The Background Field Method and the Linearization Problem for Poisson Manifolds

TL;DR

This paper explores the background field method for Poisson Sigma Models, linking it to gauge algebra linearization and Seiberg-Witten maps, providing conditions and methods for solving related Poisson structure problems.

Contribution

It establishes a connection between the background field method and the linearization problem for Poisson structures, offering new insights and solution techniques.

Findings

BFM for PSM is equivalent to a linearization problem for Poisson structures.

Provides sufficient conditions for the existence of solutions.

Offers a constructive method to derive solutions.

Abstract

The background field method (BFM) for the Poisson Sigma Model (PSM) is studied as an example of the application of the BFM technique to open gauge algebras. The relationship with Seiberg-Witten maps arising in non-commutative gauge theories is clarified. It is shown that the implementation of the BFM for the PSM in the Batalin-Vilkovisky formalism is equivalent to the solution of a generalized linearization problem (in the formal sense) for Poisson structures in the presence of gauge fields. Sufficient conditions for the existence of a solution and a constructive method to derive it are presented.