Convergence of canonical quantization series for projective-space quantizations (Kontsevich’s 2001 conjecture)

Prove that, for canonical quantizations (in the sense of Kontsevich) of quadratic Poisson structures arising from homogeneous quantizations of the homogeneous coordinate ring C[x_0, ..., x_{n-1}] of projective space P^{n−1}, the formal power series c_{ij}^{kl}(hbar) determining the quadratic defining relations converge as analytic functions of hbar in a neighborhood of hbar = 0.

Background

Kontsevich’s formality theorem provides canonical (formal) quantizations via a Feynman‑graph expansion, but extracting explicit algebras and assessing convergence for these series is difficult. In the projective case, canonical quantizations are expected to yield analytic dependence on the deformation parameter.

The authors verify this convergence up to isomorphism for a broad class of quadratic Poisson structures admitting filtered log symplectic toric degenerations, but the general conjecture remains open beyond these cases.

References

Moreover, Kontsevich conjectures in that in this case, his canonical quantization recipe should result in algebras for which the formal power series $c_{ij}{kl}(\hbar)$ determining the relations of his formal quantizations are convergent, i.e.~define analytic functions of $\hbar$ in some disk around $\hbar =0$.

— Creating quantum projective spaces by deforming q-symmetric algebras (2411.10425 - Matviichuk et al., 15 Nov 2024) in Section 1.1 (Non-commutative deformations of polynomial algebras)