Difference between revisions of "Gluing construction for Hamiltonians"
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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]]. | ||
− | * 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]] | + | * 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]] (or find reference in [[Deligne-Mumford spaces]]) <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 20:43, 7 June 2017
TODO: As cited in [expansion of expansion in point 7], construct vector-field-valued 1-forms on compatibly with the fixed Hamiltonian perturbations.
Copies from moduli spaces of pseudoholomorphic polygons:
- On the thin part near each puncture we have .
- Note that this convention together with our symmetric choice of Hamiltonian perturbations forces the vector-field-valued 1-form on in case to be -invariant, if are the Lagrangian labels for the boundary components .
- Here and in the following we denote in case , so that in case .
- The Hamiltonian perturbations should be cut off to vanish outside of the thin parts of the domains . However, there may be thin parts of a surface that are not neighborhoods of a puncture. On these, we must choose the Hamiltonian-vector-field-valued one-form compatible with gluing as in [Seidel book].
- For example, in the neighbourhood of a tree with an edge between main vertices, there are trees in which this edge is removed, the two vertices are replaced by a single vertex , and the surfaces are replaced by a single glued surface (or find reference in Deligne-Mumford spaces) for . Compatibility with gluing requires that the Hamiltonian perturbation on these glued surfaces is also given by a gluing construction (in which the two perturbations agree and hence can be matched over a long neck ).