Towards Homotopy Poisson-n Algebras from N-plectic Structures

TL;DR

This paper explores the construction of homotopy Poisson-n algebras from higher symplectic structures, aiming to extend classical Poisson algebra concepts to n-plectic geometry, with implications for quantization.

Contribution

It introduces a framework linking higher symplectic structures to homotopy Poisson-n algebras, broadening the mathematical foundation for quantization approaches.

Findings

Associates a homotopy Poisson-n algebra to higher symplectic structures.

Highlights limitations in the current algebraic closure of Poisson cotensors.

Provides a foundation for future development in higher symplectic geometry.

Abstract

We associate a homotopy Poisson-n algebra to any higher symplectic structure, which generalizes the common symplectic Poisson algebra of smooth functions. This provides robust n-plectic prequantum data for most approaches to quantization. UPDATE: It has been brought to my attention that the exterior product does not close on the exterior cotensors called Poisson cotensors in the present paper. Therefore the set of Poisson cotensors as presented, is not quite yet a homotopy Poisson-n algebra. However for those Poisson cotensor that do multiply into Poisson cotensors under the exterior prodoct, the computation remains valid and interesting, nonetheless. Therefor this paper might better be seen as a hint towards homotopy Poisson-n algebras in higher symplectic geometry, rather then a finished theory.