Derived Lagrangian Intersections and Non-commutative Hodge Structures on Shifted Symplectic Stacks

This paper investigates the deep interplay between the geometry of derived Artin stacks equipped with shifted symplectic structures and the algebraic invariants associated with their categories of perfect complexes. We construct a canonical mixed non-commutative Hodge structure on the Hochschild homology of an n-shifted symplectic derived stack X over a field of characteristic zero. This structure arises naturally from an sl₂-action on the Hochschild complex, which is induced by the circle action on the derived loop space of X. Our central result establishes a direct relationship between this non-commutative Hodge structure and the geometry of derived Lagrangian intersections within X. Specifically, we prove a homological intersection formula that computes the Euler characteristic of the derived intersection of two Lagrangian substacks as a pairing of their classes in Hochschild homology, governed by the non-commutative Hodge decomposition. This framework provides a unified perspective on enumerative invariants, such as Donaldson–Thomas invariants for Calabi–Yau 3-folds, recasting them as intersection numbers in a derived symplectic context. We leverage techniques from derived algebraic geometry, higher category theory, and cyclic homology to establish our results, thereby offering new tools for the study of moduli problems and non-commutative motives.