Difference between revisions of "Ambient space"

Revision as of 22:22, 7 June 2017

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 $d+1\geq 1$ Lagrangians $L_{0},\ldots ,L_{d}\subset M$ (indexed cyclically by $i\in \mathbb{Z } _{{d+1}}$ ) 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. The latter depend on the choice of Hamiltonian symplectomorphisms $\phi _{i}:=\phi _{{L_{{i-1}},L_{i}}}$ so that we have $\phi _{i}(L_{{i-1}})\pitchfork L_{i}$ whenever $L_{{i-1}}\neq L_{i}$ , as specified in polyfold constructions for Fukaya categories and moduli spaces of pseudoholomorphic polygons.

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. $\underline {\beta }=(\beta _{w})_{{w\in \underline {w}}}\subset \overline {\mathcal {M}}_{{0,1}}(J)$ is a tuple of sphere bubble trees $\beta _{w}\in \overline {\mathcal {M}}_{{0,1}}(J)$ indexed by the disjoint union $\underline {w}=\textstyle \bigsqcup _{{v\in V^{m}}}\underline {w}_{v}$ of sphere bubble tree attaching points.

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

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that is each $v\in V^{m}$ is labeled by a continuous map $u_{v}:\Sigma ^{v}\to M$ satisfying a Sobolev regularity, Lagrangian boundary conditions, and matching conditions as follows:

8. The generalized pseudoholomorphic polygon is stable

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in the sense that TODO: update to Ham pert.

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

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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

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):

Sphere bubble trees

The sphere bubble trees that are relevant to the compactification of the moduli spaces of pseudoholomorphic polygons are genus zero stable maps with one marked point, as described in e.g. [Chapter 5, McDuff-Salamon]. For a fixed almost complex structure $J$ , we can use the combinatorial simplification of working with a single marked point to construct the moduli space of sphere bubble trees as

$\overline {\mathcal {M}}_{{0,1}}(J):={\bigl \{}(T,\underline {z},\underline {u})\,{\big |}\,{\text{1. - 4.}}{\bigr \}}/\sim$

where

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

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

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

3. $\underline {u}=(\underline {u}_{v})_{{v\in V}}$ is a tuple of pseudoholomorphic spheres for each vertex,

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that is each $v\in V$ is labeled by a smooth map $u_{v}:S^{2}\to M$ satisfying Cauchy-Riemann equation, finite energy, and matching conditions as follows:

4. The sphere bubble tree is stable

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in the sense that

Finally, two sphere bubble trees are equivalent $(T,\underline {z},\underline {u})\sim (T',\underline {z}',\underline {u}')$ if

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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

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@@ Line 1: / Line 1: @@
 [[table of contents]]
-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>.
+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 <math>d+1\geq 1</math> Lagrangians <math>L_0,\ldots,L_d\subset M</math> (indexed cyclically by <math>i\in \Z_{d+1}</math>) and  generators <math>x_0\in\text{Crit}(L_0,L_d),</math> <math>x_1\in\text{Crit}(L_0,L_1), \ldots,</math> <math>x_d\in\text{Crit}(L_{d-1},L_d)</math> of 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]].
@@ Line 11: / Line 11: @@
 </math>
 </center>
-where
+where the data is the same as in the construction of the [[moduli spaces of pseudoholomorphic polygons]] <math>\overline\mathcal{M}(\underline{x})</math>, except for maps not necessarily being pseudoholomorphic but just of Sobolev <math>H^{3,\delta}</math>-regularity. We indicate these differences in boldface.
 .  <math>T</math> is an ordered tree with sets of vertices <math>V=V^m \cup V^c</math> and edges <math>E</math>,
@@ Line 63: / Line 63: @@
 that is each <math>v\in V^m</math> is labeled by a continuous map <math>u_v: \Sigma^v\to M</math> satisfying a '''Sobolev regularity''', Lagrangian boundary conditions, and matching conditions as follows:
 <div class="mw-collapsible-content">
-* '''TODO: Sobolev regularity - and make sure it implies uniform convergence'''
+* '''TODO: <math>H^{3,\delta}</math>-Sobolev regularity - and make sure it implies uniform convergence'''
 * The Lagrangian boundary conditions are <math>u_v(\partial \Sigma^v)\subset \underline{L}^v</math>; more precisely this requires <math>u_v\bigl( (\partial \Sigma^v)_e \bigr)\subset L^v_e</math> for each adjacent edge <math>e\in E_v</math>.
 * The finite energy condition is <math>\textstyle \int_{\Sigma^v} u_v^*\omega <\infty</math>.

Difference between revisions of "Ambient space"

Revision as of 22:22, 7 June 2017

The symplectic area function

Sphere bubble trees

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