$\DeclareMathOperator\Res{Res}$I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/abs/1701.09137) and am having trouble understanding their Section 6.3. I think the section is meant to explain the connection between the deformation of a disc $\mathbb{D}_t$ embedded in a symplectic surface $S$ and differential forms $\eta \in \mathbb{C}((z))dz$, $\Res_z(\eta) = 0$ such that \begin{equation} \forall n \geq 1, \Res_z (yx^n dx) = 0, \qquad \forall n \geq 0, \Res_z (y^2 x^ndx) = 0 \end{equation} where $x = z^2, y = z - \eta/(2zdz)$. Unfortunately, the explanation in the section is too brief for me.

Here's what I think is going on:

We choose coordinates $x$, $y$ of $S$ such that the symplectic form is $\omega = dx \wedge dy$, foliation is given by $x = \mathrm{const}$ and the disc $\mathbb{D}_t \subset S$ is embedded by $x = z^2$, $y = z$.

If the disc is deformed to $\mathbb{D}'_t$ then we can represent the deformation by a vector field $v \in T_S|_{\mathbb{D}_t}$ pointing along the direction of foliation, i.e., $\propto \partial_y$. I'm not sure how this is defined, really.

Getting a 1-form using the symplectic form, $\eta = \omega(v,\cdot) \propto dx$.

Somehow the pull-back of this form to the disc $\mathbb{D}_t$ will satisfy the residue constraints. The converse is also true. This is the part I really fail to make a connection.

I feel like this type of construction is well-known in a certain area of studies that why the explanations are so brief. But the paper cited no references for me to look further.

Could someone please explain to me in more detail what is going on or suggest me good references that will help fill the gap?