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%%%%%%% Formal DGLA and Deformation Quantization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% Milen Yakimov %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%% Title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{{\LARGE\bf{ Formal DGLA's and Deformation Quantization }}}
\author{Milen Yakimov}
\date{}
\maketitle
%\begin{abstract}
%\end{abstract}
%%%%%%%%%%%%%%%%%%%%% Introduction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{section}{-1}
\sectionnew{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In a fundamental paper \cite{BFFLS}
in 1977 Bayen, Flato, Fronsdal, Lichnerowicz, and
Sternheimer considered the passage from classical
to quantum mechanics as a deformation of the algebra of functions on a
Poisson manifold. More precisely they posed the question whether for
an arbitrary Poisson manifold $M$ with Poisson bracket $\{.,.\}$ there exists a
formal deformed multiplication $*_h$ in $C^\infty(M)$:
$$ f *_h g = fg + \sum_{k=1}^{\infty} C_k(f,g) h^k$$
with the following properties:
1) $C^\infty(M)[[h]]$ is an associative algebra under $*_h$,
2) $(f *_h g - g*_h f)/h = i\{f, g\} + O(h),$
3) $1*_h f= f*_h 1= f$ for all smooth functions $f\in C^\infty(M),$
4) $*_h$ is local, i.e. $C_k(.,.)$ is bidifferential operator.
(Sometimes one requires that $C_k(f,g)=(-1)^k C_k(g,f)$.)
Up to now the only case which is completely understood is the case of a
symplectic manifold. After De Wilde and Lecomte proved the existence
of deformation quantization, there appeared several different proofs
to this fact.
Among them, the proof of B. Fedosov is particularly interesting.
Later several authors including Nest--Tsygan,
Deligne, Bertelson--Cahen-Gutt, and Xu, using his construction,
classified completely all deformed (nonequivalent)
multiplications for a given symplectic manifold.
For more details and references the reader may consult
Alan Weinstein's Boubaki talk \cite{W} and M\'elanie Bertelson's
comprehensive review \cite{Melanie}.
Starting from a completely different point of view, M.~Kontsevich \cite{K, K2}
considered the Hochschild complex of functions on a manifold $M$.
Then he proved that if this complex is a formal DGLA (i.e.
quasi-isomorphic to its cohomology algebra as DGLA) then every Poisson
bracket on $M$ can be deformed. In Sect. 1 we make an overview of some
constructions connected with DGLA and sketch the proof of this result.
The second section of this report is divided in two parts.
In the first part we discuss briefly the idea
of Deligne, Griffiths, Morgan, and Sullivan \cite{DGMS},
showing how to use Hodge theory
to prove formality of a DGA. In particular we sketch their proof
of the fact that the de Rham algebra of
differential forms on a compact K\"ahler manifold
is a formal differential graded (commutative) algebra.
Recently A.~Voronov in \cite{V} constructed a second differential on
the Hochschild complex of a commutative algebra. It respects
the bigrading of this complex introduced by
Gerstenhaber and Schack \cite{GS2}, thus considered together with the
usual differential making it a bicomplex. In the second part of Sect. 2 we
review (very briefly) Voronov's construction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sectionnew{DGLA's and Kontsevich's Formality Conjecture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We begin this section with a discussion of
Deligne--Schlessinger--Stasheff--Goldman--Millson's approach to deformation
theory. We follow Sect. 1 and 2 of Goldman--Millson's paper \cite{GM}.
The second part of the section is entirely devoted to Kontsevich's formality
conjecture.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Differential Graded Lie Algebras}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bde{1} A differential graded Lie algebra is is a $\Zset$-graded Lie
superalgebra
$$L= \bigoplus_{i \geq 0} L^i$$
(the bracket is denoted by $[.,.]$) with a derivation of degree 1, i.e. a
map $d: L^i \rightarrow L^{i+1}$ such that
$$ d [ \a, \b ] = [ d \a , \b ] + (-1)^i[ \a , d \b ]$$
for $\a \in L^i$, $\b \in L$.
\ede
To each DGLA (and a local Artinian algebra $A$) one associates a formal
moduli space in the following way:
1) First one defines an action of $L^0$ on $L^1$ by affine transformations
by the formula:
\beq
\rho ( \l ) : \a \mapsto [ \l , \a ] -d \l
\label{1}
\eeq
for $\l \in L^0$, $\a \in L^1$, or more canonically
the degree 0 inner derivations of $L$ (i.e. $\ad L^0$) act on the degree
1 derivations of the form $d + \ad \a$ ($\a \in L^1$)
via the adjoint action
$$
[ \ad \l, d + \ad \a ] =
\ad \left( [ \l, \a] - d \l \right)
$$
or the corresponding vector field $ \rho ( \l )$ has value
$[ \l , \a ] -d \l$ at $ \a \in L^1.$
2) One defines a map
\beq
Q: L^1 \to L^2
\nn
\eeq
by
\beq
Q(\a) = d \a + \frac{1}{2} [\a , \a ].
\label{2}
\eeq
Again it can be defined in a more natural way if we consider the map
$\widetilde{Q}$ from the affine space of derivations of $L$ of the form
$d + \ad \a$ for $\a \in L^1$ to degree 2 inner derivations $\ad L^2$
given by
$$
\widetilde{Q}(d + \ad \a) =(d + \ad \a) \circ (d + \ad \a) =\ad Q(\a).
$$
3) By direct calculations one shows that the directional derivative of
$Q$ in direction of $\rho (\l )$ is 0.
4) Suppose that the action of $L^0$ on $L^1$ can be integrated to an action of
the corresponding "Lie group" $\exp (L^0):$
\beq
\exp( t \l) : \a \mapsto
\exp( t \ad \l ) (\a) +
\frac{\mathrm{Id} -exp(t \ad\l)}{\ad \l} (d \l).
\label{3}
\eeq
Step 3) shows that this action preserves $\Ker Q$.
5) If $A$ is an arbitrary local Artinian $k$-algebra with maximal ideal $\m$
and residue field $k$ we can consider the DGLA $L \otimes A$ instead of $L$.
Then the two sums in \eqref{3} are finite for $\l
\in L^0 \otimes \m$, and the corresponding action groupoid
$\LL(L,A)$ for the action of the Lie group $\exp(L^0 \otimes \m)$ on
$\Ker Q$ ($Q$ is restricted to $L^1 \otimes \m$) is well defined.
Now recall that two complexes $L$ and $L'$ are quasi-isomorphic if there
exists a chain
$$L \rightarrow L^1 \leftarrow L^2 \rightarrow \cdots
\leftarrow L^n \rightarrow L',
$$
such that all homomorphisms induce isomorphisms on
homology. For DGLA one requires
that these homomorphisms respect the Lie algebra structure.
A fundamental theorem of
Deligne, Schlessinger, Stasheff, Goldman, and Millson says:
\bth{2} If $L$ and $L'$ are quasi-isomorphic DGLA's then
the corresponding groupoids $\LL(L,A)$ and $\LL(L',A)$
are equivalent. Even more if
$\phi : L \to L'$ is a DGLA homomorphism for which the $i$-th map
$\phi_* :H^i(L) \to H^i(L') $ is an isomorphism for $i= 0$, $1$ and
a monomorphism for $i=2$ then $\phi_* : \LL(L,A) \to \LL(L',A)$ is
an equivalence of groupoids.
\eth
The {\it{proof}} proceeds by "Artinian induction" on the algebra $A$.
For every local Artinian $k$-algebra $A$ there exists a sequence of
local Artinian algebras $A= A_0,$ $A_1, \ldots ,$ $A_{r-1},$ $A_r=k$
with epimorphisms $\eta_i : A_i \to A_{i+1}$ such that
$(\Ker \eta_i) M_i =0$, where $M_i$ is the unique maximal ideal of
the local ring $A_i$. For the purposes of deformation theory
we can ignore the abstract definitions connected with Artinian
algebras and restrict ourselves to the following simple example
$$
k[h] /k[h] h^n \to k[h] /k[h] h^{n-1} \to \cdots \to k.
$$
In order to carry out the induction, first one finds the obstructions
for extending the equivalence
$$\LL(L,A_k) \to \LL(L',A_k)$$
to an equivalence
$$\LL(L,A_{k-1}) \to \LL(L',A_{k-1})$$
(Prop. 2.6 in \cite{GM})
and then checks that they are 0 using the assumptions
of \thref{2}.
Finally a differential graded Lie algebra is called formal if it is
isomorphic to the DGLA of its cohomology equipped with the trivial differential.
In the next section we will discuss the importance of this special type of
DGLA's for deformation theory. For their role in rational homotopy theory
we refer to Deligne--Griffiths--Morgan--Sullivan's paper \cite{DGMS}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Kontsevich's formality conjecture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this subsection we follow Kontsevich's Berkeley lectures \cite{K} and
paper
\cite{K2}, and Voronov's paper \cite{V}.
Consider the Hochschild complex of smooth functions on a smooth manifold $M$:
$$
HC^n(M) = \{ \phi \in \Hom(C^\infty(M)^{\otimes n},C^\infty(M)) | \phi
\mbox{--local}\}
$$
with the standard bracket (see \cite{GS1} for a very readable exposition
of Gerstenhaber's deformation theory of algebras). A theorem of
Cahen and Gutt \cite{CG} says
that the DGLA of homology of this
complex coincides with the DGLA of multivector fields on $M,$
$ \wedge^\bullet TM,$ with the Schoutens-Nijenhuis bracket
and the trivial differential.
It is a $C^\infty$ version of a theorem of
Hochschild, Kostant, and Rosenberg \cite{HKR} concerning the case of
algebraic varieties.
Now we are ready to state Kontsevich's formality conjecture \cite{K}.
\bcon{3}
$HC^\bullet (M)$ is a formal DGLA.
\econ
A remarkable theorem of Kontsevich \cite{K} connects it with the problem of
existence of deformation quantization.
\bth{4} The formality conjecture for a manifold $M$
implies the existence of deformation quantization of every Poisson
structure on it.
\eth
{\it{Sketch of the proof}}\/ Suppose $\{.,.\}$ is a Poisson bracket on
$M$. The Jacobi identity shows that it gives
an element $\g \in \LL(\wedge^\bullet TM)$ -- the groupoid described in Sect. 1
associated to the DGLA $\wedge^\bullet TM.$ Consider the path $t \g$ in
$\LL(\wedge^\bullet TM).$ Because $HC^\bullet(M)$ is
quasi-isomorphic to $\wedge^\bullet TM$ as DGLA \thref{2} shows that
there exists a corresponding path on $\LL(HC^\bullet(M))$, i.e.
a family of elements $\g' \in HC^2(X)$ such that
$d \g' + (1/2) [\g', \g'] = 0.$ Using the simple fact that the
Hochschild differential is equal to the bracket with the multiplication
2-cocycle $m \in HC^2 (M)$, $(m(f,g) = fg$
for $f,g \in C^\infty(M))$ one finally obtains that
$$
[m + \g', m+ \g' ] = 0.
$$
This is equivalent to saying that each element of the family
$m + \g' \in HC^2(M)$ defines an associative product on
$C^\infty(M)$. At the same time, this family has tangent vector at $m$
equal to $\{.,.\},$ or in other words it is the required $*$-product.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sectionnew{Hodge theory and formality of DGLA's}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the first part of this section we discuss how
Deligne, Griffiths, Morgan, and Sullivan proved that the de Rham algebra
of smooth forms on a K\"ahler manifold is a formal DGA using
Hodge theory. The second subsection is a brief review of
Voronov's construction of a $\del$-operator in the Hochschild complex
of a commutative algebra.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Formality of the de Rham complex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $M$ be a compact K\"ahler manifold. As usual $d = \del + \delbar,$
$d^c= i ( \delbar - \del).$ Denote the de Rham complex over $\Rset$
(considered as differential graded algebra) by $C_{DR} (M)$ and by $H_{DR} (M)$
the DGA of its cohomology. Deligne, Griffiths, Morgan, and Sullivan
considered the following chain of DGA's:
\beq
\{ C_{DR}(M), d\}
\stackrel{i}{\leftarrow}
\{\Ker d^c, d\}
\stackrel{p}{\rightarrow}
\{ H_{d^c}(M), d\},
\label{10}
\eeq
where $\{\Ker d^c, d\}$ is the DGA which is the kernel
of the homomorphism
$$ d^c : C_{DR}(M) \to C_{DR}(M) $$
and $H_{d^c}(M)$ is the factor-DGA $\Ker d^c / d^c(C_{DR}(M))$,
both with differentials $d$. Here $i$ is the natural inclusion and $p$ is
the natural projection.
\bth{5} (\cite{DGMS}) In \eqref{10} $i$ and $p$ induce isomorphisms on
cohomology and $d$ acts as 0 on $H_{d^c}(M)$. Therefore
$C_{DR}(M)$ is a formal DGLA.
\eth
The major ingredient of the proof is the $d d^c$-lemma for
compact K\"ahler manifold (\cite{DGMS}, see also the book \cite{GH}).
\ble{6} Let $M$ be a compact K\"ahler manifold. If
$x$ is a differential form with
1) $d x = 0 = d^c x$,
2) $x = d y$ or $x = d^c y',$
\hfill\\
then $ x= d d^c z $ for some form $z$.
\ele
The {\it{proof\/}} of the above theorem is an easy diagram chasing.
For example the proof that $i$ induces an isomorphism on cohomology
goes in the following way:
1) $i_*$ is onto: Let $\a \in H^\bullet(C_{DR}(M), d)$ $x$ be a closed
representative of $\a$. Then
$$ d^c x \in \Ker d \cap \Ker d^c \cap \Im d^c.$$
The $d d^c$-lemma implies that there exists a form $w$ such that
$d^c x = d d^c w$ or $ x + d w \in \Ker d^c$ and
$ \a = i_*([ x + dw])$.
2) $i_*$ is one-to-one: If $y$ is a closed form in $\{\Ker d^c, d\}$
which is exact in $C_{DR}(M)$ then
$$y \in \Ker d \cap \Ker d^c \cap \Im d$$ and by the $d d^c$-lemma
$y = d d^c w$ for a form $w$ or $ y= d(d^c w)$ with $ d^c(d^c w)=0$,
i.e. $[y] =0$ in $H^\bullet(\Ker d^c, d)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Hochschild complex as a bicomplex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Recently A. Voronov \cite{V} introduced a second differential in the
Hochschild complex $HC^\bullet(A,A)$ of a commutative algebra
$A$, thus making it a bicomplex. In this subsection we will sketch
his construction.
He uses a modification by Ronco, Sletsj{\o}e, and Wolfgang \cite{BW} of the
bigraded Hochschild complex of Gerstenhaber and Schack \cite{GS2}.
Let $r$ and $s$ be positive integers $n =r + s$. The shuffle product of tensors
$a_1 \otimes \cdots \otimes a_{r} \in A^{\otimes r}$ and
$a_{r+1} \otimes \cdots \otimes a_{n} \in A^{\otimes s}$
is defined by
$$
\sum \sgn (\sigma) a_{\sigma^{-1} (1)} \otimes \cdots \otimes
a_{\sigma^{-1} (n)} \in A^{\otimes n},
$$
the sum being over
$\sigma \in S_n$ for which
$\sigma(1) < \sigma (2) < \cdots < \sigma (r)$ and
$\sigma(r+1) < \sigma(r+2) <\dots < \sigma (n)$.
The image of shuffle products of $k$ elements of the tensor algebra $T(A)$
of $A$ will be denoted by $\mathrm{Sh}^k$. Finally the spaces of the bigraded
Hochschild complex are:
$$
HC^{p,q} = \Hom(\mathrm{Sh}^p \cap A^{\otimes (p+q)} /
\mathrm{Sh}^{p+1} \cap A^{\otimes (p+q)}, A).
$$
The Hochschild differential maps $HC^{p,q}$ into $HC^{p,q+1}$ and is defined
by
$$ d \phi = [ m , \phi ],$$
where $m \in C^2( A , A)$ is the multiplication cycle.
In analogy with this A. Voronov defined
a differential $d'$ mapping $HC^{p,q}$ into $HC^{p-1,q}$ by
$$ d \phi = [ 1 , \phi ],$$
where 1 is the identity element of $A$ or:
\beqa
&&(d'\phi) (a_1, \ldots, a_{n-1}) \nn \hfill \\
&&= \phi(a_1, \ldots, a_{n-1}, 1) - \phi(a_1, \ldots, a_{n-2}, 1,
a_{n-1}) + \cdots +(-1)^{n-1} \phi(1, a_1, \ldots, a_{n-1}).
\nn\hfill
\eeqa
For more details on the definition and the properties of the operators
$d$ and $d'$ we refer to Voronov's paper \cite{V}.
And finally if the $d d^c$-Lemma
holds for the bigraded Hochschild complex of the algebra of functions on
a manifold $X$, then the Kontsevich's theorem will imply the existence
of deformation quantization of any Poisson bracket on $X$.
But it seems also that it is quite possible that the $d d^c$-Lemma
not always holds.
Other ways for proving the formality conjecture are
suggested by Kontsevich in \cite{K2}.
\hfill\\
\noindent
{\bf{Acknowledgement}} I would like to thank M\'elanie Bertelson and Prof.~Alan
Weinstein for a number of suggestions which improved the paper
considerably. I am also very grateful to Prof.~Alan
Weinstein for giving me Maxim Kontsevich's recent paper \cite{K2}.
%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{small}
\begin{thebibliography}{AFMO}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{BFFLS}
F.~Bayen, M.~Flato, C.~Fronsdal, A.~Lichnerowicz, D.~Sternheimer,
{\em{Deformation theory and quantization}},
Ann. Physics {\bf 111} (1978), 61--151.
\bibitem{BW}
N.~Bergeron, H.~L.~Wolfgang,
{\em{The decomposition of Hochschild cohomology and Gerstenhaber operations}},
J. Pure Appl. Algebra {\bf 104} (1995), 243--265.
\bibitem{Melanie}
M.~Bertelson, {\em{Existence of star products, a brief history}},
term paper for Math 277, Berkeley, Spring 1997.
\bibitem{CG}
M.~Cahen, S.~Gutt,
{\em{Local cohomology of the algebra of $C^\infty$ functions on a compact
manifold}},
Lett. Math. Phys. {\bf 4} (1980), 157--167.
\bibitem{DGMS}
P.~Deligne, P.~Griffiths, J.~Morgan, and D.~Sullivan,
{\em{Real homotopy theory of K\"ahler manifolds}},
Invent. Math. {\bf 29} (1975), 245--274.
\bibitem{GS2}
M.~Gerstenhaber, S.~D.~Schack,
{\em{A Hodge-type decomposition for commutative algebra cohomology}},
J. Pure and Appl. Alg. {\bf 48} (1987), 229--247.
\bibitem{GS1}
M.~Gerstenhaber, S.~D. Schack,
{\em{Algebraic cohomology and deformation theory}}.
In: Deformation Theory of Algebras and Structures and Applications,
M.~Hazewinkel and M.~Gerstenhaber, eds.,
Kluwer Academic Publishers, 1988, 11--264.
\bibitem{GH}
Ph.~Griffiths, J.~Harris,
{\em{Principles of Algebraic Geometry}},
John Wiley \& Sons, 1978.
\bibitem{GM}
W.~M.~Goldman, J.~J.~Millson,
{\em{The deformation theory of representations of
fundamental groups of compact K\"ahler manifolds}},
Publ. IHES {\bf 67} (1988) 43--66.
\bibitem{HKR}
G.~Hochschild, B.~Kostant, A.~Rosenberg,
{\em{Differential forms on regular affine algebras}},
Trans. Amer. Math. Soc. {\bf{102}} (1962), 383--408.
\bibitem{K}
M.~Kontsevich,
{\em{Lectures on deformation theory}}, Berkeley 1995.
\bibitem{K2}
M.~Kontsevich,
{\em{Formality conjecture}}. In:
Deformation theory and symplectic geometry,
S.~Gutt, J.~Rawnsley, and D.~Sternheimer, eds.,
Mathematical Physics Studies, Kluwer, Dordrecht, 1997
(to appear).
\bibitem{V}
A.~A.~Voronov,
{\em{Quantizing Poisson manifolds}}, Preprint q-alg/9701017.
\bibitem{W}
A.~Weinstein,
{\em{Deformation quantization}},
volume no. 227, Exp. No. 789, 5 of S\'eminaire Bourbaki, Vol. 1993/94,
Asterisque, 1995.
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\end{thebibliography}
\end{small}
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\end{document}