Shifted symplectic structure on Poisson Lie algebroid and generalized complex geometry

TL;DR

This paper reformulates generalized complex geometry using homotopical and shifted symplectic structures, establishing new links between Poisson Lie algebroids, formal stacks, and branes.

Contribution

It introduces a homotopical reformulation of generalized complex manifolds and develops machinery for shifted symplectic formal stacks, connecting coisotropic intersections with shifted Poisson structures.

Findings

Reformulation of generalized complex manifolds as holomorphic symplectic formal stacks

Proof that coisotropic intersections inherit shifted Poisson structures

Study of generalized complex branes within this new framework

Abstract

Generalized complex geometry was classically formulated by the language of differential geometry. In this paper, we reformulated a generalized complex manifold as a holomorphic symplectic differentiable formal stack in a homotopical sense. Meanwhile, by developing the machinery for shifted symplectic formal stack, we prove that the coisotropic intersection inherits shifted Poisson structure. Generalized complex branes are also studied.