Invariant Star Products of Wick Type: Classification and Quantum Momentum Mappings

Abstract

We extend our investigations on g-invariant Fedosov star products and quantum momentum mappings [22] to star products of Wick type on pseudo-Kähler manifolds. Star products of Wick type can be completely characterized by a local description as given by Karabegov in [14] for star products with separation of variables. We separately treat the action of a Lie group] by (pull-backs with) diffeomorphisms and the action of a Lie algebra g on C ∞ (M ) [[ν]] by (Lie derivatives with respect to) vector fields. Within Karabegov's framework, we prove necessary and sufficient conditions for a given star product of Wick type to be invariant in the respective sense. Moreover, our results yield a complete classification of invariant star products of Wick type. We also prove a necessary and sufficient condition for (the Lie derivative with respect to) a vector field to be even a quasi-inner derivation of a given star product of Wick type. We then transfer our former results about quantum momentum mappings for g-invariant Fedosov star products to the case of invariant star products of Wick type.

On the phase form of a deformation quantization with separation of variables

“…It is particularly easy to construct a classifying form ω with a given phase form ω ph if the phase form is an invariant formal form on a homogeneous pseudo-Kähler manifold. A nondegenerate star product with separation of variables on a homogeneous pseudo-Kähler manifold is invariant if and only if its classifying form is invariant (see [11]). If * is an invariant star product with separation of variables on a homogeneous pseudo-Kähler manifold M, then its canonical trace density µ is invariant and therefore the function κ from ( 1) is a formal constant.…”

Gauge theories on noncommutative ${\mathbb C}P^N$ and BPS-like equations

“…We have to use first order differential operators as derivations to construct field theories having usual kinetic terms. In the noncommutative Kähler manifolds deformed by deformation quantization with separation of variables, it is known that vector fields are inner derivations if and only if the vector fields are the Killing vector fields [15,14]. Hence, field theories having usual kinetic terms should be constructed by using the star commutators with the Killing potentials corresponding to the Killing vectors.…”

Classification of Equivariant Star Products on Symplectic Manifolds

“…[5]. We will follow essentially the conventions from [17,18], see also [12,13]: a quantum momentum map is a formal series J ∈ C 1 (g, C ∞ (M )) ν such that for all ξ ∈ g the function J(ξ) generates the fundamental vector field by ⋆-commutators and such that one has the equivariance condition that [J(ξ), J(η)] ⋆ = νJ([ξ, η]) for ξ, η ∈ g. Here the zeroth order is necessarily an equivariant momentum map in the classical sense. We assume the zeroth order to be fixed once and for all.…”