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**Operads and motives in deformation quantization.**
*(English)*
Zbl 0945.18008

The idea that algebras of observables in quantum mechanics should be interpreted as deformations of commutative algebras of functions on certain manifolds (phase spaces) has led to the concept of deformation quantization. This viewpoint was proposed in 1978 by F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, and D. Sternheimer in their pioneering paper “Deformation theory and quantization. I: Deformations of symplectic structures” [Ann. Phys. 111, No. 1, 61-110 (1979; Zbl 0377.53024)]. In the program of deformation quantization initiated by these authors, one basic step is to construct an associative (but possibly non-commutative) multiplication law, a so-called “star product”, on the vector space \(C^\infty(X)\) of functions on a Poisson manifold \(X\), which is compatible with gauge transformations.

A rigorous mathematical proof of the existence of a canonically defined gauge equivalence class of star products on any Poisson manifold \(X\) has been given only recently by the author of the paper under review. In his preprint “Deformation quantization of Poisson manifolds. I” [cf. M. Kontsevich, \(q\)-alg/9709040] he established his so-called “formality theorem”, from which he derived the existence theorem for star products mentioned above.

His formality theorem states that in a suitably defined homotopy category of differential graded Lie algebras, two objects are equivalent. The first object is the Hochschild complex of the algebra of functions on the manifold \(X\), and the second object is a certain graded Lie superalgebra of polyvector fields on \(X\). Via an interpretation in the framework of Feynman diagrams, Kontsevich’s explicit isomorphism in the case \(X=\mathbb{R}^n\) provides a canonical way for deformation quantization.

Shortly after this break-through, Tamarkin gave another proof of the formality theorem for the case \(X=\mathbb{R}^n\) [cf. D. E. Tamarkin, “Another proof of M. Kontsevich formality theorem”, math.QA/9803025]. His approach is not only more general, but also makes the conjectured relation between the classifying space for deformation quantizations and the Grothendieck-Teichmüller group much more transparent.

The present paper is closely related to the author’s (unpublished) talk delivered at the ICM-98 Congress in Berlin, and its purpose is to further extend and generalize Tamarkin’s improvements of the author’s earlier results. Using throughout the framework of operads, and of the homotopy theory for algebraic structures, the author discusses the present state of the recent developments in deformation quantization in a unified and systematic way, including some furthergoing conjectures and speculations.

Section 1 of the letter gives a motivating introduction to the subject. Section 2 is devoted to generalities on operads and algebras over operads. However, the main topic discussed here is Deligne’s conjecture on operad actions on the Hochschild complex of an associative algebra, together with a generalization of it to higher dimensions. Section 3 provides a more general proof of Tamarkin’s formality theorem, and a sketch of its application to the author’s formality theorem in deformation quantization. Section 4 deals with the possible relations between the motivic Galois group, the Grothendieck-Teichmüller group, and various homogeneous spaces appearing in deformation quantization. In this context, the author explains five deep conjectures of his, and in the concluding Section 5 he adds two more conjectures concerning the link between higher-dimensional algebras, motives, and candidates for quantum field theories. Altogether, this letter is highly inspiring, tremendously rich of new ideas, and truly programmatic with respect to further research in this direction.

A rigorous mathematical proof of the existence of a canonically defined gauge equivalence class of star products on any Poisson manifold \(X\) has been given only recently by the author of the paper under review. In his preprint “Deformation quantization of Poisson manifolds. I” [cf. M. Kontsevich, \(q\)-alg/9709040] he established his so-called “formality theorem”, from which he derived the existence theorem for star products mentioned above.

His formality theorem states that in a suitably defined homotopy category of differential graded Lie algebras, two objects are equivalent. The first object is the Hochschild complex of the algebra of functions on the manifold \(X\), and the second object is a certain graded Lie superalgebra of polyvector fields on \(X\). Via an interpretation in the framework of Feynman diagrams, Kontsevich’s explicit isomorphism in the case \(X=\mathbb{R}^n\) provides a canonical way for deformation quantization.

Shortly after this break-through, Tamarkin gave another proof of the formality theorem for the case \(X=\mathbb{R}^n\) [cf. D. E. Tamarkin, “Another proof of M. Kontsevich formality theorem”, math.QA/9803025]. His approach is not only more general, but also makes the conjectured relation between the classifying space for deformation quantizations and the Grothendieck-Teichmüller group much more transparent.

The present paper is closely related to the author’s (unpublished) talk delivered at the ICM-98 Congress in Berlin, and its purpose is to further extend and generalize Tamarkin’s improvements of the author’s earlier results. Using throughout the framework of operads, and of the homotopy theory for algebraic structures, the author discusses the present state of the recent developments in deformation quantization in a unified and systematic way, including some furthergoing conjectures and speculations.

Section 1 of the letter gives a motivating introduction to the subject. Section 2 is devoted to generalities on operads and algebras over operads. However, the main topic discussed here is Deligne’s conjecture on operad actions on the Hochschild complex of an associative algebra, together with a generalization of it to higher dimensions. Section 3 provides a more general proof of Tamarkin’s formality theorem, and a sketch of its application to the author’s formality theorem in deformation quantization. Section 4 deals with the possible relations between the motivic Galois group, the Grothendieck-Teichmüller group, and various homogeneous spaces appearing in deformation quantization. In this context, the author explains five deep conjectures of his, and in the concluding Section 5 he adds two more conjectures concerning the link between higher-dimensional algebras, motives, and candidates for quantum field theories. Altogether, this letter is highly inspiring, tremendously rich of new ideas, and truly programmatic with respect to further research in this direction.

Reviewer: Werner Kleinert (Berlin)

### MSC:

18D50 | Operads (MSC2010) |

16S80 | Deformations of associative rings |

55P35 | Loop spaces |

14F40 | de Rham cohomology and algebraic geometry |

14F25 | Classical real and complex (co)homology in algebraic geometry |

58A50 | Supermanifolds and graded manifolds |

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

81T99 | Quantum field theory; related classical field theories |

55P48 | Loop space machines and operads in algebraic topology |