Obstructions to Deformation Quantization of Bundles

Vladimir Baranovsky; Greg Huey

mathematical physics

Obstructions to Deformation Quantization of Bundles

arXiv:2607.12910

summary

The paper investigates conditions under which a vector bundle on a symplectic algebraic variety that has a deformation quantization to a given order can be extended to higher orders, identifying an obstruction class whose vanishing is necessary and, for certain ranges, sufficient.

Abstract

Let $(M, O_{M})$ be a smooth algebraic variety over field $I^º$ of characteristic $0$ with an algebraic symplectic form $I ¨ $ , or a complex manifold with a holomorphic form $I ¨ $ . Furthermore, let $E$ be a vector bundle over $(M, O_{M})$ and $O_{ℏ}$ a deformation quantization of $O_{M}$ compatible with $I ¨ $ . Assuming that $E$ possesses a deformation quantization to order $ℏ^{k}$ we consider the problem of extending it to order $ℏ^{ℓ}$ for $ℓ > k$ , and establish triviality of an obstruction class as a necessary condition for this extension to exist. Furthermore, in the case $ℓ \leq 2 k + 1$ , we prove that this condition is also sufficient.

Topics & keywords

#deformation quantization #vector bundles #symplectic geometry #obstruction theory #algebraic geometrydeformation quantization of algebrasobstruction classsymplectic formholomorphic symplectic manifoldℏ-order extensionvector bundle quantization