Difference between revisions of "Gluing construction for Hamiltonians"

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(Created page with "table of contents '''TODO:''' As cited in http://www.polyfolds.org/index.php?title=Moduli_spaces_of_pseudoholomorphic_polygons#General_moduli_space_of_pseudoholomorphic...")
 
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'''TODO:''' As cited in [[http://www.polyfolds.org/index.php?title=Moduli_spaces_of_pseudoholomorphic_polygons#General_moduli_space_of_pseudoholomorphic_polygons  expansion of expansion in point 7]], construct vector-field-valued 1-forms <math>Y_v : {\rm T}^*\Sigma^v \times M \to {\rm T}M</math> on <math>\Sigma^v</math> compatibly with the fixed Hamiltonian perturbations.  
 
'''TODO:''' As cited in [[http://www.polyfolds.org/index.php?title=Moduli_spaces_of_pseudoholomorphic_polygons#General_moduli_space_of_pseudoholomorphic_polygons  expansion of expansion in point 7]], construct vector-field-valued 1-forms <math>Y_v : {\rm T}^*\Sigma^v \times M \to {\rm T}M</math> on <math>\Sigma^v</math> compatibly with the fixed Hamiltonian perturbations.  
  
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Copies from [[moduli space of pseudoholomorphic polygons]]:  
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Copies from [[moduli spaces of pseudoholomorphic polygons]]:  
  
 
* On the thin part <math>\iota^v_e: [0,\infty)\times[0,1] \hookrightarrow \Sigma^v</math> near each puncture <math>z^v_e</math> we have <math>(\iota^v_e)^* Y_v = X_{L^v_{e-1},L^v_e} \,{\rm d} t</math>.  
 
* On the thin part <math>\iota^v_e: [0,\infty)\times[0,1] \hookrightarrow \Sigma^v</math> near each puncture <math>z^v_e</math> we have <math>(\iota^v_e)^* Y_v = X_{L^v_{e-1},L^v_e} \,{\rm d} t</math>.  
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* The Hamiltonian perturbations <math>Y_v</math> should be cut off to vanish outside of the thin parts of the domains <math>\Sigma_v</math>. However, there may be thin parts of a surface <math>\Sigma_v</math> that are not neighborhoods of a puncture. On these, we must choose the Hamiltonian-vector-field-valued one-form <math>Y_v</math> compatible with gluing as in [[http://www.ems-ph.org/books/book.php?proj_nr=12 Seidel book]].
 
* The Hamiltonian perturbations <math>Y_v</math> should be cut off to vanish outside of the thin parts of the domains <math>\Sigma_v</math>. However, there may be thin parts of a surface <math>\Sigma_v</math> that are not neighborhoods of a puncture. On these, we must choose the Hamiltonian-vector-field-valued one-form <math>Y_v</math> compatible with gluing as in [[http://www.ems-ph.org/books/book.php?proj_nr=12 Seidel book]].
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For example, in the neighbourhood of a tree with an edge <math>e=(v,w)</math> between main vertices, there are trees in which this edge is removed, the two vertices are replaced by a single vertex <math>v\#w</math>, and the surfaces <math>\Sigma_v,\Sigma_w</math> are replaced by a single [[glued surface]]  
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* For example, in the neighbourhood of a tree with an edge <math>e=(v,w)</math> between main vertices, there are trees in which this edge is removed, the two vertices are replaced by a single vertex <math>v\#w</math>, and the surfaces <math>\Sigma_v,\Sigma_w</math> are replaced by a single [[glued surface]]  
 
<math>\Sigma_{v\#w}=\Sigma_v \#_R \Sigma_w</math> for <math>R\gg 1</math>. Compatibility with gluing requires that the Hamiltonian perturbation on these glued surfaces is also given by a gluing construction <math>Y_{v\#w}=Y_v \#_R Y_w</math> (in which the two perturbations <math>Y_v,Y_w</math> agree and hence can be matched over a long neck <math>[-R,R]\times[0,1]\subset \Sigma_{v\#w}</math>).
 
<math>\Sigma_{v\#w}=\Sigma_v \#_R \Sigma_w</math> for <math>R\gg 1</math>. Compatibility with gluing requires that the Hamiltonian perturbation on these glued surfaces is also given by a gluing construction <math>Y_{v\#w}=Y_v \#_R Y_w</math> (in which the two perturbations <math>Y_v,Y_w</math> agree and hence can be matched over a long neck <math>[-R,R]\times[0,1]\subset \Sigma_{v\#w}</math>).

Revision as of 19:42, 7 June 2017

table of contents

TODO: As cited in [expansion of expansion in point 7], construct vector-field-valued 1-forms Y_{v}:{{\rm {T}}}^{*}\Sigma ^{v}\times M\to {{\rm {T}}}M on \Sigma ^{v} compatibly with the fixed Hamiltonian perturbations.

Copies from moduli spaces of pseudoholomorphic polygons:

  • On the thin part \iota _{e}^{v}:[0,\infty )\times [0,1]\hookrightarrow \Sigma ^{v} near each puncture z_{e}^{v} we have (\iota _{e}^{v})^{*}Y_{v}=X_{{L_{{e-1}}^{v},L_{e}^{v}}}\,{{\rm {d}}}t.
  • Note that this convention together with our symmetric choice of Hamiltonian perturbations X_{{L_{i},L_{j}}}=-X_{{L_{j},L_{i}}} forces the vector-field-valued 1-form on \Sigma ^{v}\simeq \mathbb{R} \times [0,1] in case |v|=2 to be \mathbb{R} -invariant, Y_{v}=X_{{L_{0},L_{1}}}\,{{\rm {d}}}t if L_{i} are the Lagrangian labels for the boundary components \mathbb{R} \times \{i\}.
  • Here and in the following we denote X_{{L_{i},L_{j}}}:=0 in case L_{i}=L_{j}, so that (\iota _{e}^{v})^{*}Y_{v}=0 in case L_{{e-1}}^{v}=L_{e}^{v}.
  • The Hamiltonian perturbations Y_{v} should be cut off to vanish outside of the thin parts of the domains \Sigma _{v}. However, there may be thin parts of a surface \Sigma _{v} that are not neighborhoods of a puncture. On these, we must choose the Hamiltonian-vector-field-valued one-form Y_{v} compatible with gluing as in [Seidel book].
  • For example, in the neighbourhood of a tree with an edge e=(v,w) between main vertices, there are trees in which this edge is removed, the two vertices are replaced by a single vertex v\#w, and the surfaces \Sigma _{v},\Sigma _{w} are replaced by a single glued surface

\Sigma _{{v\#w}}=\Sigma _{v}\#_{R}\Sigma _{w} for R\gg 1. Compatibility with gluing requires that the Hamiltonian perturbation on these glued surfaces is also given by a gluing construction Y_{{v\#w}}=Y_{v}\#_{R}Y_{w} (in which the two perturbations Y_{v},Y_{w} agree and hence can be matched over a long neck [-R,R]\times [0,1]\subset \Sigma _{{v\#w}}).

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