On a Vanishing Theorem of Kontsevich

Notes on Kontsevich-Soibelman's theorem about cyclic A-infinity algebras

Kontsevich and Soibelman has proved a relation between a non-degenerate cyclic homology element of an A-infinity algebra A and its cyclic inner products on the minimal model of A. We find an explicit formula of this correspondence, in terms of the strong homotopy inner products and negative cyclic cohomology of A. We prove that an equivalence class of the induced strong homotopy inner product depends only on the given negative cyclic cohomology class. Also, we extend such a correspondence to the case of gapped filtered A-infinity algebras.

On a theorem of Kontsevich

In two seminal papers M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads `commutative,' `associative' and `Lie.' We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F_r) outlined by Kontsevich. In [`Infinitesimal operations on chain complexes of graphs', Mathematische Annalen, …

Combinatorics of the Modular Group II: the Kontsevich integrals

We study algebraic aspects of Kontsevich integrals as generating functions for intersection theory over moduli space and review the derivation of Virasoro and KdV constraints. 1. Intersection numbers 2. The Kontsevich integral 2.1. The main theorem 2.2 Expansion of Z on characters and Schur functions 2.3 Proof of the first part of the Theorem 3. From Grassmannians to KdV 4. Matrix Airy equation and Virasoro highest weight conditions 5. Genus expansion 6. Singular behaviour and Painlev'e equation. 7. Generalization to higher degree potentials

Kontsevich 的余循环构造与 Loday-Quillen-Tsygan 定理的量子化

Kontsevich's Cocycle Construction and Quantization of the Loday-Quillen-Tsygan Theorem

本文通过图复形和Calabi-Yau $A_\infty$-范畴，推广Loday-Quillen-Tsygan映射并构造交换方块，利用Beilinson-Drinfeld代数实现量子化，旨在用范畴方法联系枚举几何与大N规范理论。

Choix des signes pour la formalite de M. Kontsevich

The explicit realization of M. Kontsevich's formality on $R^d$ is the main step of the proof of formality theorem on any manifold. We present here a coherent choice of orientations and signs in order to write completely M. Kontsevich's proof for $R^d$, including all the signs appearing in the formality equation.

Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory

These are lecture notes for the course "Poisson geometry and deformation quantization" given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifolds, tensor fields, integration on manifolds, Stokes' theorem, de Rham's theorem and Frobenius' theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinstein's tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introd…

dg流形的Tamarkin--Tsygan演算的Duflo--Kontsevich型同构

Duflo--Kontsevich-type isomorphisms for Tamarkin--Tsygan calculi of dg manifolds

该研究为dg流形建立了Duflo--Kontsevich型定理，通过特定映射实现两类演算的同构，证实了相关上同调层面的猜想。

Relative formality theorem and quantisation of coisotropic submanifolds

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 neighbourhood of C is L-infinity-quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to the deformation quantisation of coisotropic submanifolds are given. The proof uses a duality transformation to reduce the theorem to a version of Kontsevich's theorem for supermanifolds, which we a…

Formality for algebroid stacks

We extend the formality theorem of M. Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes.

Formality theorem for gerbes

We extend the formality theorem of Maxim Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes on smooth and complex manifolds.

Covariant and Equivariant Formality Theorems

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 that it is based on the use of covariant tensors unlike Kontsevich's original proof, which is based on $\infty$-jets of polydifferential operators and polyvector fields. Using our construction we prove that if a group G acts smoothly on a manifold M and M admits a G-invariant affine connection then there exists a G-equivariant quasi-isomorphism of formality…

Triviality of the higher Formality Theorem

It is noted that the higher version of M. Kontsevich's Formality Theorem is much easier than the original one. Namely, we prove that the higher Hochschild-Kostant-Rosenberg map is already a homotopy e_{n+1}-formality quasi-isomorphism whenever n is at least 2.

Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems

Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold. In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structures coming from symplectic Lie algebroids, as well as holomorphic symplectic structures. For deformations of these structures we prove the classification theorems and a general a general index theorem.

Kontsevich 图作用于Nambu-Poisson括号，IV. 当隐形的变得关键

Kontsevich graphs act on Nambu--Poisson brackets, IV. When the invisible becomes crucial

研究Kontsevich图对Nambu括号作用的拓扑数据生成，探讨高维中隐形图对解的影响

On the LMO Invariant, the Wheeling Theorem, and the Aarhus Integral

In a previous paper, we generalized the definition of the framed Kontsevich integral initially presented by Le and Murakami. We also defined an isotopy invariant $\widetilde{Z}_f$ that is well-behaved under band sum moves. Using this invariant we study the construction of the LMO invariant, the Wheeling Theorem, and the Aarhus integral.

Characteristic classes of A-infinity algebras

A standard combinatorial construction, due to Kontsevich, associates to any A-infinity algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We describe an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology classes. Along the way we re-prove Kontsevich's theorem relating graph homology to the homology of certain infinite-dimensional Lie algebras. An application to topological conformal field theori…

BV Formality

We prove a stronger version of the Kontsevich Formality Theorem for orientable manifolds, relating the Batalin-Vilkovisky (BV) algebra of multivector fields and the homotopy BV algebra of multidifferential operators of the manifold.

Bergman's Centralizer Theorem and quantization

We prove Bergman's theorem on centralizers by using generic matrices and Kontsevich's quantization method. For any field $\textbf{k} $ of positive characteristics, set $A=\textbf{k} \langle x_1,\dots,x_s\rangle$ be a free associative algebra, then any centralizer $\mathcal{C}(f)$ of nontrivial element $f\in A\backslash \textbf{k}$ is a ring of polynomials on a single variable. We also prove that there is no commutative subalgebra with transcendent degree $\geq 2$ of $A$.