# Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms

## TL;DR

This paper proves a strictification result for isotropic distributions on derived schemes with shifted symplectic forms, enabling the construction of globally defined Lagrangian distributions as dg manifolds.

## Contribution

It introduces a strictification theorem for isotropic distributions on derived schemes with shifted symplectic structures, facilitating the global construction of Lagrangian distributions.

## Key findings

- Any derived scheme over ℂ with a -2-shifted symplectic structure and Hausdorff classical points admits a global Lagrangian distribution.
- The strictification result applies to isotropic distributions on derived schemes with negatively shifted homotopically closed 2-forms.
- The work bridges derived algebraic geometry and symplectic geometry by constructing Lagrangian distributions as dg manifolds.

## Abstract

A strictification result is proved for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed $2$-forms. It is shown that any derived scheme over $\mathbb{C}$ equipped with a $-2$-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg $\mathbb{C}^{\infty}$-manifold.