Trace Functionals of the Kontsevich Quantization
Alexander Golubev
将迹的概念推广到Kontsevich量子化代数中，证明了对于所有由幂零李群的Hamilton作用表示的辛流形的子空间所表示的Poisson流形，迹是关于幺模体积形式的积分.
We generalize the notion of trace to the Kontsevich quantization algebra and show that for all Poisson manifolds representable by quotients of a symplectic manifold by a Hamiltonian action of a nilpotent Lie group, the trace is given by integration with respect to a unimodular volume form.
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Poisson bracket, deformed bracket and gauge group actions in Kontsevich deformation quantization
本文利用形式拟同构的二阶导数表示了任意Poisson流形上Kontsevich形变量子化的泊松括号与形变括号的区别。本文还研究了Poisson形式双向量场上形式双同态作用在星积上的对应。
Kontsevich's Universal Formula for Deformation Quantization and the Campbell-Baker-Hausdorff Formula, I
我们将Maxim Kontsevich提出的任意Poisson结构的形变量子化的普适公式与Campbell-Baker-Hausdorff公式联系起来。我们的基本论点是，指数适当变形的泊松结构提供了一个原型，这样的通用公式。
Let $α$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $α$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras $S(V^*)$ and $Λ(V)$. These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on $α$, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the …
Kontsevich quantization and invariant distributions on Lie groups
研究了有限维实李代数（或超代数）g的对偶的Kontsevich形变量子化。在这种情况下，Kontsevich星积定义了S（g）上的一个新卷积，被认为是g中支持0的分布空间。对S（g）中的p，证明了卷积算子f-> f * p是一个具有解析芽的微分算子.利用这一事实证明了Kashiwara和Vergne关于李群上不变分布的一个猜想。这给出了Duflo关于李群上双不变微分算子局部可解性的结果的一个新的证明.此外，这个新的证明扩展到李超群。
A path integral approach to the Kontsevich quantization formula
给出了泊松流形上Kontsevich形变量子化公式的量子场论解释。我们证明它是由一个简单的拓扑玻色开弦理论的路径积分的微扰展开给出的。它的Batalin-Vilkovisky量子化产生一个超共形域论。星积的结合性，以及更一般的形式猜想，可以用域论方法来理解。作为应用，我们用Poisson代数的中心来计算变形代数的中心。
On the (ir)rationality of Kontsevich weights
我们计算变形量子化中Kontsevich图的权。直到有理数，结果是Zeta（3）^2/Pi^6。
Deformation quantization of submanifolds and reductions via Duflo-Kirillov-Kontsevich map
本文从已知的Poisson子流形$M$的量子化出发，给出了由方程$f_i=0$（其中$f_i$是Casimirs）定义的Poisson子流形$N$的量子化结果：利用$D(f_i)$的象考虑$M$上量子化函数的因子代数，其中$D: Fun(M) \to Fun(M)\otimes \CC[\hbar]$是Duflo-Kirillov-Kontsevich映射.我们猜想这个代数同构于从$M$继承的具有Poisson结构的$Fun(N)$的量子化.提出了关于哈密顿约化的一个类似猜想，即“形变量子化与约化互换”。在$S^2$的情况下，它可以被量化为一个子流形，作为一个减少，并使用最近发现的显式星积的结构进行检查。结果表明，所有的结构都是一致的。
On the Kontsevich and the Campbell-Baker-Hausdorff deformation quantizations of a linear Poisson structure
对于向量空间$\g^*$（其中$\g$是有限维李代数）上的Kirillov-Poisson结构，已知该结构的至少两个正则形变量子化：它们是M. Kontsevich通用公式[K]，以及由经典的Campbell-Baker-Hausdorff公式[Ka]推导出的公式。在[Ka]中证明了最后一个公式是Kontsevich公式的一部分，它包含了所有第一类顶点之间没有（定向）圈的可容许图。从CBH定理可以得出，Kontsevich公式的这一部分定义了一个结合积（在线性泊松结构的情况下）。这些注释的目的是直接证明最后的结果，使用类似于[K]的方法而不是CBH公式。构造了一个从线性系数多向量场的\dg李代数到多微分算子的\dg李代数的$L_\infty$ -态射$\U_\lin\colon[T^\ndot_\poly]_\lin\to D^\ndot_\poly$，它不等于形式$L_\infty$ -态射$\U\colon T^\ndot_\poly\to D^\ndot_\poly$ [K]对子代数$[T^\ndot_\poly]_\lin$的限制.对于具有线性系数的双向量场$α$，使得$[α,α]=0$，在调和角映射[K]的情况下，$D^\ndot_\poly$中的Maurer-Cartan方程的相应解$\U_\lin(α)$精确地定义了CBH-量化，第2节。我们证明了限制Kontsevi…
In arXiv:math/0105152, the second author used the Kontsevich deformation quantization technique to define a natural connection ω_n on the compactified configuration spaces of n points on the upper half-plane. This connection takes values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that ω_n is flat. The configuration space contains a boundary stratum at infinity which coincides with the (compactified) configuration space of n points on the complex plane. When restricted to this stratum, ω_n gives rise to a flat connection ω_n^\infty. We sh…
One way of reconciling classical and quantum mechanics is deformation quantization, which involves deforming the commutative algebra of functions on a Poisson manifold to a non-commutative, associative algebra, reminiscent of the space of quantum observables. This depends on a formal parameter $\hbar$, so that the original pointwise product is recovered when $\hbar=0$. In 1997 Kontsevich showed that a deformation quantization exists for every Poisson manifold. He furthermore gave a simple, combinatorial formula for producing a quantization of any Poisson structure on $\mathbb{R}^n$. The primar…
The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the noncommutative $\star$-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich $\star$-product up to order 4 in the deformation parameter $\hbar$. Already at this stage,…
Graph complexes in deformation quantization
Kontsevich的形式定理和随之而来的星积公式依赖于构造多向量场的DGLA和多微分算子的DGLA之间的$L_\infty$ -态射。这种构造使用了一种图形演算。在这篇文章中，我们提出了这个图形演算的细节，强调其代数特征。它是微分分次李代数中容许图的Kontsevich DGLA和多向量场与多微分算子之间的线性同态的Chevalley-Eilenberg DGLA之间的一个态射。Kontsevich的形式态射的证明是重新审查在这个光和代数框架讨论的树级减少Kontsevich的星产品描述。
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。
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…
Quantization of Poisson families and of twisted Poisson structures
我们描述了一个变形量子化的修改泊松几何封闭的3-形式。在适当的条件下，它产生一堆代数。用于此目的的基本对象是一类由Maurer-Cartan方程给出的Poisson结构族;它们很容易使用Kontsevich的形式定理进行量化。
D-branes on Group Manifolds and Deformation Quantization
最近M。Kontsevich发现了一个组合公式，定义了任何Poisson流形的变形量子化的星积。Kontsevich公式在物理上被重新解释为开弦拓扑西格玛模型的量子关联函数，以及在具有Neveu-Schwarz B场的平坦背景中的D膜。在这里，相应的Kontsevich的公式的对偶李代数推导出的形式主义的D膜的群流形。特别是，我们表明，该公式是编码在两点关联函数的Wess-Zumino-Witten有效理论与Dirichlet边界条件。进入形式主义的B场在这一推导中发挥了重要作用。
Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula.
On deformation quantization of Dirac structures
出于问题的横向变形量化的叶状流形，我们描述了狄拉克结构的量化（更确切地说，那些正规的变形）堆叠的代数体的意义上的Kontsevich。
Calabi-Yau Deformation Quantization
研究卡拉比-丘流形的变形量子化，通过记录孔采维奇形式性态射的卡拉比-丘流形版本，发现单模全纯泊松卡拉比-丘流形有典范闭变形量子化，旨在借助森-兹维巴赫弦顶点在该范畴解释孔采维奇变形量子化。
We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by grt_1, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) E_n operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Fur…
We prove a version of Kontsevich's formality theorem for two subspaces (branes) of a vector space $X$. The result implies in particular that the Kontsevich deformation quantizations of $\mathrm{S}(X^*)$ and $\wedge(X)$ associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet's recent paper on Koszul duality in deformation quantization.
Deformation quantization generates all multiple zeta values
本文证明了Kontsevich形变量子化公式中的图系数能生成所有多重Zeta值，通过开发基于多对数的积分新技术并结合Lyndon字分解完成证明。
These notes are based on the course given at the School of Geometry, University Kasdi Merbah (Ouargla) 2012. The aim of the course was the deformation quantization of Poisson Lie groups. In these notes we only review Kontsevich's formality theorem.
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 Formality. We consider the simplest example of this situation: $\mathbb{R}^2$ with the reflection symmetry $\mathbb{Z}_2$. The usual quantization leads to the Weyl algebra. While Weyl algebra is rigid, the algebra of even or twisted by $\mathbb{Z}_2$ functions has one more deformation, which was identified by Wigner and is related to Feigin's $gl_λ$ and to…
Quantization of SL(2,R)^* as Bialgebra
利用Kontsevich的积，我们将Poisson-李群SL（2，R）^* 作为双代数.余积是来自群结构的余积的变形。得到的双代数结构同构于量子泛包络代数U_hsl（2，R）.
Moyal quantization and stable homology of necklace Lie algebras
本文计算了与箭图相关的项链李代数的稳定同调，并从某些$A_\infty$ -范畴中构造了稳定同调类.我们的构造是M的曲线的模空间的同调类构造的推广。孔采维奇本文的第二部分给出了项链李代数的对称代数的Moyal型量子化。由此产生的量子化代数在通常的多项式代数的Moyal量子化中具有自然表示。
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 $A$.
We study the deformation complex of the dg wheeled properad of $\mathbb{Z}$-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmüller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of $\mathbb{Z}$-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps. In particular we show that two universal quantizations of …
Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.
On the AKSZ formulation of the Poisson sigma model
我们回顾和扩展的Alexandrov-Kontsevich-Schwarz-Zaboronsky的Batalin-Vilkovisky经典主方程的解决方案的建设。特别地，我们研究了带有边界的流形上的sigma模型的情形。我们表明，这种结构的一个特殊情况下产生的Batalin-Vilkovisky行动功能的泊松σ模型在磁盘上。正如我们在以前的论文中所示，这个模型的微扰量子化与Kontsevich的泊松流形的变形量子化和他的形式定理有关。我们还讨论了目标流形的超同态的作用。
Koszul duality in deformation quantization, I
设$α$是$\mathbb{C}$上有限维向量空间$V$上的多项式Poisson双向量。然后Kontsevich [K97]给出了代数$S(V)^*$的量子化$f\star g$的公式。利用生成元和关系给出了由$α$定义的具有PBW性质的代数的构造.也就是说，我们定义一个代数作为自由张量代数$T(V^*)$通过关系$x_i\otimes x_j-x_j\otimes x_i=R_{ij}(\hbar)$的商，其中$R_{ij}(\hbar)\in T(V^*)\otimes\hbar \mathbb{C}[[\hbar]]$，$R_{ij}=\hbar \Sym(α_{ij})+\mathcal{O}(\hbar^2)$，每对$i,j=1...\dim V$有一个关系。我们证明了所构造的代数服从PBW性质，这是Poincaré-Birkhoff-Witt定理的推广.在线性Poisson结构的情况下，我们得到PBW定理本身，对于二次Poisson结构，我们得到一个与$V$上的量子$R$ -矩阵密切相关的对象。同时得到了变形代数的自由分解（对于任意的$α$）。这种PBW代数的构造相当简单，PBW性质的证明也相当简单。主要的努力应该进行证明的猜想，这样我们得到一个代数同构的Kontsevich星代数。
From local to global deformation quantization of Poisson manifolds
基于Kontsevich的局部公式，给出了Poisson流形上函数代数的变形量子化的一个显式构造。变形函数代数被实现为具有平坦联络的向量丛的水平截面的代数。
The deformation quantization by Kontsevich [arXiv:q-alg/9709040] is a way to construct an associative noncommutative star-product $\star=\times+\hbar \{\ ,\ \}_{P}+\bar{o}(\hbar)$ in the algebra of formal power series in $\hbar$ on a given finite-dimensional affine Poisson manifold: here $\times$ is the usual multiplication, $\{\ ,\ \}_{P}\neq0$ is the Poisson bracket, and $\hbar$ is the deformation parameter. The product $\star$ is assembled at all powers $\hbar^{k\geq0}$ via summation over a certain set of weighted graphs with $k+2$ vertices; for each $k>0$, every such graph connects the two…
Poisson sigma models and deformation quantization
这是一个评论，旨在物理观众之间的关系泊松西格玛模型的表面与边界和变形量子化。这些模型是拓扑开弦论。在经典的哈密顿方法中，我们描述了约化相空间及其结构（辛广群），特别是解释了弦端点坐标的非交换性的经典起源。我们还回顾了微扰拉格朗日方法及其与Kontsevich的明星产品。最后，我们评论这两种方法之间的关系。
Traces for star products on the dual of a Lie algebra
本文给出了李代数对偶上BCH星积的所有迹。首先我们用初等的方法证明了BCH和Kontsevich星积是强闭的当且仅当李代数是幺模的。在下一步中，我们表明，BCH星积的迹是由$\ad$不变泛函。特别的例子是余伴轨道上的积分。我们表明，对于一个紧凑的李群和一个正规轨道，甚至可以实现，这种整合成为一个积极的迹功能。在这种情况下，我们明确地描述相应的GNS表示。最后，我们讨论了如何不变变形的一个组可以用来诱导变形的空间，该集团的行动。
A Canonical Quantization of Poisson Manifolds: a 2-Groupoid Scheme
该研究提出泊松流形的李2-群胚正则量子化方案，关联几何与形变量子化，给出无极化路径积分量子化映射定义，且算符可保持极化截面。
Deformation quantization with traces and vanishing of the wheels
该论文已被撤回，因为数学QA/0002057“具有迹线的变形量化”的结果仅适用于恒定体积形式。
Deformation quantization with traces
本文证明了一个与循环形式猜想密切相关的命题。特别地，我们证明了对于R^d上无发散的Poisson双向量场，Kontsevich星积与调和角函数是循环的。我们还证明了这一定理的任意泊松流形的情况下的全球化，并证明了推广的Connes-Flato-Sternheimer猜想的闭星形产品的泊松情况下。
Associative algebras, punctured disks and the quantization of Poisson manifolds
该说明的目的是提供一个介绍背后的Kontsevich-Cattaneo-Felder公式的量子化泊松结构的代数，几何和量子场论的想法。我们展示了如何自然产生的量子化公式本身时，施加以下两个要求的费曼积分：一方面，它必须重现给定的泊松结构作为其微扰展开的一阶项;另一方面，它的三点函数应描述一个结合代数。它进一步表明如何Magri-Koszul括号1-形式自然适合的泊松σ模型的理论。
We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M. Starting from a suitable Courant sigma-model on an open membrane with target space M, regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma-model on the boundary of the membrane whose target space is the cotangent bundle T^*M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields …
Star products and branes in Poisson-Sigma models
我们证明了Poisson-西格玛模型中的非共各向同性膜在量子水平上是允许的。当膜被定义为第二类约束时，模型的微扰量子化产生了与膜上的狄拉克括号相关联的Kontsevich星积。我们还讨论了第一类和第二类约束时的量化。
A combinatorial approach to coefficients in deformation quantization
利用Connes-Kreimer重整化技术，从组合的角度研究了星积的图上循环。Hochschild复形控制着结合代数的变形理论，它是由图插入定义的前李代数结构中的图的DGLA的“Kontsevich表示”。研究了它的UEA（Connes-Kreimer霍普夫代数的奇宇称模拟）的对偶性质，以求得变形方程的解.在树的层次上，初值变形问题的解是唯一的。对于线性系数，所得公式与豪斯多夫级数有关。
Hochschild cohomology of the Weyl algebra and traces in deformation quantization
本文给出了一个关于系数在对偶中的Weyl代数的生成Hochschild上同调的上圈的公式，它是由圆上序点的位形空间上的积分给出的。利用这个公式和非对易形式几何，我们得到了辛流形形变量子化中正则迹的一个显式表达式。
Based on work done by Bonechi, Cattaneo, Felder and Zabzine on Poisson sigma models, we formally show that Kontsevich's star product can be obtained from the twisted convolution algebra of the geometric quantization of a Lie 2-groupoid, one which integrates the Poisson structure. We show that there is an analogue of the Poisson sigma model which is valued in Lie 1-groupoids and which can often be defined non-perturbatively; it can be obtained by symplectic reduction using the quantization of the Lie 2-groupoid. We call these groupoid-valued sigma models and we argue that, when they exist, they…
The quantization problem for the trace-bracket algebra, derived from double Poisson brackets, is discussed. We obtain a generalization of the boundary YBE (or so-called ABCD-algebra) for the quantization of quadratic trace-brackets. A dynamical deformation is proposed on the lines of Gervais-Neveu-Felder dynamical quantum algebras.
The main goal of this paper is to compute the characteristic class of the Alekseev-Lachowska *-product on coadjoint orbits. We deduce an analogue of the Weyl dimension formula in the context of deformation quantization.
As a detailed application of the BV-BFV formalism for the quantization of field theories on manifolds with boundary, this note describes a quantization of the relational symplectic groupoid for a constant Poisson structure. The presence of mixed boundary conditions and the globalization of results is also addressed. In particular, the paper includes an extension to space-times with boundary of some formal geometry considerations in the BV-BFV formalism, and specifically introduces into the BV-BFV framework a "differential" version of the classical and quantum master equations. The quantization…
Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems
最近Kontsevich解决了分类问题的变形量子化的所有泊松结构的流形上。在本文中，我们研究的泊松结构，显式方法的Fedosov可以应用，即泊松结构来自辛李代数胚，以及全纯辛结构。对于这些结构的变形，我们证明了分类定理和一般的一般指标定理。
We develop a theory of short star-products for filtered quantizations of graded Poisson algebras, introduced in 2016 by Beem, Peelaers and Rastelli for algebras of regular functions on hyperKähler cones in the context of 3-dimensional $N=4$ superconformal field theories [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378]. This appears to be a new structure in representation theory, which is an algebraic incarnation of the non-holomorphic ${\rm SU}(2)$-symmetry of such cones. Using the technique of twisted traces on quantizations (an idea due to Kontsevi…
Etingof-Kazhdan quantization of Lie superbialgebras
对于每一个半单李代数，都可以构造Drinfeld-Jimbo代数U。该代数是由生成元和关系定义的变形霍普夫代数。为了研究U的表示论，Drinfeld利用KZ-方程构造了一个拟霍普夫代数A.他证明，特别类别的模块在代数U和A张量等价。在A-G型李超代数的情况下，存在代数U和A的类似结构。然而，Drinfeld的证明上述等价的范畴并没有推广到李超代数。本文讨论A-G型李超代数的另一种证明。我们的证明利用了李（超）双代数的Etingof-Kazhdan量子化。应该指出的是，上述等价关系非常有用。例如，它已被用于纽结理论联系量子群不变量和Kontsevich积分。
A Remark on Formal KMS States in Deformation Quantization
在形变量子化的框架下，我们将相空间中具有紧支集的函数幂级数的形变代数上的形式KMS态定义为C[[λ]]-线性泛函，它服从C^*-代数理论中已知的通常KMS条件的形式变体。我们表明，对于每个温度KMS状态总是存在的，并达到一个归一化等于跟踪的参数乘以一个正式的模拟通常的玻尔兹曼因子，一定的正式的星指数。
In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E^*, where E -> M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich's formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of E we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concr…
BRST theory without Hamiltonian and Lagrangian
我们考虑一个一般的规范系统，其物理自由度是通过约束表面上的限制，然后对规范变换的作用进行因式分解来获得的;这样做，就不存在哈密顿结构或作用原理。对于这样一个通用的规范系统，我们构建了一个一致的BRST制定，其中包括传统的BV拉格朗日和BFV哈密顿计划作为特殊情况。如果原来的流形进行弱泊松结构（一个双向量场引起的泊松括号的空间上的物理观测）的通用规范系统被证明承认变形量子化的Kontsevich形式定理。这种量化算法的西格玛模型的解释进行了简要讨论。
What is a double star-product?
本文研究了双泊松括号的变形量子化问题，提出了一种在代数表示空间上诱导星积的结构，并给出了一个具体例子和双形式定理。
Covariant and Equivariant Formality Theorems
本文利用多元微分算子代数和多元向量场的Fedosov分解证明了一般流形上的Kontsevich形式定理。我们的建设的形式准同构的主要优点是，它是基于使用共变张量不像Kontsevich的原始证明，这是基于$\infty$ -喷气机的polydifferential运营商和polyvector领域。利用我们的构造，我们证明了：如果群G光滑地作用在流形M上，且M容许G-不变的仿射联络，则存在形式上的G-等变拟同构。这个结果意味着如果流形M具有有限或紧群G的光滑作用或具有李群G的自由作用，则M容许G-等变形式拟同构。特别是，这给出了一个解决方案的变形量子化问题的任意泊松orbifold。
Creating quantum projective spaces by deforming q-symmetric algebras
通过图示计算变形 q-symmetric 代数构造大量 Koszul、Calabi-Yau 量子射影空间，证明无阻碍性，并首次显式计算验证二次 Poisson 结构的 Kontsevich 量子化收敛性。
Poisson sigma models and symplectic groupoids
我们考虑与Poisson流形相关联的Poisson sigma模型。这个模型的微扰量子化产生了Kontsevich星积公式。我们在这里研究的经典模型的哈密顿形式主义。相空间是哈密顿叶理的叶空间，具有自然的广群结构。如果它是一个流形，那么它是给定泊松流形的辛广群。我们研究了各种各样的例子。特别地，构造了一类二维Poisson域的整体辛广群。
The finite dual coalgebra as a quantization of the maximal spectrum
本文将 Heyneman–Sweedler 有限对偶余代数解释为非交换极大谱的量子化，定义完全剩余有限维代数，并刻画中心模有限素仿射代数的 Azumaya 轨迹及单位根量子平面。
Semiclassical Hodge theory for log Poisson manifolds
本文在光滑Poisson簇的拓扑K理论上构造了弱依赖紧化选择的混合Hodge结构，提供了一系列计算工具，并证明其量子参数与已知形变量子化参数一致。
Skein traces from curve counting
该研究通过全纯曲线的skein值计数定义三维流形skein间的映射，证明壁穿越满足对应公式提升，给出双分支覆叠下skein迹显式公式，特殊情形复现已有结果。
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