neq L_{i}$.
• If the edge to the root $e=(v,v_{0}^{c})$ attaches to a main vertex $v\in V^{m}$ then it is labeled by a half-infinite Morse trajectory $\underline {\gamma }_{e}\in \overline {\mathcal {M}}(L_{0},x_{0})$ if $L_{d}=L_{0}$, resp. by the constant $\underline {\gamma }_{e}\equiv x_{0}\in {{\rm {Crit}}}(L_{d},L_{0})$ in the discrete space $\phi _{0}(L_{d})\cap L_{0}$ if $L_{d}\neq L_{0}$.
• An edge $e=(v_{i}^{c},v_{j}^{c})$ between critical vertices is labeled by an infinite Morse trajectory $\underline {\gamma }_{e}\in \overline {\mathcal {M}}(x_{i},x_{j})$ (this occurs only for $d=1$ with $L_{0}=L_{1}$ and the tree with one edge $e=(v_{1}^{c},v_{0}^{c})$).
• Any edge $e=(v,w)$ between main vertices $v,w\in V^{m}$ is labeled by a finite or infinite Morse trajectory $\underline {\gamma }_{e}\in \overline {\mathcal {M}}(L_{e}^{v},L_{e}^{v})$ in case $L_{e}^{v}=L_{{e-1}}^{v}$, resp. by a constant $\underline {\gamma }_{e}\equiv x_{e}\in {{\rm {Crit}}}(L_{{e-1}}^{v},L_{{e}}^{v})$ in the discrete space $\phi _{{L_{{e-1}}^{v},L_{{e}}^{v}}}(L_{{e-1}}^{v})\cap L_{e}^{v}$ in case $L_{e}^{v}\neq L_{{e-1}}^{w}$. (Recall the matching condition $L_{e}^{v}=L_{{e-1}}^{w}$ and $L_{{e-1}}^{v}=L_{e}^{w}$ from 2.)

4. $\underline {z}=(\underline {z}_{v})_{{v\in V^{m}}}$ is a tuple of boundary points

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that correspond to the edges of $T$, are ordered counter-clockwise, and associate complex domains $\Sigma ^{v}:=D\setminus \underline {z}_{v}$ to the vertices as follows:

• For each main vertex $v$ there are $|v|$ pairwise disjoint marked points $\underline {z}_{v}=(z_{e}^{v})_{{e\in E_{v}}}\subset \partial D$ on the boundary of a disk.
• The order $E_{v}=\{e_{v}^{0},e_{v}^{1},\ldots ,e_{v}^{{|v|-1}}\}$ 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 $w\in V^{m}$
• To each main vertex $v\in V^{m}$ we associate the punctured disk $\Sigma ^{v}:=D\setminus \underline {z}_{v}$. Then the marked points $\underline {z}^{v}=(z_{e}^{v})_{{e\in E_{v}}}\subset \partial D$ partition the boundary into $|v|$ connected components $\partial \Sigma ^{v}=\textstyle \sqcup _{{e\in E_{v}}}(\partial \Sigma ^{v})_{e}$ such that the closure of each component $(\partial \Sigma ^{v})_{e}$ contains the marked points $z_{e}^{v},z_{{e+1}}^{v}$.

5. $\underline {w}=(\underline {w}_{v})_{{v\in V^{m}}}$ is a tuple of sphere bubble tree attaching points for each main vertex $v\in V^{m}$, given by an unordered subset $\underline {w}_{v}\subset \Sigma ^{v}\setminus \partial \Sigma ^{v}$ of the interior of the domain.

6. $\underline {\beta }=(\beta _{w})_{{w\in \underline {w}}}\subset \overline {\mathcal {M}}_{{0,1}}(J)$ is a tuple of not not-necessarily-pseudoholomorphic trees of sphere maps.

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More precisely, the trees of sphere maps $\beta _{w}=(T^{w},\underline {z}^{w},\underline {u}^{w})$ are indexed by the disjoint union $\underline {w}=\textstyle \bigsqcup _{{v\in V^{m}}}\underline {w}_{v}$ of sphere bubble tree attaching points, and consist of the following:

a. $T^{w}$ is a tree with sets of vertices $V^{w}$ and edges $E^{w}$, and a distinguished root vertex $v_{0}^{w}\in V^{w}$, which we use to orient all edges towards the root.

b. $\underline {z}^{w}=(\underline {z}_{v}^{w})_{{v\in V^{w}}}$ is a tuple of marked points on the spherical domains $\Sigma ^{{w,v}}=S^{2}$,

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indexed by the edges of $T^{w}$, and including a special root marked point as follows:

• For each vertex $v\neq v_{0}^{w}$ the tuple of mutually disjoint marked points $\underline {z}_{v}^{w}=(z_{e}^{{w,v}})_{{e\in E_{v}^{w}}}\subset S^{2}$ is indexed by the edges $E_{v}^{w}=\{e\in E^{w}\,|\,e=(v,\,\cdot \,)\;{\text{or}}\;e=(\,\cdot \,,v)\}$ adjacent to $v$.
• For the root vertex $v_{0}^{w}$ the tuple of mutually disjoint marked points $\underline {z}_{v}^{w}=(z_{e}^{{w,v}})_{{e\in E_{{v_{0}}}^{w}}}\subset S^{2}$ is also indexed by the edges adjacent to $v_{0}^{w}$, but is also required to be disjoint from the fixed marked point $z_{0}^{w}=0\in S^{2}\simeq \mathbb{C} \cup \{\infty \}$.
• The marked points, except for $z_{0}^{w}$, can also be denoted as $z_{e}^{{w,-}}=z_{e}^{{w,v^{-}}}$ and $z_{e}^{{w,+}}=z_{e}^{{w,v^{+}}}$ by the edges $e=(v^{-},v^{+})\in E$.

c. $\underline {u}^{w}=(u_{v}^{w})_{{v\in V^{w}}}$ is a tuple of not-necessarily-pseudoholomorphic sphere maps for each vertex, that is each $v\in V^{w}$ is labeled by a continuous map $u_{v}^{w}:S^{2}\to M$ satisfying Sobolev regularity and matching conditions as follows:

• The restriction of $u_{v}^{w}$ to the punctured sphere $S^{2}\setminus \underline {z}^{w}$ has ${\mathbf {H^{{{\mathbf {3,\delta }}}}}}$-Sobolev regularity as in [Definition 1.1, HWZ-GW]. That is, $u_{v}^{w}|_{{S^{2}\setminus \underline {z}^{w}}}$ is of class $H^{3}$ on any compact subset, and $(s,t)\mapsto e^{{\delta s}}{\bigl (}\psi ^{{-1}}{\bigl (}u_{v}^{w}(\iota (s,t){\bigr )}{\bigr )}$ in $H^{3}([0,\infty )\times S^{1},\mathbb{R} ^{{2n}})$ for any holomorphic coordinates $\iota :[0,\infty )\times S^{1}\hookrightarrow S^{2}\setminus \underline {z}^{w}$ near a marked point $z_{e}^{{w,v}}$ and a chart map $\psi :\mathbb{R} ^{{2n}}\hookrightarrow M$ with $\psi (0)=u_{v}^{w}(z_{e}^{{w,v}})$.(Independence of these conditions from the choices of charts is proven e.g. in [Section 3.3, J.Li].)
• The matching conditions are $u_{{v^{-}}}^{w}(z_{e}^{{w,-}})=u_{{v^{+}}}^{w}(z_{e}^{{w,+}})$ for each edge $e=(v^{-},v^{+})\in E$.

7. $\underline {u}=(\underline {u}_{v})_{{v\in V^{m}}}$ is a tuple of not-necessarily-pseudoholomorphic disk maps for each main vertex.

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More precisely, each $v\in V^{m}$ is labeled by a continuous map $u_{v}:\Sigma ^{v}\to M$ satisfying ${\mathbf {H^{{{\mathbf 3},{\boldsymbol {\delta }}}}}}$-Sobolev regularity, Lagrangian boundary conditions, and matching conditions as follows:

• We have $u_{v}|_{{\Sigma ^{v}\setminus \underline {w}_{v}}}$ is of class $H^{3}$ on any compact subset.
• Near each sphere bubble tree attaching point $w\in \underline {w}_{v}$ we have $(s,t)\mapsto e^{{\delta s}}{\bigl (}\psi ^{{-1}}{\bigl (}u_{v}(\iota (s,t){\bigr )}{\bigr )}$ in $H^{3}([0,\infty )\times S^{1},\mathbb{R} ^{{2n}})$ for any choice of holomorphic coordinates $\iota :[0,\infty )\times S^{1}\hookrightarrow \Sigma ^{v}\setminus \underline {w}_{v}$ near $w$ and a chart map $\psi :\mathbb{R} ^{{2n}}\hookrightarrow M$ with $\psi (0)=u_{v}(w)$.
• Near each puncture $z_{e}^{v}\in \underline {z}_{v}$ with $L_{{e-1}}^{v}=L_{e}^{v}$, the limit $u_{v}(z_{e}^{v}):=\lim _{{z\to z_{e}^{v}}}u_{v}(z)\in M$ exists and we have $H^{{3,\delta }}$-regularity as in [Definition 3.12, J.Li], that is $(s,t)\mapsto e^{{\delta s}}{\bigl (}\psi ^{{-1}}{\bigl (}u_{v}(\iota (s,t){\bigr )}{\bigr )}$ in $H^{3}([0,\infty )\times [0,\pi ],\mathbb{R} ^{{2n}})$ for any choice of holomorphic coordinates $\iota :[0,\infty )\times [0,\pi ]\hookrightarrow \Sigma ^{v}$ near $z_{e}^{v}$ and a chart map $\psi :\mathbb{R} ^{{2n}}\hookrightarrow M$ with $\psi (0)=u_{v}(z_{e}^{v})$.
• Near each puncture $z_{e}^{v}\in \underline {z}_{v}$ with $L_{{e-1}}^{v}\neq L_{e}^{v}$ and for any choice of biholomorphic coordinates $\iota :[0,\infty )\times [0,1]\hookrightarrow \Sigma ^{v}$ such that $(\iota )^{*}Y_{v}=X_{{L_{{e-1}}^{v},L_{e}^{v}}}\,{{\rm {d}}}t$, we have a uniform limit $\lim _{{s\to \infty }}u_{v}(\iota (s,t))=\phi _{{L_{{e-1}}^{v},L_{e}^{v}}}^{{t-1}}(x_{e})$ for some $x_{e}\in {\text{Crit}}(L_{{e-1}}^{v},L_{e}^{v})$, and $(s,t)\mapsto e^{{\delta s}}{\bigl (}\psi ^{{-1}}{\bigl (}\phi _{{L_{{e-1}}^{v},L_{e}^{v}}}^{{1-t}}{\bigl (}u_{v}(\iota (s,t){\bigr )}{\bigr )}{\bigr )}$ in $H^{3}([0,\infty )\times [0,1],\mathbb{R} ^{{2n}})$ for any chart map $\psi :\mathbb{R} ^{{2n}}\hookrightarrow M$ with $\psi (0)=x_{e}$. (Here the form of the Hamiltonian perturbation $Y_{v}$ that is fixed in [7. of general moduli space of pseudoholomorphic polygons] - and more precisely constructed in gluing construction for Hamiltonians - is in fact determined near the puncture by the thick-thin decomposition of the Riemann surface $\Sigma _{v}$.)
• The Lagrangian boundary conditions are $u_{v}(\partial \Sigma ^{v})\subset \underline {L}^{v}$; more precisely this requires $u_{v}{\bigl (}(\partial \Sigma ^{v})_{e}{\bigr )}\subset L_{e}^{v}$ for each adjacent edge $e\in E_{v}$.
• The matching conditions for sphere bubble trees are $u^{v}(w)={\text{ev}}_{0}(\beta _{w})$ for each main vertex $v\in V^{m}$ and sphere bubble tree attaching point $w\in \underline {w}_{v}$.
• The Sobolev regularity implies uniform convergence of $u_{v}$ near each puncture $z_{e}^{v}$, and the limits are required to satisfy the following matching conditions:
  • For edges $e\in E_{v}$ whose Lagrangian boundary conditions $L_{{e-1}}^{v}=L_{e}^{v}$ agree, the map $u_{v}$ extends smoothly to the puncture $z_{e}^{v}$, and its value is required to match with the evaluation of the Morse trajectory $\underline {\gamma }_{e}$ associated to the edge $e=(v_{e}^{-},v_{e}^{+})$, that is $u_{v}(z_{e}^{v})={{\rm {ev}}}^{\pm }(\underline {\gamma }_{e})$ for $v=v_{e}^{\mp }$.
  • For edges $e\in E_{v}$ with different Lagrangian boundary conditions $L_{{e-1}}^{v}\neq L_{e}^{v}$, the map $u_{e}^{v}:=(\iota _{e}^{v})^{*}u_{v}:(-\infty ,0)\times [0,1]\to M$ has a uniform limit $\lim _{{s\to \infty }}u^{v}(s,t)=\phi _{{L_{{e-1}}^{v},L_{e}^{v}}}^{{t-1}}(x_{e})$ for some $x_{e}\in \phi _{{L_{{e-1}}^{v},L_{e}^{v}}}(L_{{e-1}}^{v})\cap L_{e}^{v}={\text{Crit}}(L_{{e-1}}^{v},L_{e}^{v})$, and this limit intersection point is required to match with the value of the constant 'Morse trajectory' $\underline {\gamma }_{e}\in {\text{Crit}}(L_{{e-1}}^{v},L_{e}^{v})$ associated to the edge $e=(v_{e}^{-},v_{e}^{+})$, that is $x_{e}=\lim _{{s\to -\infty }}u^{v}(s,1)=\underline {\gamma }_{e}$.

8. The generalized pseudoholomorphic polygon is stable

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in the following sense:

• For any main vertex $v\in V^{m}$ with fewer than three special points $\#\underline {z}_{v}+2\#\underline {w}_{v}<3$, the map differential ${{\rm {d}}}_{z}u_{v}:{{\rm {T}}}_{z}\Sigma ^{v}\to {{\rm {T}}}_{{u_{v}(z)}}M$ must be injective on an open subset of $\Sigma ^{v}$.
• Each sphere bubble tree $\beta _{w}$ is stable in the sense that for any vertex $v\in V^{w}$ with fewer than three special points (marked points corresponding to edges or the root marked point) the map has nonzero energy $\textstyle \int {u_{v}^{w}}^{*}\omega >0$ .

Finally, two generalized pseudoholomorphic polygons are equivalent $(T,\underline {\gamma },\underline {z},\underline {w},\underline {\beta },\underline {u})\sim (T',\underline {\gamma }',\underline {z}',\underline {w}',\underline {\beta }',\underline {u}')$ if there is a tree isomorphism $\zeta :T\to T'$, a tuple of disk biholomorphisms $(\psi _{v}:D\to D)_{{v\in V^{m}}}$, and a tuple of sphere tree isomorphisms ${\bigl (}\zeta ^{w},\underline {\psi }^{w}{\bigr )}_{{w\in \underline {w}}}$,

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which preserve the tree, Morse trajectories, marked points, maps, and sphere bubble trees in the sense that

• $\zeta$ preserves the tree structure and order of edges;
• $\underline {\gamma }_{e}=\underline {\gamma }'_{{\zeta (e)}}$ for every $e\in E$;
• $\psi _{v}(z_{e}^{v})={z'}_{{\zeta (e)}}^{{\zeta (v)}}$ for every $v\in V^{m}$ and adjacent edge $e\in E_{v}$;
• the maps are related by reparametrization, $u_{v}=u'_{{\zeta (v)}}\circ \psi _{v}$ for every $v\in V^{m}$;
• the unordered sphere bubble tree attaching set is preserved $\psi _{v}(\underline {w}_{v})=\underline {w}'_{{\zeta (v)}}$ for every $v\in V^{m}$;
• for each $\overline v\in V^{m}$ and $w\in \underline {w}_{{\overline v}}$ the sphere tree isomorphism ${\bigl (}\zeta ^{w},\underline {\psi }^{w}{\bigr )}$ consists of a tree isomorphism $\zeta ^{w}:T^{w}\to {T'}^{{w'}}$ with $w':=\psi _{{\overline v}}(w)$ and a tuple of sphere biholomorphisms $\underline {\psi }^{w}=(\psi _{v}^{w}:S^{2}\to S^{2})_{{v\in V^{w}}}$ which preserve the tree, marked points, and sphere maps in the sense that
  • $\zeta ^{w}$ preserves the tree structure, in particular maps the root vertex $v_{0}$ of $T^{w}$ to the root vertex of $v_{0}'$ of ${T'}^{{w'}}$;
  • $\psi _{{v_{0}}}^{w}(0)=0$ and $\psi _{v}^{w}(z_{e}^{{w,v}})={z'}_{{\zeta ^{w}(e)}}^{{w',\zeta ^{w}(v)}}$ for every $v\in V^{w}$ and adjacent edge $e\in E_{v}^{w}$;
  • the spheres maps are related by reparametrization, $u_{v}^{w}={u'}_{{\zeta ^{w}(v)}}^{{w'}}\circ \psi _{v}^{w}$ for every $v\in V^{w}$.

The symplectic area function

The symplectic area function on the ambient space is defined in the same way as on the moduli space. In particular, if for each pair $\{i,j\}\subset \{0,\ldots ,d\}$ the Lagrangians are either identical $L_{i}=L_{j}$ or transverse $L_{i}\pitchfork L_{j}$ (so that no Hamiltonian perturbations are needed), then it is literally the sum of symplectic areas,

$\omega :{\mathcal {X}}(\underline {x})\to \mathbb{R} ,\quad b={\bigl [}T,\underline {\gamma },\underline {z},\underline {w},\underline {\beta },\underline {u}{\bigr ]}\mapsto \omega (b):=\sum _{{v\in V^{m}}}{\textstyle \int _{{\Sigma _{v}}}u_{v}^{*}\omega }\;+\;\sum _{{w\in \underline {w}}}\sum _{{v\in V^{w}}}{\textstyle \int _{{s^{2}}}{u_{v}^{w}}^{*}\omega }=\langle [\omega ],[b]\rangle .$

Again, this only depends on the total homology class of the generalized polygon

$[b]:=\sum _{{v\in V}}(\overline {u}_{v})_{*}[D]+\sum _{{w\in \underline {w}}}\sum _{{v\in V^{w}}}(u_{v}^{w})_{*}[S^{2}]\;\in \;H_{2}(M;L_{0}\cup L_{1}\ldots \cup L_{d}).$

Here $\overline {u}_{v}:D\to M$ is defined by unique continuous continuation to the punctures $z_{e}^{v}$ at which $L_{{e-1}}^{v}=L_{e}^{v}$ or $L_{{e-1}}^{v}\pitchfork L_{e}^{v}$.

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Differential Geometric TODO ( copy from Moduli spaces of pseudoholomorphic polygons )

In the presence of Hamiltonian perturbations, the definition of the symplectic area function needs to be adjusted to match with the symplectic action functional on the Floer complexes and satisfy some properties (which also need to be proven in the unperturbed case):

• $\omega (b)$ is invariant under deformations with fixed limits (used in proof of A-infty relations and invariance);
• a bound on $\omega (b)$ needs to imply Gromov-compactness ... which requires an area-energy identity for J-curves, but we are allowed (bounded!) error terms from e.g. Hamiltonian perturbations;
• invariance proofs arguing with 'upper triangular form' require contributions to $\mu ^{1}$ to be of positive symplectic area, or constant strips/disks for zero symplectic area;
• to work with the Novikov ring, rather than field, the symplectic area needs to be nonnegative for all polygons (not just the strips and other contributions to $\mu ^{1}$ for which this is automatic). It *might* be possible to achieve this by remembering that the Hamilton functions for each pair of Lagrangians are only fixed up to a constant (not see in the Hamiltonian vector field or time-1-flow), so when constructing the Hamiltonian perturbation vector fields over Deligne-Mumford spaces, it might be possible to shift e.g. the outgoing Hamiltonian in such a way that the vector field can be constructed with 'curvature terms of the correct sign'.

Topology on the ambient space

We construct the $H^{{3,\delta }}$-Gromov topology on the ambient set ${\mathcal {X}}(\underline {x})$ by specifying for each ${\hat b}\in {\mathcal {X}}(\underline {x})$, choice of representative ${\hat b}=[({\hat T},\underline {{\hat \gamma }},\underline {{\hat z}},\underline {{\hat w}},\underline {{\hat \beta }},\underline {{\hat u}})]$, and $\epsilon >0$ the $\epsilon$-neighborhood of ${\hat b}$,

${\mathcal {U}}_{\epsilon }({\hat T},\underline {{\hat \gamma }},\underline {{\hat z}},\underline {{\hat w}},\underline {{\hat \beta }},\underline {{\hat u}}):={\bigl \{}(T,\underline {\gamma },\underline {z},\underline {w},\underline {\beta },\underline {u})\,{\big |}\,{\text{1. - 8.}}{\bigr \}}/\sim \subset {\mathcal {X}}(\underline {x})$

to consist of equivalence classes of tuples $(T,\underline {\gamma },\underline {z},\underline {w},\underline {\beta },\underline {u})$ that are $\epsilon$-close to $({\hat T},\underline {{\hat \gamma }},\underline {{\hat z}},\underline {{\hat w}},\underline {{\hat \beta }},\underline {{\hat u}})$ in the following sense:

1. The tree $T$ is obtained from ${\hat T}$ by collapsing some of the glueable edges

${\hat E}^{0}=\{e\in {\hat E}\,|\,\ell (\underline {{\hat \gamma }}_{e})=0\}$ .

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More precisely, there exists a subset $E^{{{\rm {glue}}}}\subset {\hat E}^{0}$ of gluing edges such that the tree $T$ and its additional structure is given as follows:

• $E^{{{\rm {glue}}}}$ is a (possibly empty) subset of ${\hat E}^{0}$, which more precisely consists of all those edges of ${\hat T}$ that are labeled with a Morse trajectory of renormalized length $\ell (\underline {{\hat \gamma }}_{e})=0$, or with a constant in one of the discrete sets ${{\rm {Crit}}}(L_{{e-1}}^{{v}},L_{e}^{{v}})$ for edges whose associated Lagrangians $L_{{e-1}}^{{v}}\neq L_{e}^{{v}}$ do not agree.
• The edges $E={\hat E}\setminus E^{{{\rm {glue}}}}$ of $T$ are the nongluing edges, with the same orientation as in ${\hat E}$.
• The critical vertices of $T$ are the same, $V^{c}={\hat V}^{c}$, while the main vertices $v\in V^{m}$ of $T$ are the maximal subsets $v\subset {\hat V}^{m}$ of vertices that are connected by gluing edges.
• This preserves the critical root vertex $v_{0}=v_{0}^{c}={\hat v}_{0}^{c}={\hat v}_{0}$ and critical leaves $v_{i}^{c}={\hat v}_{i}^{c}$, as well as their order.
• The sets of incoming edges $E_{v}^{{{\rm {in}}}}=\textstyle \bigcup _{{{\hat v}\in v}}{\hat E}_{{{\hat v}}}^{{{\rm {in}}}}\setminus E^{{{\rm {glue}}}}$ inherit an order.

3. The generalized Morse trajectory $\underline {\gamma }_{e}$ for each edge $e\in E={\hat E}\setminus E^{{{\rm {glue}}}}$ is $\epsilon$-close to $\underline {{\hat \gamma }}_{e}$ in the sense that $d_{{\overline {\mathcal {M}}}}(\underline {\gamma }_{e},\underline {{\hat \gamma }}_{e})<\epsilon$. Here $d_{{\overline {\mathcal {M}}}}$ is the metric on the relevant compactified Morse trajectory space resp. the discrete metric on ${{\rm {Crit}}}(L_{{e-1}}^{{v}},L_{e}^{{v}})$ in case $L_{{e-1}}^{{v}}\neq L_{e}^{{v}}$ for the Lagrangians associated to the edge in either tree.

construction site

4. The boundary marked points $\underline {z}_{v}\subset \partial D$ for each main vertex $v\in V^{m}$ are $\epsilon$-close to the marked points $\textstyle \bigcup _{{{\hat v}\in v}}\underline {{\hat z}}_{{{\hat v}}}$

Katrin's work in progress here

$\underline {z}=(\underline {z}_{v})_{{v\in V^{m}}}$ is a tuple of boundary points

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that correspond to the edges of $T$, are ordered counter-clockwise, and associate complex domains $\Sigma ^{v}:=D\setminus \underline {z}_{v}$ to the vertices as follows:

• For each main vertex $v$ there are $|v|$ pairwise disjoint marked points $\underline {z}_{v}=(z_{e}^{v})_{{e\in E_{v}}}\subset \partial D$ on the boundary of a disk.
• The order $E_{v}=\{e_{v}^{0},e_{v}^{1},\ldots ,e_{v}^{{|v|-1}}\}$ 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 $w\in V^{m}$
• To each main vertex $v\in V^{m}$ we associate the punctured disk $\Sigma ^{v}:=D\setminus \underline {z}_{v}$. Then the marked points $\underline {z}^{v}=(z_{e}^{v})_{{e\in E_{v}}}\subset \partial D$ partition the boundary into $|v|$ connected components $\partial \Sigma ^{v}=\textstyle \sqcup _{{e\in E_{v}}}(\partial \Sigma ^{v})_{e}$ such that the closure of each component $(\partial \Sigma ^{v})_{e}$ contains the marked points $z_{e}^{v},z_{{e+1}}^{v}$.

5. $\underline {w}=(\underline {w}_{v})_{{v\in V^{m}}}$ is a tuple of sphere bubble tree attaching points for each main vertex $v\in V^{m}$, given by an unordered subset $\underline {w}_{v}\subset \Sigma ^{v}\setminus \partial \Sigma ^{v}$ of the interior of the domain.

6. $\underline {\beta }=(\beta _{w})_{{w\in \underline {w}}}\subset \overline {\mathcal {M}}_{{0,1}}(J)$ is a tuple of not not-necessarily-pseudoholomorphic trees of sphere maps.

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More precisely, the trees of sphere maps $\beta _{w}=(T^{w},\underline {z}^{w},\underline {u}^{w})$ are indexed by the disjoint union $\underline {w}=\textstyle \bigsqcup _{{v\in V^{m}}}\underline {w}_{v}$ of sphere bubble tree attaching points, and consist of the following:

a. $T^{w}$ is a tree with sets of vertices $V^{w}$ and edges $E^{w}$, and a distinguished root vertex $v_{0}^{w}\in V^{w}$, which we use to orient all edges towards the root.

b. $\underline {z}^{w}=(\underline {z}_{v}^{w})_{{v\in V^{w}}}$ is a tuple of marked points on the spherical domains $\Sigma ^{{w,v}}=S^{2}$,

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indexed by the edges of $T^{w}$, and including a special root marked point as follows:

• For each vertex $v\neq v_{0}^{w}$ the tuple of mutually disjoint marked points $\underline {z}_{v}^{w}=(z_{e}^{{w,v}})_{{e\in E_{v}^{w}}}\subset S^{2}$ is indexed by the edges $E_{v}^{w}=\{e\in E^{w}\,|\,e=(v,\,\cdot \,)\;{\text{or}}\;e=(\,\cdot \,,v)\}$ adjacent to $v$.
• For the root vertex $v_{0}^{w}$ the tuple of mutually disjoint marked points $\underline {z}_{v}^{w}=(z_{e}^{{w,v}})_{{e\in E_{{v_{0}}}^{w}}}\subset S^{2}$ is also indexed by the edges adjacent to $v_{0}^{w}$, but is also required to be disjoint from the fixed marked point $z_{0}^{w}=0\in S^{2}\simeq \mathbb{C} \cup \{\infty \}$.
• The marked points, except for $z_{0}^{w}$, can also be denoted as $z_{e}^{{w,-}}=z_{e}^{{w,v^{-}}}$ and $z_{e}^{{w,+}}=z_{e}^{{w,v^{+}}}$ by the edges $e=(v^{-},v^{+})\in E$.

c. $\underline {u}^{w}=(u_{v}^{w})_{{v\in V^{w}}}$ is a tuple of not-necessarily-pseudoholomorphic sphere maps for each vertex, that is each $v\in V^{w}$ is labeled by a continuous map $u_{v}^{w}:S^{2}\to M$ satisfying Sobolev regularity and matching conditions as follows:

• The restriction of $u_{v}^{w}$ to the punctured sphere $S^{2}\setminus \underline {z}^{w}$ has ${\mathbf {H^{{{\mathbf {3,\delta }}}}}}$-Sobolev regularity as in [Definition 1.1, HWZ-GW]. That is, $u_{v}^{w}|_{{S^{2}\setminus \underline {z}^{w}}}$ is of class $H^{3}$ on any compact subset, and $(s,t)\mapsto e^{{\delta s}}{\bigl (}\psi ^{{-1}}{\bigl (}u_{v}^{w}(\iota (s,t){\bigr )}{\bigr )}$ in $H^{3}([0,\infty )\times S^{1},\mathbb{R} ^{{2n}})$ for any holomorphic coordinates $\iota :[0,\infty )\times S^{1}\hookrightarrow S^{2}\setminus \underline {z}^{w}$ near a marked point $z_{e}^{{w,v}}$ and a chart map $\psi :\mathbb{R} ^{{2n}}\hookrightarrow M$ with $\psi (0)=u_{v}^{w}(z_{e}^{{w,v}})$.(Independence of these conditions from the choices of charts is proven e.g. in [Section 3.3, J.Li].)
• The matching conditions are $u_{{v^{-}}}^{w}(z_{e}^{{w,-}})=u_{{v^{+}}}^{w}(z_{e}^{{w,+}})$ for each edge $e=(v^{-},v^{+})\in E$.

7. $\underline {u}=(\underline {u}_{v})_{{v\in V^{m}}}$ is a tuple of not-necessarily-pseudoholomorphic disk maps for each main vertex.

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More precisely, each $v\in V^{m}$ is labeled by a continuous map $u_{v}:\Sigma ^{v}\to M$ satisfying ${\mathbf {H^{{{\mathbf 3},{\boldsymbol {\delta }}}}}}$-Sobolev regularity, Lagrangian boundary conditions, and matching conditions as follows:

• We have $u_{v}|_{{\Sigma ^{v}\setminus \underline {w}_{v}}}$ is of class $H^{3}$ on any compact subset.
• Near each sphere bubble tree attaching point $w\in \underline {w}_{v}$ we have $(s,t)\mapsto e^{{\delta s}}{\bigl (}\psi ^{{-1}}{\bigl (}u_{v}(\iota (s,t){\bigr )}{\bigr )}$ in $H^{3}([0,\infty )\times S^{1},\mathbb{R} ^{{2n}})$ for any choice of holomorphic coordinates $\iota :[0,\infty )\times S^{1}\hookrightarrow \Sigma ^{v}\setminus \underline {w}_{v}$ near $w$ and a chart map $\psi :\mathbb{R} ^{{2n}}\hookrightarrow M$ with $\psi (0)=u_{v}(w)$.
• Near each puncture $z_{e}^{v}\in \underline {z}_{v}$ with $L_{{e-1}}^{v}=L_{e}^{v}$, the limit $u_{v}(z_{e}^{v}):=\lim _{{z\to z_{e}^{v}}}u_{v}(z)\in M$ exists and we have $H^{{3,\delta }}$-regularity as in [Definition 3.12, J.Li], that is $(s,t)\mapsto e^{{\delta s}}{\bigl (}\psi ^{{-1}}{\bigl (}u_{v}(\iota (s,t){\bigr )}{\bigr )}$ in $H^{3}([0,\infty )\times [0,\pi ],\mathbb{R} ^{{2n}})$ for any choice of holomorphic coordinates $\iota :[0,\infty )\times [0,\pi ]\hookrightarrow \Sigma ^{v}$ near $z_{e}^{v}$ and a chart map $\psi :\mathbb{R} ^{{2n}}\hookrightarrow M$ with $\psi (0)=u_{v}(z_{e}^{v})$.
• Near each puncture $z_{e}^{v}\in \underline {z}_{v}$ with $L_{{e-1}}^{v}\neq L_{e}^{v}$ and for any choice of biholomorphic coordinates $\iota :[0,\infty )\times [0,1]\hookrightarrow \Sigma ^{v}$ such that $(\iota )^{*}Y_{v}=X_{{L_{{e-1}}^{v},L_{e}^{v}}}\,{{\rm {d}}}t$, we have a uniform limit $\lim _{{s\to \infty }}u_{v}(\iota (s,t))=\phi _{{L_{{e-1}}^{v},L_{e}^{v}}}^{{t-1}}(x_{e})$ for some $x_{e}\in {\text{Crit}}(L_{{e-1}}^{v},L_{e}^{v})$, and $(s,t)\mapsto e^{{\delta s}}{\bigl (}\psi ^{{-1}}{\bigl (}\phi _{{L_{{e-1}}^{v},L_{e}^{v}}}^{{1-t}}{\bigl (}u_{v}(\iota (s,t){\bigr )}{\bigr )}{\bigr )}$ in $H^{3}([0,\infty )\times [0,1],\mathbb{R} ^{{2n}})$ for any chart map $\psi :\mathbb{R} ^{{2n}}\hookrightarrow M$ with $\psi (0)=x_{e}$. (Here the form of the Hamiltonian perturbation $Y_{v}$ that is fixed in [7. of general moduli space of pseudoholomorphic polygons] - and more precisely constructed in gluing construction for Hamiltonians - is in fact determined near the puncture by the thick-thin decomposition of the Riemann surface $\Sigma _{v}$.)
• The Lagrangian boundary conditions are $u_{v}(\partial \Sigma ^{v})\subset \underline {L}^{v}$; more precisely this requires $u_{v}{\bigl (}(\partial \Sigma ^{v})_{e}{\bigr )}\subset L_{e}^{v}$ for each adjacent edge $e\in E_{v}$.
• The matching conditions for sphere bubble trees are $u^{v}(w)={\text{ev}}_{0}(\beta _{w})$ for each main vertex $v\in V^{m}$ and sphere bubble tree attaching point $w\in \underline {w}_{v}$.
• The Sobolev regularity implies uniform convergence of $u_{v}$ near each puncture $z_{e}^{v}$, and the limits are required to satisfy the following matching conditions:
  • For edges $e\in E_{v}$ whose Lagrangian boundary conditions $L_{{e-1}}^{v}=L_{e}^{v}$ agree, the map $u_{v}$ extends smoothly to the puncture $z_{e}^{v}$, and its value is required to match with the evaluation of the Morse trajectory $\underline {\gamma }_{e}$ associated to the edge $e=(v_{e}^{-},v_{e}^{+})$, that is $u_{v}(z_{e}^{v})={{\rm {ev}}}^{\pm }(\underline {\gamma }_{e})$ for $v=v_{e}^{\mp }$.
  • For edges $e\in E_{v}$ with different Lagrangian boundary conditions $L_{{e-1}}^{v}\neq L_{e}^{v}$, the map $u_{e}^{v}:=(\iota _{e}^{v})^{*}u_{v}:(-\infty ,0)\times [0,1]\to M$ has a uniform limit $\lim _{{s\to \infty }}u^{v}(s,t)=\phi _{{L_{{e-1}}^{v},L_{e}^{v}}}^{{t-1}}(x_{e})$ for some $x_{e}\in \phi _{{L_{{e-1}}^{v},L_{e}^{v}}}(L_{{e-1}}^{v})\cap L_{e}^{v}={\text{Crit}}(L_{{e-1}}^{v},L_{e}^{v})$, and this limit intersection point is required to match with the value of the constant 'Morse trajectory' $\underline {\gamma }_{e}\in {\text{Crit}}(L_{{e-1}}^{v},L_{e}^{v})$ associated to the edge $e=(v_{e}^{-},v_{e}^{+})$, that is $x_{e}=\lim _{{s\to -\infty }}u^{v}(s,1)=\underline {\gamma }_{e}$.