Natural star products on symplectic manifolds and quantum moment maps

Abstract

We define a natural class of star products: those which are given by a series of bidifferential operators which at order $k$ in the deformation parameter have at most $k$ derivatives in each argument. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance and give necessary and sufficient conditions for them to yield a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.

Expanded bibliography

References in corpus (2)

• Homogeneous Fedosov Star Products on Cotangent Bundles I: Weyl and Standard Ordering with Differential Operator Representation
• Quantum moment maps and invariants for G-invariant star products

Cited by in corpus (14)

• Formal symplectic groupoid of a deformation quantization
• Cotangent bundle quantization: Entangling of metric and magnetic field
• Classification of Equivariant Star Products on Symplectic Manifolds
• On Gammelgaard's formula for a star product with separation of variables
• BRST Reduction of Quantum Algebras with ${}^{\ast }$-Involutions
• On the inverse mapping of the formal symplectic groupoid of a deformation quantization
• Infinitesimal deformations of a formal symplectic groupoid
• The Fedosov deformation quantization for some induced symplectic connection
• Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds
• Symplectic connections and Fedosov's quantization on supermanifolds
• The group of Hamiltonian automorphisms of a star product
• Deformation quantization with separation of variables of an endomorphism bundle
• Quantum moment map and obstructions to the existence of closed Fedosov star products
• A heat kernel proof of the index theorem for deformation quantization