A Derived Lagrangian Fibration on the Derived Critical Locus

Abstract: We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional $f : X \rightarrow \mathbb{A}_k^1$, the derived critical locus has a natural Lagrangian fibration $\textbf{Crit}(f) \rightarrow X$. In the case where $f$ is non-degenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.