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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{deformation quantization} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry} [[!include symplectic geometry - contents]] \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{noncommutative_geometry}{}\paragraph*{{Noncommutative geometry}}\label{noncommutative_geometry} [[!include noncommutative geometry - contents]] \hypertarget{formal_geometry}{}\paragraph*{{Formal geometry}}\label{formal_geometry} [[!include formal geometry -- contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{FormalDeformationQuantization}{Formal deformation quantization}\dotfill \pageref*{FormalDeformationQuantization} \linebreak \noindent\hyperlink{ExplicitDefinitionOfTraditionalDeformationQuantization}{Explicit definition of deformation of Poisson manifolds/Poisson algebras}\dotfill \pageref*{ExplicitDefinitionOfTraditionalDeformationQuantization} \linebreak \noindent\hyperlink{FormulationAsLiftsFromPnAlgebrasToBDAlgebras}{Formulation as lifts from $P_n$-algebras to $BD_n$-algebras and $E_n$-algebras}\dotfill \pageref*{FormulationAsLiftsFromPnAlgebrasToBDAlgebras} \linebreak \noindent\hyperlink{InPerturbativeQuantumFieldTheory}{Deformation quantization in perturbative quantum field theory}\dotfill \pageref*{InPerturbativeQuantumFieldTheory} \linebreak \noindent\hyperlink{StrictDeformationQuantization}{Strict deformation quantization}\dotfill \pageref*{StrictDeformationQuantization} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{existence_results}{Existence results}\dotfill \pageref*{existence_results} \linebreak \noindent\hyperlink{deformation_of_poisson_manifolds}{Deformation of Poisson manifolds}\dotfill \pageref*{deformation_of_poisson_manifolds} \linebreak \noindent\hyperlink{deformation_of_algebraic_varieties}{Deformation of Algebraic Varieties}\dotfill \pageref*{deformation_of_algebraic_varieties} \linebreak \noindent\hyperlink{gerstenhabers_deformation_theory_by_hochschild_cohomology}{Gerstenhaber's deformation theory by Hochschild (co)homology}\dotfill \pageref*{gerstenhabers_deformation_theory_by_hochschild_cohomology} \linebreak \noindent\hyperlink{in_terms_of_differential_graded_lie_algebras}{In terms of differential graded Lie algebras}\dotfill \pageref*{in_terms_of_differential_graded_lie_algebras} \linebreak \noindent\hyperlink{the_deligne_conjecture}{The Deligne conjecture}\dotfill \pageref*{the_deligne_conjecture} \linebreak \noindent\hyperlink{RelationToUniversalEnvelopingAlgebras}{Deformation by universal enveloping algebras}\dotfill \pageref*{RelationToUniversalEnvelopingAlgebras} \linebreak \noindent\hyperlink{MotivicGaloisGroup}{Motivic Galois group action on the space of quantizations}\dotfill \pageref*{MotivicGaloisGroup} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \emph{Deformation quantization} is one formalization of the general idea of [[quantization]] of a [[classical mechanical system]]/[[classical field theory]] to a [[quantum mechanical system]]/[[quantum field theory]]. Deformation quantization focuses on the [[algebras of observables]] of a physical system (hence on the [[Heisenberg picture]]): it provides rules for how to [[deformation theory|deform]] the commutative algebra of classical observables to a non-commutative algebra of quantum observables. (This is in contrast to [[geometric quantization]], which focuses on the [[spaces of states]] and hence on the [[Schrödinger picture]].) Usually and traditionally, \emph{deformation quantization} refers to (just) \emph{formal} deformations, in the sense that it produces [[formal power series]] expansions in a formal parameter $\hbar$ (physically: [[Planck's constant]]) of the product in the deformed algebra of observables. \begin{tabular}{l|l|l|l|l} [[classical mechanics]]&[[semiclassical approximation]]&\ldots{}&[[formal deformation quantization]]&[[quantum mechanics]]\\ \hline $\mathcal{O}(\hbar^0)$&$\mathcal{O}(\hbar^1)$&$\mathcal{O}(\hbar^n)$&$\mathcal{O}(\hbar^\infty)$&\\ \end{tabular} (This is related to [[perturbation theory]], which is about [[formal power series]] in [[coupling constants]], instead of in [[Planck's constant]].) But there are refinements of this to [[C-star algebraic deformation quantization]] which studies the proper deformation to a genuine [[C-star algebra]] of observables. (This in turn is related to genuine [[geometric quantization]] via the notion of [[geometric quantization of symplectic groupoids]].) One can, therefore, argue that only [[C-star algebraic deformation quantization|strict deformation quantization]] is genuine [[quantization]]. For instance in (\hyperlink{GukovWitten09}{Gukov-Witten 09, section 1.4}) it says \begin{quote}% Generally speaking, [[physics]] is based on $[$ strict $]$ quantization, rather than $[$ formal $]$ deformation quantization, although conventional quantization sometimes leads to problems that can be treated by deformation quantization. \end{quote} As other methods of quantization, deformation quantization has as input a description of a [[classical mechanical system]], which is in this case most often a smooth [[Poisson manifold]]. The deformation quantization replaces the algebra of [[smooth functions]] on the Poisson manifold with the same [[vector space]], but equipped with new noncommutative [[associative algebra|associative unital product]] whose [[commutator]] agrees, up to order $\hbar$, with the underlying [[Poisson bracket]]. Of course the proper study of quantization of Poisson manifolds studied the appropriate notion at the level of [[sheaves]] of algebras. Gluing local solutions to the quantization problem furthermore involves [[stacks]] and specifically [[gerbes]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} If the result of deformation quantization is an algebra over the [[power series]] [[ring]] $\mathbb{R}[ [ \hbar ] ]$ of a formal parameter $\hbar$ (thought of as [[Planck's constant]]) such that the limit $\hbar \to 0$ reproduces the starting point of the deformation, then one speaks of \begin{itemize}% \item \hyperlink{FormalDeformationQuantization}{Formal deformation quantization} \end{itemize} In much of the literature this is regarded as the default meaning of ``deformation quantization''. But this is really the case corresponding to [[perturbation theory]] in [[quantum field theory]]. A ``genuine'' or ``strict'' deformation quantization \begin{itemize}% \item \hyperlink{StrictDeformationQuantization}{Strict deformation quantization} \end{itemize} is supposed to result in a non-formal deformation, which in terms of the above formal power series at least means that one can set $\hbar = 1$ such that all expressions in $\hbar$ converge, but which in general is taken to mean something stronger, such as that there is a continuous field of [[C-star algebraic deformation quantization]]. \hypertarget{FormalDeformationQuantization}{}\subsubsection*{{Formal deformation quantization}}\label{FormalDeformationQuantization} We first give the traditional \begin{itemize}% \item \hyperlink{ExplicitDefinitionOfTraditionalDeformationQuantization}{Explicit definition of traditional deformation quantization} \end{itemize} of a [[Poisson manifold]]/[[Poisson algebra]]. Thought of in terms of [[physics]] this describes a [[quantization]] of a [[physical system|system]] of [[quantum mechanics]], as opposed to full [[quantum field theory]]. More abstractly, this may be formulated and generalized in terms of lifts of [[algebras over an operad]] over a [[P-n operad]] to a [[BD-n operad]] and hence an [[E-n operad]], for $n = 1$. This we discuss in \begin{itemize}% \item \hyperlink{FormulationAsLiftsFromPnAlgebrasToBDAlgebras}{Formulation as lifts from P-n algebras to BD-n algebras} \end{itemize} In this formulation one sees that for genral $n$ the construction applies to $n$-dimensional [[quantum field theory]] (with [[quantum mechanics]] for $n = 1$ be 1-dimensional quantum field theory, for instance the [[sigma-model]] ``on the [[worldline]]'' of a [[particle]]). A formulation of deformation quantization to \emph{[[local quantum field theory|local]]} quantum field theory formulated in terms of [[factorization algebras of observables]] over [[spacetime]]/[[worldvolume]] is indicated in (\hyperlink{CostelloGwilliam}{Costello-Gwilliam, chapter 5}). \hypertarget{ExplicitDefinitionOfTraditionalDeformationQuantization}{}\paragraph*{{Explicit definition of deformation of Poisson manifolds/Poisson algebras}}\label{ExplicitDefinitionOfTraditionalDeformationQuantization} Let $M$ be a [[Poisson manifold]] and let $A = C^\infty(M)$ be the [[Poisson algebra]] of smooth functions. \begin{defn} \label{}\hypertarget{}{} A \textbf{$\ast$-product} ([[star product]]) on $A$ is a product on the [[power series]] $A [ [ t ] ]$ that is (1) bilinear over $\mathbb{R}[ [ t ] ]$, (2) associative, and (3) for $a,b \in A$ it can be written out as a [[formal power series]] \begin{displaymath} a \ast b = \sum_{n=0}^\infty B_n(a,b) t^n \end{displaymath} where $B_n$ are bilinear maps on $A$ such that $B_0(a,b) = ab$. \end{defn} \begin{defn} \label{}\hypertarget{}{} A (formal) \textbf{deformation quantization} of $M$ is a [[star product]] on $A = C^\infty(M)$ such that the [[Poisson bracket]] $\{a,b\} = B_1(a,b) - B_1(b,a)$ for $a,b \in A$; by bilinearity over $\mathbb{R}[ [ t ] ]$, this characterizes it. \end{defn} \hypertarget{FormulationAsLiftsFromPnAlgebrasToBDAlgebras}{}\paragraph*{{Formulation as lifts from $P_n$-algebras to $BD_n$-algebras and $E_n$-algebras}}\label{FormulationAsLiftsFromPnAlgebrasToBDAlgebras} (\ldots{}) (\hyperlink{CostelloGwilliam}{Costello-Gwilliam, section 2.3, 2.4}) (\ldots{}) [[!include deformation quantization - table]] \hypertarget{InPerturbativeQuantumFieldTheory}{}\paragraph*{{Deformation quantization in perturbative quantum field theory}}\label{InPerturbativeQuantumFieldTheory} Deformation quantization in [[perturbative quantum field theory]] is discussed for [[free field theory]] via [[Moyal deformation quantization]] yielding [[Wick algebras]] in (\hyperlink{DuetschFredenhagen00}{D\"u{}tsch-Fredenhagen 00}, \hyperlink{HirschfeldHenselder02}{Hirschfeld-Henselder 02}), and for interacting [[perturbative quantum field theory]] ([[perturbative AQFT]]) via [[Fedosov deformation quantization]] in (\hyperlink{Collini16}{Collini 16}), see also (\hyperlink{HawkinsRejzner16}{Hawkins-Rejzner 16}). \hypertarget{StrictDeformationQuantization}{}\subsubsection*{{Strict deformation quantization}}\label{StrictDeformationQuantization} (\ldots{}) [[C-star algebraic deformation quantization]] (\ldots{}) \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{existence_results}{}\subsubsection*{{Existence results}}\label{existence_results} [[Vladimir Drinfel'd]] has sketched a proof (and gave main ingredients) to show that every [[Poisson Lie group]] can be deformation quantized to a [[Hopf algebra]]; this proof has been completed by Etingof and Kazhdan. [[Maxim Kontsevich]] proved a certain \emph{[[Kontsevich formality|formality theorem]]} (formality is here in the sense of \emph{[[formal dg-algebra]]} in [[rational homotopy theory]]) whose main corollary (and motivation) was the statement that every [[Poisson manifold]] has a deformation quantization (\hyperlink{Kontsevich97}{Kontsevich 97}). For [[symplectic manifolds]] and those [[Poisson manifolds]] that have a [[regular foliation]] by [[symplectic leafs]], the theory of deformation quantization is much simpler; [[Boris Fedosov]] gave a construction of [[star products]] on symplectic manifolds using [[symplectic connections]] on [[smooth manifolds]] (\hyperlink{Fedosov94}{Fedosov 94}), see at \emph{[[Fedosov's deformation quantization]]}. An analogous argument was given by [[Roman Bezrukavnikov]] and [[Dmitry Kaledin]] in the context of an algebraic symplectic form (\hyperlink{BK}{BK 04}). Caution: the following are rough notes from a talk by [[J.D.S. Jones]] (Cambridge, 8.1.2013); there are probably many typos and sign errors. \hypertarget{deformation_of_poisson_manifolds}{}\paragraph*{{Deformation of Poisson manifolds}}\label{deformation_of_poisson_manifolds} \begin{theorem} \label{Kontsevich}\hypertarget{Kontsevich}{} \textbf{(Kontsevich)}. Every [[Poisson manifold]] has a (formal) deformation quantization. \end{theorem} This was shown in (\hyperlink{Kontsevich97}{Kontsevich 97}). There the deformed product is constructed by a kind of [[Feynman diagram]] [[perturbation series]]. Later this was identified as the perturbation series of the [[Poisson sigma-model]] for the given Poisson manifold. See there for more details. \hypertarget{deformation_of_algebraic_varieties}{}\paragraph*{{Deformation of Algebraic Varieties}}\label{deformation_of_algebraic_varieties} (Not from the notes of J.D.S. Jones) Let $X$ be a smooth algebraic variety over a field $\mathbb{k}$ of characteristic $0$. The analogue of the HKR Theorem here is this: \begin{theorem} \label{SwYe}\hypertarget{SwYe}{} \textbf{(Swan, \href{http://dx.doi.org/10.4153/CJM-2002-051-8}{Yekutieli}).} There is a canonical isomorphism \begin{displaymath} \operatorname{Ext}^i_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \bigoplus_q \operatorname{H}^{i - q}(X, \bigwedge^q (\mathcal{T}_X)) . \end{displaymath} \end{theorem} Here $\mathcal{T}_X$ is the tangent sheaf of $X$, and $X$ is embedded diagonally in $X^2$. This is a consequence of the following result. Let $\mathcal{C}_{cd, X}$ be the sheaf of continuous Hochschild cochains of $X$. It is a bounded below complex of quasi-coherent $\mathcal{O}_{X^2}$-modules. \begin{theorem} \label{Yek1}\hypertarget{Yek1}{} \textbf{(\href{http://dx.doi.org/10.4153/CJM-2002-051-8}{Yekutieli}).} \begin{enumerate}% \item There is a canonical isomorphism \begin{displaymath} \operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \mathcal{C}_{cd, X} \end{displaymath} in the derived category of $\mathcal{O}_{X^2}$-modules. \item There is a canonical quasi-isomorphism of complexes of $\mathcal{O}_{X^2}$-modules \begin{displaymath} \bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q] \to \mathcal{C}_{cd, X} . \end{displaymath} \item Therefore there is a canonical isomorphism \begin{displaymath} \operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q] \end{displaymath} in the derived category of $\mathcal{O}_{X^2}$-modules. \end{enumerate} \end{theorem} The relation to deformation quantization is this: $\mathcal{C}_{cd, X}$ is a shift by $1$ of the sheaf of $\mathcal{D}_{poly, X}$ of polydifferential operators (viewed only as a complex of quasi-coherent $\mathcal{O}_{X^2}$-modules). Similarly, $\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q]$ is the shift by $1$ of the sheaf $\mathcal{T}_{poly, X}$ of polyvector fields. Thus item 2 in the theorem above says that there is a canonical $\mathcal{O}_{X^2}$-linear quasi-isomorphism \begin{displaymath} \mathcal{T}_{poly, X} \to \mathcal{D}_{poly, X} . \end{displaymath} Trying to replicate the global formality theorem of Kontsevich, one would like to upgrade this to an $\mathrm{L}_{\infty}$ quasi-isomorphism. However, it seems that in general this cannot be done directly, but only after a suitable resolution. Here is the result. (See also \href{http://dx.doi.org/10.1016/j.jalgebra.2007.02.012}{Van den Bergh}.) Any quasi-coherent sheaf $\mathcal{M}$ on $X$ admits a canonical flasque resolution called the mixed resolution: \begin{displaymath} \mathcal{M} \to \operatorname{Mix}(\mathcal{M}) . \end{displaymath} This ``mixes'' the jet resolution with the Cech resolution (corresponding to an affine open covering of $X$ that we suppress). In particular there are quasi-isomorphisms of sheaves of DG Lie algebras \begin{displaymath} \mathcal{T}_{poly, X} \to \operatorname{Mix}(\mathcal{T}_{poly, X}) \end{displaymath} and \begin{displaymath} \mathcal{D}_{poly, X} \to \operatorname{Mix}(\mathcal{D}_{poly, X}) . \end{displaymath} \begin{theorem} \label{Yek2}\hypertarget{Yek2}{} \textbf{(\href{http://www.math.bgu.ac.il/~amyekut/publications/def-quant/def-quant.html}{Yekutieli}).} There is an $\mathrm{L}_{\infty}$ quasi-isomorphism \begin{displaymath} \Psi : \operatorname{Mix}(\mathcal{T}_{poly, X}) \to \operatorname{Mix}(\mathcal{D}_{poly, X}) \end{displaymath} whose $1$-st order term commutes with the HKR quasi-isomorphism above. It is independent of choices up to quasi-isomorphism. \end{theorem} A Poisson deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of Poisson $\mathbb{k}[[\hbar]]$-algebras on $X$, with an isomorphism $\mathbb{k} \otimes_{\mathbb{k}[[\hbar]]} \mathcal{A} \cong \mathcal{O}_X$ called an augmentation. Likewise an associative deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of associative unital (but noncommutative) $\mathbb{k}[[\hbar]]$-algebras on $X$, with an augmentation to $\mathcal{O}_X$. \hyperlink{Yek2}{Theorem 4} implies: \begin{theorem} \label{Yek3}\hypertarget{Yek3}{} \textbf{(Yekutieli).} Assume that the cohomology groups $\operatorname{H}^{1}(X, \mathcal{O}_X)$ and $\operatorname{H}^{2}(X, \mathcal{O}_X)$ vanish. Then there is a canonical bijection \begin{displaymath} \mathrm{quant} : \quad \frac{ \{ \text{ Poisson deformations of} \; \mathcal{O}_X \} }{\text{isomorphism}} \quad \xrightarrow{\, \simeq \,} \quad \frac{ \{ \text{ associative deformations of} \; \mathcal{O}_X \} }{\text{isomorphism}} \end{displaymath} called quantization. It preserves first order brackets. \end{theorem} When $X$ is affine this is Theorem 0.1 of \href{http://www.math.bgu.ac.il/~amyekut/publications/def-quant/def-quant.html}{this paper}. For the full statement see Corollary 11.2 of \href{http://www.math.bgu.ac.il/~amyekut/publications/twisted-defs/twisted-defs.html}{this paper}. For twisted (or stacky) deformations there is a corresponding (but much more difficult to state and prove). See the \href{http://www.math.bgu.ac.il/~amyekut/publications/twisted-defs/twisted-defs.html}{paper} and the \href{http://www.math.bgu.ac.il/~amyekut/publications/tw-defs-surv/tw-defs-surv.html}{survey}. \hypertarget{gerstenhabers_deformation_theory_by_hochschild_cohomology}{}\paragraph*{{Gerstenhaber's deformation theory by Hochschild (co)homology}}\label{gerstenhabers_deformation_theory_by_hochschild_cohomology} Let $V$ be a $k$-vector space and consider $C^p(V,V) = \Hom(V^{\otimes p}, V)$. We define a ``circle operator'' $\circ$ as follows: for $f \in C^p(V,V)$ and $g \in C^q(V,V)$, we define $f \circ g \in C^{p+q-1}(V,V)$ as the map \begin{displaymath} (f \circ g)(v_1, \ldots v_{p+q-1}) = f(v_1, \ldots, v_{i-1}, g(v_i, \ldots, v_{i+q-1}), v_{i+q}, \ldots, v_{p+q-1}). \end{displaymath} For $f \in C^\ast(V,V)$, let $A_f(g,h) = (f \circ g) \circ h - f \circ (g \circ h)$. (This is graded symmetric.) It follows that the commutator of $\circ$ is given by \begin{displaymath} [f,g] = f \circ g - (-1)^{(|f| - 1)(|g|-1)} g \circ f \end{displaymath} where $|f| = p$ when $f \in C^p(V,V)$. This defines a \emph{graded [[Lie bracket]]} of degree -1. \begin{example} \label{}\hypertarget{}{} Let $\mu \in C^2(V,V)$ ($\mu : V \otimes V \to V$). Note that $\mu$ is associative iff $\mu \circ \mu = 0$ iff $[\mu, \mu] = 0$. Let $d_\mu : C^{p}(V,V) \to C^{p+1}(V,V)$ be defined by $d_\mu(x) = \mu \otimes x \pm x \otimes \mu = [\mu, x]$. We have $d_\mu \circ d_\mu = 0$ so $(C^\ast(V,V), d_\mu)$ becomes a [[differential graded algebra]]. In fact this is the [[Hochschild cochain complex]] of the [[associative algebra]] $A = (V, \mu)$. \end{example} Apply this example to the construction of deformation quantization. The star product is uniquely determined by $\theta : A \otimes A \to A[ [ t ] ]$ given by $\theta(a,b) = ab + c(a,b)$. What we want is that \begin{displaymath} (\mu + c) \otimes (\mu + c) = 0; \end{displaymath} write this out and we get the equation \begin{displaymath} d_\mu c + c \circ c = 0, \end{displaymath} or $d_\mu c + \frac{1}{2}[c,c] = 0$; this is the [[Maurer-Cartan equation]]. Hence we are looking for solutions of the M-C equation but in the Hochschild complex $C^\ast(A,A)[ [ t ] ]$. One should note that $d_\mu$ is actually a [[derivation]] of the [[Lie bracket]], hence we have a [[dg-Lie algebra]]. \begin{theorem} \label{HKR}\hypertarget{HKR}{} \textbf{([[HKR theorem]])}. $HH^p(A,A) = \Gamma(M, \Lambda^p TM)$. \end{theorem} (Note that $C^p(C^\infty(M),C^\infty(M))$ should be interpreted as $\Hom_{diff}(C^\infty(M), C^\infty(M))$.) Under this isomorphism the Poisson bracket is mapped to the [[Poisson tensor]]: \begin{displaymath} \{ \cdot , \cdot \} \in HH^2(A,A) \quad \mapsto \quad P \in \Gamma(M, \Lambda^2 TM). \end{displaymath} The bracket in Hochschild cohomology ([[Gerstenhaber bracket]]) goes to the [[Schouten bracket]]: \begin{displaymath} [ \cdot , \cdot ]_G \quad \mapsto \quad [ \cdot, \cdot ]_S. \end{displaymath} For vector fields $\xi$ and $\eta$, the [[Schouten bracket]] satisfies (1) $[\xi,\eta]_S = [\xi,\eta]$ (the Lie bracket), and (2) $[\alpha, \beta \wedge \gamma] = [\alpha,\beta] \wedge \gamma \pm [\alpha,\gamma] \wedge \beta$; note that this completely determines it (everything is locally given by wedges\ldots{}). In the Hochschild cohomology $HH^\ast(A,A)$ of $A$, $d_\mu P \mapsto 0$ and $[P,P]_S = 0$, so \emph{we have a solution to M-C in $H^\ast(A,A)[ [ t ] ]$}. \hypertarget{in_terms_of_differential_graded_lie_algebras}{}\paragraph*{{In terms of differential graded Lie algebras}}\label{in_terms_of_differential_graded_lie_algebras} \begin{defn} \label{}\hypertarget{}{} Let $L_1$ and $L_2$ be [[differential graded Lie algebras]] (dgL). A \textbf{[[quasi-isomorphism]]} $f : L_1 \to L_2$ is a homomorphism of dgLs that induces an isomorphism on homology. $L_1$ and $L_2$ are \textbf{quasi-isomorphic} if there exists $M$ with quasi-isomorphisms $L_1 \leftarrow M \rightarrow L_2$. It can be verified that this is an equivalence relation. \end{defn} \begin{theorem} \label{KontsevichDeformationTheorem}\hypertarget{KontsevichDeformationTheorem}{} \textbf{([[Kontsevich]])}. If $L_1$ is quasi-isomorphic to $L_2$ then there is a solution to the M-C equation in $L_1$ iff there is a solution to the M-C equation in $L_2$. \end{theorem} \begin{theorem} \label{KontsevichFormality}\hypertarget{KontsevichFormality}{} \textbf{([[Kontsevich formality]])}. $C^\ast(A,A)[ [ t ] ]$ is quasi-isomorphic to $H^\ast(A,A)[ [ t ] ]$. ($A = C^\infty(M)$) \end{theorem} Hence there is a solution to M-C in $C^\ast(A,A)[ [ t ] ]$, and hence there is a deformation quantization (!). \hypertarget{the_deligne_conjecture}{}\paragraph*{{The Deligne conjecture}}\label{the_deligne_conjecture} We have $(C^*(A,A), d_\mu)$, the [[Gerstenhaber bracket]], and we also have a [[cup product]] \begin{displaymath} (f \cup g) (a_1, \ldots, a_{p+q}) = \mu(f(a_1,\ldots,a_p), g(a_{p+1},\ldots,a_{p+q})) \end{displaymath} for $f : A^{\otimes p} \to A$, $g : A^{\otimes q} \to A$; this satisfies also $d_\mu(f \cup g) = (d_\mu f) \cup g \pm f \cup d_\mu g$. The [[Deligne conjecture]] gives a relationship between these things. In $HH^*(A,A)$, we have: \begin{enumerate}% \item $[\cdot, \cdot]$ is a graded Lie bracket of degree -1. \item The cup product $\cup$ is graded commutative. \item The [[Jacobi identity]] for $[\cdot,\cdot]$. \item $[a, b \cup c] = [a,b] \cup c \pm [a,c] \cup b$. \end{enumerate} Such a thing is called a [[Gerstenhaber algebra]]. Note that we do not have these relations in $C^*(A,A)$, they are only true modulo boundaries. \begin{theorem} \label{Deligne}\hypertarget{Deligne}{} \textbf{([[Deligne conjecture]])}. $C^*(A,A)$ is a $G_\infty$-algebra, which is a Gerstenhaber algebra \emph{up to coherent homotopy}. \end{theorem} \hypertarget{RelationToUniversalEnvelopingAlgebras}{}\paragraph*{{Deformation by universal enveloping algebras}}\label{RelationToUniversalEnvelopingAlgebras} It is a classical fact that the [[universal enveloping algebra]] of a [[Lie algebra]] provides a deformation quantization of the corresponding [[Lie-Poisson structure]] (example \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below). Remarkably, this statement generalizes to more general [[polynomial Poisson algebras]] (def. \ref{PolynomialPoissonAlgebra} below) for a suitable generalized concept of universal enveloping algebra (def. \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below): it is \emph{always} true up to third order in $\hbar$, and sometimes to higher order (\hyperlink{PenkavaVanhaecke00}{Penkava-Vanhaecke 00, theorem 3.2}, prop. \ref{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3} below). In particular it also holds true for restrictions of [[Poisson bracket Lie algebras]] to their [[Heisenberg Lie algebras]] (example \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below). \begin{defn} \label{PolynomialPoissonAlgebra}\hypertarget{PolynomialPoissonAlgebra}{} \textbf{([[polynomial]] [[Poisson algebra]])} A [[Poisson algebra]] $((A,\cdot), \{-,-\})$ is called a \emph{[[polynomial Poisson algebra]]} if the underlying [[commutative algebra]] $(A,\cdot)$ is a [[polynomial algebra]], hence a [[symmetric algebra]] \begin{displaymath} Sym(V) \coloneqq T(V)/(x \otimes y - y \otimes x \vert x,y \in V) \end{displaymath} on some [[vector space]] $V$. Here \begin{displaymath} T(V) \coloneqq \underset{n \in \mathbb{N}}{\oplus} V^{\otimes^n} \end{displaymath} denotes the [[tensor algebra]] of $V$. We write \begin{displaymath} \mu \;\colon\; T(V) \longrightarrow Sym(V) \end{displaymath} for the canonical [[projection]] map (which is an algebra [[homomorphism]]) and \begin{displaymath} \sigma \;\colon\; Sym(V) \longrightarrow T(V) \end{displaymath} for its [[linear map|linear]] inverse (symmetrization, which is not in general an algebra [[homomorphism]]). Notice that by its bi-[[derivation]] property the Poisson bracket on a polynomial Poisson algebra is fixed by its restriction to linear elements \begin{displaymath} \{-,-\} \;\colon\; V \otimes V \longrightarrow Sym(V) \,. \end{displaymath} \end{defn} \begin{example} \label{PolynomialLiePoissonStructure}\hypertarget{PolynomialLiePoissonStructure}{} \textbf{(polynomial [[Lie-Poisson structure]])} Let $(C^\infty(\mathbb{R}^n), \pi)$ be a [[Poisson manifold]] whose underlying manifold is a [[Cartesian space]] $\mathbb{R}^n$. Then the restriction of its Poisson algebra $( C^\infty(\mathbb{R}^n, \cdot), \pi^{i j} \partial_i(-) \cdot \partial_j(-) )$ to the [[polynomial functions]] $\mathbb{R}[x^1, \cdots, x^n ] \ookrightarrow C^\infty(\mathbb{R}^n)$ is a polynomial Poisson algebra according to def. \ref{PolynomialPoissonAlgebra}. In particular if $(\mathfrak{g}, [-,-])$ is a [[Lie algebra]] and $(\mathfrak{g}^\ast, \{-,-\})$ the corresponding [[Lie-Poisson structure|Lie-Poisson manifold]], then the corresponding polynomial Poisson algebra is $(Sym(\mathfrak{g}), \{-,-\})$ where the restriction of the Poisson bracket to linear polynomial elements coincides with the [[Lie bracket]]: \begin{displaymath} \{x,y\} = [x,y] \,. \end{displaymath} \end{example} \begin{defn} \label{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}\hypertarget{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}{} \textbf{(universal enveloping algebra of polynomial Poisson algebra)} Given a polynomial Poisson algebra $(Sym(V), \{-,-\})$ (def. \ref{PolynomialPoissonAlgebra}), say that its \emph{universal enveloping algebra} $\mathcal{U}(V,\{-,-\})$ is the [[associative algebra]] which is the [[quotient]] of the [[tensor algebra]] of $V$ with a [[power series|formal variable]] $\hbar$ adjoined by the two-sided ideal which is generated by the the $\hbar$-Poisson bracket relation on linear elements: \begin{displaymath} \mathcal{U}(V,\{-,-\}) \;\coloneqq\; T(V)/( x \otimes y - y \otimes x - \hbar \{x,y\} \vert x,y \in V ) \,. \end{displaymath} This comes with the quotient projection linear map which we denote by \begin{displaymath} \rho \;\colon\; T(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\hbar\{-,-\}) \,. \end{displaymath} \end{defn} (\hyperlink{PenkavaVanhaecke00}{Penkava-Vanhaecke 00, def. 3.1}) \begin{example} \label{UniversalEnvelopingAlgebraOfLieAlgebra}\hypertarget{UniversalEnvelopingAlgebraOfLieAlgebra}{} \textbf{([[universal enveloping algebra]] of [[Lie algebra]])} In the case of a polynomial [[Lie-Poisson structure]] $(Sym(\mathfrak{g}), [-,-])$ (example \ref{PolynomialLiePoissonStructure}) the universal enveloping algebra $\mathcal{U}(\mathfrak{g},[-,-])$ from def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra} (for $\hbar = 1$) coincides with the standard [[universal enveloping algebra]] of the [[Lie algebra]] $(\mathfrak{g}, [-,-])$. \end{example} The combined linear projection maps from def. \ref{PolynomialPoissonAlgebra} and def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra} we denote by \begin{displaymath} \tau \coloneqq \rho \circ (\sigma/[ [ \hbar ] ]) \;\colon\; Sym(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\{-,-\}) \,. \end{displaymath} \begin{prop} \label{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3}\hypertarget{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3}{} \textbf{(universal enveloping algebra provides [[formal deformation quantization]] at least up to order 3)} Let $( Sym(V), \{-,-\} )$ be a polynomial Poisson algebra (def. \ref{PolynomialPoissonAlgebra}) such that the canonical linear map to its universal enveloping algebra (def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}) is [[injective function|injective]] up to order $n \in \mathbb{N}\cup \{\infty\}$ \begin{displaymath} \tau/(\hbar^{n+1}) \;\colon\; Sym(V)[ [ \hbar ] ]/(\hbar^{n+1}) \hookrightarrow \mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1}) \,. \end{displaymath} Then the restriction of the product on $\mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1})$ to $Sym(V)/(\hbar^{n+1})$ is a [[formal deformation quantization]] of $(Sym(V), \{-,-\})$ to order $n$ (hence a genuine deformation quantization in the case that $n = \infty$). Moreover, this is \emph{always} the case for $n = 3$, hence for every polynomial Poisson algebra its universal enveloping algebra always provides a deformation quantization of order $3$ in $\hbar$. \end{prop} (\hyperlink{PenkavaVanhaecke00}{Penkava-Vanhaecke 00, theorem 3.2 with section 2}) \begin{example} \label{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras}\hypertarget{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras}{} \textbf{([[formal deformation quantization]] of [[Lie-Poisson structures]] by universal enveloping algebras)} In the following cases the map $\tau$ in prop. \ref{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3} is injective to arbitrary order, hence in these cases the universal enveloping algebra provides a genuine [[formal deformation quantization]]: \begin{enumerate}% \item the case that the Poisson bracket is [[linear Poisson structure|linear]] in that restricts as \begin{displaymath} \{-,-\} \;\colon\; V \otimes V \longrightarrow V \hookrightarrow Sym(V) \,. \end{displaymath} This is the case of the [[Lie-Poisson structure]] from example \ref{PolynomialLiePoissonStructure} and the universal enveloping algebra that provides it deformation quantization is the standard one (example \ref{UniversalEnvelopingAlgebraOfLieAlgebra}). \item more generally, the case that the Poisson bracket restricted to linear elements has linear and constant contribution in that it restricts as \begin{displaymath} \{-,-\} \;\colon\; V \otimes V \longrightarrow \mathbb{R} \oplus V \hookrightarrow Sym(V) \,. \end{displaymath} This includes notably the Poisson structures induced by [[symplectic vector spaces]], in which case the restriction \begin{displaymath} \{-,-\} \;\colon\; (\mathbb{R} \oplus V) \otimes (\mathbb{R} \oplus V) \longrightarrow (\mathbb{R} \oplus V) \end{displaymath} is the [[Lie bracket]] of the associated [[Heisenberg Lie algebra]]. \end{enumerate} \end{example} This is (\hyperlink{PenkavaVanhaecke00}{Penkava-Vanhaecke 00, p. 26}) The first statement in itself is a classical fact (reviewed e.g. in \hyperlink{Gutt11}{Gutt 11}). \hypertarget{MotivicGaloisGroup}{}\subsubsection*{{Motivic Galois group action on the space of quantizations}}\label{MotivicGaloisGroup} In (\href{Kontsevich99}{Kontsevich 99}) it was indicated that a [[quotient group]] of the [[motivic Galois group]] apparently equivalent to the [[Grothendieck-Teichmüller group]] naturally [[action|acts]] on the space of formal deformation quantizations of a finite dimensional manifold. See also at \emph{[[cosmic Galois group]]}. This has been formalized as follows. The formal deformation quantization of (\hyperlink{Kontsevich97}{Kontsevich 97}) is all induced by the [[Kontsevich formality]] theorem, which states that ober suitable [[manifolds]]/[[varieties]] $X$ there is an [[equivalence]] of [[L-∞ algebras]] \begin{displaymath} \mathcal{X}(X) \stackrel{\simeq}{\to} C^\bullet(X) \end{displaymath} identifying the [[multivector fields]] on $X$ with the [[Hochschild cohomology]] complex (of its [[function algebra]]). Every choice of such an equivalence induces one formal deformation quantization of a [[Poisson manifold]] $X$, and the two quantizations induced by two equivalent (homotopic) equivalences are in turn equivalent. Therefore one may regard the [[∞-groupoid]] $Maps^{L_\infty}_{equiv}(\mathcal{X}(X), C^\bullet(X))$ as the ``space of formal deformation quantizations'' of $X$. In (\hyperlink{Dolgushev1109}{Dolgushev 1109, theorem 6.2}, \hyperlink{Dolgushev1111}{Dolgushev 1111, theorem 3.1}) it is shown that the set $\pi_0 Maps^{L_\infty}_{equiv}(\mathcal{X}(X), C^\bullet(X))$ of connected components of this space is, up to a choice of basepoint, the [[Grothendieck-Teichmüller group]], hence is a [[torsor]] over that group. (This is based on identifications of the GRT Lie algebra with the degree-0 [[chain cohomology]] of the [[graph complex]], due to [[Thomas Willwacher]]. See at \emph{\href{Grothendieck-Teichm%C3%BCller+tower#RelationToTheGraphComplex}{Grothendieck-Teichm\"u{}ller group -- relation to the graph complex}}. Aspects of the generalization of this statement to more general spaces then $\mathbb{R}^n$ are discussed in \hyperlink{DolgushevRogersWillwacher12}{Dolgushev-Rogers-Willwacher 12}. For discussion of [[motive|motivic structures]] in [[geometric quantization]] see at \emph{[[motivic quantization]]}. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item [[Moyal deformation quantization]] \item [[Fedosov deformation quantization]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[reductions deformations resolutions in physics]] \item [[quantization]] \begin{itemize}% \item \textbf{deformation quantization}, [[geometric quantization]] \item [[path integral]] \item [[semiclassical approximation]] \end{itemize} \item [[Jordan algebra]] \end{itemize} [[!include Isbell duality - table]] [[!include infinitesimal and local - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} Survey includes \begin{itemize}% \item [[Stefan Waldmann]] \emph{Poisson-Geometrie und Deformationsquantisierung. Eine Einf\"u{}hrung}, Springer (2007) \item Simone Gutt, \emph{Deformation quantization of Poisson manifolds}, Geometry and Topology Monographs 17 (2011) 171-220 (\href{http://msp.org/gtm/2011/17/gtm-2011-17-003p.pdf}{pdf}) \end{itemize} The concept of algebraic deformation quantization originates around \begin{itemize}% \item F. Bayen, [[Moshé Flato]], C. Fronsdal, , [[André Lichnerowicz]], [[Daniel Sternheimer]], \emph{Deformation theory and quantization. I. Deformations of symplectic structures.}, Annals of Physics (N.Y.) 111, 61, 111 (1978) (\href{http://www.sciencedirect.com/science/article/pii/0003491678902245}{publisher}) \item F. Bayen, [[Moshé Flato]], C. Fronsdal, , [[André Lichnerowicz]], [[Daniel Sternheimer]], \emph{Deformation theory and quantization. II. Physical applications}, Annals of Physics 111, 111-151 (1978) (\href{http://www.sciencedirect.com/science/article/pii/0003491678902257}{publisher}) \end{itemize} Review is in \begin{itemize}% \item [[Daniel Sternheimer]], \emph{Deformation Quantization: Twenty Years After}, AIP Conf.Proc.453:107-145,1998 (\href{https://arxiv.org/abs/math/9809056}{arXiv:math/9809056}) \end{itemize} The [[Fedosov deformation quantization]] prescription for deformation quantization of [[symplectic manifolds]] and varieties and also of [[Poisson manifolds]] that have a [[regular foliation]] by [[symplectic leaves]] is discussed in \begin{itemize}% \item [[Boris Fedosov]], \emph{Formal quantization}, Some Topics of Modern Mathematics and their Applications to Problems of Mathematical Physics (in Russian), Moscow (1985), 129-136. \item [[Boris Fedosov]], \emph{Index theorem in the algebra of quantum observables}, Sov. Phys. Dokl. 34 (1989), 318-321. \item [[Boris Fedosov]], \emph{A simple geometrical construction of deformation quantization} J. Differential Geom. Volume 40, Number 2 (1994), 213-238. (\href{http://projecteuclid.org/euclid.jdg/1214455536}{EUCLID}) \end{itemize} For algebraic forms this is discussed in \begin{itemize}% \item [[Roman Bezrukavnikov]], [[Dmitry Kaledin]], \emph{Fedosov quantization in algebraic context} Mosc. Math. J., Volume 4, Number 3, (2004) 559--592 (\href{http://www.ams.org/distribution/mmj/vol4-3-2004/bezrukavnikov-kaledin.pdf}{AMS pdf}) \end{itemize} More discussion of this approach is in \begin{itemize}% \item Claudio Emmrich, [[Alan Weinstein]], \emph{The differential geometry of Fedosov's quantization} (\href{http://arxiv.org/abs/hep-th/9311094}{arXiv:hep-th/9311094}) \end{itemize} A direct and general formula for the deformation quantization of any Poisson manifold was given in \begin{itemize}% \item [[Maxim Kontsevich]], \emph{Deformation quantization of Poisson manifolds}, Lett. Math. Phys. \textbf{66} (2003), no. 3, 157--216, (\href{http://arxiv.org/abs/q-alg/9709040}{arXiv:q-alg/9709040}). \end{itemize} see also \begin{itemize}% \item Peter Banks, Erik Panzer, Brent Pym, \emph{Multiple zeta values in deformation quantization} (\href{https://arxiv.org/abs/1812.11649}{arXiv:1812.11649}) \end{itemize} This secretly uses the [[Poisson sigma-model]] (see there for more details) induced by the given target [[Poisson Lie algebroid]]. A popular exposition of this is in \begin{itemize}% \item [[Anton Kapustin]], \emph{Quantum geometry: The reunion of math and physics} (\href{https://dl.dropboxusercontent.com/u/12796267/kapustin-talk.swf}{online slides} \href{http://www.theory.caltech.edu/~kapustin/talk.ppt}{Powerpoint}) \end{itemize} The classification of the space of such formal deformation quantization is discussed in \begin{itemize}% \item [[Vasily Dolgushev]], \emph{Stable Formality Quasi-isomorphisms for Hochschild Cochains I} (\href{http://arxiv.org/abs/1109.6031}{arXiv:1109.6031}) \item [[Vasily Dolgushev]], \emph{Exhausting formal quantization procedures} (\href{http://arxiv.org/abs/1111.2797}{arXiv:1111.2797}) \item [[Vasily Dolgushev]], [[Christopher Rogers]], [[Thomas Willwacher]], \emph{Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields} (\href{http://arxiv.org/abs/1211.4230}{arXiv:1211.4230}) \end{itemize} Deformation quantization of [[polynomial Poisson algebras]] via [[universal enveloping algebra]] (generalizing that of [[Lie-Poisson structures]]) is discussed in \begin{itemize}% \item [[Michael Penkava]], [[Pol Vanhaecke]], \emph{Deformation Quantization of Polynomial Poisson Algebras}, Journal of Algebra 227, 365\~n{}393 (2000) (\href{https://arxiv.org/abs/math/9804022}{arXiv:math/9804022}) \end{itemize} Deformation quantization of algebraic varieties is in \begin{itemize}% \item [[Amnon Yekutieli]], Deformation Quantization in Algebraic Geometry, Advances in Mathematics 198 (2005), 383-432. Erratum: Advances in Mathematics 217 (2008), 2897-2906. \end{itemize} See also \begin{itemize}% \item MO discussion , \emph{\href{http://mathoverflow.net/questions/42148/kontsevichs-conjectures-on-the-grothendieck-teichmuller-group}{Kontsevich's conjectures on the Grothendieck-Teichm\"u{}ller group?}} \end{itemize} Deformation quantization in [[perturbative quantum field theory]] is discussed for [[free field theory]] is due to \begin{itemize}% \item Joseph Dito, \emph{Star-product approach to quantum field theory: The free scalar field}. Letters in Mathematical Physics, 20(2):125--134, 1990 (\href{https://inspirehep.net/record/303898/}{spire}) \end{itemize} and further expanded on in \begin{itemize}% \item [[Michael Dütsch]], [[Klaus Fredenhagen]], section 5.1 of \emph{Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion}, Commun.Math.Phys. 219 (2001) 5-30 (\href{https://arxiv.org/abs/hep-th/0001129}{arXiv:hep-th/0001129}) \item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Perturbative algebraic quantum field theory and deformation quantization}, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) (\href{http://xxx.uni-augsburg.de/abs/hep-th/0101079}{arXiv:hep-th/0101079}) \item A. C. Hirshfeld, P. Henselder, \emph{Star Products and Perturbative Quantum Field Theory}, Annals Phys. 298 (2002) 382-393 (\href{https://arxiv.org/abs/hep-th/0208194}{arXiv:hep-th/0208194}) \end{itemize} and for interacting [[perturbative quantum field theory]] ([[perturbative AQFT]]) in \begin{itemize}% \item [[Giovanni Collini]], \emph{Fedosov Quantization and Perturbative Quantum Field Theory} (\href{https://arxiv.org/abs/1603.09626}{arXiv:1603.09626}) \item [[Eli Hawkins]], [[Kasia Rejzner]], \emph{The Star Product in Interacting Quantum Field Theory} (\href{https://arxiv.org/abs/1612.09157}{arXiv:1612.09157}) \end{itemize} The relation to [[geometric quantization]] is discussed in \begin{itemize}% \item [[Eli Hawkins]], \emph{The Correspondence between Geometric Quantization and Formal Deformation Quantization} (\href{http://arxiv.org/abs/math/9811049}{arXiv:math/9811049}) \item Christoph N\"o{}lle, \emph{Geometric and deformation quantization} (\href{http://arxiv.org/abs/0903.5336}{arXiv:0903.5336}) \end{itemize} and some remarks on the relation are also in section 1.4 of \begin{itemize}% \item [[Sergei Gukov]], [[Edward Witten]], \emph{Branes and Quantization}, Adv. Theor. Math. Phys. 13 (2009) 1--73, (\href{http://arxiv.org/abs/0809.0305}{arXiv:0809.0305}, \href{http://projecteuclid.org/euclid.atmp/1282054099}{euclid}) \end{itemize} which is about [[quantization via the A-model]]. The formulation of deformation quantization as lifts from [[P-n operads]] over [[BD-n operads]] to [[E-n operads]] is discussed in section 2.3 and 2.4 of \begin{itemize}% \item [[Kevin Costello]], [[Owen Gwilliam]], \emph{Factorization algebras in perturbative quantum field theory} (\href{http://math.northwestern.edu/~costello/factorization_public.html}{wiki}, early/partial draft \href{http://math.northwestern.edu/~gwilliam/factorization.pdf}{pdf}) \end{itemize} See also \begin{itemize}% \item \href{http://en.wikipedia.org/wiki/Weyl_quantization#Deformation_quantization}{wikipedia} \item F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, \emph{Quantum mechanics as a deformation of classical mechanics}, Lett. Math. Phys. 1 (1975/77), \href{http://www.ams.org/mathscinet-getitem?mr=674337}{MR674337}, \href{http://dx.doi.org/10.1007/BF00399745}{doi}; \emph{Deformation theory and quantization. I. Deformations of symplectic structures}, Ann. Physics 111 (1978), no. 1, 61--110, \href{http://www.ams.org/mathscinet-getitem?mr=496157}{MR496157}; \emph{Deformation theory and quantization. II. Physical applications}, Ann. Physics 111 (1978), no. 1, 111--151, \href{http://www.ams.org/mathscinet-getitem?mr=496158}{MR496158} \item M. Flato, A. Lichnerowicz, D. Sternheimer, \emph{Deformations of Poisson brackets, Dirac brackets and applications}, J. Math. Phys. \textbf{17} (1976), no. 9, 1754--1762, \href{http://www.ams.org/mathscinet-getitem?mr=420723}{MR420723}, \href{http://dx.doi.org/10.1063/1.523104}{doi} \item D. Arnal, J.-C. Cortet, \emph{$\ast$-products in the method of orbits for nilpotent groups}, J. Geom. Phys. 2 (1985), no. 2, 83--116, \end{itemize} On the [[stack]] of deformation quantizations: \begin{itemize}% \item [[Maxim Kontsevich]], \emph{Deformation quantization of algebraic varieties} (\href{http://arxiv.org/abs/math/0106006}{arXiv:math/0106006}) \item Pietro Polesello, [[Pierre Schapira]], \emph{Stacks of quantization-deformation modules on complex symplectic manifolds} (\href{http://arxiv.org/abs/math/0305171}{arXiv:math/0305171}) \item [[Amnon Yekutieli]], \emph{Twisted Deformation Quantization of Algebraic Varieties} , \href{http://arxiv.org/abs/0905.0488}{arxiv:0905.0488} \end{itemize} The action of a [[motivic Galois group]] (``[[cosmic Galois group]]'') on the space of deformation quantization was observed in \begin{itemize}% \item [[Maxim Kontsevich]], \emph{Operads and Motives in Deformation Quantization}, Lett. Math. Phys.48:35-72,1999 (\href{http://arxiv.org/abs/math/9904055}{arXiv:math/9904055}) \end{itemize} See also at \emph{[[motives in physics]]}. 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