An $S^1$-Reduction of Non-Formal Star Product

Abstract

Starting from the Moyal product on eight-dimensional canonical Euclidean phase space $T^* \mathbb{R}^4$ with an $S^1$-symplectic action, we construct a non-formal star product, i.e., the deformation parameter is a real number, on the cotangent bundle of three-dimensional Euclidean space except the origin $T^* (\mathbb{R}^3\backslash\{\rm 0\})$ which is the reduced symplectic manifold by the $S^1$-action.