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%  Algebraic Aspects of Deformation Quantization
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%  Amnon Yekutieli
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\title[Deformation Quantization]
{Algebraic Aspects of Deformation Quantization}
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\author[Amnon Yekutieli]{Amnon Yekutieli}
\institute[BGU]
{Department of Mathematics\\
Ben Gurion University\\[3em]
Notes available at http://www.math.bgu.ac.il/$\sim$amyekut}
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\date[19 May 2007]{}
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\begin{document}


% % ** top matter for notes **
% \begin{center}
% {\large \bfseries Algebraic Aspects of Deformation Quantization}
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% \vspace{1ex}
% Lecture Notes
% \footnote[1]{written: 13 May 2007}
% 
% \vspace{1ex}
% Amnon Yekutieli\\
% Ben Gurion University, ISRAEL
% 
% \texttt{\small http://www.math.bgu.ac.il/$\sim$amyekut}
% \end{center}
% % ** end of top matter for notes **


\begin{frame}
\titlepage
\end{frame}

\begin{frame}
Here is the plan of my lecture:
\pause
\begin{enumerate}
\item[1.] What is Deformation Quantization? \pause
\item[2.] The Solution by Kontsevich \pause
\item[3.] Deforming Algebraic Varieties \pause
\item[4.] About the Proofs \pause
\item[5.] Final Remarks
\end{enumerate}
\end{frame}


\section{1. What is Deformation Quantization?}
\begin{frame}

\sect{1. What is Deformation Quantization?}

\pause
Let $\K$ be a field of characteristic $0$ and let $C$ be a 
commutative $\K$-algebra. 

\pause
A \hilite{drkblue}{Poisson bracket} on $C$ is a 
$\K$-bilinear function
\[ \alpha : C \times C \to C \]
which makes $C$ into a Lie algebra, and is a biderivation (i.e.\ 
a derivation in each argument). 

\pause
We usually write
\[ \{ f, g \}_{\alpha} := \alpha(f,g) \]
for $f, g \in C$. 

\pause
The pair $(C, \alpha)$ is called a \hilite{rust}{Poisson algebra}.
\end{frame}

\begin{frame}
Poisson brackets arise in several ways. 

\pause
\begin{exa} \label{exa1}
Classical Hamiltonian mechanics. Here $\K = \mbb{R}$,
$X$ is an even dimensional differentiable manifold (the
phase space), and $C = \mrm{C}^{\infty}(X)$, the ring of
differentiable $\mbb{R}$-valued functions on $X$.

\pause
There is a symplectic (i.e.\ nondegenerate) Poisson
bracket $\{ -,- \}$ on $C$. 

\pause
Given a function $H \in C$ (the energy), the Hamiltonian
vector field $\{ H, - \}$ on $X$ determines 
how points $x \in X$ move. See the book \cite{GS} for a discussion.  
\end{exa}
\end{frame}

\begin{frame}
\begin{exa} \label{exa4}
Here is an explicit formula. Take $C := \K[t_1, \ldots, t_{2m}]$,
the polynomial algebra in $2m$ variables. We may view $C$ as
$\mcal{O}(X)$, the ring of algebraic function on the affine variety
$X := \mbf{A}^{2m}$.

\pause
Define a biderivation $\alpha$ by the formula 
\[ \{ f, g \}_{\alpha} = 
{\smfrac{1}{2}} \sum_{i = 1}^m
\bigl( {\smfrac{\partial f}{\partial t_i}}  \cdot 
\smfrac{\partial g}{\partial t_{i + m}} - 
\smfrac{\partial g}{\partial t_i}  \cdot 
\smfrac{\partial f}{\partial t_{i + m}} \bigr)  .  \]

\pause
This is a symplectic Poisson bracket on $C$.
\end{exa}
\end{frame}

\begin{frame}
Let $\K[[\hbar]]$ be the ring of formal power series in the
variable $\hbar$. We call $\hbar$ the deformation parameter.

\pause
\begin{dfn} 
Let $C[[\hbar]]$ be the set of formal power series with
coefficients in $C$, which we view only as a 
$\K[[\hbar]]$-module. 

\pause
A \hilite{drkblue}{star product} on $C[[\hbar]]$ is an
associative,
unital, $\K[[\hbar]]$-bilinear multiplication
\[ \star : C[[\hbar]] \times C[[\hbar]] \to C[[\hbar]] , \]
such that 
\[ f \star g \equiv f g \ \opn{mod}\ \hbar \]
for any $g, f \in C$. 

\pause
The pair $\bigl( C[[\hbar]], \star \bigr)$ is also called an 
\hilite{rust}{associative deformation} of $C$.
\end{dfn}
\end{frame}

\begin{frame}
\begin{exa} \label{exa3}
Suppose $\bigl( C[[\hbar]], \star \bigr)$ is an associative
deformation of $C$. 

\pause
Given $f, g \in C$, we know that 
\[ f \star g - g \star f \equiv 0 \ \opn{mod}\ \hbar . \]

\pause
Hence there is a unique element 
\[ \{ f, g \}_{\star} \in C \]
such that
\[ \smfrac{1}{2 \hbar} \bigl( f \star g - g \star f \bigr)
\equiv \{ f, g \}_{\star}  \ \opn{mod}\ \hbar . \]

\pause
It is quite easy to show that $\{ -,- \}_{\star}$ is a Poisson
bracket on $C$. 

\pause
Thus an associative deformation of $C$ induces a
Poisson bracket on $C$.
\end{exa}
\end{frame}

\begin{frame}
Deformation quantization seeks to reverse Example \ref{exa3}.

\pause
\begin{dfn}
Given a Poisson bracket $\alpha$ on the algebra $C$, a 
\hilite{brtgreen}{deformation quantization} of $\alpha$ is an
associative
deformation $\bigl( C[[\hbar]], \star \bigr)$ of $C$ such that
\[ \{ f,g \}_{\star} = \{ f,g \}_{\alpha}  \]
for all $f, g \in C$.
\end{dfn}
\end{frame}

\begin{frame}
In physics $\hbar$ is the \hilite{drkblue}{Planck constant}. 

\pause
For a quantum
phenomenon depending on $\hbar$, the limit as $\hbar \to 0$ is thought
of the as the classical limit of this phenomenon.

\pause
The original idea by the physicists Flato et.\ al.\ (\cite{BFFLS},
1978) was that in the setup of Example \ref{exa1}, deformation
quantization should model the transition from classical to
quantum mechanics.

\pause
For a symplectic manifold $X$ and $C = \mrm{C}^{\infty}(X)$ 
the problem was solved by De Wilde and Lacomte (\cite{DL}, 1983).
A more geometric solution was discovered by Fedosov (\cite{Fe},
1994). 

\pause
The general case, i.e.\ $C = \mrm{C}^{\infty}(X)$ for a 
Poisson manifold $X$, was solved by Kontsevich (\cite{Ko1}, 1997).
See surveys in the book \cite{CKTB} and the article \cite{CI}.
\end{frame}


\section{2. The Solution by Kontsevich}

\begin{frame}
\sect{2. The Solution by Kontsevich}

\pause
Kontsevich, in his paper \cite{Ko1}, actually did more than proving
existence of deformation quantization -- he in fact classified all of
them. 

\pause
To state Kontsevich's result we need the notion of
\hilite{drkblue}{gauge 
equivalence}. 

\pause
A gauge equivalence of the $\K[[\hbar]]$-module 
$C[[\hbar]]$ is a $\K[[\hbar]]$-linear automorphism 
\[ \gamma : C[[\hbar]] \iso C[[\hbar]] \]
that is the identity modulo $\hbar$; namely
\[ \gamma(f) \equiv f  \ \opn{mod}\ \hbar \]
for all $f \in C$.
\end{frame}

\begin{frame}
Two associative deformations $\bigl( C[[\hbar]], \star \bigr)$ and 
$\bigl( C[[\hbar]], \star' \bigr)$ of $C$ are called
\hilite{drkblue}{gauge
equivalent} if there is some gauge equivalence $\gamma$ such that 
\[ \gamma(f \star g) = \gamma(f) \star' \gamma(g) \]
for all $f, g \in C$.

\pause
\begin{dfn}
A \hilite{rust}{formal Poisson bracket} on $C[[\hbar]]$ is a 
$\K[[\hbar]]$-bilinear Poisson bracket $\alpha$ on the commutative
algebra $C[[\hbar]]$ that vanishes modulo $\hbar$, i.e.\
\[ \{ f,g \}_{\alpha}  \equiv 0  \ \opn{mod}\ \hbar . \]

\pause
We also call the pair $\bigl(C[[\hbar]], \alpha \bigr)$ a 
\hilite{drkblue}{Poisson deformation} of $C$.
\end{dfn}

\pause
There is a similar notion of gauge equivalence between Poisson
deformations.
\end{frame}

\begin{frame}
Given a formal Poisson bracket $\alpha$, it can be expanded into a
power series
\[ \{ f,g \}_{\alpha}  = \alpha(f, g) = 
\sum_{j = 1}^{\infty} \alpha_j(f, g) \hbar^j  \]
for $f, g \in C$.

\pause
The first order term $\alpha_1$ turns out to be a Poisson bracket on
$C$. Conversely:

\pause
\begin{exa} \label{exa5}
If $\alpha_1$ is a Poisson bracket on $C$, then
$\alpha := \alpha_1 \hbar$ is a formal Poisson bracket on
$C[[\hbar]]$.
\end{exa}

\pause
Thus a Poisson bracket on $C$ is a special case of a Poisson
deformation of $C$.
\end{frame}

\begin{frame}
\begin{thm}\tup{(Kontsevich)} \label{thm4}
\hilite{brtgreen}{
Let $X$ be a differentiable $\mbb{R}$-manifold and 
$C := \mrm{C}^{\infty}(X)$.}

\pause
\hilite{brtgreen}{
Then there is a canonical bijection 
\[ \opn{quant} : \frac{ \{ \tup{Poisson deformations of $C$} \} }
{\tup{gauge equivalence}} \iso
\frac{ \{ \tup{associative deformations of $C$} \} }
{\tup{gauge equivalence}} \]
preserving first order terms.}
\end{thm}

\pause
The function $\opn{quant}$ is called the \hilite{drkblue}{quantization
map}. 

\pause
Here is what I mean by ``preserving first order terms'': given a
formal Poisson bracket 
$\alpha  = \sum_{j = 1}^{\infty} \alpha_j \hbar^j $ 
and corresponding star product 
$\star = \opn{quant}(\alpha)$, one has
\[ \{ -,- \}_{\star} = \{ -,- \}_{\alpha_1} \] 
as Poisson brackets on $C$.
\end{frame}


\section{3. Deforming Algebraic Varieties}
\begin{frame}
\sect{3. Deforming Algebraic Varieties}

\pause
Suppose $C =  \mcal{O}(X)$, the ring of functions on an affine smooth
algebraic variety over the field $\K$. It is natural to ask whether
a Poisson bracket on $C$ can be quantized. 

\pause
This sort of question is
central in current ring theory -- cf.\ the fundamental paper by
Artin, Tate and Van den Bergh (\cite{ATV}, 1990), and the survey
paper \cite{SV}.

\pause
Regarding complete varieties, there are classical (i.e.\ mid 20-th
century) results on commutative deformations.
But very little has been done on noncommutative deformations. 
\end{frame}

\begin{frame}
A more recent paper by Kontsevich (\cite{Ko3}, 2001) is about
noncommutative 
deformations of algebraic varieties. It contains many new ideas, but
still it does not settle the affine quantization problem mentioned
above.

\pause
I would like to state our main result, which goes in a different
direction than \cite{Ko3}. 

\pause
In order to state this result I must use
some fancy algebro-geometric language in the next two slides.
\end{frame}

\begin{frame}
Let $X$ be an $n$-dimensional smooth variety over the
field $\K$, with structure sheaf $\mcal{O}_X$. 

\pause
A \hilite{drkblue}{Poisson
deformation of $\mcal{O}_X$} is a sheaf $\mcal{A}$ of flat, complete,
commutative Poisson $\K[[\hbar]]$-algebras on $X$, such that 
\[ \mcal{A} / (\hbar) \cong \mcal{O}_X . \]

\pause
An \hilite{rust}{associative deformation
of $\mcal{O}_X$} is a sheaf $\mcal{A}$ of flat, complete,
associative, unital $\K[[\hbar]]$-algebras on $X$, such that 
\[ \mcal{A} / (\hbar) \cong \mcal{O}_X . \]
\end{frame}

\begin{frame}
Here is our main result from \cite{Ye2}, with enhancement in
\cite{LY}:

\pause
\begin{thm}\tup{(Y, Leitner-Y)} \label{thm3}
\hilite{brtgreen}{
Assume $\mbb{R} \subset \K$, and let $X$ be a smooth algebraic variety
satisfying the cohomological vanishing condition
\[ \mrm{H}^1(X, \mcal{O}_X) = \mrm{H}^2(X, \mcal{O}_X) = 0 . \]}

\pause
\hilite{brtgreen}{
Then there is a canonical bijection
\[ \opn{quant} : \frac{ \{ \tup{Poisson deformations of $\mcal{O}_X$}
\} }
{\tup{gauge equivalence}} \iso
\frac{ \{ \tup{associative deformations of $\mcal{O}_X$} \} }
{\tup{gauge equivalence}} \]
which preserves first order terms.}
\end{thm}
\end{frame}

\begin{frame}
As corollary we solve the affine quantization problem:

\pause
\begin{cor}
\hilite{drkblue}{
Assume $\mbb{R} \subset \K$. Let $X$ be a smooth affine algebraic
variety and let $C := \mcal{O}(X)$. 
Then there is a canonical bijection 
\[ \opn{quant} : \frac{ \{ \tup{Poisson deformations of $C$} \} }
{\tup{gauge equivalence}} \iso
\frac{ \{ \tup{associative deformations of $C$} \} }
{\tup{gauge equivalence}} \]
preserving first order terms.}
\end{cor}

\pause
Theorem \ref{thm3} applies also to a large class of complete
varieties: the \hilite{rust}{flag varieties} $X = G / P$, where $G$
is a connected reductive algebraic group and $P$ is a 
parabolic subgroup. This class of varieties includes the projective
spaces $\mbf{P}_{\K}^n$.
\end{frame}




\section{4. About the Proofs}
\begin{frame}
\sect{4. About the Proofs}

\pause
The proofs of Theorems \ref{thm4} and \ref{thm3}
are extremely long and technical, so I will only mention a
few highlights in order to give the flavor. 

\pause
The key to it all is the \hilite{drkblue}{Kontsevich Formality
Theorem}
(\cite{Ko1}, 1997). 

\pause
It gives an explicit quantization map 
\[ \opn{quant} : \{ \tup{Poisson deformations of $C$} \} 
\to \{ \tup{associative deformations of $C$} \}  \]
for the commutative $\K$-algebra
\[ C := \K[[t_1, \ldots, t_n]] , \]
the ring of formal power series in $n$ variables.
\end{frame}

\begin{frame}
Here is a list of ``buzz words'' that occur in the proof of the
Formality Theorem: 
\begin{itemize}
\pause
\item DG Lie algebras and the Maurer-Cartan equation,
\pause
\item $\mrm{L}_{\infty}$ morphisms,
\pause
\item poly vector fields and poly differential operators,
\pause
\item trees and configuration spaces.
\end{itemize}
\end{frame}

\begin{frame}
The passage from the local setup (the power series ring 
$\K[[t_1, \ldots, t_n]]$) to the global setup (an
$n$-dimensional smooth algebraic variety $X$, or a $\mrm{C}^{\infty}$
manifold $X$) is done using \hilite{drkblue}{formal geometry}. 

\pause
There is an infinite dimensional bundle 
\[ \pi : \opn{Coor} X \to X , \]
called the \hilite{rust}{coordinate bundle} of $X$, 
which is a moduli space for coordinate systems on $X$. 

\pause
Given a point $x \in X$, the points in the fiber $\pi^{-1}(x)$
correspond to coordinate systems at the point $x$. 

See Figure
\ref{fig1} for an illustration. 
\end{frame}

\begin{frame}
\begin{figure}

\only<1>{\input{diagram_1_a.tex}}
%  \psset{xunit=0.2pt,yunit=0.2pt,runit=0.2pt}

\only<2>{\input{diagram_1_b.tex}}

\only<3>{\input{diagram_1_c.tex}}

\caption{The coordinate bundle of $X$}
\label{fig1}
\end{figure}
\end{frame}

\begin{frame}
On $\opn{Coor} X$ there is a universal coordinate system. So working
on $\opn{Coor} X$ we ``pretend that we are working locally on $X$'',
and invoke the Formality Theorem. (All this can be made precise!) 

\pause
The next step is to try to find a section of the coordinate bundle.
Having such a section will finish the proof. 

\pause
We are allowed to replace the coordinate bundle $\opn{Coor} X$ with a
slightly modified bundle, denoted by $\opn{LCC} X$. One can show that
in the $\mrm{C}^{\infty}$ case this new bundle has a global section.
This is where Kontsevich's proof of Theorem \ref{thm4} ends.

\pause
In the case of an algebraic variety we must work harder. 
\end{frame}

\begin{frame}
The crucial fact is that the bundle $\opn{LCC} X$ is (almost) a
torsor under a pro-unipotent group. 

\pause
Using an averaging process for pro-unipotent group actions \cite{Ye3}
we prove that there exists a \hilite{drkblue}{simplicial section} of
the bundle
$\opn{LCC} X$.

See Figure \ref{fig2} for an illustration.
\end{frame}

\begin{frame}
\begin{figure}

\only<1>{\input{diagram_2_a.tex}}

\only<2>{\input{diagram_2_b.tex}}

\only<1>{\caption{From two usual sections \ldots}}
\only<2>{\caption{\ldots we pass to a single simplicial section.}}
\label{fig2}
\end{figure}
\end{frame}

\begin{frame}
By combining the simplicial section with a technique called
\hilite{drkblue}{mixed
resolutions} \cite{Ye4} we can finish the proof of Theorem \ref{thm3}.


As usually happens when dealing with open coverings and resolutions, 
cohomological obstructions crop up. This is why we need the
cohomological vanishing condition in Theorem \ref{thm3}.
\end{frame}



\section{5. Final Remarks}
\begin{frame}
\sect{5. Final Remarks}

\pause
{\bf Feynman Diagrams.} 
The explicit formula for the universal quantization map (in the
Kontsevich Formality Theorem) contains countably many real constants
(the weights). These arise as integrals on configuration spaces of
graphs. 

\pause
Kontsevich \cite{Ko2} says that the idea of the proof came
from considerations of string theory: Feynman diagrams of some
quantum field theories.
\end{frame}

\begin{frame}
{\bf Number Theory.} 
The Formality Theorem has an intriguing arithmetic side to
it. The weights mentioned above belong 
the ring of \hilite{drkblue}{motivic periods}, a subring of
$\mbb{C}$. 
They are the reason we required $\mbb{R} \subset \K$ in the
quantization results.

\pause
The
structure of the ring of motivic periods is related to some very deep
conjectures in number theory. For a discussion see Kontsevich's paper 
\cite{Ko2}.

\pause
It is claimed that because of certain results of Drinfeld (on the
Grothendick-Teichmuller group) it is
possible to find an alternative universal quantization map that is
defined over $\mbb{Q}$. But I don't know of a published proof. 
\end{frame}

\begin{frame}
{\bf Algebroids.} 
The last remark I want to make is about quantizing an algebraic
variety $X$ for which the cohomological vanishing condition in Theorem
\ref{thm3} does not hold; for instance an abelian surface.

\pause
As suggested in \cite{Ko3}, and from ongoing research of mine with
Leitner \cite{LY}, it seems that in these more difficult cases there
will be deformation quantization, but in a twisted way, involving
\hilite{drkblue}{stacks of algebroids} and \hilite{drkblue}{nonabelian
gerbes}. 

\pause
Strangely, the latter are now studied mostly by mathematical
physicists, in the context of \hilite{rust}{higher gauge
theory}. See \cite{BS}
and \cite{BM}. At present there does seem to be any direct link
between that and deformation quantization. 

\pause
\begin{center}
\red \large 
END
\end{center}


\end{frame}






\begin{thebibliography}{BFFLS}
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  {\bf 86}, Birkh\"auser Boston.
\bibitem[BFFLS]{BFFLS} F. Bayen, M. Flato, C. Fronsdal, 
    A. Lichnerowicz and D. Sternheimer,
    Deformation theory and quantization I, II, Ann.\ Physics
    {\bf 111} (1978), 61-110, 111-151.
\bibitem[BS]{BS} J. Baez and U. Schreiber,
  Higher Gauge Theory: 2-Connections on 2-Bundles, 
  eprint hep-th/0412325v1 at http://arxiv.org.
\bibitem[BM]{BM} L. Breen and W. Messing,
  Differential Geometry of Gerbes,
  Adv.\ Math.\ {\bf 198}, no.\ 2 (2005). 
\bibitem[CFT]{CFT} S. Cattaneo, G. Felder and L. Tomassini, 
    From local to global deformation quantization of Poisson 
    manifolds, Duke Math.\ J.\ \textbf{115} (2002), no.\ 2, 329-352.
\bibitem[CI]{CI} A.S. Cattaneo and D. Indelicato, Formality and Star
	Products, in ``Poisson Geometry, Deformation Quantisation and 
    Group Representations'', London Mathematical Society Lecture
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\bibitem[CKTB]{CKTB} A.S. Cattaneo, B. Keller, C. Torossian and A.
	Brugui\`eres, ``Deformation, Quantification, Theorie de Lie'',
	Panoramas et Synth\`ese {\bf 20} (2005), Soc.\ Math.\ France.
\bibitem[DL]{DL} M. De Wilde and P.B.A. Lacomte, 
    Existence of star-products and of formal deformations in 
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    Math.\ Phys.\ {\bf 7} (1983), 487-496.
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