$(-1)$-Shifted Darboux theorem of derived schemes in characteristic $p>2$

The derived geometry approach to Donaldson–Thomas theory (over $\mathbb{C}$) is built on Pantev–Toën–Vezzosi–Vaquié’s existence theorem of $(-1)$-shifted symplectic forms \cite{pantev2013shifted} and Brav–Bussi–Joyce’s shifted Darboux theorem \cite{brav2019darboux}. In this paper, we prove a Darboux theorem in characteristic $p>2$ for the $(-1)$-shifted symplectic forms endowed with an $\mathit{\text{infinitesimal}}\mathit{\text{structure}}$. A key ingredient is Antieau’s derived infinitesimal cohomology \cite{antieau2025filtrations}, which enjoys a Poincaré-type lemma. Our argument is in fact characteristic-free and provides a conceptual understanding of the Brav–Bussi–Joyce theorem. Moreover, we extend the existence theorem of Pantev–Toën–Vaquié–Vezzosi by constructing a de Rham $(-1)$-shifted symplectic form on ${\mathrm{Map}}_k(X,\underset{}{\mathrm{Perf}})$, where $X$ is a Calabi–Yau $3$-fold over a field $k$ in characteristic $p>2$. We conjecture that this $(-1)$-shifted symplectic form admits an infinitesimal structure.