Gluing construction for Hamiltonians

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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 (or find reference in Deligne-Mumford space) \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}}).