Star products on generalised complex manifolds

TL;DR

This paper explores how the *-product on classical phase space, viewed as a generalized complex manifold, transforms under B-field changes, revealing insights into phase-space quantum mechanics with magnetic backgrounds.

Contribution

It introduces a framework for understanding the B-transformation properties of the *-product on generalized complex manifolds, connecting geometric transformations to quantum mechanics.

Findings

C*-algebra of smooth functions transforms as expected

C*-algebra of holomorphic functions transforms nontrivially

B-transformed *-product encodes phase-space quantum mechanics with magnetic fields

Abstract

We regard classical phase space as a generalised complex manifold and analyse the B-transformation properties of the *-product of functions. The C*-algebra of smooth functions transforms in the expected way, while the C*-algebra of holomorphic functions (when it exists) transforms nontrivially. The B-transformed *-product encodes all the properties of phase-space quantum mechanics in the presence of a background magnetic field.