Deformation quantization and localization of noncommutative
algebras and vertex algebras
TOSHIRO KUWABARA1

The present paper is a report for the proceeings of 59th Algebra Symposium
held at the University of Tokyo on 8–11th September 2014.
0. Introduction
We discuss certain infinite-dimensional noncommutative associative algebras.
One well-known class of infinite-dimensional associative algebras is universal enveloping algebras of Lie algebras. For a semi-simple Lie algebra g, the central
quotient of its universal enveloping algebra U (g)/Z(g) is naturally isomorphic to
the algebra of differential operators D(G/B) on the corresponding flag manifold
G/B. Such an isomorphism connects the representation theory of the Lie algebra g and geometrical properties of the flag manifold G/B and/or its cotangent
bundle T ∗ (G/B). Namely, the isomorphism leads an equivalence of categories between the category of (U (g)/Z(g))-modules and the category of DG/B -modules
where DG/B is the sheaf of differential operators. The equivalence is known as theBeilinson-Bernstein correspondence and it is one of the most fundamental facts of
the representation theory of semi-simple Lie algebras.
The above situation can be generalized, and it achieve the notion of quantizations. Note that, in the above, the universal enveloping algebra U (g) is equippedwith a natural filtration, and the associated graded algebra gr U (g)/Z(g) with respect to the filtration is naturally isomorphic to the coordinate ring C[T ∗ (G/B)].
The commutator [a, b] = ab−ba on U (g) induces a Poisson bracket on gr U (g)/Z(g),while C[T ∗ (G/B)] is also equipped with a Poisson bracket induced from the symplectic structure of the cotangent bundle T ∗ (G/B). The isomorphism between
gr U (g)/Z(g) and C[T ∗ (G/B)] is not just an isomorphism of commutative algebras,
but it is an isomorphism of Poisson algebras. We say that U (g)/Z(g) is a quantization of C[T ∗ (G/B)] (or T ∗ (G/B)). Namely, in general cases, for a symplectic
manifold X, a filtered associative algebra is called a quantization of C[X] (or ofX), if the associated graded algebra is isomorphic to C[X] as a Poisson algebra.
In the past decades, several noncommutative algebras which give quantizationsof certain symplectic manifolds were introduced. These algebras are constructedby using so called the quantum Hamiltonian reduction, which is noncommutative
analogue of Hamiltonian reduction in symplectic geometry. These algebras, constructed by the quantum Hamiltonian reduction, include finite W-algebras andrational Cherednik algebras, which play important and fundamental roles in the
representation theory of semi-simple and/or affine Lie algebras and Ariki-Koike
algebras.
When an associative algebra A is a quantization of a certain symplectic manifold
X, a natural question is if we can localize A as a sheaf of noncommutative algebras
on X: Namely, does there exist a sheaf of noncommutative algebras on X whose
algebra of global sections is isomorphic to A? And if one exists, is there an equivalence of categories between their module categories like the Beilinson-Bernsteincorrespondence? If the algebra A is constructed by the quantum Hamiltonian reduction, such problems were well studied and now we know when such a methodof localization works well.
1

National Research University Higher School of Economics, Department of Mathematics andInternational Laboratory of Representation Theory and Mathematical Physics, 20 Myasnitskaya
Ulitsa, Moscow 101000, RUSSIA. email: toshiro.kuwa@gmail.com
1

2

The present paper is a brief and introductory survey which is intended to show
an outline of the constructions of quantizations and their localization, and also
known results and applications. We also discuss quantizations of certain infinitedimensional manifolds, called arc spaces, and their localization. Such a quantization
is equipped with a certain infinite-dimensional algebraic structure, called a vertex
algebra. On the other hand, while we will discuss the construction by quantum
Hamiltonian reduction in a general setting, the detail of constructions of certain
concrete algebras, e.g. the rational Cherednik algebras, will not be included in thepresent survey. In [K3], we have a catalog of several algebras obtained by quantum
Hamiltonian reduction such as the rational Cherednik algebras and quantized toric
algebras. Also, for the finite W-algebras, we have a nice survey [Lo1]. Refer themfor these examples.
1. Quantization of Poisson structure
Consider a complex symplectic manifold X and let OX be its structure sheaf.
Its symplectic form makes OX a sheaf of Poisson algebras on X. Namely, OX
is equipped with a C-bilinear form { , }, called a Poisson bracket, such that{ , } is a Lie bracket on OX and {f, } is a derivation on OX for any f ∈ OX .
In particular, the coordinate ring C[X] = Γ(X, OX ) is a Poisson algebra. In thissurvey, we consider noncommutative algebras connected with such a Poisson algebra
structure.
Let A be a noncommutative associative algebra over C, and we assume that
there exists a filtration {Fi A}i≥0 of algebras, satisfying the condition:(1)[a, b] = ab − ba ∈ Fi+j−1 A,
def

for any a ∈ Fi A, b ∈ Fj A.

In this situation, the associated graded algebra grF A turns out to be a Poisson
algebra whose the Poisson bracket is given by{ā, b̄} = [a, b]
def

mod Fi+j−2 A,

for a ∈ Fi A, b ∈ Fj A

where ā (resp. b̄) is a class in grF A in which a ∈ A (resp. b) belongs. Then,
we say that the noncommutative algebra A is a quantization, i.e. noncommutative
analogue, of the Poisson algebra C[X] if there exists an isomorphism grF A andC[X] as Poisson algebras with the above Poisson bracket.
1.1. Examples. Since Lie algebras are an algebraic structure which appears as analgebra consisting of vector fields on a certain manifold, such a situation appears
frequently in Lie algebra theory. For example, here we see two classical and typicalexamples of quantization:
1. The algebra of differential operators with polynomial coefficient (the Weyl
algebra) D(Cd ) on Cd . The algebra D(Cd ) is equipped with a filtration given byorder of differential operators. Namely, the filtration given by deg xi = 0 anddeg ∂/∂xi = 1 where x1 , . . . , xd is the standard coordinates of Cd . The associatedgraded algebra is naturally isomorphic to the coordinate ring C[T ∗ Cd ] of the cotangent bundle T ∗ Cd , and moreover the isomorphism is an isomorphism of Poisson
algebras. Thus we regard the noncommutative algebra D(Cd ) as a quantization of
the Poisson algebra C[T ∗ Cd ]. Note that we can choose a different filtration given
by deg xi = 1/2 and deg ∂/∂xi = 1/2, and this filtration also induces the same
associated graded Poisson algebra C[T ∗ Cd ].
2. The universal enveloping algebra U (g) of a finite-dimensional simple algebra g. Let X = G/B be the flag manifold associated with a Lie group G
with the Lie algebra g (where B is its Borel subgroup). We have an action of
G on X = G/B by left-multiplication. This action induces a homomorphismof Lie algebras µD : g −→ Vect(X) where Vect(X) is a Lie algebra consisting3of vector fields on X. This homomorphism induces an isomorphism of algebras∼→ D(X), where Z(g) is the center of U (g). That is, the quotientµD : U (g)/Z(g) −−
algebra U (g)/Z(g) is a quantization of the coordinate ring C[T ∗ X] ' C[N ] whereN ⊂ g is the nilpotent cone of the Lie algebra g. Note that there exists a resolutionof singularities T ∗ X −→ N called the Springer resolution, and this resolution induces the isomorphism between the coordinate rings C[T ∗ X] and C[N ] as Poisson
algebras. Namely, U (g)/Z(g) is a quantization of C[T ∗ X].
2. Localization of quantized algebras
Let X be a symplectic manifold with structure sheaf OX . Assume that we havea noncommutative algebra A which gives a quantization of the coordinate ring
C[X] = Γ(X, OX ). Theory of algebraic geometry allows us to consider localizationof the commutative algebra C[X], that is the structure sheaf OX . Namely, localization of C[X] is a sheaf of commutative algebras whose algebra of global sections
turns out to be C[X]. Since our quantization A is a natural noncommutative analogue of the coordinate ring C[X], it is a natural question if there exists a sheaf
of noncommutative algebras on X whose algebra of global sections turns out tobe our algebra A. We call it localization of the noncommutative algebra A. We
will require that localization A of A is a sheaf of noncommutative algebra on X,
equipped with a filtration satisfying the condition (1), and the associated gradedalgebra is isomorphic to the structure sheaf OX as a sheaf of Poisson algebras. We
also require that the following diagram commutes:
AO
Γ(X,

gr

)

A

/ C[X]
O
Γ(X,

gr

)

/ OX

In such a situation, we also call A a quantization of the structure sheaf OX .
To be precise, we need to consider a sheaf of C[[~]]-algebras on X equipped witha certain C∗ -action, and consider the C∗ -finite part of the algebra of its global
sections to extract a C-algebra from it. In the following example, we see why weneed to consider a sheaf of C[[~]]-algebras, not a sheaf of C-algebras.
2.1. Microlocalization of sheaves of differential operators. Consider the second example of the quantizations. If we consider the flag manifold X = G/B associated with the simple Lie algebra g = Lie(G), the algebra of differential operators
D(X) on X is isomorphic to the quotient algebra of the universal enveloping algebra
U (g)/Z(g). Thus, if we consider the sheaf of differential operators DX on X, wehave a natural isomorphism Γ(X, DX ) ' U (g)/Z(g). Nevertheless, the sheaf DX is
not localization of U (g)/Z(g) in the above sense, because the sheaf DX is a sheaf
on X, not on its cotangent bundle X = T ∗ X. Thus, to construct localization ofdef

U (g)/Z(g) in the above sense, we need to construct a sheaf on X, i.e. we need to
localize DX in the cotangent direction. Now we consider how such a localization Aof D(X) on the cotangent bundle X = T ∗ X looks like.
Assume that we have local coordinates (x1 , . . . , xd ) of X. Let ∂1 , . . . , ∂d ∈ DX
be the partial differential operators ∂i = ∂/∂xi associated with this coordinates.
We denote the equivalent class of ∂i in gr DX ' OT ∗ X by yi and the equivalent classof xi by the same symbol xi . Then (x1 , . . . , xd , y1 , . . . , yd ) gives local symplectic
coordinates of X = T ∗ X. On the open subset {yi 6= 0} ⊂ X, the localization sheaf
A has a local section ∂i−1 , which is the inverse of ∂i . Since A is localization of
D(X), it is natural that the multiplication · in A satisfies the Leibniz rule. Then,4by the Leibniz rule, we have−1−2
−3
∂i−1 · x−1
= x−1
+ x−2
+ 2x−3+ ...,
i
i ∂i
i ∂i
i ∂i

and RHS turns an infinite sum. Here each term of RHS means symbols of differential
operators. Thus, to construct localization of D(X) on X = T ∗ X, we need to allow
such infinite sums. Algebraically, it can be obtained by introducing a Rees algebrawith respect to the filtration of DX and taking completion of it.
Now we introduce the precise definition of deformation-quantization of the structure sheaf OX . Let ~ be an indeterminate and let A be an associative C[[~]]-algebrasuch that the quotient algebra A/~A is a commutative C-algebra. Then, for any
a, b ∈ A, [a, b] = ab − ba lies in ~A, and hence {ā, b̄} = ~1 [a, b] is well-defined on
A/~A, where ā (resp. b̄) be the class to which a (resp. b) belongs. It is easy to seethat { , } is a Poisson bracket on the quotient algebra A/~A.
Definition 2.1. For a symplectic manifold X, a deformation-quantization of OX(or of X) is a sheaf A of associative C[[~]]-algebras, which is flat over C[[~]] andcomplete in ~-adic topology, such that
(A/~A, h−1 [ ,]) ' (OX , { ,})as a sheaf of Poisson algebras.
A basic example of deformation-quantization is the following ~-deformed Weyl
algebra AT ∗ Cd . As a vector space, we defineAT ∗ Cd = C[[~]][x1 , . . . , xd , y1 , . . . , yd ],and its defining relations are given by [yi , xj ] = yi xj − xj yi = ~δij , [xi , xj ] =[yi , yj ] = 0 where δij is Kronecker’s delta. With localizing it in a straight-forward
way, we have a sheaf AT ∗ Cd 'C OT ∗ Cd ⊗C C[[~]], and this sheaf gives a deformationquantization of the structure sheaf OT ∗ Cd . Note that in AT ∗ Cd , we have a local
section x−1
and yi−1 , and their product is given by the infinite sum
i
−1
−2 −2
2 −3 −3yi−1 · x−1= x−1
ii yi + ~xi yi + 2~ xi yi + . . . ,which is well-defined in AT ∗ Cd .
To construct the localization of D(X) for the manifold X on its cotangent bundle X = T ∗ X, we patch up AT ∗ Cd with local symplectic coordinates by gluing.
Let {Uα }α be an affine open covering of X. Then we have the open covering{T ∗ Uα }α of X = T ∗ X, and on each T ∗ Uα , we have local symplectic coordinates
α
α α
∗
(xα
1 , . . . , xd , y1 , . . . , yd ). Now we can introduce a sheaf AT ∗ Uα on T Uα , which is a
restriction of AT ∗ Cd with respect to the symplectic coordinates. These sheaves canbe glued together into a sheaf of C[[~]]-algebras AT ∗ X 'C OT ∗ X ⊗C C[[~]]. This
sheaf AT ∗ X gives a deformation-quantization of OT ∗ X .
One natural and fundamental problem of deformation-quantizations is if there
exists a deformation-quantization of OX and how many deformation-quantizations
exist for given complex symplectic manifold X. In this survey, we do not care thisproblem and consider deformation-quantizations which can be constructed explicitly. For the existence and classification problem of deformation-quantization, refer[BeKa], [Lo2] and references therein.
2.2. Quantum Hamiltonian reduction. As the above example, for a symplectic
manifold X which is the cotangent bundle X = T ∗ X of a complex manifold X, wehave a deformation-quantization of OX . But an important point of deformationquantization is that we can also construct a deformation-quantization on a symplectic manifold which is not a cotangent bundle of a certain manifold. Quantum
Hamiltonian reduction is a method to construct a noncommutative algebra which
is a quantization of a certain Poisson algebra.

5First we review the notion of Hamiltonian reduction. It is a method to construct
a symplectic variety. The quantum Hamiltonian reduction is noncommutative analogue of the Hamiltonian reduction.
Let X be a complex manifold. We assume that an algebraic/Lie group M acts on
X. Then this action naturally induces an action of M on T ∗ X and on its structure
sheaf OT ∗ X , and the action preserves the Poisson bracket on OT ∗ X . Moreover, bydifferentiating the induced action of M on the structure sheaf OX of X (not T ∗ X),
we have a homomorphism of Lie algebrasµD : m = LieM −→ Vect(X) ⊂ D(X),and a homomorphism of C-algebras,µD : U (m) −→ D(X).
The homomorphism µD induces a homomorphism of commutative algebras between
their associated graded algebrasµ∗ : S(m) = C[m∗ ] −→ C[T ∗ X]
where S(m) is the symmetric algebra over the vector space m, and m∗ is the dual
vector space of m. By considering associated morphism with µ∗ between algebraic
varieties, we have a morphism µ : T ∗ X −→ m∗ , called a moment map associatedwith the M -action on T ∗ X. By the construction, this morphism µ is compatible
with the M -action on T ∗ X. Namely, the subset µ−1 (χ) ⊂ T ∗ X is closed under theaction of M for χ ∈ m∗ . Then, we set
Xχ0 = µ−1 (χ)//M = Spec C[µ−1 (χ)]M .
def

The affine scheme Xχ0 may have singularities, but it is a Poisson variety, whose∗

Poisson bracket is naturally induced from the Poisson bracket on C[T X]. We may
also construct nonsingular symmetric manifold associated with the M -action byusing geometric invariant theory. Here we do not care details of geometric invariant
theory, and we assume that we can take a Zariski open subset of µ−1 (χ) denotedby µ−1 (χ)ss such that the group M acts free on µ−1 (χ)ss . The subset µ−1 (χ)ss
is called semistable locus with respect to the action of M . Since the M -action onµ−1 (χ)ss is free, the quotient with respect to the M -action
Xχ = µ−1 (χ)ss /Mturns out to be a nonsingular scheme. Moreover, the symplectic form on T ∗ X
induces a symplectic form on Xχ and thus Xχ is a symplectic manifold. The
symplectic manifold Xχ is called Hamiltonian reduction of T ∗ X with respect tothe M -action. Note that Xχ is a quotient of subset of the original symplectic
manifold T ∗ X with respect to the M -action.
Important known algebras are obtained by quantum Hamiltonian reduction forthe case where X is a linear representation of the group M (thus is a C-vector
space), and χ = 0. For known examples of such algebras, refer [K3, Section 5].
Below we consider such a case and denote X = X0 simply.
The structure sheaf of the new symplectic manifold X can be constructed explicitly from the structure sheaf of T ∗ X. While X is a quotient of subset of T ∗ X,
its structure sheaf OX can be written in the form of the M -invariant subalgebra of
the quotient of OT ∗ X as follows:M
OX ' p∗ O(T ∗ X)ss O(T ∗ X)ss µ∗ (m)where p : µ−1 (0)ss −→ X is the projection. Namely, the structure sheaf OX is thealgebra of M -invariants of M -coinvariants of the structure sheaf OT ∗ X .

6The quantum Hamiltonian reduction is quantization of the Hamiltonian reduction. Namely, we consider noncommutative analogue of the above construction byreplacing the structure sheaf OT ∗ X by its deformation-quantization AT ∗ X . Letc : m −→ C be a linear function on m which is invariant under the adjoint actionof m. This function c is a parameter of the quantization. Consider the following
sheaf of C[[~]]-algebras on the symplectic manifold X, which is a subquotient of thedeformation-quantization A(T ∗ X)ss :AX,c = p∗ A(T ∗ X)ssM
A(T ∗ X)ss (µD − ~c)(m).

The construction naturally implies that the sheaf of C[[~]]-algebras AX,c is adeformation-quantization of OX . Indeed, under the following geometric conditions,
we can show that AX,c is a deformation-quantization of OX :(1) µ−1 (0)ss 6= ∅,(2) the moment map µ is a flat morphism,(3) the action of M on µ−1 (0)ss is free,(4) the morphism X −→ X0 is birational and X0 has only normal singularities.
2.3. Algebra of global sections. Now we make precise the connection between
the quantization of the coordinate ring C[X] and the deformation-quantization of
the structure sheaf OX .
Let X be a symplectic manifold obtained by Hamiltonian reduction, and let AX,cbe a deformation-quantization of OX as in Section 2.2. Note that AX,c is a sheafof C[[~]]-algebras and thus its algebra of global sections Γ(X, AX,c ) is also a C[[~]]algebra. By construction, we have Γ(X, AX,c )/~Γ(X, AX,c ) ' C[X]. In Section 1,
we defined a quantization of the Poisson algebra C[X] as a filtered C-algebra. Thus
the algebra of global sections Γ(X, AX,c ) is not a quantization in that sense. But
we can obtain a quantization of C[X] by modifying the notion of “the algebra ofglobal sections” a little.
Recall that we assumed that X is a linear representation of M and thus X is a
C-vector space. Consider a C∗ -action on T ∗ X such that the induced equivariantC∗ -action on OT ∗ X makes the coordinate functions xi , yi of T ∗ X semi-invariant
elements of weight one. By letting ~ be also a semi-invariant element of weight two,
the C∗ -action lift to an equivariant action on the deformation-quantization AT ∗ X .
These C∗ -actions induce a C∗ -action on the symplectic manifold X, and also an
equivariant C∗ -action on the sheaf AX,c on X. Then the algebra of global sections
bc = Γ(X, AX,c ) is a C[[~]]-algebra with a C∗ -action.
A
def

bc is decomposed into a direct product of weight spaces with respect
The algebra A∗
bc = Q (Ab ) where (Abc )m is the weight space of weight m.
to the C -action: A
Lm bc mbc . We callConsider the direct sum m (Ac )m . Then, it is a C[~]-subalgebra of A∗bb
it the finite part of Ac with respect to the C -action, and denote (Ac )C∗ -fin . Now
consider the specialization ~ = 1, and we obtain a C-algebrabc )C∗ -fin /(~ − 1)(Abc )C∗ -fin .
Ac = ΓF (X, AX,c ) = (Adefbc is graded by the power of ~, and the grading induces a
The C[[~]]-algebra A
filtration of the C-algebra Ac .
Proposition 2.2. Under the conditions (1)–(4) in Section 2.2, the algebra Ac =
ΓF (X, AX,c ) is a quantization of the Poisson algebra C[X].

7

We also have the following commutative diagram:
AO c
ΓF (X,gr/ C[X]
O

)

AX,c

Γ(X,
~=0

)

/ OX

Similarly, we can construct “the algebra of global sections” also for the deformationquantization AT ∗ X of the cotangent bundle T ∗ X for a manifold X.
3. Deformation-quantization of arc spaces
3.1. Arc spaces. The notion of deformation-quantization and quantum Hamiltonian reduction works also for certain infinite-dimensional manifolds, not only
finite-dimensional ones. For a finite-dimensional manifold X, consider an infinitedimensional manifold J∞ X, called an arc space or a ∞-Jet scheme on X. We
regard X as a scheme of finite type over C. Then the arc space J∞ X is a scheme
whose R-valued points for a C-algebra R is given by
J∞ X(R) = Hom(Spec R, J∞ X) = Hom(Spec R[[t]], X) = X(R[[t]]).
def

Namely, the arc space J∞ X is a scheme, whose C-valued points are infinitesimalarcs on X. Then, for an R-valued point x(t) of J∞ X, we can regard x(t) as an
infinite-dimensional arc on X and its head x(0) is an R-valued point of X. Thus,
we have a canonical projection J∞ X −→ X, x(t) 7→ x(0).
We consider a finite-dimensional symplectic manifold X. Then its structure sheaf
OX is equipped with a Poisson bracket { , }. Along with it, the structure sheaf
OJ∞ X has an additional structure, called a vertex Poisson algebra. Vertex Poisson
algebras are vertex algebra analogue of Poisson algebras, which are naturally obtained as an associated graded algebra of a certain vertex algebra. First, we review
briefly the definition of vertex algebras.
A vertex algebra is a quadruple of data (V, 1, T, Y ( , z)), where• V — a vector space,• 1 ∈ V — a vector in V , called the vacuum vector• T : V −→ V — a linear operator, called the translation operator• Y−→ End V [[z, z −1 ]] — a linear operator, where Y (A, z) =P( , z) : V−n−1
is a formal series of linear operators A(n) on V for eachn∈Z A(n) zvector A ∈ V . The operator Y (A, z) is called the vertex operator associatedwith A.
These data are subject to the following axioms:• (vacuum axiom) The vacuum vector satisfies Y (1, z) = idV , and for any
A ∈ V , we have Y (A, z)1 ∈ V [[z]], so that Y (A, z)1 can be specialized at
z = 0. Then, we have Y (A, z)1|z=0 = A. In other words, we have A(n) 1 = 0
for any n ≥ 0 and A(−1) 1 = A in V .
• (translation axiom) For any A ∈ V , we have
[T, Y (A, z)] = ∂z Y (A, z),
and T 1 = 0.
• (locality axiom) For any A, B ∈ V , the vertex operators Y (A, z) andY (B, w) are mutually local: Namely, there exists N ∈ Z≥0(z − w)N [Y (A, z), Y (B, w)] = 0.

8

From the above definition of vertex algebras, we obtain the following identity between coefficients of vertex operators Y (A, z) and Y (B, z) for A, B ∈ V , so called
Borcherds’ identity:
X  m
(A(n+j) B)(m+l−j)
j
j≥0
 
X
n
=
(−1)j
{A(m+n−j) B(l+j) − (−1)n B(n+l−j) A(m+j) }jj≥0

for any l, m and n ∈ Z.
A vertex algebra V is called a commutative vertex algebra if A(n) = 0 for any
A ∈ V and n ∈ Z≥0 . In this situation, operators A(−m) and B(−n) commute for any
A, B ∈ V and m, n ∈ Z≥1 , and we can identify the commutative vertex algebra V
as a commutative C-algebra in infinitely many variables with the derivation T . For
any manifold X, the structure sheaf OJ∞ X of the arc space J∞ X has a structureof a sheaf of commutative vertex algebras with variables f(−n) = (1/n!)T n f for
f ∈ OX and n ∈ Z≥1 . The canonical projection J∞ X −→ X induces an embedding
OX ,→ OJ∞ X given by f 7→ f(−1) for f ∈ OX .
A vertex Poisson algebra is a tuple (V, 1, T, Y+ ( , z), Y− ( , z)), such that thequadruple (V, 1, P
T, Y+ ( , z)) is a commutative vertex algebra withP
the vertex operator Y+ (A, z) = n≤−1 A(n) z −n−1 , and coefficients of Y− (A, z) = n≥0 A(n) z −n−1satisfies a truncation of the Borcherds’ identity. It is vertex-algebraic analogue ofPoisson algebras in the following sense; if (V, 1, T, Y ( )) is a vertex algebra withfiltration and the associated graded vertex algebra gr V with respect to the filtration is commutative, we can define the structure of a vertex Poisson algebra on theassociated graded vertex algebra gr V .
Assume that X is a symplectic manifold. Consider the arc space J∞ X, andthen the structure sheaf OJ∞ X is a sheaf of commutative vertex algebras as above.
Moreover, the Poisson bracket { , } on OX induces a structure of a vertex Poisson
algebra on OJ∞ X which satisfies({f, g} if n = 0,f(n) g =0
if n ≥ 1,
for f , g ∈ OX and also satisfies the (truncated) Borcherds’ identity.
3.2. Deformation-quantization of J∞ X. As deformation-quantization of thevertex Poisson algebras, we obtain the notion of ~-adic vertex algebras, which are introduced by H. Li in [Li]. An ~-adic vertex algebra is a quadruple (V, 1, T, Y ( , z))where V is a flat C[[~]]-module which is complete in ~-adic topology, such that
(V /~N V, 1, T, Y ( , z)) is a vertex algebra over C for any N ≥ 0. Note that a~-adic vertex algebra need not a vertex algebra over C[[~]].
Assume that, for an ~-adic vertex algebra (V, 1, T, Y ( , z)), the quotient vertex
algebra (V /~V, 1, T, Y ( , z)) is a commutative vertex algebra. Then, V /~V has
naturally the structure of vertex Poisson algebra defined byā(n) b̄ = ~−1 a(n) b

mod ~for a, b ∈ V and n ∈ Z≥0 , and where ā = a mod ~ ∈ V /~V is the equivalent classof a.
Now we introduce the notion of deformation-quantization for vertex algebras.
Let X be a symplectic manifold, and consider the structure sheaf OJ∞ X of the arc
space J∞ X as a sheaf of vertex Poisson algebras. We say that the sheaf of ~-adicvertex algebras Ach
X on J∞ X is a deformation-quantization of OJ∞ X (or of J∞ X) if9ch
its quotient Ach
X /~AX is isomorphic to OJ∞ X as a sheaf of vertex Poisson algebras
on J∞ X.
As the usual deformation-quantization, we can construct a deformation-quantizationof OJ∞ X as a sheaf of ~-adic vertex algebras in the following two cases:(1) When X is the cotangent bundle of a certain manifold X, i.e. X = T ∗ X, we
may have a sheaf of vertex algebras called an algebra of chiral differential operatorsch, which was introduced in [BD] and [GMS] indepen(CDO) on X, denoted by DX
dently. Note that there exists an obstruction for existence of such a sheaf, but itchis known that if the second Chern class ch2 (TX ) vanishes, such a sheaf DX
exists.
See [GMS] for details.
By a similar method for localizing the sheaf of differential operators DX on T ∗ X
as a deformation-quantization of X = T ∗ X, we can construct localization of thechsheaf of vertex algebras DX
as a sheaf of ~-adic vertex algebras on X = T ∗ X, andmoreover, as a sheaf on its arc space J∞ X. This construction gives a deformationquantization of the structure sheaf OJ∞ X in the above sense. We denote it Ach
X,~ .
(2) When a symplectic manifold X is constructed by Hamiltonian reduction
X = µ−1 (χ)ss /M as above for an action of a certain unipotent Lie group M on achexists, we can construct a deformation-quantization
manifold X and the CDO DX
of OJ∞ X as quantum Hamiltonian reduction of Ach
X,~ . In [AKM], with using acertain cohomological and quantum Hamiltonian reduction, called a BRST cohomology, a deformation-quantization of OJ∞ X is constructed for a Slodowy variety
X, a symplectic manifold which is obtained by Hamiltonian reduction of a flag
variety.
As “a vertex algebra of global sections” with respect to a certain C∗ -action, we
obtain (1) a vertex algebra at critical level associated with the affine Lie algebra corresponding to the flag variety, or (2) an affine W-algebra at critical level associatedwith the Slodowy variety X, respectively. Namely, these deformation-quantizations
are regarded as localization of such vertex algebras.

4. Applications of localization to representation theory
When an associative algebra A is localized by a deformation-quantization of acertain symplectic manifold X, the representation theory of A may be connectedwith the geometrical structure of the underlying manifold X. And indeed many
applications of such a localization to the representation theory are known. In thissection, we summarize some of such results which are recently studied for algebras
obtained by quantum Hamiltonian reduction.
Throughout this section, we use the following notation. Let X be a symplectic
manifold, and let AX,c be a deformation-quantization of the structure sheaf OX
with a parameter of quantization c. Set Ac = Γ(X, AX,c )f in |~=1 be an associative
C-algebra which is obtained as “the algebra of global sections”. In this section, we
may assume that the symplectic manifold X is obtained by Hamiltonian reduction,and the Hamiltonian reduction satisfies the assumptions (1)–(4).
4.1. Beilinson-Bernstein type correspondence. The most fundamental results
are certain equivalences of categories between the category of Ac -modules and thecategory of AX,c -modules with an equivariant torus action, which the functor of taking “global sections” gives. In the classical case of the universal enveloping algebraof simple Lie algebra U (g), such an equivalence of categories essentially coincides
with an equivalence of abelian categories known as the Beilinson-Bernstein correspondence. In the case of quantum Hamiltonian reduction, we have (1) equivalenceof triangulated categories between derived categories of module categories and (2)10equivalence of abelian categories which is direct analogue of the Beilinson-Bernsteincorrespondence.
(1) Let Ac -mod be the abelian category of finitely-generated Ac -modules, andlet AX,c -modC∗ -equiv be the abelian category of coherent AX,c -modules with the C∗ equivariant action, which are torsion-free over C[[~]]. Then an object of AX,c -modC∗ -equiv
are a sheaf on X and thus we can consider the module of its global sections. For an
object M ∈ AX,c -modC∗ -equiv , the module of global sections Γ(X, M) is a C[[~]]module with C∗ -action, since M is equipped with the equivariant C∗ -action. Then,
taking finite part with respect to C∗ -action, denoted Γ(X, M)C∗ -fin , we have aC[~]-module. Finally, substituting ~ = 1, we obtain a C-vector spaceΓF (X, M) = Γ(X, M)C∗ -fin |~=1 = Γ(X, M)C∗ -fin ⊗C[~] C1 ,defwhere C1 is the C[~]-module by augmentation at ~ = 1. Since ΓF (X, AX,c ) ' Ac ,
ΓF (X, M) is an Ac -module. This construction gives a functor of abelian categoriesΓF (X,) : AX,c -modC∗ -equiv −→ Ac -mod.

Consider the derived functor, and we haveRΓF (X,) : Db (AX,c -modC∗ -equiv ) −→ Db (Ac -mod)

where Db ( ) is a bounded derived category.
The following result is due to I. Gordon and I. Losev [GL]. Independently,
K. McGerty and T. Nevins also studied essentially the same result independently
in a little different manner in [MN1].
Theorem 4.1. If the algebra Ac has finite global dimension, we have an equivalenceof triangulated categories
RΓF (X,∼→ Db (A -mod)) : Db (AX,c -modC∗ -equiv ) −−cwith the quasi-inverse functor AX,c ⊗L
Ac ( ).
(2) Under a certain condition, the above functor ΓF (X, ) also gives an equivalence of abelian categories, not only the derived equivalence. Note that the Hamiltonian reduction X is defined as a projective variety over X 0 :M0−1
X = Proj
C[µ−1 (0)]M(0)]M
θ m −→ X = Spec C[µm≥0where θ is a certain character of the groupa lineLwe have
L M . From the definition,
−1
M
C[µ
(0)]
.
bundle O(1) which is associated with the m C[µ−1 (0)]M-moduleθmm
θ (m+1)θMoreover, we can construct a sheaf AX,c which gives a quantization of this line bundle O(1) in the sense that AθX,c ⊗C[[~]] C0 ' O(1) where C0 is a C[[~]]-module on C
by augmentation at ~ = 0. This sheaf is an (AX,c+dθ , AX,c )-bimodule where dθ isa character of the Lie algebra m obtained by differentiating θ. By considering themtensor product with AθX,c over AX,c , we have a functormAθX,c ⊗AX,c ( ) : AX,c -modC∗ -equiv −→ AX,c+mdθ -modC∗ -equivmfor each m ∈ Z. We have AθX,c ⊗C[[~]] C0 ' O(m) ' O(1)⊗m , and the functor is−man equivalence of categories whose quasi-inverse functor is given by AθX,c+mdθ . By
applying “the functor of taking global sections” ΓF (X, ), we have functorsmΓF (X, AθX,c ) ⊗Ac ( ) : Ac -mod −→ Ac+mdθ -modfor each m ∈ Z.

11m

Theorem 4.2. Assume that the functor ΓF (X, AθX,c ) ⊗Ac ( ) is an equivalence of−mabelian categories (with the quasi-inverse given by ΓF (X, AθX,c+mdθ ) ⊗Ac+mdθ ( ))for all m ∈ Z≥0 . Then,ΓF (X,∼→ A -mod) : AX,c -modC∗ -equiv −−cis an equivalence of abelian categories with the quasi-inverse AX,c ⊗Ac ( ).
The theorem is analogue of well-known theorem in the representation theory ofsimple Lie algebras, so called the Beilinson-Bernstein correspondence, studied in[BB1, BB2], and also in [BrKa]. For algebras obtained by quantum Hamiltonian
reduction, it was first established for the rational Cherednik algebra of the symmetric group Sn by M. Kashiwara and R. Rouquier in [KR], and later for someother cases in [BeKu], [DK] separately. Recently, K. McGerty and T. Nevins in[MN2] gave a general criterion when such an abelian equivalence holds by using
Kashiwara’s equivalence and Kirwan-Ness stratification.
In known cases of quantum Hamiltonian reduction, the sheaf of C[[~]]-algebras
AX,c is locally isomorphic to the ~-deformed Weyl algebra
DCd ,~ = C[[~]][x1 , . . . , xd , y1 , . . . , yd ]with defining relation given by [yi , xj ] = δij ~, [xi , xj ] = [yi , yj ] = 0. Thus theabove equivalences connect the representation theory of Ac with microlocal analysis
on the symplectic manifold X through the sheaf AX,c , and hence we have many
applications in the representation theory of Ac .
First, for an Ac -module M , the support of the corresponding sheaf of modules
AX,c ⊗Ac M in the symplectic manifold X is an invariant of modules. It is analogueof characteristic varieties in D-module theory. We also have analogue of characteristic cycles, cycles on X (with multiplicities) associated with the module. For
certain quantum Hamiltonian reduction Ac , characteristic cycles of some importantAc -modules were studied in [GS] (for the rational Cherednik algebra for Sn ) andin [K1] (for the rational Cherednik algebra for Z/lZ).
Moreover, for the rational Cherednik algebra for Z/lZ, we can construct explicitly
sheaves of modules corresponding to irreducible modules and standard modules
in the category O of Ac , a highest weight category analogous to the BernsteinGelfand-Gelfand category for a simple Lie algebra. ([K2]) As a consequence, itfollows that sheaves of modules corresponding to modules in the category O are
regular holonomic in the sense of microlocal analysis. Conjecturally the same factholds for the category O of other type of rational Cherednik algebra, but it is still
an open problem.
4.2. BRST cohomologies. The construction of the algebra Ac as quantum Hamiltonian reduction associated with the M -action on AT ∗ X induces a certain cohomology, called a BRST cohomology. The BRST cohomology (or so called BRST
reduction) is first introduced by theoretical physicist in the area of quantum field
theory. Mathematically, it is known that the BRST cohomology gives a cohomological interpretation of the (quantum) Hamiltonian reduction (cf. [KS], [F]). In[K3], the explicit description of the BRST cohomology associated with the quantum
Hamiltonian reduction is given in the case where X is a linear representation of thegroup M . We denote the BRST cohomology associated with the action of M on•
the ring of differential operators D(X) by HBRST,c(m, D(X)), where m is the Liealgebra of M and c is the parameter of the quantum Hamiltonian reduction Ac .
We can also define the sheaf version of the BRST cohomology, and it is denoted•
HBRST,c(m, A(T ∗ X)ss ). Then, we have the following two isomorphisms of graded12

algebras (with respect to the degree of cohomologies) under certain conditions:(2)••
HBRST,c(m, A(T ∗ X)ss ) ' AX,c ⊗C HDR
(M ),

(3)

•
•
(M ),
HBRST,c(m, D(X)) ' Ac ⊗C HDR•
where HDR
(M ) is the de Rham cohomology of M . The former isomorphism follows from the geometrical conditions in Section 2.2, and indeed it is essentially ageometrical fact. On the other hand, to prove the latter isomorphism we need tomake advantage of the representation theory of Ac . Indeed, by using the abelian
and derived equivalences of categories in Section 4.1, we obtain (3) from (2).

4.3. Isomorphism of quantizations. Some noncommutative algebras give quantizations of the same Poisson algebra. In such a case, it is a natural problem if thesequantizations are isomorphic with each other. But usually it is not easy to compare
their different constructions directly. On the other hand, the deformation theory for
deformation-quantization of a symplectic manifold is studied by R. Bezrukavnikov
and D. Kaledin in [BeKa]. As an application of their result, I. Losev proved isomorphisms between certain noncommutative algebras, which are constructed byquantum Hamiltonian reduction in [Lo2]. For example, deformed preprojective
algebras introduced in [CBH] and finite W-algebras associated with a subregular
nilpotent orbit in simple Lie algebra of type ADE are isomorphic with each other.
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Phys., available online, (DOI) 10.1007/s00220-014-2183-x.
[BB1] A. Beilinson, J. Bernstein, Localisation de g-modules (French), C. R. Acad. Sci. Paris Se’r.
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[BB2] A. Beilinson, J. Bernstein, A proof of Jantzen conjectures, I. M. Gelfand Seminar, Adv.
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[BD]
A. Beilinson, V. Drinfeld, Chiral algebras, American Mathematical Society Colloquium
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Mosc. Math. J. 4 (2004), 559-592.
[BeKu] G. Bellamy and T. Kuwabara, On deformation quantizations of hypertoric varieties, Pacific J. Math. 260 (2012), no. 1, 89-127.
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[CBH] W. Crawley-Boevey and M. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605-635.
[DK] C. Dodd and K. Kremnizer, A Localization Theorem for Finite W-algebras, preprint,
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[F]
B. Feigin, Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras (Russian),
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[GL]
I. Gordon and I. Losev, On category O for cyclotomic rational Cherednik algebras, J. Eur.
Math. Soc 16 (2014), no. 5, 1017-1079.
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Vertex algebroids, Invent. Math. 155 (2004), no. 3, 605-680.
[GS]
I.G. Gordon and J.T. Stafford, Rational Cherednik algebras and Hilbert schemes II: representations and sheaves, Duke Math. J 132 (2006), 73-135.
[K1]
T. Kuwabara, Characteristic cycles of standard modules for the rational Cherednik algebraof type Z/lZ, J. Math. Kyoto Univ. 48 (2008), vol. 1, Kyoto University, 167-217.
[K2]
T. Kuwabara, Representation theory of the rational Cherednik algebras of type Z/lZ via
microlocal analysis, Publ. Res. Inst. Math. Sci. 49 (2013), no. 1, 87-110.
[K3]
T. Kuwabara, BRST cohomologies for symplectic reflection algebras and quantizations ofhypertoric varieties, preprint, arXiv:1311.1787v1.
[KR]
M. Kashiwara and R. Rouquier, Microlocalization of the rational Cherednik algebras, Duke
Math. J. 144 (2008), no. 3, 525-573.
[KS]
B. Kostant, S. Sternberg, Symplectic reduction, BRS cohomology, and infinitedimensional Clifford algebras, Ann. Physics 176, no.1, 49-113 (1987)

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[Li]

H. Li, Vertex algebras and vertex Poisson algebras, Contemp. Math., 6 (2004), no. 1,
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[Lo1] I. Losev, Finite W-algebras, Proceedings of the International Congress of Mathematicians,
Volume III (2010), Hindustan Book Agency, New Delhi, 1281-1307.
[Lo2] I. Losev, Isomorphisms of quantizations via quantization of resolutions, Adv. Math. 231(2012), vol. 3-4, 1216-1270.
[MN1] K. McGerty and T. Nevins, Derived equivalence for quantum symplectic resolutions, Selecta Math. (N.S.) 20 (2014), no. 2, 675-717.
[MN2] K. McGerty and T. Nevins, Compatibility of t-structures for quantum symplectic resolutions, preprint, arXiv:1312.7180v1.