Deformation Quantizations

• Deformation quantizations are formal frameworks that replace the pointwise product with a star-product, where the commutator recovers the Poisson bracket.
• They employ universal deformation formulas, Hochschild cohomology, and jet space methods to guarantee associativity and classify deformations.
• Extensions include applications to orbifolds, superspaces, and nonassociative algebras, unifying algebraic, geometric, and analytic techniques.

Deformation quantization is a formal mathematical framework that systematically constructs noncommutative deformations of commutative associative algebras, typically algebras of functions on Poisson manifolds, controlled by algebraic, geometric, and cohomological data. At its heart is the replacement of the pointwise product with an associative star-product whose commutator recovers the Poisson bracket to leading order.

1. Universal Deformation Formulas: Algebraic Foundations

The archetypal construction is the Universal Deformation Formula (UDF), which associates to a given associative algebra $\mathcal{A}$ over $\mathbb{Q}$ a family of deformed associative products parameterized by a formal variable $\hbar$. A central role is played by Hochschild 2-cocycles $F$ expressible in terms of commuting derivations $D_i$ and central coefficients $a_{ij}$: $F = \sum_{i,j=1}^n a_{ij} D_i \smile D_j,$ where $[D_i, D_j]=0$ and $a_{ij} \in Z(\mathcal{A})$. The cup product $\smile$ encodes the action $F(x, y) = \sum_{i,j} a_{ij} D_i(x) D_j(y)$. Such an $F$ is a Hochschild 2-cocycle and defines a class in $HH^2(\mathcal{A},\mathcal{A})$. The deformation is then explicitly given by exponentiating $F$: $$x * y = \sum_{k=0}^\infty \frac{\hbar^k}{k!} F^{\smile k}(x, y),$$ resulting in the star-product. Associativity is guaranteed by cocycle conditions and the graded-commutativity of involved derivations, ensuring all higher-order associators vanish (Gerstenhaber, 2018).

A prototypical example is the Moyal--Weyl product on a constant Poisson manifold, where the $D_i$ are coordinate derivatives and $a_{ij}$ are the entries of a Poisson bivector, leading to the familiar formula: $$f * g = \sum_{k=0}^\infty \frac{\hbar^k}{k!} \pi^{a_1 b_1} \dots \pi^{a_k b_k} (\partial_{a_1} \dots \partial_{a_k} f) (\partial_{b_1} \dots \partial_{b_k} g).$$

2. Deformation Quantization on Manifolds and Jet Spaces

Deformation quantization on manifolds generalizes the UDF by encoding star-products as formal power series in bidifferential operators determined by Poisson or symplectic structures. In jet manifold formalism, multidifferential operators are realized as smooth functions on infinite order jet spaces, and the star-product is determined covariantly by the formal exponential

$$f \star g = m \circ \exp[h \widehat{\epsilon}](f, g)$$

where $$\widehat{\epsilon} = \frac12 w^{\mu\nu}(x) d_\mu d'_\nu$$ is a biderivation encoded via total derivatives $d_\mu$, and $w^{\mu\nu}$ is the Poisson tensor or symplectic form. Associativity is a consequence of the flatness of the Cartan connection (i.e., $[d_\mu, d_\nu]=0$), and deformation classes are classified cohomologically by the DGLA of multidifferential operators, which is quasi-isomorphic to the DGLA of multivector fields—the formal manifestation of the Kontsevich formality theorem (Sardanashvily et al., 2015).

3. Symmetry, Orbifolds, and Quantum Principal Bundles

When the underlying manifold carries discrete symmetries, new structures appear. For instance, on $\mathbb{R}^2/\mathbb{Z}_2$ the standard Weyl quantization (Moyal–Weyl star product) is rigid, but extra deformations exist on the algebra of $\mathbb{Z}_2$-invariant or twisted functions, governed by Hochschild cohomology with coefficients in appropriately twisted modules. Homological perturbation theory constructs explicit higher-order terms, extending Kontsevich formality to Poisson orbifolds and generating rich algebraic structures such as symplectic reflection algebras and fuzzy spheres, with classical and quantum structures intertwined (Sharapov et al., 2022). In the context of principal bundles, Drinfeld twist deformations extend to Hopf–Galois extensions, yielding noncommutative principal bundles with quantum group fibers and noncommutative bases (Aschieri, 2016).

4. Supergeometry and Noncommutative Superspaces

Deformation quantization extends naturally to supermanifolds, with additional structures and phenomena. On split super-Kähler manifolds, deformation quantizations with separation of variables are constructed using super-Fedosov techniques and controlled by closed nondegenerate $(1,1)$-superforms. The star-product is given locally by conjugating anti-Wick products via exponentials of nilpotent superpotentials. Classification is governed by formal deformations of the superform in $H^{1,1}(\Pi E)$, and all quantizations admit canonical Berezin supertrace densities synthesized from bosonic and fermionic data (Karabegov, 2016). For algebraic symplectic supervarieties, deformation quantization is classified by period maps into super de Rham cohomology, directly generalizing Bezrukavnikov–Kaledin theory; this enables a full quantum orbit method for basic Lie superalgebras (Xiao, 28 Jul 2024).

5. Constraints, Cohomology, and Classification

The entire machinery of deformation quantization is underpinned by cohomological constraints. The existence and classification of star-products are determined by Hochschild and Poisson cohomology and associated formality theorems. For smooth symplectic manifolds, equivalence classes of star-products correspond to formal power series in the second de Rham cohomology $H^2_{dR}(M)[[\hbar]]$. In mixed characteristic (positive characteristic or relative to Witt vectors), the Hochschild cohomology of deformation quantizations aligns with the de Rham–Witt complex, and centers of reductions modulo powers of $p$ are described schematically in terms of Witt vectors (Tikaradze, 2014). For quadratic Poisson structures, the deformation complex is shown to be quasi-isomorphic to the even Kontsevich graph complex, and the full classification is encoded by the action of the Grothendieck–Teichmüller group, as universal quantizations correspond bijectively to Drinfeld associators (Khoroshkin et al., 2021).

6. Generalizations and Alternative Deformation Types

Deformation quantization frameworks are broad enough to accommodate several generalizations:

• Nonassociative algebras: Twisted associativity, Lie-admissible and anti-associative algebras, as well as nonassociative Poisson-type algebras, all admit formal deformation procedures paralleling Poisson quantization, replacing the Hochschild cocycle condition with twisted versions appropriate to the structure. These quantizations capture “polarizations” into commutative and skew parts and unify via generalized operads and cohomological classifications (Remm, 2022).
• Transposed Poisson and Novikov structures: Novikov deformations quantize so-called transposed Poisson algebras, where the classical limit satisfies a “transposed Leibniz” identity. For all Novikov–Poisson type structures (including all unital cases), canonical quantizations exist, and explicit classifications in dimension 2 are available (Chen et al., 21 Oct 2024).
• Formal graded bases and $A_\infty$-structures: Considering base algebras with parameters of non-positive degrees naturally generates $A_\infty$-algebras with higher operations, and the formal moduli are classified by degree-2 elements in cohomology with respect to the graded parameter ring. This extends the scope of deformation quantization to encompass Maurer–Cartan-type deformation spaces, and allows for the appearance of higher Massey operations (Altinay-Ozaslan et al., 2017).

7. Analytical and Non-Archimedean Extensions

The analytic, $C^*$-algebraic, and non-Archimedean realms are included by generalizing oscillatory integral quantization formulas and convolution-type products. Rieffel's approach, extended to non-Lie abelian groups over $p$-adic local fields, realizes deformation quantizations at the level of $C^*$-algebras, with $p$-adic Weyl quantization and Moyal-type formulas, ensuring continuity, associativity, and functional-analytic properties parallel to those seen in the classical real case (Gayral et al., 2014). For actions of supergroups, the machinery is adapted to graded contexts, developing $C^*$-superalgebras and universal deformation formulas compatible with supergeometry and applications to quantum field theory (Bieliavsky et al., 2010).

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Deformation quantizations unify algebraic and analytic deformation techniques for associative algebras, Poisson and noncommutative geometry, quantum groups, and supergeometry. The theory's versatility is demonstrated through explicit formulas, classification theorems, generalizations to orbifolds and superspaces, and connections to homological and operadic structures (Gerstenhaber, 2018, Sardanashvily et al., 2015, Sharapov et al., 2022, Aschieri, 2016, Karabegov, 2016, Xiao, 28 Jul 2024, Much, 2021, Altinay-Ozaslan et al., 2017, Tikaradze, 2014, Remm, 2022, Chen et al., 21 Oct 2024, Bieliavsky et al., 2010, Gayral et al., 2014, Khoroshkin et al., 2021).