The derived Picard group of Fukaya categories of closed surfaces

Abstract

Let $Σ_g$ be a closed oriented surface of genus $g\geq2$, and let $\mathscr F_g$ be its split-closed, strictly unobstructed, two-periodic Fukaya category over the complex Novikov field $Λ$. We determine the derived Picard group \[ \operatorname{DPic}_Λ(\mathscr F_g) \cong \bigl(H^1(Σ_g;Λ^\times) \rtimesπ_0\operatorname{Diff}^+(Σ_g)\bigr) \times\mathbb Z/2\mathbb Z. \] 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.