Shifted Poisson unfoldings and quantum anomalies

Abstract: Families of classical field theories often depend on geometric parameters. A basic question is whether the corresponding classical observables can be compared by flat parallel transport, and whether this comparison survives quantization. We study this problem for families of shifted Poisson structures. To such a family we attach a Poisson transverse controller $\mathbb U_π$, a homotopy stabilizer encoding transverse symmetries which preserve the Poisson Maurer--Cartan element up to coherent homotopy. Its flat splittings are precisely transversal shifted Poisson unfoldings, and they act on vertical symmetries, Poisson cohomology and local deformation theory. We then formulate the $\hslash$-adic lifting problem for quantized objects; its degree-two obstruction classes are the transport anomalies. The construction is realized for star-products, BV observables, factorization algebras and AKSZ theories. For the Poisson sigma model, anomaly-free quantization of the flat transport makes the Cattaneo--Felder/Kontsevich boundary product horizontal over the parameter space.

• The paper presents a derived geometric framework for flat parallel transport in families of shifted Poisson structures, emphasizing the emergence of quantum transport anomalies.
• It constructs the Poisson transverse controller as a derived L∞ algebroid, linking classical observables with their deformation quantization and providing a bridge between classical and quantum frameworks.
• It systematically characterizes obstruction classes in the quantization process, thereby offering clear criteria for when flat transport persists or fails in global field theories.

Shifted Poisson Unfoldings and Quantum Anomalies

Introduction and Problem Statement

The paper "Shifted Poisson unfoldings and quantum anomalies" (2607.05918) presents a rigorous, derived-geometric treatment of the parallel transport problem for families of shifted Poisson structures, with a particular focus on the compatibility of classical and quantized field-theoretic data. The main technical objective is to identify the precise homotopical structure controlling flat comparison of classical observables along the base of a family, and to describe the obstruction classes arising in the passage to quantized (deformation-theoretic) settings—quantum transport anomalies.

Building from motivations in classical and quantum field theory, especially the Poisson sigma model and its perturbative quantization (yielding Kontsevich’s star-product as boundary observable), the authors address the structural question of when such families admit flat identifications of (quantum) observables, and what obstructs such horizontal transport. The approach employs the formalism of shifted Poisson geometry, derived foliations, and filtered deformation theory.

Poisson Controllers and Transversal Unfoldings

The central technical innovation is the construction of the Poisson transverse controller, a derived Lie (or $L_\infty$) algebroid, denoted $\mathbb{U}_\pi$, attached to a family of shifted Poisson manifolds $(X/S, \pi)$. This object encodes projectable transversal symmetries of the Hamiltonian derived foliation $\mathcal{F}_\pi/S$ that preserve the Poisson Maurer–Cartan element up to coherent homotopy.

A transversal shifted Poisson unfolding is then defined as a flat splitting of $\mathbb{U}_\pi$. Notably, these unfoldings are not merely deformations of the Poisson structure (which would be controlled by the tangent complex to the Maurer–Cartan space of bivectors), but rather data that identify a flat action of the base’s tangent directions as transverse Poisson symmetries, null-homotopic with respect to infinitesimal variation of the Maurer–Cartan element. The precise classification theorem shows an equivalence:

$$\mathrm{Unf}^{tr}_{\mathrm{Pois}_n}(X/S,\pi) \simeq \mathrm{Flat}^{der}_S(T_S, \mathbb{U}_\pi)$$

where $\mathrm{Flat}^{der}$ denotes flat sections in the derived category. In effective (classical) situations, this reduces to the familiar moduli of unfoldings modulo tangent data, while in the general crossed or $L_\infty$-module case, it retains higher, central isotropy.

Quantization, Lifting, and Transport Anomalies

The passage from classical to quantized data is systematized via the filtered quantum lifting principle. Given a filtered quantization $\mathcal{Q}$ of the Poisson algebra (e.g., as in star-products, quantum BV observables, or quantized factorization algebras), and a filtered quantum symmetry extension of $\mathbb{U}_\pi$, the obstruction theory for lifting a flat classical unfolding to a quantum counterpart is described recursively via obstruction classes—the transport anomalies. More specifically, for a flat splitting $\mathbb{U}_\pi$0 and a partial quantum lift modulo $\mathbb{U}_\pi$1, the order $\mathbb{U}_\pi$2 anomaly lives in

$\mathbb{U}_\pi$3

where $\mathbb{U}_\pi$4 is the quantum vertical kernel and $\mathbb{U}_\pi$5 denotes the Chevalley–Eilenberg complex.

These anomalies refine and generalize classical monodromy: even when fibrewise quantization exists, global parallel transport may be obstructed by higher cohomology. The order-by-order filtered lifting formalism encapsulates both extension and automorphism classes in $\mathbb{U}_\pi$6 and $\mathbb{U}_\pi$7 respectively.

Realizations and Theoretical Implications

The paper provides explicit geometric realizations across several domains:

• Star-product deformations: In the symplectic setting, the key obstruction is horizontality of the Fedosov characteristic class, and the star-product derivation connection is constructed via horizontal lifts. For general Poisson families, Kontsevich formality is required to be equivariant with respect to the controller to produce a compatible transport on the Hochschild complex.
• BV, AKSZ, and Factorization Algebras: The $\mathbb{U}_\pi$8-shifted symplectic (BV) case is addressed analogously, with the homotopy fixing both the BV symplectic structure and action. Locality hypotheses permit the transport of factorization algebras of both classical and quantum observables, with transport anomalies again described in controller terms.
• AKSZ field theory: The structure transports to mapping spaces via integration, leveraging the compatibility of the controller formalism with the AKSZ transgression. The Poisson sigma model is treated as the prototypical example; flat transport precisely explains when the Cattaneo–Felder/Kontsevich boundary star-product varies horizontally over the parameter space. The obstruction cohomology classifies when the desired Leibniz rule for the quantized boundary product is satisfied.

Effective vs. Crossed Models and Central Isotropy

A persistent theme is the need for crossed or $\mathbb{U}_\pi$9-module presentations to fully capture the derived nature of the problem. In classical (“effective”) settings, one may work with quotient controllers and ordinary flat splittings; in general, central isotropy is retained, corresponding to higher automorphisms of the unfolding. The formal moduli of unfoldings, their automorphisms, and their obstructions are all governed by the corresponding (shifted, possibly abelian) Chevalley–Eilenberg cohomology.

Consequences and Future Directions

This work provides a comprehensive derived-geometric architecture for compatibility problems in deformation quantization, field theory, and global Poisson geometry. By situating transport anomalies as primary cohomological obstructions, it enables a systematic description of when quantization interacts well with parameter variations.

Potential Developments:

• Applications to Derived Moduli: Since shifted Poisson and symplectic structures underpin much of the modern theory of derived moduli stacks, these results suggest directions for constructing globally compatible quantizations and invariants therein.
• Field Theory and Higher Categories: The formalism’s compatibility with BV/AKSZ quantization, factorization structures, and boundary conditions opens new avenues for the global analysis of (extended) TQFTs and their deformation quantizations.
• Equivariant and Stacked Settings: The necessity of controller-equivariant formality links these questions to equivariant quantization, descent theory, and categorified symmetry.

Conclusion

"Shifted Poisson unfoldings and quantum anomalies" (2607.05918) formalizes, in the setting of derived algebraic and Poisson geometry, the obstruction theory for flat parallel transport of both classical and quantized field-theoretic data across parameter spaces. The derived Poisson controller and its quantized analogues serve as the organizing principle, with precise cohomological obstructions—the transport anomalies—governing the quantizability of flat transport. The framework unites and generalizes diverse instances in deformation quantization, quantum field theory, and derived foliation theory, providing both refined theoretical insights and concrete computational tools for future studies in geometry and mathematical physics.