Deformation quantization and localization of noncommutative algebras and vertex algebras

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1 Deformaton quantzaton and localzaton of noncommutatve algebras and vertex algebras TOSHIRO KUWABARA 1 The present paper s a report for the proceengs of 59th Algebra Symposum held at the Unversty of Tokyo on 8 11th September Introducton We dscuss certan nfnte-dmensonal noncommutatve assocatve algebras. One well-known class of nfnte-dmensonal assocatve algebras s unversal envelopng algebras of Le algebras. For a sem-smple Le algebra g, the central quotent of ts unversal envelopng algebra U(g)/Z(g) s naturally somorphc to the algebra of dfferental operators D(G/B) on the correspondng flag manfold G/B. Such an somorphsm connects the representaton theory of the Le algebra g and geometrcal propertes of the flag manfold G/B and/or ts cotangent bundle T (G/B). Namely, the somorphsm leads an equvalence of categores between the category of (U(g)/Z(g))-modules and the category of D G/B -modules where D G/B s the sheaf of dfferental operators. The equvalence s known as the Belnson-Bernsten correspondence and t s one of the most fundamental facts of the representaton theory of sem-smple Le algebras. The above stuaton can be generalzed, and t acheve the noton of quantzatons. Note that, n the above, the unversal envelopng algebra U(g) s equpped wth a natural fltraton, and the assocated graded algebra gr U(g)/Z(g) wth respect to the fltraton s naturally somorphc to the coordnate rng C[T (G/B)]. The commutator [a, b] = ab ba on U(g) nduces a Posson bracket on gr U(g)/Z(g), whle C[T (G/B)] s also equpped wth a Posson bracket nduced from the symplectc structure of the cotangent bundle T (G/B). The somorphsm between gr U(g)/Z(g) and C[T (G/B)] s not just an somorphsm of commutatve algebras, but t s an somorphsm of Posson algebras. We say that U(g)/Z(g) s a quantzaton of C[T (G/B)] (or T (G/B)). Namely, n general cases, for a symplectc manfold X, a fltered assocatve algebra s called a quantzaton of C[X] (or of X), f the assocated graded algebra s somorphc to C[X] as a Posson algebra. In the past decades, several noncommutatve algebras whch gve quantzatons of certan symplectc manfolds were ntroduced. These algebras are constructed by usng so called the quantum Hamltonan reducton, whch s noncommutatve analogue of Hamltonan reducton n symplectc geometry. These algebras, constructed by the quantum Hamltonan reducton, nclude fnte W-algebras and ratonal Cherednk algebras, whch play mportant and fundamental roles n the representaton theory of sem-smple and/or affne Le algebras and Ark-Koke algebras. When an assocatve algebra A s a quantzaton of a certan symplectc manfold X, a natural queston s f we can localze A as a sheaf of noncommutatve algebras on X: Namely, does there exst a sheaf of noncommutatve algebras on X whose algebra of global sectons s somorphc to A? And f one exsts, s there an equvalence of categores between ther module categores lke the Belnson-Bernsten correspondence? If the algebra A s constructed by the quantum Hamltonan reducton, such problems were well studed and now we know when such a method of localzaton works well. 1 Natonal Research Unversty Hgher School of Economcs, Department of Mathematcs and Internatonal Laboratory of Representaton Theory and Mathematcal Physcs, 20 Myasntskaya Ultsa, Moscow , RUSSIA. emal: toshro.kuwa@gmal.com 1

2 2 The present paper s a bref and ntroductory survey whch s ntended to show an outlne of the constructons of quantzatons and ther localzaton, and also known results and applcatons. We also dscuss quantzatons of certan nfntedmensonal manfolds, called arc spaces, and ther localzaton. Such a quantzaton s equpped wth a certan nfnte-dmensonal algebrac structure, called a vertex algebra. On the other hand, whle we wll dscuss the constructon by quantum Hamltonan reducton n a general settng, the detal of constructons of certan concrete algebras, e.g. the ratonal Cherednk algebras, wll not be ncluded n the present survey. In [K3], we have a catalog of several algebras obtaned by quantum Hamltonan reducton such as the ratonal Cherednk algebras and quantzed torc algebras. Also, for the fnte W-algebras, we have a nce survey [Lo1]. Refer them for these examples. 1. Quantzaton of Posson structure Consder a complex symplectc manfold X and let O X be ts structure sheaf. Its symplectc form makes O X a sheaf of Posson algebras on X. Namely, O X s equpped wth a C-blnear form {, }, called a Posson bracket, such that {, } s a Le bracket on O X and {f, } s a dervaton on O X for any f O X. In partcular, the coordnate rng C[X] = Γ(X, O X ) s a Posson algebra. In ths survey, we consder noncommutatve algebras connected wth such a Posson algebra structure. Let A be a noncommutatve assocatve algebra over C, and we assume that there exsts a fltraton {F A} 0 of algebras, satsfyng the condton: (1) [a, b] = ab ba F +j 1 A, for any a F A, b F j A. In ths stuaton, the assocated graded algebra gr F A turns out to be a Posson algebra whose the Posson bracket s gven by {ā, b} = [a, b] mod F +j 2 A, for a F A, b F j A where ā (resp. b) s a class n grf A n whch a A (resp. b) belongs. Then, we say that the noncommutatve algebra A s a quantzaton,.e. noncommutatve analogue, of the Posson algebra C[X] f there exsts an somorphsm gr F A and C[X] as Posson algebras wth the above Posson bracket Examples. Snce Le algebras are an algebrac structure whch appears as an algebra consstng of vector felds on a certan manfold, such a stuaton appears frequently n Le algebra theory. For example, here we see two classcal and typcal examples of quantzaton: 1. The algebra of dfferental operators wth polynomal coeffcent (the Weyl algebra) D(C d ) on C d. The algebra D(C d ) s equpped wth a fltraton gven by order of dfferental operators. Namely, the fltraton gven by deg x = 0 and deg / x = 1 where x 1,..., x d s the standard coordnates of C d. The assocated graded algebra s naturally somorphc to the coordnate rng C[T C d ] of the cotangent bundle T C d, and moreover the somorphsm s an somorphsm of Posson algebras. Thus we regard the noncommutatve algebra D(C d ) as a quantzaton of the Posson algebra C[T C d ]. Note that we can choose a dfferent fltraton gven by deg x = 1/2 and deg / x = 1/2, and ths fltraton also nduces the same assocated graded Posson algebra C[T C d ]. 2. The unversal envelopng algebra U(g) of a fnte-dmensonal smple algebra g. Let X = G/B be the flag manfold assocated wth a Le group G wth the Le algebra g (where B s ts Borel subgroup). We have an acton of G on X = G/B by left-multplcaton. Ths acton nduces a homomorphsm of Le algebras µ D : g Vect(X) where Vect(X) s a Le algebra consstng

3 of vector felds on X. Ths homomorphsm nduces an somorphsm of algebras µ D : U(g)/Z(g) D(X), where Z(g) s the center of U(g). That s, the quotent algebra U(g)/Z(g) s a quantzaton of the coordnate rng C[T X] C[N ] where N g s the nlpotent cone of the Le algebra g. Note that there exsts a resoluton of sngulartes T X N called the Sprnger resoluton, and ths resoluton nduces the somorphsm between the coordnate rngs C[T X] and C[N ] as Posson algebras. Namely, U(g)/Z(g) s a quantzaton of C[T X]. 2. Localzaton of quantzed algebras Let X be a symplectc manfold wth structure sheaf O X. Assume that we have a noncommutatve algebra A whch gves a quantzaton of the coordnate rng C[X] = Γ(X, O X ). Theory of algebrac geometry allows us to consder localzaton of the commutatve algebra C[X], that s the structure sheaf O X. Namely, localzaton of C[X] s a sheaf of commutatve algebras whose algebra of global sectons turns out to be C[X]. Snce our quantzaton A s a natural noncommutatve analogue of the coordnate rng C[X], t s a natural queston f there exsts a sheaf of noncommutatve algebras on X whose algebra of global sectons turns out to be our algebra A. We call t localzaton of the noncommutatve algebra A. We wll requre that localzaton A of A s a sheaf of noncommutatve algebra on X, equpped wth a fltraton satsfyng the condton (1), and the assocated graded algebra s somorphc to the structure sheaf O X as a sheaf of Posson algebras. We also requre that the followng dagram commutes: Γ(X, ) A A gr gr C[X] O X Γ(X, ) In such a stuaton, we also call A a quantzaton of the structure sheaf O X. To be precse, we need to consder a sheaf of C[[ ]]-algebras on X equpped wth a certan C -acton, and consder the C -fnte part of the algebra of ts global sectons to extract a C-algebra from t. In the followng example, we see why we need to consder a sheaf of C[[ ]]-algebras, not a sheaf of C-algebras Mcrolocalzaton of sheaves of dfferental operators. Consder the second example of the quantzatons. If we consder the flag manfold X = G/B assocated wth the smple Le algebra g = Le(G), the algebra of dfferental operators D(X) on X s somorphc to the quotent algebra of the unversal envelopng algebra U(g)/Z(g). Thus, f we consder the sheaf of dfferental operators D X on X, we have a natural somorphsm Γ(X, D X ) U(g)/Z(g). Nevertheless, the sheaf D X s not localzaton of U(g)/Z(g) n the above sense, because the sheaf D X s a sheaf on X, not on ts cotangent bundle X = T X. Thus, to construct localzaton of U(g)/Z(g) n the above sense, we need to construct a sheaf on X,.e. we need to localze D X n the cotangent drecton. Now we consder how such a localzaton A of D(X) on the cotangent bundle X = T X looks lke. Assume that we have local coordnates (x 1,..., x d ) of X. Let 1,..., d D X be the partal dfferental operators = / x assocated wth ths coordnates. We denote the equvalent class of n gr D X O T X by y and the equvalent class of x by the same symbol x. Then (x 1,..., x d, y 1,..., y d ) gves local symplectc coordnates of X = T X. On the open subset {y 0} X, the localzaton sheaf A has a local secton 1, whch s the nverse of. Snce A s localzaton of D(X), t s natural that the multplcaton n A satsfes the Lebnz rule. Then, 3

4 4 by the Lebnz rule, we have 1 x 1 = x x x , and RHS turns an nfnte sum. Here each term of RHS means symbols of dfferental operators. Thus, to construct localzaton of D(X) on X = T X, we need to allow such nfnte sums. Algebracally, t can be obtaned by ntroducng a Rees algebra wth respect to the fltraton of D X and takng completon of t. Now we ntroduce the precse nton of ormaton-quantzaton of the structure sheaf O X. Let be an ndetermnate and let A be an assocatve C[[ ]]-algebra such that the quotent algebra A/ A s a commutatve C-algebra. Then, for any a, b A, [a, b] = ab ba les n A, and hence {ā, b} = 1 [a, b] s well-ned on A/ A, where ā (resp. b) be the class to whch a (resp. b) belongs. It s easy to see that {, } s a Posson bracket on the quotent algebra A/ A. Defnton 2.1. For a symplectc manfold X, a ormaton-quantzaton of O X (or of X) s a sheaf A of assocatve C[[ ]]-algebras, whch s flat over C[[ ]] and complete n -adc topology, such that as a sheaf of Posson algebras. (A/ A, h 1 [, ]) (O X, {, }) A basc example of ormaton-quantzaton s the followng -ormed Weyl algebra A T Cd. As a vector space, we ne A T C d = C[[ ]][x 1,..., x d, y 1,..., y d ], and ts nng relatons are gven by [y, x j ] = y x j x j y = δ j, [x, x j ] = [y, y j ] = 0 where δ j s Kronecker s delta. Wth localzng t n a straght-forward way, we have a sheaf A T C d C O T C d C C[[ ]], and ths sheaf gves a ormatonquantzaton of the structure sheaf O T C d. Note that n A T Cd, we have a local secton x 1 and y 1, and ther product s gven by the nfnte sum y 1 x 1 = x 1 y 1 + x 2 y x 3 y , whch s well-ned n A T C d. To construct the localzaton of D(X) for the manfold X on ts cotangent bundle X = T X, we patch up A T C wth local symplectc coordnates by glung. d Let {U α } α be an affne open coverng of X. Then we have the open coverng {T U α } α of X = T X, and on each T U α, we have local symplectc coordnates (x α 1,..., x α d, yα 1,..., yd α). Now we can ntroduce a sheaf A T U α on T U α, whch s a restrcton of A T C wth respect to the symplectc coordnates. These sheaves can d be glued together nto a sheaf of C[[ ]]-algebras A T X C O T X C C[[ ]]. Ths sheaf A T X gves a ormaton-quantzaton of O T X. One natural and fundamental problem of ormaton-quantzatons s f there exsts a ormaton-quantzaton of O X and how many ormaton-quantzatons exst for gven complex symplectc manfold X. In ths survey, we do not care ths problem and consder ormaton-quantzatons whch can be constructed explctly. For the exstence and classfcaton problem of ormaton-quantzaton, refer [BeKa], [Lo2] and references theren Quantum Hamltonan reducton. As the above example, for a symplectc manfold X whch s the cotangent bundle X = T X of a complex manfold X, we have a ormaton-quantzaton of O X. But an mportant pont of ormatonquantzaton s that we can also construct a ormaton-quantzaton on a symplectc manfold whch s not a cotangent bundle of a certan manfold. Quantum Hamltonan reducton s a method to construct a noncommutatve algebra whch s a quantzaton of a certan Posson algebra.

5 5 Frst we revew the noton of Hamltonan reducton. It s a method to construct a symplectc varety. The quantum Hamltonan reducton s noncommutatve analogue of the Hamltonan reducton. Let X be a complex manfold. We assume that an algebrac/le group M acts on X. Then ths acton naturally nduces an acton of M on T X and on ts structure sheaf O T X, and the acton preserves the Posson bracket on O T X. Moreover, by dfferentatng the nduced acton of M on the structure sheaf O X of X (not T X), we have a homomorphsm of Le algebras and a homomorphsm of C-algebras, µ D : m = LeM Vect(X) D(X), µ D : U(m) D(X). The homomorphsm µ D nduces a homomorphsm of commutatve algebras between ther assocated graded algebras µ : S(m) = C[m ] C[T X] where S(m) s the symmetrc algebra over the vector space m, and m s the dual vector space of m. By consderng assocated morphsm wth µ between algebrac varetes, we have a morphsm µ : T X m, called a moment map assocated wth the M-acton on T X. By the constructon, ths morphsm µ s compatble wth the M-acton on T X. Namely, the subset µ 1 (χ) T X s closed under the acton of M for χ m. Then, we set X 0 χ = µ 1 (χ)//m = Spec C[µ 1 (χ)] M. The affne scheme X 0 χ may have sngulartes, but t s a Posson varety, whose Posson bracket s naturally nduced from the Posson bracket on C[T X]. We may also construct nonsngular symmetrc manfold assocated wth the M-acton by usng geometrc nvarant theory. Here we do not care detals of geometrc nvarant theory, and we assume that we can take a Zarsk open subset of µ 1 (χ) denoted by µ 1 (χ) ss such that the group M acts free on µ 1 (χ) ss. The subset µ 1 (χ) ss s called semstable locus wth respect to the acton of M. Snce the M-acton on µ 1 (χ) ss s free, the quotent wth respect to the M-acton X χ = µ 1 (χ) ss /M turns out to be a nonsngular scheme. Moreover, the symplectc form on T X nduces a symplectc form on X χ and thus X χ s a symplectc manfold. The symplectc manfold X χ s called Hamltonan reducton of T X wth respect to the M-acton. Note that X χ s a quotent of subset of the orgnal symplectc manfold T X wth respect to the M-acton. Important known algebras are obtaned by quantum Hamltonan reducton for the case where X s a lnear representaton of the group M (thus s a C-vector space), and χ = 0. For known examples of such algebras, refer [K3, Secton 5]. Below we consder such a case and denote X = X 0 smply. The structure sheaf of the new symplectc manfold X can be constructed explctly from the structure sheaf of T X. Whle X s a quotent of subset of T X, ts structure sheaf O X can be wrtten n the form of the M-nvarant subalgebra of the quotent of O T X as follows: O X ( ( / p O(T X) ss O(T X) ssµ (m) )) M where p : µ 1 (0) ss X s the projecton. Namely, the structure sheaf O X s the algebra of M-nvarants of M-convarants of the structure sheaf O T X.

6 6 The quantum Hamltonan reducton s quantzaton of the Hamltonan reducton. Namely, we consder noncommutatve analogue of the above constructon by replacng the structure sheaf O T X by ts ormaton-quantzaton A T X. Let c : m C be a lnear functon on m whch s nvarant under the adjont acton of m. Ths functon c s a parameter of the quantzaton. Consder the followng sheaf of C[[ ]]-algebras on the symplectc manfold X, whch s a subquotent of the ormaton-quantzaton A (T X) ss: A X,c = ( p ( A(T X) ss / A(T X) ss(µ D c)(m) )) M. The constructon naturally mples that the sheaf of C[[ ]]-algebras A X,c s a ormaton-quantzaton of O X. Indeed, under the followng geometrc condtons, we can show that A X,c s a ormaton-quantzaton of O X : (1) µ 1 (0) ss, (2) the moment map µ s a flat morphsm, (3) the acton of M on µ 1 (0) ss s free, (4) the morphsm X X 0 s bratonal and X 0 has only normal sngulartes Algebra of global sectons. Now we make precse the connecton between the quantzaton of the coordnate rng C[X] and the ormaton-quantzaton of the structure sheaf O X. Let X be a symplectc manfold obtaned by Hamltonan reducton, and let A X,c be a ormaton-quantzaton of O X as n Secton 2.2. Note that A X,c s a sheaf of C[[ ]]-algebras and thus ts algebra of global sectons Γ(X, A X,c ) s also a C[[ ]]- algebra. By constructon, we have Γ(X, A X,c )/ Γ(X, A X,c ) C[X]. In Secton 1, we ned a quantzaton of the Posson algebra C[X] as a fltered C-algebra. Thus the algebra of global sectons Γ(X, A X,c ) s not a quantzaton n that sense. But we can obtan a quantzaton of C[X] by modfyng the noton of the algebra of global sectons a lttle. Recall that we assumed that X s a lnear representaton of M and thus X s a C-vector space. Consder a C -acton on T X such that the nduced equvarant C -acton on O T X makes the coordnate functons x, y of T X sem-nvarant elements of weght one. By lettng be also a sem-nvarant element of weght two, the C -acton lft to an equvarant acton on the ormaton-quantzaton A T X. These C -actons nduce a C -acton on the symplectc manfold X, and also an equvarant C -acton on the sheaf A X,c on X. Then the algebra of global sectons  c = Γ(X, A X,c ) s a C[[ ]]-algebra wth a C -acton. The algebra Âc s decomposed nto a drect product of weght spaces wth respect to the C -acton:  c = m (Âc) m where (Âc) m s the weght space of weght m. Consder the drect sum m (Âc) m. Then, t s a C[ ]-subalgebra of Âc. We call t the fnte part of Âc wth respect to the C -acton, and denote (Âc) C -fn. Now consder the specalzaton = 1, and we obtan a C-algebra A c = Γ F (X, A X,c ) = (Âc) C -fn/( 1)(Âc) C -fn. The C[[ ]]-algebra Âc s graded by the power of, and the gradng nduces a fltraton of the C-algebra A c. Proposton 2.2. Under the condtons (1) (4) n Secton 2.2, the algebra A c = Γ F (X, A X,c ) s a quantzaton of the Posson algebra C[X].

7 7 We also have the followng commutatve dagram: Γ F (X, ) A c gr A X,c =0 C[X] O X Γ(X, ) Smlarly, we can construct the algebra of global sectons also for the ormatonquantzaton A T X of the cotangent bundle T X for a manfold X. 3. Deformaton-quantzaton of arc spaces 3.1. Arc spaces. The noton of ormaton-quantzaton and quantum Hamltonan reducton works also for certan nfnte-dmensonal manfolds, not only fnte-dmensonal ones. For a fnte-dmensonal manfold X, consder an nfntedmensonal manfold J X, called an arc space or a -Jet scheme on X. We regard X as a scheme of fnte type over C. Then the arc space J X s a scheme whose R-valued ponts for a C-algebra R s gven by J X(R) = Hom(Spec R, J X) = Hom(Spec R[[t]], X) = X(R[[t]]). Namely, the arc space J X s a scheme, whose C-valued ponts are nfntesmal arcs on X. Then, for an R-valued pont x(t) of J X, we can regard x(t) as an nfnte-dmensonal arc on X and ts head x(0) s an R-valued pont of X. Thus, we have a canoncal projecton J X X, x(t) x(0). We consder a fnte-dmensonal symplectc manfold X. Then ts structure sheaf O X s equpped wth a Posson bracket {, }. Along wth t, the structure sheaf O J X has an addtonal structure, called a vertex Posson algebra. Vertex Posson algebras are vertex algebra analogue of Posson algebras, whch are naturally obtaned as an assocated graded algebra of a certan vertex algebra. Frst, we revew brefly the nton of vertex algebras. A vertex algebra s a quadruple of data (V, 1, T, Y (, z)), where V a vector space, 1 V a vector n V, called the vacuum vector T : V V a lnear operator, called the translaton operator Y (, z) : V End V [[z, z 1 ]] a lnear operator, where Y (A, z) = n Z A (n)z n 1 s a formal seres of lnear operators A (n) on V for each vector A V. The operator Y (A, z) s called the vertex operator assocated wth A. These data are subject to the followng axoms: (vacuum axom) The vacuum vector satsfes Y (1, z) = d V, and for any A V, we have Y (A, z)1 V [[z]], so that Y (A, z)1 can be specalzed 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 n V. (translaton axom) For any A V, we have [T, Y (A, z)] = z Y (A, z), and T 1 = 0. (localty axom) For any A, B V, the vertex operators Y (A, z) and Y (B, w) are mutually local: Namely, there exsts N Z 0 (z w) N [Y (A, z), Y (B, w)] = 0.

8 8 From the above nton of vertex algebras, we obtan the followng dentty between coeffcents of vertex operators Y (A, z) and Y (B, z) for A, B V, so called Borcherds dentty: j 0 ( m j ) (A (n+j) B) (m+l j) = ( ) n ( 1) j {A (m+n j) B (l+j) ( 1) n B (n+l j) A (m+j) } j j 0 for any l, m and n Z. A vertex algebra V s called a commutatve vertex algebra f A (n) = 0 for any A V and n Z 0. In ths stuaton, operators A ( m) and B ( n) commute for any A, B V and m, n Z 1, and we can dentfy the commutatve vertex algebra V as a commutatve C-algebra n nfntely many varables wth the dervaton T. For any manfold X, the structure sheaf O J X of the arc space J X has a structure of a sheaf of commutatve vertex algebras wth varables f ( n) = (1/n!)T n f for f O X and n Z 1. The canoncal projecton J X X nduces an embeddng O X O J X gven by f f ( 1) for f O X. A vertex Posson algebra s a tuple (V, 1, T, Y + (, z), Y (, z)), such that the quadruple (V, 1, T, Y + (, z)) s a commutatve vertex algebra wth the vertex operator Y + (A, z) = n 1 A (n)z n 1, and coeffcents of Y (A, z) = n 0 A (n)z n 1 satsfes a truncaton of the Borcherds dentty. It s vertex-algebrac analogue of Posson algebras n the followng sense; f (V, 1, T, Y ( )) s a vertex algebra wth fltraton and the assocated graded vertex algebra gr V wth respect to the fltraton s commutatve, we can ne the structure of a vertex Posson algebra on the assocated graded vertex algebra gr V. Assume that X s a symplectc manfold. Consder the arc space J X, and then the structure sheaf O J X s a sheaf of commutatve vertex algebras as above. Moreover, the Posson bracket {, } on O X nduces a structure of a vertex Posson algebra on O J X whch satsfes { {f, g} f n = 0, f (n) g = 0 f n 1, for f, g O X and also satsfes the (truncated) Borcherds dentty Deformaton-quantzaton of J X. As ormaton-quantzaton of the vertex Posson algebras, we obtan the noton of -adc vertex algebras, whch are ntroduced by H. L n [L]. An -adc vertex algebra s a quadruple (V, 1, T, Y (, z)) where V s a flat C[[ ]]-module whch s complete n -adc topology, such that (V/ N V, 1, T, Y (, z)) s a vertex algebra over C for any N 0. Note that a -adc vertex algebra need not a vertex algebra over C[[ ]]. Assume that, for an -adc vertex algebra (V, 1, T, Y (, z)), the quotent vertex algebra (V/ V, 1, T, Y (, z)) s a commutatve vertex algebra. Then, V/ V has naturally the structure of vertex Posson algebra ned by ā (n) b = 1 a (n) b mod for a, b V and n Z 0, and where ā = a mod V/ V s the equvalent class of a. Now we ntroduce the noton of ormaton-quantzaton for vertex algebras. Let X be a symplectc manfold, and consder the structure sheaf O J X of the arc space J X as a sheaf of vertex Posson algebras. We say that the sheaf of -adc vertex algebras A ch X on J X s a ormaton-quantzaton of O J X (or of J X) f

9 ts quotent A ch X / Ach X s somorphc to O J X as a sheaf of vertex Posson algebras on J X. As the usual ormaton-quantzaton, we can construct a ormaton-quantzaton of O J X as a sheaf of -adc vertex algebras n the followng two cases: (1) When X s the cotangent bundle of a certan manfold X,.e. X = T X, we may have a sheaf of vertex algebras called an algebra of chral dfferental operators (CDO) on X, denoted by DX ch, whch was ntroduced n [BD] and [GMS] ndependently. Note that there exsts an obstructon for exstence of such a sheaf, but t s known that f the second Chern class ch 2 (T X ) vanshes, such a sheaf DX ch exsts. See [GMS] for detals. By a smlar method for localzng the sheaf of dfferental operators D X on T X as a ormaton-quantzaton of X = T X, we can construct localzaton of the sheaf of vertex algebras DX ch as a sheaf of -adc vertex algebras on X = T X, and moreover, as a sheaf on ts arc space J X. Ths constructon gves a ormatonquantzaton of the structure sheaf O J X n the above sense. We denote t A ch X,. (2) When a symplectc manfold X s constructed by Hamltonan reducton X = µ 1 (χ) ss /M as above for an acton of a certan unpotent Le group M on a manfold X and the CDO DX ch exsts, we can construct a ormaton-quantzaton of O J X as quantum Hamltonan reducton of A ch X,. In [AKM], wth usng a certan cohomologcal and quantum Hamltonan reducton, called a BRST cohomology, a ormaton-quantzaton of O J X s constructed for a Slodowy varety X, a symplectc manfold whch s obtaned by Hamltonan reducton of a flag varety. As a vertex algebra of global sectons wth respect to a certan C -acton, we obtan (1) a vertex algebra at crtcal level assocated wth the affne Le algebra correspondng to the flag varety, or (2) an affne W-algebra at crtcal level assocated wth the Slodowy varety X, respectvely. Namely, these ormaton-quantzatons are regarded as localzaton of such vertex algebras Applcatons of localzaton to representaton theory When an assocatve algebra A s localzed by a ormaton-quantzaton of a certan symplectc manfold X, the representaton theory of A may be connected wth the geometrcal structure of the underlyng manfold X. And ndeed many applcatons of such a localzaton to the representaton theory are known. In ths secton, we summarze some of such results whch are recently studed for algebras obtaned by quantum Hamltonan reducton. Throughout ths secton, we use the followng notaton. Let X be a symplectc manfold, and let A X,c be a ormaton-quantzaton of the structure sheaf O X wth a parameter of quantzaton c. Set A c = Γ(X, A X,c ) fn =1 be an assocatve C-algebra whch s obtaned as the algebra of global sectons. In ths secton, we may assume that the symplectc manfold X s obtaned by Hamltonan reducton, and the Hamltonan reducton satsfes the assumptons (1) (4) Belnson-Bernsten type correspondence. The most fundamental results are certan equvalences of categores between the category of A c -modules and the category of A X,c -modules wth an equvarant torus acton, whch the functor of takng global sectons gves. In the classcal case of the unversal envelopng algebra of smple Le algebra U(g), such an equvalence of categores essentally concdes wth an equvalence of abelan categores known as the Belnson-Bernsten correspondence. In the case of quantum Hamltonan reducton, we have (1) equvalence of trangulated categores between derved categores of module categores and (2)

10 10 equvalence of abelan categores whch s drect analogue of the Belnson-Bernsten correspondence. (1) Let A c -mod be the abelan category of fntely-generated A c -modules, and let A X,c -mod C -equv be the abelan category of coherent A X,c -modules wth the C - equvarant acton, whch are torson-free over C[[ ]]. Then an object of A X,c -mod C -equv are a sheaf on X and thus we can consder the module of ts global sectons. For an object M A X,c -mod C -equv, the module of global sectons Γ(X, M) s a C[[ ]]- module wth C -acton, snce M s equpped wth the equvarant C -acton. Then, takng fnte part wth respect to C -acton, denoted Γ(X, M) C -fn, we have a C[ ]-module. Fnally, substtutng = 1, we obtan a C-vector space Γ F (X, M) = Γ(X, M) C -fn =1 = Γ(X, M) C -fn C[ ] C 1, where C 1 s the C[ ]-module by augmentaton at = 1. Snce Γ F (X, A X,c ) A c, Γ F (X, M) s an A c -module. Ths constructon gves a functor of abelan categores Γ F (X, Consder the derved functor, and we have RΓ F (X, ) : A X,c -mod C -equv A c -mod. ) : D b (A X,c -mod C -equv) D b (A c -mod) where D b ( ) s a bounded derved category. The followng result s due to I. Gordon and I. Losev [GL]. Independently, K. McGerty and T. Nevns also studed essentally the same result ndependently n a lttle dfferent manner n [MN1]. Theorem 4.1. If the algebra A c has fnte global dmenson, we have an equvalence of trangulated categores RΓ F (X, wth the quas-nverse functor A X,c L A c ( ). ) : D b (A X,c -mod C -equv) D b (A c -mod) (2) Under a certan condton, the above functor Γ F (X, ) also gves an equvalence of abelan categores, not only the derved equvalence. Note that the Hamltonan reducton X s ned as a projectve varety over X 0 : X = Proj m 0 C[µ 1 (0)] M θ m X0 = Spec C[µ 1 (0)] M where θ s a certan character of the group M. From the nton, we have a lne bundle O(1) whch s assocated wth the m C[µ 1 (0)] M θ m-module m C[µ 1 (0)] M. θ (m+1) Moreover, we can construct a sheaf A θ X,c whch gves a quantzaton of ths lne bundle O(1) n the sense that A θ X,c C[[ ]] C 0 O(1) where C 0 s a C[[ ]]-module on C by augmentaton at = 0. Ths sheaf s an (A X,c+dθ, A X,c )-bmodule where dθ s a character of the Le algebra m obtaned by dfferentatng θ. By consderng the tensor product wth A θm X,c over A X,c, we have a functor A θm X,c AX,c ( ) : A X,c -mod C -equv A X,c+mdθ -mod C -equv for each m Z. We have A θm X,c C[[ ]] C 0 O(m) O(1) m, and the functor s an equvalence of categores whose quas-nverse functor s gven by A θ m X,c+mdθ. By applyng the functor of takng global sectons Γ F (X, ), we have functors for each m Z. Γ F (X, A θm X,c) Ac ( ) : A c -mod A c+mdθ -mod

11 Theorem 4.2. Assume that the functor Γ F (X, A θm X,c ) A c ( ) s an equvalence of abelan categores (wth the quas-nverse gven by Γ F (X, A θ m X,c+mdθ ) A c+mdθ ( )) for all m Z 0. Then, Γ F (X, ) : A X,c -mod C -equv A c -mod s an equvalence of abelan categores wth the quas-nverse A X,c Ac ( ). The theorem s analogue of well-known theorem n the representaton theory of smple Le algebras, so called the Belnson-Bernsten correspondence, studed n [BB1, BB2], and also n [BrKa]. For algebras obtaned by quantum Hamltonan reducton, t was frst establshed for the ratonal Cherednk algebra of the symmetrc group S n by M. Kashwara and R. Rouquer n [KR], and later for some other cases n [BeKu], [DK] separately. Recently, K. McGerty and T. Nevns n [MN2] gave a general crteron when such an abelan equvalence holds by usng Kashwara s equvalence and Krwan-Ness stratfcaton. In known cases of quantum Hamltonan reducton, the sheaf of C[[ ]]-algebras A X,c s locally somorphc to the -ormed Weyl algebra D C d, = C[[ ]][x 1,..., x d, y 1,..., y d ] wth nng relaton gven by [y, x j ] = δ j, [x, x j ] = [y, y j ] = 0. Thus the above equvalences connect the representaton theory of A c wth mcrolocal analyss on the symplectc manfold X through the sheaf A X,c, and hence we have many applcatons n the representaton theory of A c. Frst, for an A c -module M, the support of the correspondng sheaf of modules A X,c Ac M n the symplectc manfold X s an nvarant of modules. It s analogue of characterstc varetes n D-module theory. We also have analogue of characterstc cycles, cycles on X (wth multplctes) assocated wth the module. For certan quantum Hamltonan reducton A c, characterstc cycles of some mportant A c -modules were studed n [GS] (for the ratonal Cherednk algebra for S n ) and n [K1] (for the ratonal Cherednk algebra for Z/lZ). Moreover, for the ratonal Cherednk algebra for Z/lZ, we can construct explctly sheaves of modules correspondng to rreducble modules and standard modules n the category O of A c, a hghest weght category analogous to the Bernsten- Gelfand-Gelfand category for a smple Le algebra. ([K2]) As a consequence, t follows that sheaves of modules correspondng to modules n the category O are regular holonomc n the sense of mcrolocal analyss. Conjecturally the same fact holds for the category O of other type of ratonal Cherednk algebra, but t s stll an open problem BRST cohomologes. The constructon of the algebra A c as quantum Hamltonan reducton assocated wth the M-acton on A T X nduces a certan cohomology, called a BRST cohomology. The BRST cohomology (or so called BRST reducton) s frst ntroduced by theoretcal physcst n the area of quantum feld theory. Mathematcally, t s known that the BRST cohomology gves a cohomologcal nterpretaton of the (quantum) Hamltonan reducton (cf. [KS], [F]). In [K3], the explct descrpton of the BRST cohomology assocated wth the quantum Hamltonan reducton s gven n the case where X s a lnear representaton of the group M. We denote the BRST cohomology assocated wth the acton of M on the rng of dfferental operators D(X) by H BRST,c (m, D(X)), where m s the Le algebra of M and c s the parameter of the quantum Hamltonan reducton A c. We can also ne the sheaf verson of the BRST cohomology, and t s denoted H BRST,c (m, A (T X) ss). Then, we have the followng two somorphsms of graded 11

12 12 algebras (wth respect to the degree of cohomologes) under certan condtons: (2) (3) H BRST,c(m, A (T X) ss) A X,c C H DR(M), H BRST,c(m, D(X)) A c C H DR(M), where H DR (M) s the de Rham cohomology of M. The former somorphsm follows from the geometrcal condtons n Secton 2.2, and ndeed t s essentally a geometrcal fact. On the other hand, to prove the latter somorphsm we need to make advantage of the representaton theory of A c. Indeed, by usng the abelan and derved equvalences of categores n Secton 4.1, we obtan (3) from (2) Isomorphsm of quantzatons. Some noncommutatve algebras gve quantzatons of the same Posson algebra. In such a case, t s a natural problem f these quantzatons are somorphc wth each other. But usually t s not easy to compare ther dfferent constructons drectly. On the other hand, the ormaton theory for ormaton-quantzaton of a symplectc manfold s studed by R. Bezrukavnkov and D. Kaledn n [BeKa]. As an applcaton of ther result, I. Losev proved somorphsms between certan noncommutatve algebras, whch are constructed by quantum Hamltonan reducton n [Lo2]. For example, ormed preprojectve algebras ntroduced n [CBH] and fnte W-algebras assocated wth a subregular nlpotent orbt n smple Le algebra of type ADE are somorphc wth each other. References [AKM] T. Arakawa, T. Kuwabara, F. Malkov, Localzaton of affne W-algebras, Comm. Math. Phys., avalable onlne, (DOI) /s x. [BB1] A. Belnson, J. Bernsten, Localsaton de g-modules (French), C. R. Acad. Sc. Pars Se r. I Math. 292 (1981), no. 1, [BB2] A. Belnson, J. Bernsten, A proof of Jantzen conjectures, I. M. Gelfand Semnar, Adv. Sovet Math. 16 (1993), Part 1, Amer. Math. Soc., Provdence, RI, [BD] A. Belnson, V. Drnfeld, Chral algebras, Amercan Mathematcal Socety Colloquum Publcatons 51, Amercan Mathematcal Socety, Provdence, RI (2004) [BeKa] R. Bezrukavnkov and D. Kaledn, Fedosov quantzaton n algebrac context, Mosc. Math. J. 4 (2004), [BeKu] G. Bellamy and T. Kuwabara, On ormaton quantzatons of hypertorc varetes, Pacfc J. Math. 260 (2012), no. 1, [BrKa] J.-L. Brylnsk, M. Kashwara, Kazhdan-Lusztg conjecture and holonomc systems, Inv. Math. 64 (1981), no. 3, [CBH] W. Crawley-Boevey and M. Holland, Noncommutatve ormatons of Klenan sngulartes, Duke Math. J. 92 (1998), no. 3, [DK] C. Dodd and K. Kremnzer, A Localzaton Theorem for Fnte W-algebras, preprnt, arxv:math.rt/ v1. [F] B. Fegn, Sem-nfnte homology of Le, Kac-Moody and Vrasoro algebras (Russan), Uspekh Mat. Nauk 39 (1984), no. 2, [GL] I. Gordon and I. Losev, On category O for cyclotomc ratonal Cherednk algebras, J. Eur. Math. Soc 16 (2014), no. 5, [GMS] V. Gorbounov, F. Malkov, V. Schechtman, Gerbes of chral dfferental operators. II. Vertex algebrods, Invent. Math. 155 (2004), no. 3, [GS] I.G. Gordon and J.T. Stafford, Ratonal Cherednk algebras and Hlbert schemes II: representatons and sheaves, Duke Math. J 132 (2006), [K1] T. Kuwabara, Characterstc cycles of standard modules for the ratonal Cherednk algebra of type Z/lZ, J. Math. Kyoto Unv. 48 (2008), vol. 1, Kyoto Unversty, [K2] T. Kuwabara, Representaton theory of the ratonal Cherednk algebras of type Z/lZ va mcrolocal analyss, Publ. Res. Inst. Math. Sc. 49 (2013), no. 1, [K3] T. Kuwabara, BRST cohomologes for symplectc reflecton algebras and quantzatons of hypertorc varetes, preprnt, arxv: v1. [KR] M. Kashwara and R. Rouquer, Mcrolocalzaton of the ratonal Cherednk algebras, Duke Math. J. 144 (2008), no. 3, [KS] B. Kostant, S. Sternberg, Symplectc reducton, BRS cohomology, and nfntedmensonal Clfford algebras, Ann. Physcs 176, no.1, (1987)

13 13 [L] H. L, Vertex algebras and vertex Posson algebras, Contemp. Math., 6 (2004), no. 1, [Lo1] I. Losev, Fnte W-algebras, Proceedngs of the Internatonal Congress of Mathematcans, Volume III (2010), Hndustan Book Agency, New Delh, [Lo2] I. Losev, Isomorphsms of quantzatons va quantzaton of resolutons, Adv. Math. 231 (2012), vol. 3-4, [MN1] K. McGerty and T. Nevns, Derved equvalence for quantum symplectc resolutons, Selecta Math. (N.S.) 20 (2014), no. 2, [MN2] K. McGerty and T. Nevns, Compatblty of t-structures for quantum symplectc resolutons, preprnt, arxv: v1.