Quantized reduction as a tensor product

TL;DR

This paper reinterprets symplectic reduction within a categorical framework of Poisson manifolds and explores its quantization via bimodules, establishing a deep analogy with operator algebra categories.

Contribution

It introduces a categorical perspective on symplectic reduction and proposes a quantization approach using bimodules, linking classical and quantum structures.

Findings

Categorical description of symplectic reduction as composition of bimodules

Quantization of reduction via Hilbert bimodules and von Neumann algebra correspondences

Establishment of analogies between Poisson geometry and operator algebra categories

Abstract

Symplectic reduction is reinterpreted as the composition of arrows in the category of integrable Poisson manifolds, whose arrows are isomorphism classes of dual pairs, with symplectic groupoids as units. Morita equivalence of Poisson manifolds amounts to isomorphism of objects in this category. This description paves the way for the quantization of the classical reduction procedure, which is based on the formal analogy between dual pairs of Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with correspondences between von Neumann algebras. Further analogies are drawn with categories of groupoids (of algebraic, measured, Lie, and symplectic type). In all cases, the arrows are isomorphism classes of appropriate bimodules, and their composition may be seen as a tensor product. Hence in suitable categories reduction is simply composition of arrows, and Morita equivalence is isomorphism of objects.