Towards resurgence of Joyce structures

Given a Joyce structure, we show that the associated $\mathbb{C}^*$-family of non-linear connections $\mathcal{A}^ε$ can be gauged to a standard form $\mathcal{A}^{ε,\text{st}}$ by a gauge transformation $\hat{g}$, formal in $ε$. 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.

💡 Research Summary

This paper investigates the formal classification and resurgence properties of Joyce structures, a geometric framework originally introduced by Bridgeland to encode Donaldson‑Thomas (DT) invariants of a 3‑dimensional Calabi‑Yau category. A Joyce structure on a holomorphic symplectic manifold ((M,\Omega)) consists of a family of non‑linear connections (\mathcal A^{\varepsilon}) on the projection (\pi:T M\to M), parametrised by a non‑zero complex parameter (\varepsilon). The connections are required to be flat, symplectic and to have the schematic form \