View source for Gluing construction for Hamiltonians

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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. 

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>. 

* Note that this convention together with our symmetric choice of Hamiltonian perturbations <math>X_{L_i,L_j}= - X_{L_j,L_i}</math> forces the vector-field-valued 1-form on <math>\Sigma^v\simeq\R\times[0,1]</math> in case  <math>|v|=2</math> to be <math>\R</math>-invariant, <math>Y_v = X_{L_0,L_1} \,{\rm d} t</math> if <math>L_i</math> are the Lagrangian labels for the boundary components <math>\R\times\{i\}</math>.

* Here and in the following we denote <math>X_{L_i,L_j}:=0</math> in case <math>L_i=L_j</math>, so that <math>(\iota^v_e)^* Y_v=0</math> in case <math>L^v_{e-1}=L^v_e</math>.

* 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]] (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>).