Noncommutative gauge theory for Poisson manifolds

TL;DR

This paper constructs a noncommutative gauge theory for Poisson manifolds, providing explicit semi-classical and quantum Seiberg-Witten maps for Abelian gauge theories using Kontsevich's formality theorem.

Contribution

It introduces a comprehensive method to derive noncommutative gauge theories from Abelian gauge theories on Poisson manifolds, including explicit all-order quantum maps.

Findings

Explicit semi-classical Seiberg-Witten map for Poisson manifolds

Quantum Seiberg-Witten map based on Kontsevich's formality theorem

Applicable to all Abelian gauge theories on Poisson manifolds

Abstract

A noncommutative gauge theory is associated to every Abelian gauge theory on a Poisson manifold. The semi-classical and full quantum version of the map from the ordinary gauge theory to the noncommutative gauge theory (Seiberg-Witten map) is given explicitly to all orders for any Poisson manifold in the Abelian case. In the quantum case the construction is based on Kontsevich's formality theorem.