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 proposed 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 differential chains on the Lagrangian. However, such push-pull constructions require transversality of evaluation maps to the differential chains, so that a rigorous construction of the <math>A_\infty</math>-structure in this setting would require a complicated infinite iteration.
 
One could - as proposed 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 differential chains on the Lagrangian. However, such push-pull constructions require transversality of evaluation maps to the differential chains, so that a rigorous construction of the <math>A_\infty</math>-structure in this setting would require a complicated infinite iteration.
  
−
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.
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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.
 
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 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 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>
 
<math> \overline\mathcal{M}(L,L) , \overline\mathcal{M}(p^-,L), \overline\mathcal{M}(L,p^+), \overline\mathcal{M}(p^-,p^+) </math>
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  [https://math.berkeley.edu/~katrin/papers/disktrees.pdf JL]
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  [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf JL]]
  
 
<center>
 
<center>

Revision as of 07:54, 21 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, 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 proposed 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 differential chains on the Lagrangian. However, such push-pull constructions require transversality of evaluation maps to the differential chains, so that a rigorous construction of the A_{\infty }-structure in this setting would require 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 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). 


[JL]




TODO: compact if J with no spheres ... otherwise add spheres as below

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

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