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arXiv:2607.12910

Obstructions to Deformation Quantization of Bundles

Vladimir Baranovsky, Greg Huey

math.AGmath-phmath.DGquant-phSubmitted Jul 14, 2026• Announced Jul 15, 2026
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Abstract

Let $\left(M, \mathcal{O}_M \right)$ be a smooth algebraic variety over field $κ$ of characteristic $0$ with an algebraic symplectic form $ω$, or a complex manifold with a holomorphic form $ω$. Furthermore, let $E$ be a vector bundle over $\left(M, \mathcal{O}_M \right)$ and $\mathcal{O}_{\hbar}$ a deformation quantization of $\mathcal{O}_M$ compatible with $ω$. Assuming that $E$ possesses a deformation quantization to order $\hbar^k$ we consider the problem of extending it to order $\hbar^\ell$ for $\ell > k$, and establish triviality of an obstruction class as a necessary condition for this extension to exist. Furthermore, in the case $\ell \le 2k+1$, we prove that this condition is also sufficient.

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•Jan 1, 1994•4 citations
Let (M,OM)\left(M, \mathcal{O}_M \right)(M,OM​) be a smooth algebraic variety over field κκκ of characteristic 000 with an algebraic symplectic form ωωω, or a complex manifold with a holomorphic form ωωω. Furthermore, let EEE be a vector bundle over (M,OM)\left(M, \mathcal{O}_M \right)(M,OM​) and Oℏ\mathcal{O}_{\hbar}Oℏ​ a deformation quantization of OM\mathcal{O}_MOM​ compatible with ωωω.
Assuming that EEE possesses a deformation quantization to order ℏk\hbar^kℏk we consider the problem of extending it to order ℏℓ\hbar^\ellℏℓ for ℓ>k\ell > kℓ>k, and establish triviality of an obstruction class as a necessary condition for this extension to exist. Furthermore, in the case ℓ≤2k+1\ell \le 2k+1ℓ≤2k+1, we prove that this condition is also sufficient.
We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations we find a compatibility condition restricting the set of admissible connections. In the special case when phase space is a cotangent bundle this compatibility condition has many solutions, and the resulting quantum theory not only reproduces the well-known geometric quantization scheme, but also allows to quantize all interesting observables. For Kähler manifolds there is no compatibility condition, but a canonical choice for the parameters. The explicit form of the observables however remains undetermined.
Starting from a (possibly infinite dimensional) pre-symplectic space (E, σ) , we study a class of modified Weyl quantizations. For each value of the real Planck parameter ℏ , we have a C*-Weyl algebra \mathcal W(E,ℏ_σ) , which altogether constitute a continuous field of C*-algebras, as discussed in previous works. For ℏ = 0 , we construct a Fréchet–Poisson algebra, densely contained in \mathcal W(E, 0) , as the classical observables to be quantized. The quantized Weyl elements are decorated by so-called quantization factors, indicating the kind of normal ordering in specific cases. Under some assumptions on the quantization factors, the quantization map may be extended to the Fréchet–Poisson algebra. It is demonstrated to constitute a strict and continuous deformation quantization, equivalent to the Weyl quantization, in the sense of Rieffel and Landsman. Realizing the C*-algebraic quantization maps in regular and faithful Hilbert space representations leads to quantizations of the unbounded field expressions.
It is known that in the WKB approximation of multicomponent systems like Dirac equation or Born-Oppenheimer approximation, an additional phase appears apart from the Berry phase. So far, this phase was only examined in special cases, or under certain restrictive assumptions, namely that the eigenspaces of the matrix or endomorphism valued symbol of the Hamiltonian form trivial bundles. We give a completely global derivation of this phase which does not depend on any choice of local trivializing sections. This is achieved using a star product approach to quantization. Furthermore, we give a systematic and global approach to a reduction of the problem to a problem defined completely on the different ``polarizations''. Finally, we discuss to what extent it is actually possible to reduce the problem to a really scalar one, and make some comments on obstructions to the existence of global quasiclassical states.
The authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial.
It is shown that q-deformed quantum mechanics (systems with q-deformed Heisenberg commutation relations) can be interpreted as an ordinary quantum mechanics on Kähler manifolds, or as a quantum theory with second (or first)-class constraints. 1. The q-deformed Heisenberg-Weyl algebras [1], [2] exhibiting the quantum group symmetries [3],[4] have attracted much attention of physicists and mathematicians. In particular, some works have been devoted to establishing potentials for ordinary quantum systems which exhibit a q-deformed energy spectrum (see, for example, [5] and references therein) in order to obtain classical dynamical systems associated with the q-deformed Heisenberg-Weyl algebras. In the present paper, we analyze the correspondence limit, when ¯h → 0 (rather than q → 1), of multimode q-deformed Heisenberg-Weyl algebras [1], [2] regardless of the specific form of a Hamiltonian. We show that classical systems associated with the q-deformed Heisenberg-Weyl algebras in this way possess a non-trivial symplectic structure which, in turn, is related to the Kähler symplectic structure [6]. From the other hand, a natural (physical) reason for a dynamical system to have a nontrivial symplectic structure is the existence of ”frozen ” , non-dynamical degrees of freedom or, in other words, constraints [7]. Dirac pointed out that a nontrivial symplectic structure might naturally occur in dynamical systems through second-class constraints [7]. We shall demonstrate below how constrained dynamics can be associated with the q-deformed Heisenberg-Weyl algebra. As was shown in [8], a non-commutative phase space does not necessarily emerges in the formal classical limit of q-deformed quantum mechanics, provided the deformation parameter q is a function of the Planck constant. In the latter case, q-deformed Heisenberg-Weyl algebras yield quadratic symplectic structures on a commutative phase space. For example, the one-mode q-deformed Heisenberg-Weyl algebra defined by the commutation relation ˆ b ˆ b − q