Towards resurgence of Joyce structures

Abstract

Given a Joyce structure, we show that the associated ${\mathbb{C}}^{\ast }$-family of non-linear connections ${\mathcal{A}}^{\hat{I}}\mu$ can be gauged to a standard form ${\mathcal{A}}^{\hat{I}\mu ,\text{st}}$ by a gauge transformation $\hat{g}$, formal in $\hat{I}\mu$. 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 hyperkähler structure associated to the Joyce structure, and show a similar result about convergence of the Borel transform of the formal twistor Darboux coordinates.

36 pages