Eli Hawkins
Using the classification of formal deformation quantizations, and the formal, algebraic index theorem, I give a simple proof as to which formal deformation quantization (modulo isomorphism) is derived from a given geometric quantization.
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This paper has been withdrawn due to non-clearness of some technical points, as well as lack of a reasonable statement of quantization conjecture.
On the Quantization-Dequantization Correspondence for (co)Poisson Hopf Algebras
本文在广义范畴框架内构建了(co)Poisson Hopf代数的函子性量化，并引入Drinfeld-Yetter模块作为自然范畴，定义了逆向去量化函子，同时恢复了Etingof和Kazhdan的经典结果，并探讨了变形量化应用。
Asymptotic localization of symbol correspondences for spin systems and sequential quantizations of $S^2$
本文研究自旋系统的符号对应关系，提出投影算子符号经典局域化的几何判据以判定Poisson代数的渐近涌现，还定义了$S^2$上光滑函数的序列量子化并建立其与符号对应渐近局域化的关联。
Quantization of the Momentum Map via $\frak{g}$-adapted Formalities
本文利用形式化定理的等变版本，证明了泊松流形上具有量子动量映射的 $G$-不变星乘积的存在性并给出完全分类。
Noncommutative Hamiltonian structures and quantizations on preprojective algebras
本文证明非交换哈密顿空间上“量子化与约化交换”猜想成立，构造半直积代数并建立等变层与模的对应，进而给出预投射代数与箭图簇量子化间的显式对应。
Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al.. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered…
We recall the construction of non-formal deformation quantization of the Poincare Group ISO(1,1) on its coadjoint orbit and exhibit the associated non-formal star-exponentials.
We give a new construction of strict deformation quantization of symplectic manifolds equipped with a proper Lagrangian fiber bundle structure, whose representation spaces are the quantum Hilbert spaces obtained by geometric quantization. The construction can be regarded as a "lattice approximation of the correspondence between differential operators and principal symbols". We analyze the corresponding formal deformation quantization. We also investigate into relations between our construction and Berezin-Toeplitz deformation quantization.
In the framework of formal deformation quantization, we apply our formal moment map construction on the space of almost complex structures to recover the Donaldson-Fujiki moment map picture of the Hermitian scalar curvature. In the integrable case, it yields a formal moment map deforming the scalar curvature moment map.
In the framework of deformation quantization we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[λ]]-linear functionals obeying a formal variant of the usual KMS condition known in the theory of C^*-algebras. We show that for each temperature KMS states always exist and are up to a normalization equal to the trace of the argument multiplied by a formal analogue of the usual Boltzmann factor, a certain formal star exponential.
In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the *-representation theory of such *-algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C^*-algebras. Throughout this paper the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C^*-algebras, we show that the GNS construction of *-representations can be understood as Rieff…
Using theory of props we prove a formality theorem associated with universal quantizations of (strongly homotopy) Lie bialgebras.
The aim of this paper is two-fold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly, we discuss the existence of products of KMS states. As an application, we discuss the correspondence between quantum and classical Hamiltonians in spin systems and we provide a relation between the resolvent of Schödinger operators for non-interacting many-particle systems and quantization maps.
In this paper we prove the existence of a formal dynamical twist quantization for any triangular and non-modified formal classical dynamical $r$-matrix in the reductive case. The dynamical twist is constructed as the image of the dynamical $r$-matrix by a $L_\infty$-quasi-isomorphism. This quasi-isomorphism also allows us to classify formal dynamical twist quantizations up to gauge equivalence.
A Holomorphic perspective of Strict Deformation Quantization
该研究提出复分析方法以克服形式形变量子化的形式特征，提出严格形变量子化一般概念并结合复分析结果分析相关对象，还将方案应用于局部凸向量空间常泊松结构及李群余切丛正则力学的严格形变量子化。
We review recent works concerning deformation quantization of abelian supergroups. Indeed, we expose the construction of an induced representation of the Heisenberg supergroup and an associated pseudodifferential calculus by using Kirillov's orbits method. Then, a star-product is built on the abelian supergroup R^{m|n} together with a universal deformation formula for its actions. Using topological Hopf algebras, we reformulate this deformation as a continuous twist on comodule-algebras. We also introduce the notion of C*-superalgebra which is natural and compatible with the deformation. Final…
We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid `with separation of variables' over an arbitrary Kaehler-Poisson manifold.
Let $(g,δ_\hbar)$ be a Lie bialgebra. Let $(U_\hbar(g),Δ_\hbar)$ a quantization of $(g,δ_\hbar)$ through Etingof-Kazhdan functor. We prove the existence of a $L_\infty$-morphism between the Lie algebra $C(\g)=Λ(g)$ and the tensor algebra $TU=T(U_\hbar(g)[-1])$ with Lie algebra structure given by the Gerstenhaber bracket. When $(g,δ_\hbar,r)$ is a coboundary Lie bialgebra, we deduce from the formality morphism the existence of a quantization $R$ of $r$.
In this note, we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a Lie group action on Rd. This family of quantizations is parametrized by the formal G-systems introduced by the authors in a previous paper and allows us to obtain classical invariant Hamiltonians that quantize without anomalies with respect to the quantizations of the action prescribed by the formal G-systems.
In paper arXiv:1109.6031 the author introduced stable formality quasi-isomorphisms and described the set of its homotopy classes. This result can be interpreted as a complete description of formal quantization procedures. In this note we give a brief exposition of stable formality quasi-isomorphisms and prove that every homotopy class of stable formality quasi-isomorphisms contains a representative which admits globalization. This note is loosely based on the talk given by the author at the XXX Workshop on Geometric Methods in Physics in Bialowieza, Poland.
We describe a globalization construction for the Rozansky-Witten model in the BV-BFV formalism for a source manifold with and without boundary in the classical and quantum case. After having introduced the necessary background, we define an AKSZ sigma model, which, upon globalization through notions of formal geometry extended appropriately to our case, is shown to reduce to the Rozansky-Witten model. The relations with other relevant constructions in the literature are discussed. Moreover, we split the model as a $BF$-like theory and we construct a perturbative quantization of the model in th…
Formal Deformation quantization as a Fréchet algebra
本文定义了形式光滑函数空间上的弗雷歇拓扑，证明了星积运算在该拓扑下连续，并展示了量子多项式在形式光滑函数中的局部稠密性及迹的连续性。
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…
To each natural star product on a Poisson manifold $M$ we associate an antisymplectic involutive automorphism of the formal neighborhood of the zero section of the cotangent bundle of $M$. If $M$ is symplectic, this mapping is shown to be the inverse mapping of the formal symplectic groupoid of the star product. The construction of the inverse mapping involves modular automorphisms of the star product.
In deformation quantization one can associate five characteristic functions to (stable) formality morphisms on cochains and chains and to "two-brane" formality morphisms. We show that these characteristic functions agree.
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 …
The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semi-formal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semi-formal deformations are Noetherian and have polynomial growth. We propose constructions of semi-formal quantizations of…
We express the difference between Poisson bracket and deformed bracket for Kontsevich deformation quantization on any Poisson manifold by means of second derivative of the formality quasi-isomorphism. The counterpart on star products of the action of formal diffeomorphisms on Poisson formal bivector fields is also investigated.
We realize quantized anti de Sitter space black holes, building Connes spectral triples, similar to those used for quantized spheres but based on Universal Deformation Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symplectic symmetric space. More precisely we first obtain a UDF for Lie subgroups acting on a symplectic symmetric space M in a locally simply transitive manner. Then, observing that a curvature contraction canonically relates anti de Sitter geometry to the geometry of symplectic symmetric spaces, we use that UDF to define what we call Di…
In this contribution we review the formal GNS construction developped in a previous preprint (q-alg/9607019), and formulate the usual WKB-expansion in flat 2n-dimensional phase space in terms of a GNS construction with a positive linear functional with support on a projectable Lagrangean submanifold defined as a graph of an exact one form dS. The main trick is a suitable form of the star-exponential of S.
Following [14] and [12], we formalize the notion of an oscillatory integral interpreted as a functional on the amplitudes supported near a fixed critical point $x_0$ of the phase function with zero critical value. We relate to an oscillatory integral two objects, a formal oscillatory integral kernel and the full formal asymptotic expansion at $x_0$. The formal asymptotic expansion is a formal distribution supported at $x_0$ which is applied to the amplitude. In [12] this distribution itself is called a formal oscillatory integral (FOI). We establish a correspondence between the formal oscillat…
Motivated by the correspondence between the vertex and IRF models in statistical mechanics, we define and study a notion of vertex-IRF transformation for dynamical twists that generalizes a usual gauge transformation. We use vertex-IRF transformations to quantize all completely degenerate dynamical r-matrices on finite-dimensional Lie algebras.
In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well-known GNS construction for complex $C^*$-algebras: in this approach the corresponding positive linear functionals (`states') take their values not in the field of complex numbers, but in (a suitable extension field of) the field of formal complex Laurent series in the formal parameter. By using the algebraic and topological properties of these fields we prove that this construction makes sense and show in…
In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \cite{cm:deformation}. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \cite{z:deformation} of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author \cite{t1:def-gpd}, we reconstruct a universa…
To each natural deformation quantization on a Poisson manifold M we associate a Poisson morphism from the formal neighborhood of the zero section of the cotangent bundle to M to the formal neighborhood of the diagonal of the product M x M~, where M~ is a copy of M with the opposite Poisson structure. We call it dequantization of the natural deformation quantization. Then we "dequantize" Fedosov's quantization.
Symmetry in Deformation quantization and Geometric quantization
本文研究 Kähler 流形的量子化，对 Berezin-Toeplitz 形变量子化中的 1 次形式可量子化函数进行分类，并在 Hamiltonian $G$-对称性下建立了形变量子化与几何量子化的无穷小对称性之间的兼容性。
We give a new construction of symbols of the differential operators on the sections of a quantum line bundle $L$ over a Kaehler manifold $M$ using the natural contravariant connection on $L$. These symbols are the functions on the tangent bundle $TM$ polynomial on fibres. For high tensor powers of $L$, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on $TM$ corresponding to some pseudo-Kaehler structure on $TM$. Surprisingly, this star product is intimately related to the formal symplectic groupoid with se…
Quantizable functions on Kähler manifolds and non-formal quantization
应用先前构造的 Fedosov connections，在 Kähler 流形上找到了允许非形式形变量子化的函数子层，并在特定条件下证明其可量子化为与 prequantum line bundle 关联的 twisted differential operators 层。
Poisson algebraic structures on current manifolds (of maps from a finite dimensional Riemannian manifold into a 2-dimensional manifold) are investigated in terms of symplectic geometry. It is shown that there is a one to one correspondence between such current manifolds and Poisson current algebras with three generators. A geometric meaning is given to q-deformations of current algebras. The geometric quantization of current algebras and quantum current algebraic maps is also studied.
We give a simple geometric description of all formal deformation quantizations on a Kähler manifold $M$ which enjoy the following property of separation of variables into holomorphic and antiholomorphic ones. For each open subset $U\subset M$, $\star$-multiplication from the left by a holomorphic function and from the right by an antiholomorphic function on $U$ coincides with the pointwise multiplication by these functions. These quantizations are in 1-1 correspondence with formal deformations of the original Kähler metrics on $M$. It has been shown in [Ka] that the formal deformation quantiza…
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