Deformations of Poisson structures by closed 3-forms

TL;DR

This paper demonstrates how arbitrary Poisson structures can be deformed using closed 3-forms, resulting in new local Poisson structures and graded epsilon-deformations on loop spaces.

Contribution

It introduces a method to generate local Poisson structures from given Poisson structures and closed 3-forms, and constructs graded epsilon-deformations.

Findings

Generated local Poisson structures from arbitrary Poisson structures and closed 3-forms.

Established a graded epsilon-deformation of Poisson structures using closed 3-forms.

Provided explicit formulas relating deformations to loop space structures.

Abstract

We prove that an arbitrary Poisson structure omega^{ij}(u) and an arbitrary closed 3-form T_{ijk}(u) generate the local Poisson structure A^{ij}(u,u_x) = M^i_s(u,u_x)omega^{sj}(u), where M^i_s(u,u_x)(delta^s_j + omega^{sp}(u)T_{pjk}(u)u^k_x) = delta^i_j, on the corresponding loop space. We obtain also a special graded epsilon-deformation of an arbitrary Poisson structure omega^{ij}(u) by means of an arbitrary closed 3-form T_{ijk}(u).