Derived operadic centers in algebraic geometry and deformation quantization

Abstract: In algebraic geometry, it is well known that Hochschild cohomology and, in particular, the algebraic structure it carries, plays a central role in studying the infinitesimal noncommutative deformations of geometric spaces. This thesis provides, for the first time, an explicit interface between J. Lurie's work on higher centers and the Hochschild cohomology of an algebraic $\mathbb{k}$-scheme within the framework of formal deformation quantization. Our motivation stems from the mysterious appearance of the square root of the Todd genus in Kontsevich's formality theorem for algebraic varieties, as well as the conjectural relationships between these objects and the motivic Galois group. Our main result is a canonical solution to Deligne's conjecture for Hochschild cochains in the affine and global cases, even for singular schemes, by exhibiting the Hochschild complex as an $\infty$-operadic center. We show that this equips the Hochschild complex with a universal $\mathbb{E}_2$-algebra structure that precisely agrees with the classical Gerstenhaber bracket and cup product on cohomology in the affine and smooth cases. Finally, we interpret this universal $\mathbb{E}_2$-algebra structure in terms of operadic formal moduli problems by exhibiting a relationship between $\infty$-operadic centers and formal automorphism groups.

• The paper constructs the Hochschild cochain complex as the derived operadic center, yielding a canonical E2-algebra structure free from auxiliary choices.
• It employs an ∞-categorical Eckmann-Hilton argument to uniquely recover the Gerstenhaber bracket and cup product in cohomology.
• The construction globalizes to schemes, extending deformation quantization techniques and operadic moduli perspectives to singular algebraic geometry.

Derived Operadic Centers in Algebraic Geometry and Deformation Quantization

Introduction and Motivation

The study of Hochschild cohomology in algebraic geometry, particularly through its algebraic structure, has been central to understanding the infinitesimal noncommutative deformation theory of schemes and varieties. Historically, the structure of Hochschild cochains as a $C_\ast(D_2)$-algebra—where $D_2$ is the little 2-disks operad—has been anticipated to explain the presence of Gerstenhaber algebra structures on their cohomology. Yet, explicit canonical constructions in the derived and higher categorical settings, reconciling all cohomological operations and universal properties, have remained open, especially in the context of singular schemes and in comparison to the known constructions reliant on choices, such as associators.

This work provides a systematic and universal answer to this gap. It establishes the Hochschild cochain complex, both affinely and globally, as the derived operadic center in the sense of $\infty$-categories, and demonstrates that this construction canonically recovers the desired universal $E_2$-algebra structure, including the Gerstenhaber algebra on cohomology, for smooth and singular cases. The thesis further connects these results to formal deformation quantization, the mysterious appearance of the Todd genus in Kontsevich's formality theorem, and advances the operadic approach to moduli problems in noncommutative geometry.

Main Results

The central achievements can be organized as follows:

1. Canonical Solution to Deligne’s Conjecture: By realizing the Hochschild complex as the $\infty$-operadic center of an associative algebra in a suitable derived $\infty$-category, the author constructs a canonical $C_\ast(D_2)$-algebra structure on the Hochschild complex, in contrast to earlier constructions which depended on auxiliary choices (e.g., Drinfeld associators). This result extends to singular schemes, not only to smooth or affine cases.
2. Identification of the $E_2$-Structure: Employing Lurie's theory of centers in monoidal $\infty$-categories, the Hochschild complex $C^\ast(A, A)$ is shown to inherit a universal $D_2$0-algebra structure. In the case of an affine algebra, this structure uniquely lifts the classical Gerstenhaber bracket and the cup product via the convolution and composition operations on endomorphism objects in module categories. An $D_2$1-categorical version of the Eckmann-Hilton argument is deployed to characterize the $D_2$2-structure in terms of explicit coherence data, precisely identifying the bracket in homology as the Gerstenhaber bracket.
3. Globalization to Schemes: For a scheme $D_2$3, even when singular, the Hochschild complex is characterized as the derived center of $D_2$4 as an associative algebra in the $D_2$5-category of dg sheaves. On affine patches, this construction descends to the affine (algebraic) case, enabling gluing via derived functoriality and hence validating this approach for global geometry.
4. Compatibility with Classical Structures: In the case of smooth schemes, the canonical $D_2$6-algebra structure on the center matches the homotopy Gerstenhaber algebra structure on the sheaf of polydifferential operators, both in terms of cup product and Gerstenhaber bracket, and hence matches the cohomological structures appearing in the formality theorems of Kontsevich and Tamarkin.
5. Operadic Formal Moduli Perspectives: The $D_2$7-algebra structure on the derived center is given an interpretation in terms of formal moduli problems governed by operads: centers control $D_2$8-deformations of module categories, and the structure is related to the automorphism group valued in derived loop spaces, with consequences for the understanding of Galois–Teichmüller symmetries in deformation quantization.
6. Technical Developments: The work develops the formalism of $D_2$9-operadic centers in detail, including a direct analysis of the universal properties, transfer of structures under base change and descent, and precise comparison theorems between local and global (affine and sheaf-theoretic) constructions.

Summary Table: Results and Implications

Aspect	Canonical Construction	Recovers Classical Structures	Extends to Singular	Relates to Moduli/Automorphisms	$\infty$0-Algebra Universality
Hochschild Cochains	Yes	Yes (Gerstenhaber, cup)	Yes	Yes (def. theory)	Yes
Polydifferential Operators	Yes (smooth varieties)	Yes	N/A	Yes	Yes
Singular Schemes	Yes	N/A	Yes	Yes	Yes

Technical Highlights

• Derived Center Construction: For an algebra $\infty$1 in a symmetric monoidal $\infty$2-category, the derived center $\infty$3 is characterized as a universal $\infty$4-algebra acting on $\infty$5; in the derived category, this realizes $\infty$6 as the endomorphism object $\infty$7, combining Yoneda composition and convolution products.
• $\infty$8-Categorical Eckmann-Hilton: The classical Eckmann-Hilton argument (commutativity of compatible monoid structures) is elevated to compute specific coherence data in $\infty$9-algebras arising from centers, crucial for realizing the Gerstenhaber bracket.
• Globalization and Descent: The analysis of how the affine construction of the derived center glues to the global Hochschild complex of a scheme is carried out via functoriality of (co)sheafification and compatibility with derived tensor product operations.
• Gerstenhaber Algebra Compatibility: The comparison of the canonical $E_2$0-algebra structure to the existing homotopy Gerstenhaber structures from polydifferential (Braces) operads is made explicit, confirming not only structural but numerical consistency on cohomology.
• Operadic Moduli and Formal Groups: The higher (operadic) deformation theory picture is advanced, tying the center to the linearized automorphism formal group of an algebra (analogous to the tangent space at the identity), and relating the appearance of the Todd genus in deformation quantization to derived group-theoretic structures.

Implications and Future Directions

• Universality and Canonicity: The presented construction removes dependence on choices (e.g., associators) in the $E_2$1-algebra structure on the Hochschild complex, thus providing a canonical lift of algebraic deformations. This is significant for applications where ambiguity in higher operations would otherwise lead to non-invariant structures, especially relevant for motivic and Galois-theoretic contexts.
• Extension to Singular Geometry: By not relying on smoothness, this theory offers a path to studying deformation quantization and higher centers in much broader settings, potentially impacting the understanding of noncommutative and singular algebraic geometry.
• Operators, Motives, and Galois Action: The relationship between higher centers, Grothendieck–Teichmüller symmetries, and motivic Galois theory is made more tractable via a universal, canonical structure. Although only partially addressed here, this establishes groundwork for new invariants of algebraic varieties and for tracing Galois symmetries in deformation contexts.
• Deformation Quantization and Physics: The generality and rigor of this construction can impact mathematical physics, especially in quantization and field theory approaches where operadic and higher categorical structures are fundamental.
• Future Developments: The thesis notes in-progress work on connections between these canonical centers and the Grothendieck–Teichmüller group action, potentially leading to new insights into the arithmetic and motivic aspects of deformation quantization and formality morphisms. Additionally, the operadic moduli perspectives suggest further generalizations for noncommutative and derived stacks.

Conclusion

This thesis establishes a universal, canonical construction of the derived operadic center in algebraic geometry, resolves Deligne's conjecture in generality, and systematically connects Hochschild cohomology, deformation quantization, and operadic moduli theory. The techniques and results unify and clarify the higher algebraic structures underlying deformations of algebras and schemes and pave the way for future work on arithmetic, motivic, and physical applications involving higher centers and operadic actions (2607.05455).