Towards resurgence of Joyce structures

TL;DR

This paper demonstrates that certain gauge transformations associated with Joyce structures can be formally normalized and their Borel transforms are convergent, advancing the understanding of resurgent properties and twistor coordinates in complex hyperkähler geometry.

Contribution

It establishes the resurgent nature of gauge transformations and twistor Darboux coordinates related to Joyce structures, providing new insights into their formal and analytic properties.

Findings

Gauge transformations can be formally normalized to a standard form.

The Borel transforms of these transformations are convergent under certain conditions.

Formal twistor Darboux coordinates have convergent Borel transforms, indicating resurgent behavior.

Abstract

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.