Shifted coisotropic structures for differentiable stacks

TL;DR

This paper develops a new notion of coisotropic structures on 1-shifted symplectic Lie groupoids, extending concepts from derived algebraic geometry to differentiable stacks and providing new tools for symplectic reduction.

Contribution

It introduces 1-shifted coisotropic structures using twisted Dirac structures, demonstrating their properties and transferability via Morita equivalences for differentiable stacks.

Findings

Intersections of 1-coisotropics are 0-shifted Poisson.

Coisotropic structures transfer through Morita equivalences.

Examples include Hamiltonian actions and generalized symplectic reduction.

Abstract

We introduce a notion of coisotropics on 1-shifted symplectic Lie groupoids (i.e. quasi-symplectic groupoids) using twisted Dirac structures and show that it satisfies properties analogous to the corresponding derived-algebraic notion in shifted Poisson geometry. In particular, intersections of 1-coisotropics are 0-shifted Poisson. We also show that 1-shifted coisotropic structures transfer through Morita equivalences, giving a well-defined notion for differentiable stacks. Most results are formulated with clean-intersection conditions weaker than transversality while avoiding derived geometry. Examples of 1-coisotropics that are not necessarily Lagrangians include Hamiltonian actions of quasi-symplectic groupoids on Dirac manifolds, and this recovers several generalizations of Marsden-Weinstein-Meyer's symplectic reduction via intersection and Morita transfer.