Gluing construction for Hamiltonians

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 $[-R,R]\times [0,1]\subset \Sigma _{{v\#w}}$).