A homological approach to singular reduction in deformation quantization

  title={A homological approach to singular reduction in deformation quantization},
  author={Martin Bordemann and Hans-Christian Herbig and Markus J. Pflaum},
We use the method of homological quantum reduction to construct a deformation quantization on singular symplectic quotients in the situation, where the coefficients of the moment map define a complete intersection. Several examples are discussed, among others one where the singularity type is worse than an orbifold singularity. 
Remarks on Singular Symplectic Reduction and Quantization of the Angular Moment
A direct algebraic method of symplectic reduction is demonstrated for some singular problems. The problem of quantization of singular surfaces is discussed. Mathematics Subject Classification (2010).
Algebraic symplectic reduction and quantization of singular spaces
The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular
Remarks on Singular Symplectic Reduction and Quantization of the Angular Moment
A direct algebraic method of symplectic reduction is demonstrated for some singular problems. The problem of quantization of singular surfaces is discussed.
Quantization of Whitney functions and reduction
For a possibly singular subset of a regular Poisson manifold we construct a deformation quantization of its algebra of Whitney functions. We then extend the construction of a deformation quantization
Morita theory in deformation quantization
Various aspects of Morita theory of deformed algebras and in particular of star product algebras on general Poisson manifolds are discussed. We relate the three flavours ring-theoretic Morita
On the Existence of Star Products on Quotient Spaces of Linear Hamiltonian Torus Actions
We discuss BFV deformation quantization (Bordemann et al. in A homological approach to singular reduction in deformation quantization, singularity theory, pp. 443–461. World Scientific, Hackensack,
Equivariant quantization of orbifolds
Abstract Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the
Quantization of Whitney functions
We propose to study deformation quantizations of Whitney functions. To this end, we extend the notion of a deformation quantization to algebras of Whitney functions over a singular set, and show the
Coisotropic Triples, Reduction and Classical Limit
Coisotropic reduction from Poisson geometry and deformation quantization is cast into a general and unifying algebraic framework: we introduce the notion of coisotropic triples of algebras for which


A simple geometrical construction of deformation quantization
A construction, providing a canonical star-product associated with any symplectic connection on symplectic manifold, is considered. An action of symplectomorphisms by automorphisms of star-algebra is
On the deformation quantization of symplectic orbispaces
Abstract In the first part of this article we provide a geometrically oriented approach to the theory of orbispaces originally introduced by G. Schwarz and W. Chen. We explain the notion of a vector
Non-Abelian Reduction in Deformation Quantization
We consider a G-invariant star-product algebra A on a symplectic manifold (M,ω) obtained by a canonical construction of deformation quantization. Under assumptions of the classical Marsden–Weinstein
Applications of perturbation theory to iterated fibrations
For a class of spaces including simply connected spaces and classifying spaces of nilpotent groups, relatively small differential graded algebras are constructed over commutative rings with 1 which
Deformation Quantization and Index Theory
Elements of Differential Geometry Weyl Quantization Introduction to Index Theory Deformation Quantization Index Theorem for Quantum Algebra Asymptotic Operator Representation.
The Jacobian module of a Lie algebra
There is a natural way to associate to the commuting variety C(A) of an algebra A a module over a polynomial ring. It serves as a vehicle to study the arithmetical properties of C(A) , particularly
BRST Cohomology and Phase Space Reduction in Deformation Quantization
Abstract:In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework
Geometric and Algebraic Reduction for Singular Momentum Maps
This paper concerns the reduction of singular constraint sets of symplectic manifolds. It develops a “geometric” reduction procedure, as well as continues the work of Sniatycki and Patrick on
Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras
Abstract This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical finite dimensional BRS procedure and relate it to Marsden-Weinstein
Homological reduction of constrained Poisson algebras
Reduction of a Hamiltonian system with symmetry and/or constraints has a long history. There are several reduction procedures, all of which agree in “nice” cases [AGJ]. Some have a geometric emphasis