Towards resurgence of Joyce structures

Given a Joyce structure, we show that the associated $\mathbb{C}^*$-family of non-linear connections $\mathcal{A}^{\epsilon}$ can be gauged to a standard form $\mathcal{A}^{\epsilon,\text{st}}$ by a gauge transformation $\hat{g}$, formal in $\epsilon$. We show that the corresponding infinitesimal gauge transformation $\dot{g}=\log(\hat{g})$ has a convergent Borel transform, provided $\dot{g}$ vanishes on the base of the Joyce structure. This establishes the first step in showing that such a $\dot{g}$ is resurgent. We also use $\hat{g}$ to produce formal twistor Darboux coordinates for the complex hyperk\"{a}hler structure associated to the Joyce structure, and show a similar result about convergence of the Borel transform of the formal twistor Darboux coordinates.