Operads and Motives in Deformation Quantization

@article{Kontsevich1999OperadsAM,
  title={Operads and Motives in Deformation Quantization},
  author={Maxim Kontsevich},
  journal={Letters in Mathematical Physics},
  year={1999},
  volume={48},
  pages={35-72}
}
  • M. Kontsevich
  • Published 13 April 1999
  • Mathematics
  • Letters in Mathematical Physics
The algebraic world of associative algebras has many deep connections with the geometric world of two-dimensional surfaces. Recently, D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Deligne's conjecture on Hochschild complexes follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois… 

Deformation theory of bialgebras, higher Hochschild cohomology and Formality

A first goal of this paper is to precisely relate the homotopy theories of bialgebras and E 2-algebras. For this, we construct a conservative and fully faithful ∞-functor from pointed conilpotent

Diagrams for primitive cycles in spaces of pure braids and string links

The based loop space of the configuration space of m points in R can be viewed as the space of pure braids on m strands in R. For n > 2, Cohen and Gitler calculated its homology, relating it to

On the rational homotopy type of embedding spaces of manifolds in $R^n$

We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we

Little discs operads, graph complexes and Grothendieck-Teichmüller groups

This paper is a survey on the homotopy theory of $E_n$-operads written for the new handbook of homotopy theory.

Symmetric multiplicative formality of the Kontsevich operad

In his famous paper entitled "Operads and motives in deformation quantization", Maxim Kontsevich constructed (in order to prove the formality of the little d-disks operad) a topological operad, which

Kontsevich’s Swiss cheese conjecture

We prove a conjecture of Kontsevich which states that if $A$ is an $E_{d-1}$ algebra then the Hochschild cohomology object of $A$ is the universal $E_d$ algebra acting on $A$. The notion of an $E_d$

Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves M_g,n

We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves Mg,n (which, according to Penner, is controlled by the ribbon graph

Gerstenhaber algebras and the homology of spaces of long knots and long links

The goal of this thesis is to better understand the homology of spaces of long knots and long links. Our approach is based on the one hand on the Goodwillie and Weiss calculus of functors, and on the

HOLONOMIES FOR CONNECTIONS WITH VALUES IN L∞-ALGEBRAS

Given a flat connection α on a manifoldM with values in a filtered L∞-algebra g, we construct a morphism hol ∞ α : C•(M)→ BÛ∞(g), which generalizes the holonomy map associated to a flat connection

Holohonies for connections with values in $L_\infty$-algebras

Given a flat connection on a manifold with values in a filtered L-infinity-algebra, we construct a morphism of coalgebras that generalizes the holonomies of flat connections with values in Lie
...

References

SHOWING 1-10 OF 31 REFERENCES

Homotopy algebra and iterated integrals for double loop spaces

This paper provides some background to the theory of operads, used in the first author's papers on 2d topological field theory (hep-th/921204, CMP 159 (1994), 265-285; hep-th/9305013). It is intended

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the

Singletons, Physics in AdS Universe and Oscillations of Composite Neutrinos

The study starts with the kinematical aspects of singletons and massless particles. It extends to the beginning of a field theory of composite elementary particles and its relations with conformal

Another proof of M. Kontsevich formality theorem

Thisisadraftofpaperin which weannounceaplan ofan alternativeproofofM . Kontsevich form ality theorem [7]. The basic idea is to equip Hochschild cochains ofan associativealgebra A with a

Values of Zeta Functions and Their Applications

Zeta functions of various sorts are all-pervasive objects in modern number theory, and an ever-recurring theme is the role played by their special values at integral arguments, which are linked in

Homotopy Invariant Algebraic Structures on Topological Spaces

Motivation and historical survey.- Topological-algebraic theories.- The bar construction for theories.- Homotopy homomorphisms.- Structures on based spaces.- Iterated loop spaces and actions on