The derived Picard group of Fukaya categories of closed surfaces

Let ${\mathrm{\Sigma }}_g$ be a closed oriented surface of genus $g\ge 2$, and let ${\mathcal{F}}_g$ be its split-closed, strictly unobstructed, two-periodic Fukaya category over the complex Novikov field $\mathrm{\Lambda }$. We determine the derived Picard group $${\mathrm{DPic}}_{\mathrm{\Lambda }}({\mathcal{F}}_g)\cong (H^1({\mathrm{\Sigma }}_g;{\mathrm{\Lambda }}^{\times })⋊{\pi }_0{\mathrm{Diff}}^{+}({\mathrm{\Sigma }}_g))\times \mathbb{Z}\mathrm{/}2\mathbb{Z}\mathrm{.}$$ This proves the enhanced form of the conjecture of arXiv:2006.09689. As a corollary, we obtain that the derived Picard group itself determines the genus of the surface.