Difference between revisions of "Ambient space"

Revision as of 00:30, 8 June 2017

table of contents Katrin's work in progress here

This section constructs 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 depend only on the choice of a Sobolev decay constant $\delta >0$ .

Thus 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. The latter depend on the choices of Hamiltonian symplectomorphisms $\phi _{i}:=\phi _{{L_{{i-1}},L_{i}}}$ so that $\phi _{i}(L_{{i-1}})\pitchfork L_{i}$ whenever $L_{{i-1}}\neq L_{i}$ , as specified in polyfold constructions for Fukaya categories and moduli spaces of pseudoholomorphic polygons.

TODO: mention choice of Morse functions (? also in other places previously ?)

Then - as set - we define the ambient space as

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

where the data is the same as in the construction of 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:

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:

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:

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:

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. $\underline {{\hat \beta }}=({\hat \beta }_{w})_{{w\in \underline {w}}}\subset \overline {\mathcal {M}}_{{0,1}}(J)$ is a tuple of not not-necessarily-pseudoholomorphic trees of sphere maps ${\hat \beta }_{w}=(T^{w},\underline {z}^{w},\underline {u}^{w})$ indexed by the disjoint union $\underline {w}=\textstyle \bigsqcup _{{v\in V^{m}}}\underline {w}_{v}$ of sphere bubble tree attaching points, and consisting 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:

c. $\underline {u}^{w}=(\underline {u}_{v}^{w})_{{v\in V^{w}}}$ 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:

TODO: $H^{{3,\delta }}$ -Sobolev regularity ... on punctured spheres ?!?
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, that is each $v\in V^{m}$ is labeled by a continuous map $u_{v}:\Sigma ^{v}\to M$ satisfying the TODO: $H^{{3,\delta }}$ -Sobolev regularity - and make sure it implies uniform convergence

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Moreover, each $u_{v}$ satisfies Lagrangian boundary conditions, and matching conditions as follows:

8. The generalized pseudoholomorphic polygon is stable in the following sense:

TODO: update to Ham pert, sphere bubble tree

Any main vertex $v\in V^{m}$ whose map has zero energy $\textstyle \int u_{v}^{*}\omega =0$ has enough special points to have trivial isotropy, that is the number of boundary marked points $|v|=\#\underline {z}_{v}$ plus twice the number of interior marked points $\#\underline {w}_{v}$ is at least 3.
Each sphere bubble tree is stable in the sense that

any vertex $v\in V$ whose map has zero energy $\textstyle \int u_{v}^{*}\omega =0$ (which is equivalent to $u_{v}$ being constant) has valence $|v|\geq 3$ . Here the marked point $z_{0}$ counts as one towards the valence $|v_{0}|$ of the root vertex; in other words the root vertex can be constant with just two adjacent edges.

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 $\zeta :T\to T'$ and a tuple of disk biholomorphisms $(\psi _{v}:D\to D)_{{v\in V^{m}}}$ which preserve the tree, Morse trajectories, marked points, and maps in the sense that

TODO: add sphere tree iso: Finally, two sphere bubble trees are equivalent $(T,\underline {z},\underline {u})\sim (T',\underline {z}',\underline {u}')$ if there is a tree isomorphism $\zeta :T\to T'$ and a tuple of sphere biholomorphisms $(\psi _{v}:S^{2}\to S^{2})_{{v\in V}}$ which preserve the tree, marked points, and pseudoholomorphic curves in the sense that

$\zeta$ preserves the tree structure, in particular maps the root $v_{0}$ to the root $v_{0}'$ ;
$\psi _{{v_{0}}}(0)=0$ and $\psi _{v}(z_{e}^{v})={z'}_{{\zeta (e)}}^{{\zeta (v)}}$ for every $v\in V$ and adjacent edge $e\in E_{v}$ ;
the pseudoholomorphic spheres are related by reparametrization, $u_{v}=u'_{{\zeta (v)}}\circ \psi _{v}$ for every $v\in V$ .

$\zeta$ 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 $e\in E_{v}$ ;
$\psi _{v}(\underline {w}_{v})=\underline {w}'_{{\zeta (v)}}$ for every $v\in V^{m}$ ;
$\beta _{w}=\beta '_{{\psi _{v}(w)}}$ for every $v\in V^{m}$ and $w\in \underline {w}_{v}$ ;
the maps are related by reparametrization, $u_{v}=u'_{{\zeta (v)}}\circ \psi _{v}$ for every $v\in V^{m}$ .

The symplectic area function

If no Hamiltonian perturbations are involved in the construction of a moduli space of generalized pseudoholomorphic polygons - i.e. if for each pair $\{i,j\}\subset \{0,\ldots ,d\}$ the Lagrangians are either identical $L_{i}=L_{j}$ or transverse $L_{i}\pitchfork L_{j}$ - then the symplectic area function on the moduli space is defined by

$\omega :\overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})\to \mathbb{R} ,\quad b={\bigl [}T,\underline {\gamma },\underline {z},\underline {w},\underline {\beta },\underline {u}{\bigr ]}\mapsto \omega (b):=\sum _{{v\in V^{m}}}\textstyle \int _{{\Sigma _{v}}}u_{v}^{*}\omega \;+\;\sum _{{\beta _{w}\in \underline {\beta }}}\omega (\beta _{w})=\langle [\omega ],[b]\rangle ,$

which - since $\omega |_{{L_{i}}}\equiv 0$ only depends on the total homology class of the generalized polygon

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

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:

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):

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@@ Line 3: / Line 3: @@
 This section constructs topological ambient spaces <math>\mathcal{X}(\underline{x})</math> of the [[moduli spaces of pseudoholomorphic polygons]] <math>\overline\mathcal{M}(\underline{x})=\overline\partial_{J,Y}^{-1}(0)</math>, within which polyfold theory will construct the [[regularized moduli spaces]] <math>\overline\mathcal{M}(\underline{x};\nu)</math>. These will depend only on the choice of a Sobolev decay constant <math>\delta>0</math>.
-Thus we are given Lagrangians <math>L_i\subset M</math> indexed cyclically by <math>i\in \Z_{d+1}</math> and a tuple of generators <math>\underline{x}=(x_0, \ldots, x_d) \in \text{Crit}(L_0,L_d)\times \ldots\times\text{Crit}(L_{d-1},L_d)</math> in their morphism spaces. The latter depend on the choice of Hamiltonian symplectomorphisms <math>\phi_i:=\phi_{L_{i-1},L_i}</math> so that we have <math>\phi_i(L_{i-1})\pitchfork L_i</math> whenever <math>L_{i-1}\neq L_i</math>, as specified in [[polyfold constructions for Fukaya categories]] and [[moduli spaces of pseudoholomorphic polygons]].
+Thus we are given Lagrangians <math>L_i\subset M</math> indexed cyclically by <math>i\in \Z_{d+1}</math> and a tuple of generators <math>\underline{x}=(x_0, \ldots, x_d) \in \text{Crit}(L_0,L_d)\times \ldots\times\text{Crit}(L_{d-1},L_d)</math> in their morphism spaces. The latter depend on the choices of Hamiltonian symplectomorphisms <math>\phi_i:=\phi_{L_{i-1},L_i}</math> so that <math>\phi_i(L_{i-1})\pitchfork L_i</math> whenever <math>L_{i-1}\neq L_i</math>, as specified in [[polyfold constructions for Fukaya categories]] and [[moduli spaces of pseudoholomorphic polygons]].
 '''TODO: mention choice of Morse functions (? also in other places previously ?)'''

Difference between revisions of "Ambient space"

Revision as of 00:30, 8 June 2017

The symplectic area function

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