Convergence up to gauge equivalence for quadratic Poisson structures (weaker version of Kontsevich’s conjecture)

Establish that for any quadratic Poisson bivector σ on V = C^n, the canonical quantization in the sense of Kontsevich is gauge‑equivalent to a convergent star product on the polynomial algebra C[V], i.e., all star products of polynomial generators yield convergent power series in the deformation parameter.

Background

The paper introduces a slightly weaker version of Kontsevich’s convergence conjecture that allows convergence 'up to isomorphism' (gauge equivalence). In this formulation, one seeks a gauge transformation making the canonical star product convergent, rather than requiring the original canonical series to converge directly.

Using recent results identifying canonical quantizations for toric brackets and their own deformation theory, the authors verify this weaker conjecture for a class of quadratic Poisson structures admitting filtered log symplectic toric degenerations, leaving the general case open.