Difference between revisions of "Moduli spaces of pseudoholomorphic polygons"

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(→‎Disk trees for a fixed Lagrangian)
(→‎Disk trees for a fixed Lagrangian)
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One could - as in the approach by Fukaya et al - regularize such a space and then interpret disk bubbling as contribution to <math>\widehat\mu\circ \widehat\mu</math> - where the composition is via a push-pull construction on some space of chains, currents, or differential forms on the Lagrangian. However, such push-pull constructions require transversality of the chains to the evaluation maps from the regularized moduli spaces, so that a rigorous construction of the <math>A_\infty</math>-structure in this setting requires a complicated infinite iteration.
 
One could - as in the approach by Fukaya et al - regularize such a space and then interpret disk bubbling as contribution to <math>\widehat\mu\circ \widehat\mu</math> - where the composition is via a push-pull construction on some space of chains, currents, or differential forms on the Lagrangian. However, such push-pull constructions require transversality of the chains to the evaluation maps from the regularized moduli spaces, so that a rigorous construction of the <math>A_\infty</math>-structure in this setting requires a complicated infinite iteration.
  
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We will resolve this issue as in [https://math.berkeley.edu/~katrin/papers/disktrees.pdf JL] by following another earlier proposal by Fukaya-Oh to allow disks to flow apart along a Morse trajectory.
+
We will resolve this issue as in [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf JL]] by following another earlier proposal by Fukaya-Oh to allow disks to flow apart along a Morse trajectory.
 
Here we work throughout with the Morse function <math>f:L\to\R</math> chosen in the setup of the morphism space <math>\text{Hom}(L,L)=\textstyle\sum_{x\in\text{Crit}(f)} \Lambda x</math>. We also choose a metric on <math>L</math> so that the gradient vector field <math>\nabla f</math> satisfies the Morse-Smale conditions and an additional technical assumption in [https://arxiv.org/abs/1205.0713] which guarantees a natural smooth manifold-with-boundary-and-corners structure on the [[compactified Morse trajectory spaces]]
 
Here we work throughout with the Morse function <math>f:L\to\R</math> chosen in the setup of the morphism space <math>\text{Hom}(L,L)=\textstyle\sum_{x\in\text{Crit}(f)} \Lambda x</math>. We also choose a metric on <math>L</math> so that the gradient vector field <math>\nabla f</math> satisfies the Morse-Smale conditions and an additional technical assumption in [https://arxiv.org/abs/1205.0713] which guarantees a natural smooth manifold-with-boundary-and-corners structure on the [[compactified Morse trajectory spaces]]
 
<math> \overline\mathcal{M}(L,L) , \overline\mathcal{M}(p^-,L), \overline\mathcal{M}(L,p^+), \overline\mathcal{M}(p^-,p^+) </math> for <math>p^\pm\in\text{Crit}(f)</math>.
 
<math> \overline\mathcal{M}(L,L) , \overline\mathcal{M}(p^-,L), \overline\mathcal{M}(L,p^+), \overline\mathcal{M}(p^-,p^+) </math> for <math>p^\pm\in\text{Crit}(f)</math>.
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'''TODO:''' if J with no spheres, then compact and trivial isotropy [https://math.berkeley.edu/~katrin/papers/disktrees.pdf JL,Prop.2.5]
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'''TODO:''' if J with no spheres, then compact and trivial isotropy [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf JL,Prop.2.5]]
  
  

Revision as of 22:00, 23 May 2017

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 moduli space \overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}). We will do this below by combining two special cases.

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.

For a sequence of such maps (modulo reparametrization by automorphisms of the disk), energy concentration at a boundary point is usually captured in terms of a disk bubble attached via a boundary node. This yields to a compactification of the moduli space of pseudoholomorphic disks modulo reparametrization that is given by adding boundary strata consisting of fiber products of moduli spaces of disks. One could - as in the approach by Fukaya et al - regularize such a space and then interpret disk bubbling as contribution to \widehat \mu \circ \widehat \mu - where the composition is via a push-pull construction on some space of chains, currents, or differential forms on the Lagrangian. However, such push-pull 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 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. Here 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 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 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})_{{v\in V^{m}}} such that

  • T 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}.



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

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 =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).



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\}

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