Related papers: A proof of the Tsygan formality conjecture for cha…

The formality morphism $\boldsymbol{\mathcal{F}}=\{\mathcal{F}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to…

We prove Kontsevich's cyclic formality conjecture.

Recently the first two authors constructed an L-infinity morphism using the S^1-equivariant version of the Poisson Sigma Model (PSM). Its role in deformation quantization was not entirely clear. We give here a "good" interpretation and show…

We prove a relative version of Kontsevich's formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich's theorem if C=M. It states that the DGLA of multivector fields on an infinitesimal…

In this paper we prove a conjecture of B. Shoikhet. This conjecture states that the tangent isomorphism on homology, between the Poisson homology associated to a Poisson structure on $\mathbb{R}^d$ and the Hochschild homology of its…

Whenever a given Poisson manifold is equipped with discrete symmetries the corresponding algebra of invariant functions or the algebra of functions twisted by the symmetry group can have new deformations, which are not captured by…

Kontsevich's formula for a deformation quantization of Poisson structures involves a Feynman series of graphs, with the weights given by some complicated integrals (using certain pullbacks of the standard angle form on a circe). We explain…

We prove the 0-(co)homology part of the conjecture on the cup-products on tangent cohomology in the Tsygan formality [Sh2]. We discuss its applications to the Duflo formula.

Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of…

We start with a short exposition of developments in physics and mathematics that preceded, formed the basis for, or accompanied, the birth of deformation quantization in the seventies. We indicate how the latter is at least a viable…

For a finite dimensional Lie algebra $\mathfrak{g}$, the Duflo map $S\mathfrak{g}\rightarrow U\mathfrak{g}$ defines an isomorphism of $\mathfrak{g}$-modules. On $\mathfrak{g}$-invariant elements it gives an isomorphism of algebras.…

In 1979, M. Kashiwara and M. Vergne formulated a conjecture on a Lie group G which implies that the Duflo isomorphism of Z(g) and S(g)^g extends to a natural module isomorphism between the spaces of germs of invariant distributions on G and…

The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative. We prove that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the…

We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is…

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the…

We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string…

In the paper "Formality conjecture" (1996) Kontsevich designed a universal flow $\dot{\mathcal{P}}=\mathcal{Q}_{a:b}(\mathcal{P})=a\Gamma_{1}+b\Gamma_{2}$ on the spaces of Poisson structures $\mathcal{P}$ on all affine manifolds of…

The third author recently proved that the Shoikhet-Dolgushev L-infinity-morphism from Hochschild chains of the algebra of smooth functions on manifold to differential forms extends to cyclic chains. Localization at a solution of the…

We state the analogs of Kontsevich's formality conjecture for Hochschild and cyclic chains, as well as their

Kontsevich constructed a map from suitable cocycles in the graph complex to infinitesimal deformations of Poisson bi-vector fields. Under the deformations, the bi-vector fields remain Poisson. We ask, are these deformations trivial,…