Lagrangian structures on the derived moduli of constructible sheaves

TL;DR

This paper demonstrates that the moduli spaces of constructible and perverse sheaves on a stratified manifold possess shifted Lagrangian structures, linking geometric and categorical properties through Calabi--Yau structures.

Contribution

It establishes the shifted Lagrangian nature of these moduli spaces using Calabi--Yau structures and introduces a lax gluing technique for categorical cubes with Calabi--Yau structures.

Findings

Moduli of constructible sheaves are (2-n)-shifted Lagrangian.

Moduli of perverse sheaves are (2-n)-shifted Lagrangian.

Identification of symplectic leaves with prescribed monodromy.

Abstract

Given a compact oriented manifold of dimension $n$ with a conically smooth stratification, we show that the moduli of $\mathcal{D}(k)$-valued constructible sheaves and the moduli of perverse sheaves are $(2-n)$-shifted Lagrangian. The former statement follows from the construction of a relative left $n$-Calabi--Yau structure on the stable $\infty$-category of $\mathcal{D}(k)$-valued constructible sheaves. This is achieved via a lax gluing result for categorical cubes equipped with cubical Calabi--Yau structures. Given a codimension $2$ submanifold, we further identify symplectic leaves corresponding to perverse sheaves with prescribed monodromy.