A Counterexample to the Quantizability of Modules

TL;DR

This paper provides a counterexample demonstrating that certain natural representations of Poisson algebras cannot be quantized, highlighting limitations in the quantization process of modules associated with Poisson structures.

Contribution

The paper constructs a specific counterexample showing the non-quantizability of evaluation representations at points where the Poisson structure vanishes.

Findings

Evaluation at zero cannot be quantized in the given example.

The non-quantizability persists under certain formal deformations.

The result highlights fundamental limitations in Poisson module quantization.

Abstract

Let a Poisson structure on a manifold M be given. If it vanishes at a point m, the evaluation at m defines a one dimensional representation of the Poisson algebra of functions on M. We show that this representation can, in general, not be quantized. Precisely, we give a counterexample for M=R^n, such that: (i) The evaluation map at 0 can not be quantized to a representation of the algebra of functions with product the Kontsevich product associated to the Poisson structure. (ii) For any formal Poisson structure extending the given one and vanishing at zero up to second order in epsilon, (i) still holds. We do not know whether the second claim remains true if one allows the higher order terms in epsilon to attain nonzero values at zero.