Shifted Symplectic Fibrations and Derived Thurston Theorem

Abstract

In classical symplectic geometry, under mild conditions, Thurston proved that one can construct a compatible symplectic form on the total space of a symplectic fibration with a connected symplectic base. Here we prove a derived symplectic analog of this result. More precisely, we show that if a morphism $π: X \rightarrow S$ of derived stacks has a shifted symplectic fibration structure and the target stack $S$ admits a shifted symplectic structure, then, under certain conditions, one can construct a shifted symplectic structure on the source stack $X$, compatible with $π$ in a sense similar to the classical case. In this derived context, an affine model construction for shifted symplectic fibrations is also developed. Along the way, we present numerous examples of shifted symplectic fibrations and provide applications of the derived Thurston theorem.