Poisson bivectors and Poisson brackets on affine derived stacks

TL;DR

This paper proves the existence of a canonical map linking shifted Poisson structures on affine derived stacks to homotopy Poisson algebra structures, using operad theory and explicit resolutions.

Contribution

It provides two proofs of a canonical map between moduli spaces of shifted Poisson structures and homotopy Poisson algebra structures, advancing understanding of Poisson geometry in derived algebraic geometry.

Findings

Two proofs of the canonical map established

Explicit resolutions of the Poisson operad constructed

Enhanced understanding of Poisson structures on derived stacks

Abstract

Let Spec(A) be an affine derived stack. We give two proofs of the existence of a canonical map from the moduli space of shifted Poisson structures (in the sense of Pantev-To\"en-Vaqui\'e-Vezzosi, see http://arxiv.org/abs/1111.3209 ) on Spec(A) to the moduli space of homotopy (shifted) Poisson algebra structures on A. The first makes use of a more general description of the Poisson operad and of its cofibrant models, while the second in more computational and involves an explicit resolution of the Poisson operad.