# Gluing construction for Hamiltonians

From Polyfolds.org

**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 space) 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 ).