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\newcommand{\CC}{C^{\infty}({\Bbb R}/{\Bbb Z}, \Bbb C)}
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\newcommand{\htt}{ \hat{{\frak g}} }
\newcommand{\hp}{{L{\frak g}_1}^*}
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\newcommand{\g}{{\frak g}}
\newcommand{\n}{{\frak n}}
\newcommand{\bb}{{\frak b}}
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\newcommand{\K}{\w\g\otimes F(M)}
\newcommand{\Ko}{\w(\g +\g^*) \otimes F(M)}
\newcommand{\gt}{\hat{{\frak g}}}
\newcommand{\LL}{\Lambda}
\newcommand{\Z}{\Bbb Z}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\author{Alexander Kogan}
\title{Drinfeld-Sokolov reduction and $W$-algebras.}
\maketitle
\section{Introduction.}
$W$-algebras first appeared in physics as symmetries of conformal field
theories \cite{Bo-Sc}. For example, the symplest nontrivial classical $W$-algebra $W({\frak sl}_2)$ is roughly a Virasoro algebra.
From physics $W$-algebras naturally drifted into mathematics. The following
questions arose:
\begin{enumerate}
\item How to define classical $W$-algebras?
\item How to quantize them?
\item Are there any nice applications?
\end{enumerate}
Below we briefly explain the answers to the above questions.
\begin{enumerate}
\item The answer to this question is explained in
details in the section 3. Let $\g$ be a semisimple Lie algebra, $G$ -
corresponding Lie group, $N$ - unipotent subgroup of $G$.
We consider the Poisson coadjoint action of the unipotent loop group $LN$ on the certain hyperplane $\hp$. Performing the symplectic
reduction, which is called (classical) Drinfeld-Sokolov reduction in this case,
we obtain a manifold $B$. The space of smooth functions on $B$ is called $W$-
algebra associated to the algebra $\g$. This $W$-algebra is of great importance because the manifold $B$ can be identified with the space of certain
differential operators modulo the gauge equivalence (the action of $LN$).
It was noticed by V. Drinfeld and proved by B.~Feigin and E.~Frenkel
\cite{Fe-Fr, Fr:thesis} that the
classical $W$-algebra $W(\g)$ is isomorphic to the center of the completion of
the universal enveloping algebra $U(\htt)_{cr}$ at the critical level, i.e.
when the central element of the affine algebra $\htt$ acts by multiplication
by the dual Coxeter number of $\g$. This approach gave another description of the Poisson structure of the classical $W$-algebras.
\item In the geometric setting $W$-algebras can be quantized using
BRST quantization procedure. This is the subject of sections 4 and 5. BRST
quantization is usually applied to a manifold which is obtained in the
process of symplectic reduction from another easily quantizable manifold. This exacly happens in our case since we start with a hyperplane, which is easy to
quantize.
The drawback of this approach is that it is hard to describe the quantum $W$-algebra in terms of generators and relations, what could be used in constructing
the representation theory. For this purpose the second approach is more useful.
In the classical case there is a well-known Wakimoto realization of $U(\htt)_{cr}$ \cite{Fr-Re}
inside the product of a Heisenberg and commutative algebras. The center
of $U(\htt)_{cr}$ is mapped into the commutative part. The restriction to the
center has several names, such as: Miura transformation, bosonization,
free field realization.
There are deformed (with one parameter $q$) \cite{Fr-Re, Ko}
and quantized (with parameters $q$ and $t$) \cite{Fr-Re:2}
versions of the above picture. First one defines the deformed
(but still commutative) $W$-algebra $W_q(\g)$ as a center of the quantized
universal enveloping algebra $U_q(\htt)_{cr}$. Then one defines the $q$-deformation of the Wakimoto realization. As a result we have a map, called
$q$-Miura transformation, from $W_q(\g)$ to a certain commutative Poisson
algebra $H_q(\g)$. The geometric interpretation is similar to the classical
one, only one has to apply symplectic reduction to the space of
$q$-difference operators \cite{Fr-Re-Se, Se-Se}
The quantization of the above picture \cite{Fr-Re:2}
consists of deforming the algebra
$H_q(\g)$ into the algebra $H_{q,t}(\g)$. Then one introduces the $W$-algebra
$W_{q,t}(\g)$ as a subalgebra of $H_{q,t}(\g)$ which commutes with certain
operators, called the screening operators.
\item There are numerous application of $W$-algebras. We can now
quantize the Virasoro algebra, which is isomorphic to $W({\frak sl}_2)$.
For various $q$ and $t$ $W_{q,t}(\g)$ can be identified vith various algebras
arising in the theory of integrable systems \cite{Fr-Re:2}.
Also $W$-algebras play an important role in the study of the geometric
Langlands-Drinfeld correspondence \cite{Fr:Langlands}. Finally
the formulas for the $q$-deformed $W$-algebra $W_q(\g)$ cojecturally coinsides with
the formulas for the eigenvalues of transfer matrices of $\htt$-invariant models of statistical mechanics \cite{Fr-Re}. There must be many other applications which are not known to the author.
The purpose of this paper is to review the desctiption of the classical
$W$-algebras via Poisson geometry and the BRST quantization of this picture.
The paper is organized as follows. In Section 2 we give the motivation for the
definition of $W$-algebras. In Section 3 we give the definition using
Drinfeld-Sokolov reduction. BSRT cohomology is explained in Section 4. In
Section 5 we explain how to quantize $W$-algebras using BRST quantization.
\end{enumerate}
{\bf Acknoledgements.} The author is grateful to professor Alan Weinstein for giving a very interesting course in noncommutative geometry in Spring 1997.
\section{On the classification of differential operators.}
The main result of this section is the well-known correspondence between differential equations and certain matrices.
Let $(-\partial_{t}^2 +u)\psi =0$ be a second order linear differential equation, where $u\in \CC$. If instead of $\psi$
we consider the column vector $\phi$, \[ \phi = \left( \begin{array}{c} \psi \\
\psi_{t} \end{array} \right) \]
then the equation becomes a linear first order matrix \eq
\[\partial_t \phi = A\phi, A=\left( \begin{array}{cc} 0 & 1\\u & 0 \end{array} \right). \]
Thus we associated the matrix $A$ to our differential \eq. But the process is not quite canonical since we can
choose a vector different from $\phi.$ For example, if we replace $\phi$ by $v \cdot \phi$, where \[v=
\left( \begin{array}{cc} 1 & 0\\ \alpha & 1 \end{array} \right), \]
where $\alpha \in \CC$, then $A$ gets replaced by \[ \tilde{A} = vAv^{-1} + (\partial_{t}v)v^{-1}.\]
The transformation $A \rightarrow \tilde{A}$ is called a gauge transformation.
Thus we have a bijective correspondence between 1) second order differential {\eq}s with coefficients in $\CC$, first
coefficient 1, and 2) $2\times 2$ matrices with entries in $\CC$ modulo the gauge transformation
by the lower triangular unipotent matrices. We will denote this set $M^f/LN$. The space of $C^{\infty}$-
functions on $M^f/LN$ is called the classical $W$-algebra $W({\frak gl}(2))$.\\
If we consider in 1) only the differential equations with second coefficient equal to zero, then in 2) ${\frak gl}(2)$
gets replaced by ${\frak sl}(2)$, where ${\frak gl}(2)$ (resp. ${\frak sl}(2)$) is the Lie algebra of
$GL(2, {\Bbb R})$ (resp. $SL(2, {\Bbb R})$). \\
The above facts are true in general for differential equations of $n$-th order. Drinfeld and Sokolov have observed
in \cite{Dr-So} that this story can be put in the hamiltonian setting. This is the subject of the next section.
\section{Symplectic reduction and the classical $W$-algebras.}
Drinfeld and Sokolov have observed \cite{Dr-So} that the space of differential operators mentioned above can be given the structure
of a Poisson manifold. Let us briefly explain this result.
Let $G$ be a complex semisimple Lie group with Lie algebra $\g$. Let $LG = C^{\infty}(S^1, G)$ be the loop group and
$L\g = C^{\infty}(S^1, \g)$ the corresponding loop algebra. $L\g$ has the inner product
\[ (X,Y)_L = \int_{S^1}(X(t), Y(t))dt ,\] where $(,)$ is the Killing form on $\g$, extended to $\hat{\g}$ by the formula
\[ (at^n, bt^m) = (a, b)t^{n+m}, a, b \in \g. \]
Let $\htt$ be the central extension of $L\g$ with respect to the $2$-cocycle
\[ c(X,Y) = (X, \partial_{t}Y)_L , X, Y \in L\g.\]
The standard coadjoint Poisson action $Ad^*$ of $LG$ on $L\g^*$ can be extended to a Poisson action
$\widehat{Ad}$
on $\htt^*$. Let us explain the notation first. We identify $\htt^*$ with $L\g\oplus {\Bbb C}$. Under this identification the
coadjoint action becomes the adjoint action $Ad$. Let $\vartheta$ be the standard Maurer-Cartan form on $G$. Let
\[ \widehat{Ad}_{g}(l,k) = (Ad_{g}(l) - ki_{\partial_t}g^{*}\vartheta, k), \]
where $l\in L\g, k \in \CC$.
Denote the hyperplane $(L\g, 1)$ by $\hp$. This hyperplane is preserved under the new action, therefore we have
momentum map $\hp \rightarrow
\hp$, where we trivially identify $\hp$ with $(L\g,0)$. Calculations show that this momentum map
is just the identity. \\
The identity map is not very interesting. To remedy the situation, we want to restrict the action of $LG$ to the action of
its subgroup. Let $\n$ be the nilpotent subalgebra of upper triangular matrices of $\g$, $N= exp(\n), LN = C^\infty(S^1,
N)$ as usual. The restriction of $\widehat{Ad}$ to $LN$ results in the new momentum map $\mu : \hp \rightarrow L\n^*$. It is
easy to describe this map. If we identify $\hp$ with $(L\g,0)$ , then $\mu$ becomes the trivial projection onto $(L\n,0)$.
There is a coadjoint action of $LN$ on $L\n^*$, let $f^*$ be a fixed point under this action
, $f$ - the corresponding element of $L\n$. Let $M^f = \mu^{-1}(f)$ be the
level surface of the momentum map. It is easy to see that
\[ M^f=\{(l,1)\in \hp : l \in f + L\bb\}, \] where $\bb$ is the Borel subalgebra of $\g$.
Since $f$ is fixed under the coadjoint action, $M^f$ is preserved under the action of $LN$. Thus we obtain the Poisson
manifold $M^f/LN$ as a result of symplectic reduction. We have the following
\begin{theorem} \cite{Dr-So, Se-Se}
\begin{enumerate}
\item The action of $LN$ on $M^f$ is free.
\item The quotient space $B = M^f/LN$ is an infinite-dimensional Poisson manifold. Its symplectic leaves have finite
codimension.
\end{enumerate}
\end{theorem}
{\bf Example.} Let $\g = {\frak sl}(n)$. Let \[ f^*=\left( \begin{array}{ccccccc}
0 & 1 & 0 & ... & 0 & 0 & 0 \\
0 & 0 & 1 & ... & 0 & 0 & 0 \\
... & ... & ... & ... & ... & ... & ...\\
0 & 0 & 0 & ... & 0 & 0 & 1 \\
0 & 0 & 0 & ... & 0 & 0 & 0
\end{array} \right). \]
Then $f$ is preserved under the action of $LN$. The manifold $B = M^f/LN$ is the one that we had in the previous
section for $n=2$. The classical $W$-algebra $W({\frak sl}(2))$ is the Virasoro algebra, i.e. the algebra of
diffeomorphisms of a circle. \\
The process of going from $\hp$ to $B$ is just a symplectic reduction. In this specific case it is called (classical)
Drinfeld-Sokolov reduction. The Poisson algebra of functions on $B$ is called the classical $W$-algebra associated to the
Lie algebra $\g$. In the rest of the paper we describe the quantization of this algebra using BRST cohomology.
\section{BRST cohomology.}
In this section I will define the BRST cohomology of a Poisson manifold obtained by a process of symplectic reduction.
Then we will be able to express the space of functions on this manifold in cohomological terms. Thus the quantization of
the BRST cohomology will give us the quantization of our manifold. A good introduction to this subject
can be found in \cite{Ko-St}\\
Let us introduce some notation first.
Let $M$ be a Poisson manifold, $G$ - a complex Lie group, $\g$ - its Lie algebra, $G: M \rightarrow M$ - Poisson action,
$\Phi : M \rightarrow \g^*$ - momentum map. Suppose that $0$ is a regular value of $\Phi$. Let $C= \Phi^{-1}(0)$ and
$B = C/G.$ Let $\dd :\g \rightarrow F(M)$ be the map used in the construction of the momentum map, so that
$\Phi(m)(\eta) = \dd(\eta)(m)$, where $m\in M, \eta \in \g$.\\
Let us consider the complex $\w\g\otimes F(M)$, which we will denote by $K$:
\[ ...\rightarrow \w^{q}\g\otimes F(M) \rightarrow \w^{q-1}\g\otimes F(M)
\rightarrow ...\]
with the differential $\dd$ defined on generators below and extended as a {superderivation:}
\[ \dd(\xi\otimes 1)= 1\otimes\dd(\xi), \xi \in \g \]
\[ \dd(1\otimes f)=0, \ \ f\in F(M)\]
It is easy to see that $\dd^2=0$.
$H^0_{\dd}(\w\g\otimes F(M))= F(M)/{F(M)\cdot \dd(F(M)} = F(C)$ by definition of $C$. \\
{\it Remark}: $K$, considered as a tensor product of the exterior algebra of $\g$ and $F(M)$, is a $\g$-module in a natural way.
Recall that $\w\g$ is a super-Poisson algebra, which has the adjoint action of $\g$ on it. On the second component
of $K$ action of $\g$ comes from the action of $G$ on $M$.
Let's consider the complex $\w\g^*\otimes K$:
\[ ...\rightarrow \w^{p}\g^*\otimes K \rightarrow \w^{p+1}\g^*\otimes K
\rightarrow ...\]
with the differential $d$ defined on $K$ as follows: if we identify $\g^*\otimes K$ with $Hom(\g, K)$, then
\[dk(\xi) = \xi(k), \]
and we extend the action of $D$ by the following formula:
\[ d(\omega\otimes k) = d\omega\otimes k + (-1)^p\omega\wedge dk, \]
It's easy to see that $d^2=0$.
From the definition of this complex we can see that $H^0_d(\w\g^*\otimes K) =$ the space of $\g$-invariants in $K$. This is true for any $\g$-module $K$.
Thus the complex $\w\g^*\otimes \w\g\otimes F(M) = \w(\g +\g^*)\otimes F(M)$ has the structure of a bicomplex.
Let us define the {\it total degree} of an element in $\w^p \g^*\otimes \w^q \g\otimes F(M)$ to be $p-q$.
We create a single complex out of our bicomplex by summing up elements with the same total degree.
A new differential is defined by \[ D = d\otimes 1 + 2(-1)^{deg}1\otimes \dd \]
Then $D^2=0$ and $D$ is called the classical BRST operator.
\begin{corollary} $H^0_D = H^0_dH^0_\dd(\w(\g +\g^*)\otimes F(M)) = F(C)^G = F(B).$
\end{corollary}
Thus we have expressed the space of functions on $B$ as the zeroth cohomology of the BRST complex.
In order to quantize $B$, we will find quantum BSRT complex and take its zeroth cohomology.
\section{Quantization of the BRST complex.}
Let $A$ be a Poisson algebra. Then by quantization we mean a linear map from $A$ to
the space of operators on some Hilbert space, such that the Poisson bracket of two
elements of $A$ is mapped to the commutator of the corresponding operators.
Here is the plan for BRST quantization:
\begin{enumerate}
\item Quantize $F(M)$, i.e. realize it as operators on some Hilbert space $V$.
\item Introduce a super-Poisson bracket on $\w(\g + \g^*)$ using the Clifford algebra $C(\g + \g^*)$.
Recall that the Clifford algebra $C(V)$, associated to a vector space $V$, over field $k$, with a scalar product $(,)$ is
defined to be a unital algebra
\[ C(V) = (k \oplus V \oplus V^2 \oplus V^3 \oplus ...)/(uv + vu = (u,v)\cdot 1_{C(V)}). \]
$C(V)$ is an algebra in a natural way. We can naturally define a supercommutant on it. $C(V)$ is naturally graded by total degrees of
elements. Thus we have an increasing filtration on $C(V)$.
It is easy to see that the associated graded algebra (which is defined to be a direct sum of quotients of
successive pieces of the filtration) ${\frak gr}(C(V))$ is just $\w\g$. Under the natural map from $C(V)$ to $\w\g$ the commutant
becomes the super-Poisson bracket, so we can think of $C(V)$ as a quantization of $\w\g$. For more information on this subject the reader
should consult \cite{Ko-St}.
\item Find $\theta \in \Ko$, such that $D = ad\theta = \{\theta, *\}$
\item Quantize $\w(\g + \g^*)$ using the corresponding
Clifford algebra $C(\g + \g^*)$ and its ${\Bbb Z}$-graded spin-representation $T$.
\item Find an element $Q$ in $End(T\otimes V)$, such that $Q$ is a quantization of $\theta$, $Q^2=0$ and $Q$ is of degree 1
(elements of $V$ have degree $0$).
Define quantization of $F(B)$ to be $H^0_Q(T\otimes V)$.
\end{enumerate}
Because of the lack of space we do not consider the realization of this plan in general, but just remark on it.
{\it Remarks}:
\begin{enumerate}
\item The quantization can be performed only if $M$ can be quantized, so $M$ should be simpler than $B$.
\item Clifford algebera $C(\g + \g^*)$ (or rather its representations) should be considered as a quantization
of $\w(\g + \g^*)$ \cite{Ko-St}.
\item If $\g$ is finite-dimensional then the Clifford algebra has only one representation and we can always find
$Q$ such that $Q^2=0$. If $\g$ is infinite-dimensional this is not always true. The cohomology class of $Q^2$ is called
the obstruction to the quantization.
\end{enumerate}
In the case of Drinfeld-Sokolov reduction the quantization can be performed. Let us describe the quantum BRST complex
following \cite{Fr}.
Let $\gt$ be the affine algebra corresponding to $\g$ as before. Let $V_k$ be the vacuum representation of $\gt$, i.e. it
is generated by a vacuum vector, annihilated by $\g \otimes {\Bbb C}[t]$;
the generator of the center acts by multiplication
by $k$. \\
Remark: From now on we consider ${\gt}$ to be just the polynomial maps from $S^1$ to $\g$, so we can put
${\gt} = \g \otimes {\Bbb C}[t]$.
Let us introduce the Clifford algebra $C(\n + \n^*)$. It is generated by
$\psi_{\alpha}(m), \psi^*_{\alpha}(m),
m \in {\Bbb Z}, \alpha \in \Delta_+$. The relations are: \[ [\psi_{\alpha}(m), \psi^*_{\beta}(n)]=
\delta_{\alpha, \beta}\delta_{n, -m} \]
where [ , ] is an anticommutator.
Let $\LL$ be the irreducible representation of this Clifford algebra, generated by the vacuum vector, annihilated
by $\psi (m), m \geq 0, {\psi}^*(m), m >0$. It is ${\Bbb Z}$-graded: deg$\psi (m) = -1,$ deg$\psi^*(m) = 1.$ \\
Let us introduce the following notation. If $a \in \g$, then \[ a(z) = \sum_{m \in \Z}{a(m)z^{-m-1}},\]
\[ \psi_{\alpha}(z) = \sum_{m \in \Z}{\psi_{\alpha}(m)z^{-m-1}}, \]
\[ \psi^*_{\alpha}(z) = \sum_{m \in \Z}{\psi^*_{\alpha}(m)z^{-m}}, \alpha \in \Delta_+.\] \\
The space $\LL \otimes V_k$ has the structure of a complex. The $\Z$-grading of $\LL$
is defined above, the degree of elements
of $V_k$ is $0$, and the degree of the product of two elements is the sum of their degrees. \\
The differential is defined by $Q = d_{st} + p$, where
\[ d_{st} = \int{(\sum_{\alpha \in \Delta_+}e_{\alpha}(z)\psi^*_{\alpha}(z) - \frac{1}{2}
\sum_{\alpha, \beta, \gamma \in
\Delta_+}c^{\gamma}_{\alpha \beta}\psi^*_{\alpha}(z)\psi^*_{\beta}(z)\psi_{\gamma}(z))dz} \]
where $c^{\gamma}_{\alpha \beta}$ are the structure constants of $\hat{\n}$, is the standard differential of Lie
algebra cohomology of $\hat{\n}$, and
\[ p = \int\sum_{simple\ roots}\psi^*_{\alpha}(z)dz.\]
Then $Q^2 = 0$. This complex is the quantum BRST complex, associated to the Drinfeld-Sokolov reduction. Its zeroth
cohomology is called the quantum $W$-algebra $W(\g).$
\ifx\undefined\bysame
\newcommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
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\end{document}