Obstructions to Deformation Quantization of Bundles

Technical Summary: Obstructions to Deformation Quantization of Bundles

Problem Statement
The paper addresses the problem of extending a deformation quantization of a vector bundle $E$ over a smooth algebraic variety or complex manifold $M$ (equipped with a symplectic or holomorphic form $\omega$) from a finite order $k$ to a higher order $\ell$ ($\ell > k$). Specifically, given an order $k$ quantization $E_k$ (a locally free sheaf of modules over the quantized structure sheaf $\mathcal{O}_\hbar / \hbar^{k+1}\mathcal{O}_\hbar$), the authors investigate the cohomological obstructions to finding an order $\ell$ quantization $E_\ell$ that extends $E_k$. The study distinguishes between two regimes: the "abelian" range where $k+1 \leq \ell \leq 2k+1$, and the "non-abelian" range where $\ell \geq 2k+2$ (including the infinite order case $\ell = \infty$).

Methodology
The authors employ a geometric approach based on Harish-Chandra torsors and formal Darboux theorems.

1. Local Trivialization and Torsors: Utilizing a formal Darboux lemma (Theorem 7), the authors establish that locally, the pair $(\mathcal{O}_\hbar, E_k)$ is isomorphic to a standard pair involving the formal Weyl algebra $D$ and a free module $M_k = (D/\hbar^{k+1}D)^{\oplus e}$. The collection of all such local isomorphisms forms a torsor $P(k)$ over $M$ with structure group $F_k = \text{Aut}(D, M_k)$.
2. Harish-Chandra Structure: This torsor is upgraded to a transitive Harish-Chandra torsor over the pair $(F_k, \mathfrak{g}_k)$, where $\mathfrak{g}_k$ is the Lie algebra of infinitesimal automorphisms of the pair. This structure endows associated vector bundles with flat algebraic connections.
3. Lie Algebra Extensions: The problem of extending the quantization is translated into the problem of lifting the torsor $P(k)$ to a torsor $P(\ell)$ over an extended Harish-Chandra pair. This involves analyzing extensions of the Lie algebras $\mathfrak{g}_k \to \mathfrak{g}_\ell$.
   • For $k+1 \leq \ell \leq 2k+1$, the kernel of the extension is abelian, leading to a standard cohomological obstruction in $H^2$.
   • For $\ell \geq 2k+2$, the kernel is non-abelian. The authors abelianize the kernel by quotienting out the commutator subalgebra to derive a necessary condition (an "abelian shadow") for the existence of the lift.
4. Gelfand-Fuchs Map: The authors utilize the Gelfand-Fuchs map to relate Lie algebra cohomology classes (representing the extension cocycles) to de Rham cohomology classes of the associated flat bundles on $M$.
5. Fedosov Connections: In the holomorphic case, the results are reinterpreted using Fedosov-type connections on jet bundles, providing a concrete differential geometric realization of the obstruction classes.

Key Contributions and Results

• Theorem 1 (Abelian Range): For $k+1 \leq \ell \leq 2k+1$, the authors define a sheaf $\text{End}(E)^{(k, \ell]}$ (a quotient of endomorphisms of $E_k$). They prove that:

  • There exists an obstruction class in $H^2(M, \text{End}(E)^{(k, \ell]})$.
  • This class vanishes if and only if an extension $E_\ell$ exists.
  • If the obstruction vanishes, the set of isomorphism classes of such extensions is a torsor over $H^1(M, \text{End}(E)^{(k, \ell]})$.
  • This result is established via the vanishing of the image of the extension cocycle under the Gelfand-Fuchs map.

• Theorem 2 (Non-Abelian Range): For $\ell \geq 2k+2$, the authors define a sheaf $\text{End}^+_{\mathcal{O}_\hbar}(E_k)$ related to the abelianized kernel of the extension. They prove that:

  • The existence of an extension $E_\ell$ implies the vanishing of an obstruction class in $H^2(M, \text{End}^+_{\mathcal{O}_\hbar}(E_k))$.
  • This condition is necessary but not sufficient, as the full problem involves non-abelian cohomology.

• Theorem 8: Establishes the equivalence between the quantization $E_k$ and the sheaf of flat sections of the vector bundle associated with the Harish-Chandra torsor $P(k)$. It provides a quasi-isomorphism between the quantization sheaf and the de Rham complex of the associated bundle.

• Corollaries on Existence and Uniqueness:

  • If $H^2(M, \text{End}(E)) = 0$ (e.g., $M$ is a union of two affine open sets), any order $k$ quantization extends to all orders $\ell \geq k+1$.
  • If $H^1(M, \text{End}(E)) = 0$ (e.g., $M$ is affine), the extension is unique.
  • If $M$ is affine and $E$ is trivial, any quantization of order $k \geq 1$ is trivial.

• Lifts of Morphisms (Section 7): The paper extends the analysis to lifting morphisms $\phi_k: E_k \to F_k$ to order $\ell$. An obstruction $O \in H^1(M, \text{Hom}_{\mathcal{O}_\hbar}(E_s, F_s))$ is identified. The authors show that even if $O$ does not vanish, one can adjust the choices of the target quantizations $E_\ell$ and $F_\ell$ (by adding elements from $H^1$) to make the obstruction vanish, thereby allowing the lift of the morphism.

Significance and Claims
The paper claims to provide a complete characterization of the obstructions to extending deformation quantizations of vector bundles in the "abelian" range ($\ell \leq 2k+1$), where the obstruction is both necessary and sufficient. For the more complex non-abelian range, it provides a rigorous framework for identifying necessary conditions via abelianization.

The authors emphasize that their approach unifies the study of quantization obstructions through the lens of Harish-Chandra torsors and flat connections, offering a systematic way to recover quantized bundles from their associated geometric data. The work generalizes previous results on the quantization of the structure sheaf to the case of vector bundles and provides a spectral sequence approach to analyzing the cohomology of the obstruction sheaves. The paper does not claim to solve the non-abelian existence problem in full generality but rather isolates the abelian obstructions that must vanish for a solution to exist.