Deformation quantization of submanifolds and reductions via Duflo-Kirillov-Kontsevich map

Abstract

We propose the following receipt to obtain the quantization of the Poisson submanifold $N$ defined by the equations $f_i=0$ (where $f_i$ are Casimirs) from the known quantization of the manifold $M$: one should consider factor algebra of the quantized functions on $M$ by the images of $D(f_i)$, where $D: Fun(M) \to Fun(M)\otimes \CC[\hbar]$ is Duflo-Kirillov-Kontsevich map. We conjecture that this algebra is isomorphic to quantization of $Fun(N)$ with Poisson structure inherited from $M$. Analogous conjecture concerning the Hamiltonian reduction saying that "deformation quantization commutes with reduction" is presented. The conjectures are checked in the case of $S^2$ which can be quantized as a submanifold, as a reduction and using recently found explicit star product. It's shown that all the constructions coincide.

arXiv comment20 pages

References in corpus (7)

• Deformation quantization of Poisson manifolds, I
• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• The Origin of Chiral Anomaly and the Noncommutative Geometry
• A New Noncommutative Product on the Fuzzy Two-Sphere Corresponding to the Unitary Representation of SU(2) and the Seiberg-Witten Map
• Deformation quantization of compact Kähler manifolds via Berezin-Toeplitz operators
• The Moyal product is the matrix product
• Invariant star-products on S^2 and the canonical trace

Cited by in corpus (3)

• A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold
• Decomplexification of the Capelli identities and holomorphic factorization
• Quantizing Derived Mapping Stacks