ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaModi�ed Regular Representationsof AÆne and Virasoro Algebras, VOA Stru tureand Semi{In�nite CohomologyIgor B. FrenkelKonstantin Styrkas

Vienna, Preprint ESI 1622 (2005) Mar h 10, 2005Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

MODIFIED REGULAR REPRESENTATIONSOF AFFINE AND VIRASORO ALGEBRAS, VOA STRUCTUREAND SEMI-INFINITE COHOMOLOGY.IGOR B. FRENKEL AND KONSTANTIN STYRKASAbstra t. We �nd a ounterpart of the lassi al fa t that the regular representation R(G)of a simple omplex group G is spanned by the matrix elements of all irredu ible represen-tations of G. Namely, the algebra of fun tions on the big ell G0 � G of the Bruhatde omposition is spanned by matrix elements of big proje tive modules from the ategoryO of representations of the Lie algebra g of G, and has the stru ture of a g� g-module.The standard regular representation R(G) of the aÆne group G has two ommutinga tions of the Lie algebra g with total entral harge 0, and arries the stru ture of a onformal �eld theory. The modi�ed versions R0(G) and R0(G0), originating from the loopversion of the Bruhat de omposition, have two ommuting g-a tions with entral hargesshifted by the dual Coxeter number, and a quire vertex operator algebra stru tures derivedfrom their Fo k spa e realizations, given expli itly for the ase G = SL(2; C ).The quantum Drinfeld-Sokolov redu tion transforms the representations of the aÆne Liealgebras into their W-algebra ounterparts, and an be used to produ e analogues of themodi�ed regular representations. When g = sl(2; C ) the orresponding W-algebra is the Vi-rasoro algebra. We des ribe the Virasoro analogues of the modi�ed regular representations,whi h are vertex operator algebras with the total entral harge equal to 26.The spe ial values of the total entral harges in the aÆne and Virasoro ases lead to thenon-trivial semi-in�nite ohomologywith oeÆ ients in the modi�ed regular representations.The inherited vertex algebra stru ture on this ohomology degenerates into a super ommuta-tive asso iative superalgebra. We des ribe these superalgebras in the ase when the entral harge is generi , and identify the 0th ohomology with the Grothendie k ring of �nite-dimensional G-modules. We onje ture that for the integral values of the entral hargethe 0th semi-in�nite ohomology oin ides with the Verlinde algebra and its ounterpartasso iated with the big proje tive modules.0. Introdu tion.The study of the regular representation of a simple omplex Lie group G is at the foun-dation of representation theory of G. Realized as the spa e of regular fun tions on G, theregular representation R(G) arries two ompatible stru tures of a G-bimodule and of a ommutative asso iative algebra. An algebro-geometri version of the Peter-Weyl theoremestablishes the de omposition of R(G) into a dire t sum of subspa es, spanned by matrixelements of all irredu ible �nite-dimensional representations V� of G, indexed by integraldominant highest weights � 2 P+. In other words, we have an isomorphism of G-bimodulesR(G) = M�2P+ V� V �� : (0.1)1

2 IGOR B. FRENKEL AND KONSTANTIN STYRKASwhere V �� is the dual representation of G. The multipli ation in R(G) an be des ribed inrepresentation-theoreti terms as a pairing of intertwining operators for the left and right g-a tions with appropriate stru tural oeÆ ients. Thus the algebra stru ture on R(G) en odesthe information about the tensor ategory of �nite-dimensional g-modules.The representations of G an also be viewed as modules over the simple omplex Liealgebra g asso iated with G. In the ase of the Lie algebra g it is natural to onsider alarger olle tion of modules - namely, the Bernstein-Gelfand-Gelfand ategory O. In�nite-dimensional g-modules from the ategory O are not integrable, and therefore their matrixelements annot be regarded as fun tions on G. However, one an interpret them as fun tionson the open dense subset Go � G, given by the Gauss de ompositionGo = N� � T �N+; (0.2)where N� is the upper and lower triangular unipotent subgroup of G, and T is the diagonalmaximal abelian subgroup. The spa e R(Go) of regular fun tions on Go does not have thestru ture of a representation of G. Nevertheless, the left and right in�nitesimal a tions ofthe Lie algebra g on this spa e are well-de�ned, and an be expressed in terms of expli itdi�erential operators in the parameters of the Gauss de omposition (0.2). The enlargedregular representation R(Go) de omposes into the dire t sum of bimodules spanned by thematrix oeÆ ients of all "big" proje tive g-modules P�, indexed by stri tly anti-dominanthighest weights � 2 �P++ = �(P+ + �), where � is the half-sum of all positive roots of g.Thus we obtain an isomorphism of g-bimodulesR(Go) �= M�2�P++ (P� P �� )�I�; (0.3)where P �� is the module dual to P�, and I� is the sub-bimodule of the matrix oeÆ ientswhi h vanish identi ally on the universal enveloping algebra U(g).It is important to note that the dual modules P �� do not belong to the ategory O, butto its \mirror image", in whi h all highest weight modules are repla es by lowest weightmodules. In order to stay in the ategory O we repla e the open subset Go oming from theGauss de omposition by the maximal ell in the Bruhat de ompositionG0 = N+ �w0 � T �N+; (0.4)where w0 is the longest element of the Weyl group W . Then we obtain a version of theisomorphism (0.3), R(G0) �= M�2�P++ (P� P ?� )�I�; (0.5)where the 'twisted' duals P ?� di�ers from P �� by the automorphism ! of g, whi h is indu edby w0 and inter hanges the positive and negative roots.The theorems of Peter-Weyl type and the Gauss de omposition an be extended to the entral extension of the loop group G asso iated to G, and to the orresponding aÆneLie algebra g and its universal enveloping algebra U(g). In this in�nite-dimensional asethe spa e R(G) of regular fun tions on G is de omposed into the dire t sum of subspa esRk(G), orresponding to the value k 2 Zof the entral harge. Using the version of theGauss de omposition (0.2) known as the Birkho� de omposition, one an show (see [PS℄)

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 3that for any k 2Zthere is an isomorphismRk(G) �= M�2Pk+ V�;k V ��;k; (0.6)where � runs over the trun ated al ove P+k � P+, depending on k, and V�;k are the orre-sponding irredu ible modules. Similarly, one an obtain de ompositions of Rk(Go) analogousto (0.3), where Go is the maximal ell in the Birkho� de omposition. Viewing the de ompo-sition (0.6) in terms of the Lie algebra g allows to extend it for all values of k, with P+k = P+for k =2 Q.To transform the dual module V ��;k into a module from the ategory O for g, one mightapply again an automorphism of g whi h inter hanges the positive and negative aÆne roots.However, it no longer belongs to the aÆne Weyl group, and the Bruhat de omposition for Gdoes not have a maximal ell. To over ome this obsta le we onsider instead an intermediatebetween the Birkho� and the aÆne Bruhat de ompositions - the loop version of the �nite-dimensional Bruhat de omposition, and the orresponding big ellG0 = LN+ �w0 � LT � LN+; (0.7)where LN� denote the loop groups with values in n�, and LT is the entral extension ofthe loop group with values in T . The de omposition (0.7) is espe ially useful for expli itrealizations of the left and right regular g-a tions in terms of di�erential operators. However,we are still in \semi-in�nite" distan e from the ategory O, and need to further apply awell-known pro edure of \ hanging the va uum", whi h has originated from the free �eldrealizations of the Wakimoto modules and the irredu ible representations V�;k (see [FeFr1,BF℄). As a result of this pro edure we obtain a modi�ed aÆne version of the extendedregular representation (0.5),R0k(G0) �= M�2�P++ �P�;k�h_ P ?�;�k�h_��I�;k; (0.8)where P�;k�h_ and P ?�;�k�h_ are the proje tive g-modules and their 'twisted' duals, I�;k areappropriate sub-bimodules, and we assume k =2 Q. The levels are shifted by the dual Coxeternumber h_, so that the diagonal g-a tion has the level �2h_.Like the Wakimoto modules, the bimodule R0k(G0) is realized as a ertain Fo k spa e,with two ommuting g-a tions des ribed expli itly ( f. [FeP℄). This realization is similar tothe standard realization of the Wakimoto modules, but the a tions of g ontain a ru ialnew ingredient - the vertex operators, dire tly related to the s reening operators used to onstru t intertwining operators for the aÆne Lie algebra. We also establish that for k =2 Qthe stru ture of the so le �ltration of the non-semisimple bimodule R0k(G0) is the same as inthe �nite-dimensional ase. In parti ular, R0k(G0) ontains the distinguished sub-bimoduleR0k(G) �= M�2P+ V�;k�h_ V ?�;�k�h_ : (0.9)The shifts of the entral harge by the dual Coxeter number no longer allow the interpretationof the bimodules in (0.8) and (0.9) as spa es of matrix elements of g-modules. Nevertheless,the stru tures of these bimodules are ompletely analogous to those of bimodulesR(G0) andR(G)!

4 IGOR B. FRENKEL AND KONSTANTIN STYRKASThe va uum module V0;k of the aÆne Lie algebra g arries an extremely ri h additionalstru ture of a vertex operator algebra (VOA); other g-modules V�;k be ome its representa-tions (see [FZ℄). We show in this paper that the bimodules R0k(G) and R0k(G0) also admit avertex operator stru ture, ompatible with g-a tions. In the proof we use the expli it Fo kspa e realization of these bimodules. An alternative approa h is given by the geometri onstru tion of the algebras of hiral di�erential operators ( do) over G, studied in [GMS℄and [AG℄. 1 Thus besides the va uum modules there is a lass of vertex operator algebrasasso iated to aÆne Lie algebras with �xed entral harges. On the other hand, the originalregular representation Rk(G) does not seem to have a VOA stru ture. Instead it has thestru ture of a two-dimensional onformal �eld theory, whi h is an obje t of a di�erent na-ture despite having lo al properties similar to those of a VOA. The bimodule Rk(G0) has astru ture of a generalized (non-semisimple!) onformal �eld theory.It is well-known that the representation theory of g is losely related to the representationtheory of the orresponding W-algebra via a fun torial ohomologi al onstru tion, alledthe quantum Drinfeld-Sokolov redu tion [FeFr2℄. For the generi entral harge, this fun torgives an equivalen e of ategories, and all VOA onstru tions and stru tural results in theaÆne ase are automati ally transformed into their W-algebra ounterparts. In parti ular,we get W-algebra analogues of the Peter-Weyl theorem and its non-semisimple extension.In this paper we onsider in full detail the simplest ase of g = sl(2; C ), when the orre-spondingW-algebra is the in�nite-dimensional Virasoro algebra. We give expli it realizationsof its modi�ed regular representations, analogous to (0.8) and (0.9), and equip them with ompatible VOA stru tures. We des ribe their bimodule stru tures, whi h are ompletelyparallel to those in the �nite-dimensional and aÆne ases. Using the approa h presented inthis paper, it is easy to generalize the bimodule and vertex algebra onstru tions to the aseof aÆne Lie algebras of higher rank and the asso iated W-algebras. However, the hara ter-ization of R(G0) as the algebra of matrix elements of all modules from the ategory O, andthe des ription of the orresponding bimodule stru tures, require more general methods andwill be given in the forth oming paper [S2℄.A remarkable feature of all the modi�ed bimodules is that the entral harge for the diag-onal a tion is always equal to the spe ial values that appear in the semi-in�nite ohomologytheory [Fe, FGZ℄ - namely, �2h_ for the aÆne Lie algebras and 26 for Virasoro. More-over, thanks to a general result of [LZ1℄, the orresponding semi-in�nite ohomology with oeÆ ients in the modi�ed regular representations inherits a VOA stru ture; in our ase itdegenerates into a super ommutative asso iative superalgebra. For generi entral hargewe establish isomorphisms between the semi-in�nite ohomology spa es for the aÆne andVirasoro algebras and their �nite-dimensional ounterparts. Moreover, their superalgebrastru tures retain the information about the fusion tensor ategories of aÆne and Virasoromodules, built into the orresponding modi�ed regular representations. In parti ular, weshow that the 0th semi-in�nite ohomologies of the aÆne and Virasoro algebras are isomor-phi to the Grothendie k ring of �nite-dimensional representations of G. We onje ture thatfor integral k they lead to the Verlinde algebra and its proje tive ounterpart.This paper is organized as follows. In Se tion 1 we onsider the Bruhat de omposition andthe Peter-Weyl theorems in the �nite-dimensional ase with G = SL(2; C ). We give a Fo kspa e realization of the algebra R(G0), and obtain expli it formulas for the g-a tions and1We were informed by F. Malikov and S. Arkhipov that their do algebras ould be shown to have thesame bimodule stru ture as R0k(G), establishing the equivalen e of our realization and their onstru tions.

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 5de omposition theorems, whi h will later be used as prototypes of the in�nite-dimensional ase. In the last subse tion we ompute the Lie algebra ohomology with oeÆ ients inR(G) and R(G0). In Se tion 2 we study the aÆne ase, and use the loop version of the�nite-dimensional Bruhat de omposition to obtain the modi�ed Peter-Weyl theorems forthe spa es R0k(G) and R0k(G0). The Fo k spa e realization of these spa es equips them withVOA stru tures ompatible with the regular g-a tions. The semi-in�nite ohomology of gwith oeÆ ients in the modi�ed regular representations R0k(G) and R0k(G0) for generi en-tral harge is shown to be isomorphi to its �nite-dimensional ounterpart. In Se tion 3we onstru t the analogues of the modi�ed regular representations of the Virasoro algebrausing the quantum Drinfeld-Sokolov redu tion. We ompute the orresponding semi-in�nite ohomology groups using methods developed in string theory, and prove that they are iso-morphi to their aÆne ounterparts. Finally, in Se tion 4 we des ribe another lass of vertexoperator algebras obtained by the pairing of bsl(2; C ) and Virasoro modules. We also dis ussgeneralizations of our results to Lie algebras of other types, and to the integral values of the entral harge. We on lude with onje tures on relations of the semi-in�nite ohomologyof R0k(G) and R0k(G0) for k 2Z>0 with the Verlinde algebra, its proje tive ounterpart andtwisted equivariant K-theory.We thank G. Zu kerman for sharing his expertise on semi-in�nite ohomology. We aregrateful to F. Malikov, who told us about the papers [FeP, GMS, AG℄ when this work wasnearing ompletion. We also thank S. Arkhipov and F. Malikov for valuable omments.I.B.F. is supported in part by NSF grant DMS-0070551.1. Regular representation of sl(2; C ) on the big ell.1.1. Regular representations of sl(2; C ). Let G = SL(2; C ). We de�ne the left and rightregular a tions of G on the spa e R(G) of regular fun tions on G by(�l(g) )(h) = (g�1h); (�r(g) )(h) = (h g); g; h 2 G: (1.1)The multipli ation in R(G) intertwines both left and right regular a tions.Let T;N+ denote the diagonal and unipotent upper-triangular subgroups of G. The groupW = Norm(T )=T is alled the Weyl group. The Bruhat de omposition G = N+ �W � T �N+implies that every g 2 G an be fa tored as g = n �w �t �n0 for some n; n0 2 N+; t 2 T;w 2 W .We denote by G0 the big ell of the Bruhat de omposition, orresponding to the longestWeyl group element w0. Expli itly, G0 is the dense open subset of G, onsisting of g 2 G;g = �1 x0 1��0 �11 0 ��� 00 ��1��1 y0 1� (1.2)for some x; y 2 C and � 2 C � . The variables x; y; � an be viewed as oordinates on G0, andthus the algebra R(G0) of regular fun tions on G0 is identi�ed with the spa e C [x; y; ��1℄.Let g = sl(2; C ) be the Lie algebra of G, with the standard basise = �0 10 0� ; h = �1 00 �1� ; f = �0 01 0� ;satisfying the ommutation relations [h; e℄ = 2e; [h; f ℄ = �2f ; [e; f ℄ = h: The nilpotentsubalgebras n� and the Cartan subalgebra h of g are de�ned by n+ = C e; h = Ch; n� = C f :The elementw0 2 W determines a Lie algebra involution ! of g, su h that !(n�) = n� and

6 IGOR B. FRENKEL AND KONSTANTIN STYRKAS!(h) = h, de�ned by !(e) = �f ; !(h) = �h; !(f) = �e: (1.3)The in�nitesimal regular a tions of g on R(G), orresponding to (1.1), are given by(�l(x) )(g) = ddt (e�t xg)����t=0; (�r(x) )(g) = ddt (g et x)����t=0; x 2 g; g 2 G: (1.4)These formulas also de�ne left and right in�nitesimal a tions of g on the spa e R(G0).(These a tions annot be lifted to the group G, be ause G0 is not invariant under left andright shifts). Elementary al ulations yield the following expli it des ription of the regularg-a tions ( f. [FeP℄).Proposition 1.1. The regular g-a tions on R(G0) are given by�l(e) = ��x;�l(h) = ��� � 2x�x;�l(f) = �x��� + x2�x � ��2�y: (1.5)�r(e) = �y;�r(h) = ��� � 2y�y;�r(f) = y��� � y2�y + ��2�x: (1.6)1.2. Bosoni realizations. We now reformulate the onstru tions of the previous se tionin terms of Fo k modules for ertain Heisenberg algebras. These realizations admit gen-eralizations to the aÆne and Virasoro ases, where the geometri approa h to the regularrepresentations be omes more subtle.The operators � = x; = ��x a ting on polynomials in y give a representation of theHeisenberg algebra with generators �; and relation [�; ℄ = 1. The polynomial spa e C [y℄ isthen identi�ed with its irredu ible representation F (�; ), generated by a ve tor 1 satisfying 1 = 0.The operators �� = �y; � = �y generate a se ond Heisenberg algebra, a ting irredu ibly inthe spa e F ( ��; � ) �= C [x℄. Here and everywhere else in this paper the 'bar' notation is usedto denote the se ond opies of algebras and their generators; it does not denote the omplex onjugation.We identify h� �= C so that P �=Z. Whenever possible, we use the more invariant notationin order to avoid possible numeri oin iden es. The operators 1� = �� and a = ��� gives riseto the semi-dire t produ t C [a℄nC [P℄, with C [a℄ a ting on C [P℄ by derivations: a1� = �1�.Thus, we get a realization of the algebra R(G0) of regular fun tions on G0, with theg� g-a tion des ribed by the abstra t versions of the formulas (1.5), (1.6).Theorem 1.2. The spa e F = F (�; ) F ( ��; � ) C [P℄ gives a realization of the algebraR(G0). In parti ular,(1) The spa e F has a g� g-module stru ture, given bye = ;h = 2� + a;f = ��2 � � a� � 1�2; (1.7)

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 7�e = � ;�h = 2 ��� + a;�f = ���2� � �� a� 1�2: (1.8)(2) The spa e F has a ompatible ommutative algebra stru ture (i.e. the multipli ationin F intertwines the g� g-a tion).By spe ializing the a tion (1.7) to the subspa e ker � � F, we get the following well-knownrealizations of g-a tion in the spa es F� = F (�; ) C 1�:e = ;h = 2� + �;f = ��2 � ��: (1.9)Remark 1. Simultaneous res aling of the extra terms in (1.7),(1.8), involving the shift1�2, by any multiple � would preserve all the g� g ommutation relations. For � = 0 su hg� g-a tion degenerates into the produ t of two standard g-a tions (1.9). However, themultipli ation in this na��ve bimodule loses mu h of its ri h stru ture, and no longer en odesthe information about the fusion rules in the tensor ategory of �nite-dimensional g-modules.1.3. g� g-module stru ture of the modi�ed regular representation. In this subse -tion we des ribe the so le �ltration of the g� g-module F.For any � 2 h�, we denote by V� the irredu ible g-module, generated by a highest weightve tor v� satisfying e v� = 0 and h v� = � v�.Re all that a g-module V is said to have a weight spa e de omposition, ifV =M�2h� V [�℄; V [�℄ = �v 2 V ��h v = � v :The restri ted dual spa e V 0 =L�2h� V [�℄0 an be equipped with a g-a tion, de�ned byhg v0; vi = �hv0; !(g) vi; g 2 g; v 2 V; v0 2 V 0;where ! is as in (1.3). We denote the resulting dual module V ?.We have an involution � 7! �? of h�, determined by the ondition (V�)? �= V�?. Thisinvolution an also be de�ned by �? = �w0(�), where w0 is the longest Weyl group element.For g = sl(2; C ), we have �? = �. However, we keep the notation �?, to indi ate how our onstru tions generalize to Lie algebras of higher rank, where the involution is nontrivial.Theorem 1.3. There exists a �ltration0 � F(0) � F(1) � F(2) = F (1.10)of g� g-submodules of F, su h thatF(2)=F(1) �= M�2P+ V���2 V ?���2; (1.11)F(1)=F(0) �= M�2P+ �V� V ?���2 � V���2 V ?� � ; (1.12)F(0) �=M�2P V� V ?� : (1.13)

8 IGOR B. FRENKEL AND KONSTANTIN STYRKASProof. We introdu e a �ltration of g� g-submodules of F� � � � F��2 � F��1 � F�0 � F�1 � F�2 � : : : ; (1.14)satisfying T�2P F�� = 0 and S�2P F�� = F; whereF�� = F (�; ) F ( ��; � )M��� C 1� ; � 2 P:It is lear that F��=F<� �= F� F�?; moreover, for � < 0 we have F� �= V�, and for � � 0there is a short exa t sequen e 0 ! V� ! F� ! V���2 ! 0. The linking prin iple forg-modules implies that the su essive quotients F��=F<� and F��=F<� of this �ltration maybe non-trivially linked only if � = ��� 2.Thus we see that the g� g-module F splits into the dire t sum of blo ksF = F(�1) � M�2P+ F(�); (1.15)where F(�1) �= V�1 V ?�1, and F(�) �= �V���2 V ?���2� + (F� F�?) for � 2 P+; anotherway to obtain the de omposition (1.15) is by using the Casimir operator.It remains to des ribe the stru ture of F(�) for ea h � 2 P+. By onstru tion, F(�) anbe in luded in a short exa t sequen e 0 ! V���2 V ?���2 ! F(�) ! F� F�? ! 0: We on lude that there exists a �ltration 0 � F(�)(0) � F(�)(1) � F(�)(2) = F(�); su h thatF(�)(2)=F(�)(1) �= V���2 V ?���2;F(�)(1)=F(�)(0) �= �V� V ?���2�� (V���2 V ?� ) ;F(�)(0) �= �V���2 V ?���2�+ (V� V ?� ) :In fa t, the linking prin iple implies that the sum in F(�)(0) is dire t:F(�)(0) �= �V���2 V ?���2�� (V� V ?� ) :Finally, we onstru t the �ltration (1.10) by settingF(0) = F(�1) � M�2P+ F(�)(0) ; F(1) = F(�1) � M�2P+ F(�)(1);whi h obviously satis�es the required onditions (1.11), (1.12), (1.13). �Remark 2. The analogous �ltration for a Lie algebra g of higher rank has blo ks F(�),whose stru ture is governed by the endomorphism ring Endg(P (�)) of the orrespondingproje tive generator P (�) =L�2W ��P�.The natural in lusion of algebrasR(G) � R(G0) an be seen in the Fo k spa e realizations.Corollary 1.4. There exists a subspa e F � F satisfying the following properties.(1) F is a subalgebra of F, and is generated by the elements from the submodule V1V ?1 , orresponding to the matrix elements of the anoni al representation of G.(2) F is a g� g-submodule of F, and is generated by the ve tors f1�g�2P+. We haveF = M�2P+F(�) �= M�2P+ V� V ?� : (1.16)(3) The spa e F is a realization of the algebra R(G).

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 9In the polynomial realization, the generators of F from V1V ?1 are identi�ed with fun tions 11 = �; 12 = x�; 21 = y�; 22 = xy� + ��1;whi h satisfy the relation 11 22 � 12 21 = 1. This establishes a very dire t onne tionwith the spa e of regular fun tions on the group G = SL(2; C ).1.4. The generalized Peter-Weyl theorem. In this se tion we interpret the spa e R(G0)of regular fun tions on G0 and its Fo k spa e realization F as the algebra of matrix elementsof all modules from the ategory O.Re all that the Bernstein-Gelfand-Gelfand ategory O onsists of all �nitely generated,lo ally n+-nilpotent g-modules. In parti ular, V� 2 O for any �. If V 2 O, then V ? 2 O.For any g-module V we de�ne M (V ) to be the subspa e of U(g)0, spanned by fun tionals�v;v0(x) = hv0; x vi; v 2 V; v0 2 V 0; x 2 U(g); (1.17)where h�; �i stands for the natural pairing between V and V 0: The fun tionals (1.17) are alledmatrix elements of the representation V:Proposition 1.5. Introdu e a g� g-module stru ture on the restri ted dual U(g)0 by(�l(g)�)(x) = �(xg); (�r(g)�)(x) = ��(!(g)x) (1.18)for any � 2 U(g)0; g 2 g; x 2 U(g). Then(1) For any g-module V , the spa e M (V ) is a g� g-submodule of U(g)0.(2) For any ' 2 U(g)0; there exists a g-module V , su h that ' 2 M (V ): Moreover, if 'is n+ � n+-nilpotent, then V an be hosen from the ategory O:Proof. To show that M (V ) is invariant under the left a tion of g, we ompute(�l(g)�v;v0)(x) = �v;v0(xg) = hv0; xg vi = �gv;v0(x);for any x 2 U(g); g 2 g; v 2 V; v0 2 V 0: This shows that y�v;v0 2 M (V ): The invarian eunder the right a tion follows from the omputation(�r(g)�v;v0)(x) = ��v;v0(!(g)x) = �hv0; !(g)x vi = hyv0; x vi = �v;yv0(x):For the se ond part, assume ' 2 U(g)0. Denote by V the subspa e of U(g)0, generatedfrom ' by the left a tion of g. Let '0 be the restri tion to V of the unit 1 2 U(g) = U(g)00.Equivalently, '0 is the linear fun tional on V , determined by h'0; i = (1); for any 2V � U(g)0. We laim that ' = �';'0 2 M (V ). Indeed, for any x 2 U(g) we have�';'0(x) = h'0; x'i = (x')(1) = '(x):Finally, if ' is left-n+-nilpotent, then V is lo ally n+-nilpotent. Sin e V is generated bya single element '; it belongs to ategory O: The right-n+-nilpoten y ondition guaranteesthat '0 belongs to the restri ted dual spa e V 0. �The elements of the universal enveloping algebra U(g) may be regarded as the di�erentialoperators, a ting onR(G). This gives an interpretation of the regular fun tions on G (or evenon G0) as linear fun tionals on U(g), and thus to identi� ations of the spa esR(G) andR(G0)with ertain subspa es of U(g)0. In the expli it realizations F and F this orresponden e is onstru ted using the algebrai analogue of the " o-unit" element of the Hopf algebra R(G)- the linear fun tional h�i : F ! C , de�ned byh�m ��n1�i = Æm;0Æn;0: (1.19)

10 IGOR B. FRENKEL AND KONSTANTIN STYRKASProposition 1.6. The linear map # : F ! U(g)0, de�ned by v 7! #v,#v(x) = h�l(x)vi; v 2 F; x 2 U(g): (1.20)is an inje tive g� g-homomorphism.Proof. In terms of the polynomial realization, h�i orresponds to evaluating a fun tion (x; y; �) 2 R(G0) at the element w0: h i = (0; 0; 1). This implies that for any v 2 Fhevi = �h�fvi; hhvi = �h�hvi; hfvi = �h�evi: (1.21)Therefore, for any g 2 g and x 2 U(g) we have#gv(x) = hx gvi = #v(xg) = (�l(g)#v)(x);#�gv(x) = hx �gvi = h�g xvi = �h!(g)xvi = �#v(!(g)x) = (�r(g)#v)(x):We on lude that the map # is a g � g-homomorphism. To prove that it is inje tive, we needto show that for any nonzero v 2 F there exists an element x 2 g� g su h that hxvi 6= 0:Sin e F is lo ally n+-nilpotent, we an pi k k � 0 su h that ekv 6= 0; but ek+1v = 0:Repla ing v by ekv; we see that it suÆ es onsider the ase of v 6= 0 su h that ev = 0:Similarly, we may assume that �ev = 0:A ve tor v satisfying ev = 0 = �ev must have the form v = P�2P �1� with only �nitelymany � 6= 0: Using the formula for the Vandermonde determinant and the fa t thathhmvi =X�2P ��m; m � 0;we on lude that hhkvi = 0 for all k � 0 if and only if all � vanish. Thus, �v = 0 isequivalent to v = 0, whi h means that # is an inje tion. �The following statement is an algebrai version of the lassi al Peter-Weyl theorem.Theorem 1.7. The spa e R(G) of regular fun tions on G is spanned by the matrix elementsof �nite-dimensional irredu ible g-modules,R(G) �= M�2P+M (V�):The de omposition of R(G) as a g� g-module is given byR(G) �= M�2P+ V� V ?� :Remark 3. The subspa e of U(g)0, orresponding to R(G), is invariantly hara terized asthe restri ted Hopf dual U(g)0Hopf � U(g)0, de�ned byU(g)0Hopf = f� 2 U(g)0��9 two-sided ideal J � U(g) su h that �(J) = 0 and odimJ <1g:The extended spa e R(G0) orresponds to a larger subalgebra of U(g)0, spanned by thematrix elements of all modules in the ategory O.Re all that the ategory O has enough proje tives; we denote by P� the inde omposableproje tive over of the irredu ible module V�. It is known that every inde omposable modulein the ategory O with integral weights is isomorphi to a subfa tor of the proje tive module, orresponding to some anti-dominant integral weight �. In parti ular, this means that itsuÆ es to onsider the matrix elements of the big proje tive modules fP�g�<0.The following result an be regarded as a non-semisimple generalization of the Peter-Weyltheorem.

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 11Theorem 1.8. The spa eR(G0) of regular fun tions on G0 is spanned by the matrix elementsof all big proje tive modules in the ategory O,R(G0) �= M�2P+M (P�):As a g� g-module, R(G0) is given byR(G0) �= M�2�P++ (P� P ?� )�I�where I�'s are the g� g-submodules of P�P ?� , orresponding to identi ally vanishing matrixelements.Proof. We use the realization of R(G0) in the Fo k spa e F. The in lusion (1.20) provides theidenti� ation of F with a subspa e of U(g)0. Sin e F is lo ally n+�n+-nilpotent, Proposition1.5 implies that for any v 2 F there exists a g-module W 2 O su h that #v 2 M (W ):Let W = W1 �W2 � � � � �Wm be the de omposition of W into a dire t sum of inde om-posable submodules. Ea h inde omposable omponent Wi; i = 1; : : : ;m; is a subfa tor ofsome big proje tive module P�i. Then M (Wi) � M (P�i ), and therefore we haveM (W ) = M (W1) + M (W2) + � � �+ M (Wm) � M�2�P++M (P�);whi h shows that #(F) � L�2�P++ M (P�): To prove that in fa t #(F) = L�2�P++ M (P�),we ompare the hara ters of the two spa es, and show that they have the same size.For any � 2 P+ the g� g-module M (P���2) is isomorphi to the quotient of the produ tP���2 P ?���2 by the kernel of the map�� : P���2 P ?���2 ! U(g)0; ��(v v0) = �v;v0 : (1.22)Obviously, I� = ker�� is a g� g-submodule of P���2P ?���2; we des ribe it more expli itly.It is known that the module P���2 has a �ltration 0 � P (0) � P (1) � P���2 su h thatP (0) �= V���2; P (1)=P (0) �= V�; P���2=P (1) �= V���2;and the dual �ltration of the module P ?���2 is given by0 � Ann(P (1)) � Ann(P (0)) � P ?���2:They determine a �ltration of the tensor produ t0 � P (0) Ann(P (1)) � P (0) Ann(P (0)) + P (1) Ann(P (1)) �� P (0) P ?���2 + P (1) Ann(P (0)) + P���2 Ann(P (1)) �� P (1) P ?���2 + P���2 Ann(P (0)) � P���2 P ?���2:If v 2 P (0) and v0 2 Ann(P (0)), then �v;v0 is the zero fun tional, sin e for any x 2 U(g) wehave x v 2 P (0) and �v;v0(x) = hv0; x vi = 0. Hen e the submodule P (0) Ann(P (0)) lies inthe kernel of the map ��, and similarly does P (1) Ann(P (1)). One an easily see that�� �P (1) Ann(P (0))� = M (V�);�� �P (0) P ?���2� = �� �P���2 Ann(P (1))� = M (V���2):

12 IGOR B. FRENKEL AND KONSTANTIN STYRKASIt follows that the g � g-module M (P���2) has a �ltration0 � M (0) � M (1) � M (2) = M (P���2)su h that M (2)=M (1) �= V���2 V ?���2;M (1)=M (0) �= (V� V ?���2) � (V���2 V ?� );M (0) �= (V� V ?� )� (V���2 V ?���2):Thus, the blo k F(�) of (1.15) is identi�ed with the subspa e, spanned by the matrixelements of the big proje tive module P���2. Taking dire t sums over all � 2 P+, addingthe g � g-module M (P�1) �= V�1 V ?�1, and omparing with Theorem 1.3, we see thatL�2�P++ M (P�) and F have the same hara ters. The statement of the theorem follows. �1.5. Cohomology of gwith oeÆ ients in regular representations. The algebraR(G) ontains the subalgebra R(G)G of the onjugation-invariant fun tions on G, whi h is lin-early spanned by the hara ters of the irredu ible �nite-dimensional representations. Thesubalgebra R(G)G is thus isomorphi to the Grothendie k ring of the �nite-dimensionalrepresentations of G.There is an isomorphism R(G)G �= C [P℄W , obtained by restri ting the group hara tersto h and taking its Fourier expansion. Finally, the algebra R(G)G also admits a ohomo-logi al interpretation, whi h will be instrumental for further generalizations to the regularrepresentations of the aÆne and Virasoro algebras. We brie y re all the de�nition of the ohomology of g.Proposition 1.9. Let � = Vg0 be the exterior algebra of g0 with unit 1. Then(1) The Cli�ord algebra, generated by f�(g); "(g0)gg2g;g02g0 with relationsf�(x); �(y)g = f"(x0); "(y0)g = 0; f�(x); "(y0)g = hy0; xi; (1.23)a ts irredu ibly on �, so that for any ! 2 � we have�(g)1 = 0; "(g0)! = g0 ^ !; g 2 g; g0 2 g0; ! 2 �:(2) � is a ommutative superalgebra,!1 ^ !2 = (�1)j!1j�j!2j !2 ^ !1; !1; !2 2 �;where j � j is the natural grading on � satisfying j1j = 0; j�(g)j = �1; j"(g0)j = 1:(3) The g-module stru ture on � is given by��(x) =Xi "(g0i)�([gi; x℄);where fgig is any basis of g, and fg0jg is the orresponding dual basis of g0.De�nition 1. The ohomology H�(g;V ) of g with oeÆ ients in a g-module V is the oho-mology of the graded omplex C�(g;V ) = �� V , with the di�erentiald =Xi "(g0i)�V (gi)� 12Xi;j "(g0i)"(g0j)�([gi; gj ℄); (1.24)where fgig is any basis of g, and fg0ig is the dual basis of g0.The following is one of the fundamental results in Lie algebra ohomology, (see e.g. [HS℄).

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 13Theorem 1.10. For any �nite-dimensional g-module V we haveH�(g;V ) �= V g H�DR(G); (1.25)where H�DR(G) denotes the holomorphi de Rham ohomology H�DR(G) of the Lie group G.If V is a ommutative algebra with a ompatible g-a tion, then its ohomology inheritsthe multipli ation from V and �, and H�(g;V ) be omes itself a ommutative superalgebra.Moreover, the isomorphism (1.25) be omes an isomorphism of superalgebras, with respe tto the up produ t in H�DR(G).The diagonal g-a tion in F orresponds to the oadjoint a tion of G in R(G); thus, we getCorollary 1.11. There is an isomorphism of ommutative superalgebrasH�(g;F) = C [P℄W H�DR(G):Our next goal is to study the ohomology of g with oeÆ ients in the extended regularrepresentation F �= R(G0). For in�nite-dimensional g-modules Theorem 1.10 does not hold,and the ohomology H�(g;F) does not redu e to Fg H�DR(G). We have insteadTheorem 1.12. There is an isomorphism of ommutative superalgebrasH�(g;F) �= C [P℄W ^� C 2 :Proof. It is easy to show using the results of [W℄ that for � � �1Hn(g;V� V ?���2) = Hn(g;V���2 V ?� ) = (C ; n = 1; 20; otherwiseand that for � � 0 we have Hn(g;V���2 V ?���2) = 0 for all n. The spe tral sequen easso iated with the �ltration of Theorem 1.3 an be used to show thatHn(g;F(�1)) = (C ; n = 1; 20; otherwise ; Hn(g;F(�)) = 8><>:C ; n = 0; 2C 2 ; n = 10; otherwise ; � � 0: (1.26)Also, this spe tral sequen e shows that we have a natural isomorphismH0(g;F) �= H0(g;F):To expli itly get the generators of the ommutative superalgebra H�(g;F), we pi k nonzeroelements � 2 H0(g;F(1)); ��1 2 H1(g;F(�1)); �0 2 H1(g;F(0));su h that �0 is not proportional to � ��1. It is known that H0(g;F) �= C [P℄W is isomorphi to the polynomial algebra C [�℄. It is also lear that H�(g;F) is a free C [�℄-module. For ea h� � 0, the set B�� = f��1; ���1; : : : ; ��+1 ��1g[f�0; � �0; : : : ; �� �0g onsists of 2�+3 linearly independent elements, and in view of (1.26) is a basis of H1(g;F��).Finally, one an he k that �0 ��1 6= 0, and thus the elements f�0 ��1; � �0 ��1; : : : ; ��+1 �0 ��1ggive a basis of H2(g;F��) for ea h � � �1.It follows that H�(g;F) �= C [�℄ V�[��1; �0℄, and the theorem is proven. �Remark 4. One of the ingredients in the exterior algebra part of the ohomology H�(g;F)is the exterior algebra V� h, orresponding to V�[�0℄ above. It would be interesting to obtainan invariant hara terization of the remaining part of H�(g;F) for arbitrary g.

14 IGOR B. FRENKEL AND KONSTANTIN STYRKASRemark 5. In ea h of the two-dimensional spa es H1(g;F(�)) there is a unique up toproportionality ohomology lass �� divisible by ��1; the elements ����1 onstitute a basis ofH0(g;F) �= C [P℄W , asso iated with the hara ters of big proje tive modules ( f. [La℄).2. Modified regular representations of the affine Lie algebra bsl(2; C ).2.1. Regular representations of bsl(2; C ). Let G be the entral extension of the loopgroup LG, asso iated with G = SL(2; C ) (see [PS℄), and let g be the orresponding Liealgebra. As we dis ussed in the introdu tion, there is no maximal ell in the aÆne Bruhatde omposition, and thus we will use the loop version (0.7) of the �nite-dimensional one.An additional advantage is that we get an expli it realization of the left and right regularg-a tions, analogous to the �nite-dimensional ase.The standard basis of g onsists of the elements fen;hn; fngn2Zand the entral elementk, subje t to the ommutation relations[hm; en℄ = 2em+n; [hm; fn℄ = �2fm+n; [hm;hn℄ = 2mÆm+n;0k;[em; fn℄ = hm+n +mÆm+n;0k; [em; en℄ = [fm; fn℄ = 0:The Lie algebra g has a Z-grading g =Ln2Zg[n℄, determined bydeg fn = deghn = deg en = �n; degk = 0;We introdu e subalgebras g� =L�n>0 g[n℄; the �nite-dimensional Lie algebra g is naturallyidenti�ed with a subalgebra in g[0℄.The element w0 of the lassi al Weyl group de�nes an involution ! of g, su h that!(en) = �fn; !(hn) = �hn; !(fn) = �en; !(k) = �k: (2.1)We use the loop version of the �nite-dimensional Bruhat de omposition (0.4), and fa torizethe entral extension LT into the produ t of loops that extend holomorphi ally inside andoutside of the unit ir le. The analogue of (1.2) is the formal de ompositiong = exp Xn2Zxnen! w0 �k exp Xm<0 �mhm! �h0 exp Xm>0 �mhm! exp Xn2Zynen! :The polynomial algebraR0(G0) = C [fxng; fyng; f�n6=0g; ��1℄ an be thought of as the algebraof regular fun tions on the big ell of the loop group, and for R(G0) we getR(G0) = R0(G0) C [��1 ℄ =M{2ZR{(G0); R{(G0) = R0(G0) C �{Note that for ea h { the subspa e R{(G0) is a g� g-submodule of R(G0), but it is not asubalgebra of R(G0) when { 6= 0 ! It is easy to see that the in�nitesimal regular g-a tionsof the entral element k on R{(G0) are given by�l(k) = �{ � Id; �r(k) = { � Id : (2.2)As ve tor spa es, all R{(G0) are identi�ed with the same polynomial spa e, and one an ompute the in�nitesimal regular a tions of g by treating { as a omplex parameter. Inparti ular, the regular a tions of g make sense for arbitrary { 2 C . Computations yield thefollowing des ription, analogous to Proposition 1.1.

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 15Theorem 2.1. The regular a tions of g on R{(G0) are given by (2.2) and�l(en) = ��xn;�l(hn) = �2Xi2Zxi�in+n +8><>:��n + 2n{ ��n; n > 0� ��; n = 0��n; n < 0 ;�l(fn) = Xi;i02Zxixi0�xi+i0+n �Xj<0 xj�n��j � x�n� �� �Xj>0 xj�n ���j + 2j{ ��j��� { nx�n + ��2 Xj;j0>0 sj0(�2�1;�2�2; : : : ) sj(�2��1;�2��2; : : : )�yn�j+j0 ; (2.3)�r(en) = �yn;�r(hn) = �2Xi2Zyi�yi+n +8><>:��n n > 0;� ��; n = 0;��n � 2n{ ��n; n < 0: ;�r(fn) = �Xi;i02Zyiyi0�yi+i0+n +Xj>0 yj�n��j + y�n� �� +Xj<0 yj�n ���j � 2j{ ��j��� { ny�n � ��2 Xj;j0>0 sj0(�2��1;�2��2; : : : ) sj(�2�1;�2�2; : : : )�xn+j�j0 ; (2.4)where the S hur polynomials sk(�1; �2; : : : ) are de�ned bysm(�1; �2; : : : ) = Xl1;l2;:::�0l1+2l2+:::=m �l11 �l22 : : :l1!l2! : : : :Proof. The presen e of the entral extension requires the use of some elementary ases of theCampbell-Hausdor� formula in our omputations; we use the identityexp(B) exp(tA) mod t2� exp0�t 1Xj=0 1j! [B; : : : ; [B; [B;A℄℄ : : : ℄| {z }j ommutators 1A exp(B):

16 IGOR B. FRENKEL AND KONSTANTIN STYRKASFor example, to derive the last of (2.4), we use the formulas:exp Xi2Zyiei! exp (tfn) mod t2� exp (tfn) exp �tny�nk+ tXi2Zyihi+n!�� exp �tXi;i02Zyiyi0ei+i0+n! exp Xi2Zyiei! ;exp Xm>0 �mhm! exp (tfn) mod t2� exp tXj>0 sj(�2�1;�2�2; : : : )fn+j! exp Xm>0 �mhm! ;�h0 exp (tfn) mod t2� exp �t��2 fn� �h0;exp Xm<0 �mhm! exp (tfn) mod t2� exp tXj0>0 sj0(�2��1;�2��2; : : : )fn�j0! exp Xm<0 �mhm! ;w0 exp (tfn) mod t2� exp (�t en)w0:Combining these equations, we get the desired formulas. We leave the te hni al al ulationsto the reader. �2.2. Vertex operator algebras: review and useful examples. We aim to endowR{(G0)(or its modi� ation) with a stru ture similar to that of an asso iative ommutative algebra.The relevant formalism is provided by the vertex algebra theory.We re all the de�nitions of vertex and vertex operator algebras in the most onvenient tous form. For more details and equivalent alternative de�nitions, we refer the reader to thebooks on the subje t [FLM, FrB℄.Let V be a ve tor spa e, equipped with a linear orresponden ev 7! Y(v; z) =Xn2Zv(n)z�n�1; v(n) 2 End(V): (2.5)We refer to su h formal End(V)-valued generating fun tions as 'quantum �elds'.We say that V satis�es the lo ality property, if for any a; b 2 V(z � w)N [Y(a; z);Y(b; w)℄ = 0 for N � 0 (2.6)in the ring of End(V)-valued formal Laurent series in two variables z;w.A ve tor 1 2 V is alled the va uum ve tor, if it satis�esY(1; z) = IdV ; Y(v; z)1��z=0 = v: (2.7)An element D 2 End(V), is alled the in�nitesimal translation operator, if it satis�esD 1 = 0; [D;Y(v; z)℄ = ddzY(v; z); for all v 2 V: (2.8)De�nition 2. The spa e V is alled a vertex algebra, if it is equipped with a linear map(2.5), va uum ve tor 1, and in�nitesimal translation operator D, satisfying the axioms (2.6),(2.7), (2.8) above.

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 17Vertex superalgebras are de�ned as usual by inserting � signs a ording to parity. Avertex superalgebra V is alled bi-graded, if it has Z-gradings, j � j and deg,V = Mm;n2ZVm[n℄; Vm[n℄ = �v 2 V ���� jvj = m and deg v = n� ;su h that the parity in superalgebra is determined by j � j, and for any homogeneous vv 7! Y(v; z) =Xn2Zv(n)z�n�1; with jv(n)j = jvj and deg v(n) = deg v � n� 1:In parti ular, for the va uum we must have j1j = deg 1 = 0. Also, we write jY(v; z)j = jvjand degY(v; z) = deg v for the quantum �eld Y(v; z), if the above onditions are satis�ed.A vertex algebra V is alled a vertex operator algebra (VOA) of rank 2 C , if there existsan element ! 2 V, usually alled the Virasoro element, su h that the operators fLngn2Zde�ned by Y(!; z) =Xn2ZLnz�n�2;satisfy L�1 = D, and the Virasoro ommutation relations[Lm;Ln℄ = (m� n)Lm+n + Æm+n;0m3 �m12 :We de�ne the the normal ordered produ t of two quantum �elds X(z) and Y (z) by: X(z)Y (w) := X�(z)Y (w) + Y (w)X+(z);where X�(z) are the regular and prin ipal parts of X(z) =Pn2ZX(n)z�n�1,X+(z) =Xn�0X(n)z�n�1; X�(z) =Xn<0X(n)z�n�1:For produ ts of three or more quantum �elds, the normal ordered produ t is de�ned indu -tively, starting from the left. In general, the normal ordered produ t is neither ommutativenor asso iative.The following 're onstru tion theorem' is an e�e tive tool for onstru ting vertex algebras.Proposition 2.2. Let V be a ve tor spa e with a distinguished ve tor 1 and a family ofpairwise lo al End(V)-valued quantum �elds fX�(z) = Pn2ZX�(n)z�n�1g�2I: Suppose V isgenerated from 1 by the a tion of the Laurent oeÆ ients of quantum �elds X�(w), and thatthe ve tors fX�(z)1��z=0g�2I are linearly independent in V. Then the operatorsY �X�1(�n1�1) : : :X�k(�nk�1)1; z� = : X�1(z)(n1) : : :X�k(z)(nk) :;where X(z)(n) = 1n! dndznX(z), satisfy (2.6) and (2.7).If a linear operator D 2 End(V) satis�es D1 = 0 and [D;X�(z)℄ = ddzX�(z) for every� 2 I; then [D;Y(v; z)℄ = ddzY(v; z) for any v 2 V.We say that a vertex algebra V has a PBWbasis, asso iated with quantum�elds fX�(z)g�2I,if the index set I is ordered, and we have a linear basis of V, formed by the ve tors�X�1(�n1�1) : : :X�k(�nk�1)1 ����n1 � n2 � � � � � nk � 0; and if ni = ni+1; then �i � �i+1� :

18 IGOR B. FRENKEL AND KONSTANTIN STYRKASFor two mutually lo al quantum �elds X(z); Y (w) we introdu e the operator produ texpansion (OPE) formalism, and writeX(z)Y (w) �Xj Cj(w)(z �w)j ;if for a �nite olle tion of quantum �elds fCj(w)gj=1;2;::: we have the equalityX(z)Y (w) =Xj Cj(w)(z �w)j + : X(z)Y (w) :where 1(z�w)j should be expanded into the Laurent series in non-negative powers of wz . Theimportan e of OPE lies in the fa t that all ommutators [Xm; Yn℄ of Laurent oeÆ ients ofquantum �elds X(z); Y (w) are ompletely en oded by the olle tion fCj(w)g.The remainder of this subse tion presents some examples of vertex algebras, whi h willbe used in this paper. All of these algebras are bi-graded and have a PBW basis asso iatedwith given quantum �elds, for whi h we spe ify the OPEs.Example 1. We denote by F (�; ) the vertex algebra generated by quantum �elds�(z) =Xn2Z�nz�n; j�(z)j = 0; deg �(z)= 0; (z) =Xn2Z nz�n�1; j (z)j = 1; deg (z)= 0;with the operator produ t expansions�(z) (w) � 1z � w; �(z)�(w) � (z) (w) � 0: (2.9)The ommutation relations for the underlying Heisenberg algebra are[�m; n℄ = Æm+n;0; [�m; �n℄ = [ m; n℄ = 0: (2.10)Example 2. We denote by �( ; �) the vertex superalgebra generated by quantum �elds (z) =Xn2Z nz�n�1; j (z)j = �1; deg (z) = 1; �(z) =Xn2Z �nz�n; j �(z)j = 1; deg �(z) = 0;with the operator produ t expansions (z) (w) � �(z) �(w) � 0; (z) �(w) � 1z � w:The (anti)- ommutation relations for the underlying Cli�ord algebra aref m; ng = f �m; �ng = 0; f m; �ng = Æm+n;0: (2.11)

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 19Example 3. We denote by gk the vertex algebra generated by quantum �eldsXn =Xn2ZXnz�n�1; jX(z)j = 0; degX(z) = 1; X 2 g;with the operator produ t expansionsX(z)Y (w) � [X;Y ℄(w)z � w + k hX;Y i(z �w)2 ; k 2 Cwhere h�; �i is the Killing form on g. The number k is alled the level of gk.We note that a module for the vertex algebra gk is aZ-graded g-module V =Ln�n0 V [n℄,su h that �V (k) = k � IdV and g[m℄V [n℄ � V [m+ n℄ for any m;n 2Z.Example 4. We denote by Vir the vertex algebra generated by the quantum �eldL(z) =Xn2ZLnz�n�2; jL(z)j = 0; degL(z) = 2;with the operator produ t expansionL(z)L(w) � =2(z � w)4 + 2L(w)(z � w)2 + L0(w)z � w; 2 C :The number is alled the entral harge of Vir .A module for the vertex algebra Vir is a Z-graded Vir-module ~V = Ln�n0 ~V [n℄, su hthat �~V ( ) = � Id~V and L�m ~V [n℄ � ~V [m+ n℄ for any m;n 2Z.Example 5. We denote by F{(a) the vertex algebra generated by the quantum �elda(z) =Xn2Zanz�n�1; ja(z)j = 0; deg a(z) = 1;with the operator produ t expansiona(z)a(w) � 2{(z � w)2 ; { 2 C : (2.12)The ommutation relations for the underlying Heisenberg algebra H(a) are[am; an℄ = 2{mÆm+n;0: (2.13)Note that the operator a0 is entral and kills the va uum.Below we give the onstru tion of a vertex algebra, whi h will be ru ial for our future onsiderations. Let F�{(�a) be de�ned similarly to F{(a), so that[�am; �an℄ = �2{mÆm+n;0; �a(z)�a(w) � � 2{(z � w)2 : (2.14)Theorem 2.3. Let { 6= 0. The spa e ~F{ = F{(a) F�{(�a) C [P℄ has a vertex algebrastru ture, extending those of F{(a) and F�{(�a), and su h that a01� = �a01� = �1�.

20 IGOR B. FRENKEL AND KONSTANTIN STYRKASProof. Introdu e the quantum �elds fY(�;w)g�2P byY(�; z) = exp �2{Xn<0 an�nz�n! exp �2{Xn>0 an�nz�n!�� exp � �2{Xn<0 �an�nz�n! exp � �2{Xn>0 �an�nz�n! 1�: (2.15)Straightforward omputations lead to the operator produ t expansionsa(z)�a(w) � �a(z)a(w) �Y(�; z)Y(�;w)� 0;a(z)Y(�;w)� �a(z)Y(�;w)� �Y(�;w)z � w ;and establish mutual pairwise lo ality for the quantum �elds a(z); �a(z);Y(�; z).The va uum is, of ourse, the ve tor 1 1 10 2 F{ . We set Y(1�; z) = Y(�; z) for any� 2 P. The spanning and linear independen e onditions of Proposition 2.2 are immediate.Finally, we set D1� = �2{ (a�1 � �a�1)1�. The onditions on D amount toY0(�; z) = �2{ � : a(z)Y(�; z) : � : �a(z)Y(�; z) :�; (2.16)whi h is he ked dire tly. Applying Proposition 2.2, we get the desired statement. �Theorem 2.3 should be ompared with the onstru tion of latti e vertex algebras. It isknown that the spa e F{(a) C [P℄ arries a vertex algebra stru ture only for spe ial valuesof {, satisfying ertain integrality onditions.2.3. Bosoni realizations. We now pro eed to study the generalizations of the algebraR(G0). As in the �nite-dimensional ase, we study modules for the Lie algebra g� g, whi his equivalent to having two ommuting a tions of g on the same spa e.As in the lassi al ase, the regular g-a tions on R{(G0), des ribed in Theorem 2.1, an bereformulated in terms of representations of Heisenberg algebras. We note that the operators�n = �y�n n = �yn ; an =8><>:��n; n > 0���; n = 0��n � 2n{ ��n; n < 0satisfy the ommutation relations (2.10),(2.13), and similarly for��n = x�n;� n = ��xn; �an = 8><>:��n + 2n{ ��n n > 0���; n = 0��n n < 0 :Note also that C [��1 ℄ �= C [P℄, and a0 = �a0 = a a t on C [P℄ by derivations a1� = �1�.The formulas of Theorem 2.1 are parti ularly simple, when written for the generatingseries e(z);h(z); f(z). For example, (2.4) be omes

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 21�r(e(z)) = (z);�r(h(z)) = 2�(z) (z) + a(z);�r(f(z)) = ��(z)2 (z)� �(z)a(z)� {� 0(z) + exp 1{Xn6=0 an � �ann z�n! � (z)1�2: (2.17)Note that in this polynomial realization the onstants are annihilated by fengn2Zandfhngn�0. The vertex algebra formalism requires a di�erent hoi e of va uum, and the intro-du tion of normal ordering to make produ ts of quantum �elds well-de�ned. This pro edureis well-known in the theory of Wakimoto modules (see [FrB℄ and referen es therein), for whi hthe g-a tion is onstru ted by modifying the formulas originating from the semi-in�nite agvariety. In parti ular, one expe ts the shifts of the levels of the representations by the dualCoxeter number h_ = 2.The modi� ations of the formulas (2.17) leads to the following result.Theorem 2.4. Let { 6= 0; and let k = { � h_ and �k = �{ � h_, and letF{ = F (�; ) F ( ��; � ) ~F{ :(1) The spa e F{ has a gk � g�k-module stru ture, de�ned bye(z) = (z);h(z) = 2 : �(z) (z) : +a(z);f(z) = � : �(z)2 (z) : ��(z)a(z)� k� 0(z) +Y(�2; z)� (z); (2.18)�e(z) = � (z);�h(z) = 2 : ��(z)� (z) : +�a(z);�f(z) = � : ��(z)2� (z) : ���(z)�a(z)� �k �� 0(z) +Y(�2; z) (z): (2.19)(2) The spa e F{ has a ompatible VOA stru ture with rank F{ = 6. (Compatible meansthat the operators Y(v; z) are gk � g�k-intertwining operators in the VOA sense).Similar formulas for the two ommuting a tions of g were suggested in [FeP℄, by analogywith the �nite-dimensional Gauss de omposition of G. However, in order to get a mean-ingful VOA stru ture - and the orresponding semi-in�nite ohomology theory! - one mustin orporate the twist by w0, built into the Bruhat de omposition.One an re over the original Wakimoto realization from (2.18) by properly dis arding the'bar' variables. We use supers ripts 'W' to distinguish the Wakimoto gk-a tion from (2.18).Corollary 2.5. The spa e W�;k = F (�; ) F{(a) C 1� has the stru ture of a gk-modulewith k = { � h_, de�ned by the formulaseW (z) = (z);hW (z) = 2 : �(z) (z) : +a(z);fW (z) = � : �(z)2 (z) : ��(z)a(z)� k � 0(z): (2.20)The gk-module W�;k is alled the Wakimoto module.

22 IGOR B. FRENKEL AND KONSTANTIN STYRKASProof of Theorem 2.4. It suÆ es to show that modifying the standard Wakimoto a tions bythe extra terms Æf(z) = f(z)� fW (z) =Y(�2; z)� (z);Æf(z) = �f(z)� �fW (z) =Y(�2; z) (z);does not destroy the operator produ t expansions.We begin by showing that the ommutation relations for gk hold. Only those involvingthe modi�ed quantum �eld f(z) need to be onsidered. We have:eW (z) Æf(w) = (z)Y(�2; w)� (w) � 0;hW (z) Æf(w) = (2 : �(z) (z) : +a(z))Y(�2; w)� (w) �� a(z)Y(�2; w)� (w) � �2Y(�2; w)z � w � (w) = �2 Æf(w)z � w ;fW (z) Æf(w) = ��(z)a(z) (z)Y(�2; w)� (w) � 2Y(�2; w)z � w �(w)� (w);Æf(z) Æf(w) = Y(�2; z)� (z)Y(�2; w)� (w) � 0:Using the operator produ t expansions above we immediately he k thate(z)f(w) = eW (z)fW (w) + eW (z) Æf(w) � � k(z � w)2 + hW (w)z � w �+ 0 = k(z � w)2 + h(w)z �w;h(z)f(w) = hW (z)fW (w) + hW (z) Æf(w) � �2 fW (w)z � w + 0 = �2 f(w)z � w ;f(z)f(w) = fW (z)fW (w) + fW (z) Æf(w) + Æf(z) fW (w) + Æf(z) Æf(w) �� 0 + 2Y(�2; w)z � w �(w)� (w) + 2Y(�2; z)w � z �(z)� (z) + 0 � 0;and sin e the operator produ t expansions not involving f(z) are un hanged, we have provedthe ommutation relations for the (left) gk-a tion. Similarly, one veri�es the ommutationrelations for the (right) g�k-a tion.We now prove that the two a tions of gk and g�k ommute. We have�eW (z) Æf(w) = � (z)Y(�2; w)� (w) � 0;�hW (z) Æf(w) = �2 : ��(z)� (z) : +�a(z)�Y(�2; w)� (w) �� 2 � (z)z �wY(�2; w)� 2Y(�2; w)z � w � (w) � 0;

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 23whi h implies that �e(z)f(w) � �h(z)f(w) � 0: Finally, we ompute�fW (z) Æf(w) =�� : ��(z)2� (z) : ��k ��0(z)� ��(z)�a(z)��Y(�2; w)� (w)� �� �2 ��(z)� (z)z � w Y(�2; w) + �k(z �w)2Y(�2; w)����2Y(�2; w)z � w : ��(z)� (w) : +: �a(w)Y(�2; w) :z � w � 2Y(�2; w)(z � w)2 � �� (�k + 2)Y(�2; w)(z � w)2 � : �a(w)Y(�2; w) :z � w = �{Y(�2; w)(z � w)2 � : �a(w)Y(�2; w) :z � w :and similarly Æf(z)fW (w) � {Y(�2; w)(z � w)2 + : �a(w)Y(�2; w) :z � w � ��fW (z) Æf(w):Therefore,�f (z)f(w) = �fW (z)fW (w) + �fW (z) Æf(w) + Æf(z) fW (w) + Æf(z) Æf(w) � 0;and we have established the ommutativity of the two g-a tions.Proposition 2.2 implies that F{ is a vertex algebra. The formulas (2.18) an be written ase(z) = Y( �110; z);h(z) = Y(2�0 �110 + a�110; z);f(z) = Y(�(�0)2 �110 � a�1�010 � k��110 � � �11�2; z);whi h means that the quantum �elds (2.18) are spe ial ases of the operators Y(�; z). Thesame is true for the quantum �elds (2.19). Therefore, the vertex algebra stru ture is om-patible (in the vertex algebra sense) with the gk � g�k-module stru ture on F{ .To give F{ a VOA stru ture we need to introdu e the Virasoro element. The Sugawara onstru tion for the aÆne algebra gk gives a Virasoro quantum �eld with entral harge = 3kk+h_ = 3� 6{ :L(z) = 12{ � : h2(z) :2 + : e(z)f(z) : + : f(z)e(z) :� == : a(z)2 :4{ � a0(z)2{ � : � 0(z) (z) : + 1{ Y(�2; z) (z)� (z): (2.21)We also note that L(z) = LW (z)� 1{ Y(�2; z) (z)� (z);where LW (z) is the Virasoro quantum �eld given by the Sugawara onstru tion for thestandard Wakimoto realization (2.20).Similarly, the aÆne algebra g�k produ es another Virasoro quantum �eld with entral harge � = 3�k�k+h_ = 3 + 6{ :�L(z) = � 1{ � : �h2(z) :2 + : �e(z)�f(z) : + : �f(z)�e(z) :� == � : �a(z)2 :4{ + �a0(z)2{ � : �� 0(z)� (z) : � 1{ Y(�2; z) (z)� (z): (2.22)

24 IGOR B. FRENKEL AND KONSTANTIN STYRKASWe set L(z) = L(z) + �L(z) = LW (z) + �LW (z). To show that L�1 = D, we he k thatY(L�1v; z) = ddzY(v; z); v 2 F{ ; (2.23)whi h for all the generating quantum �elds follows from straightforward omputations.Finally, the rank of the VOA F{ is equal torank F{ = + � = �3 � 6{�+�3 + 6{� = 6:This on ludes the proof of the theorem. �2.4. gk � g�k-module stru ture of F{ for generi {. We now prove the analogue of theTheorem 1.3, des ribing the stru ture of the gk � g�k-module F{ for generi values of theparameter {.For � 2 h�; k 2 C we denote by V�;k the irredu ible gk-module, generated by a ve tor vsatisfying g+v = n+v = 0 and hv = � v.For any gk-module V , the restri ted dual spa e V 0 an be equipped with a gk-a tion byhgn v0; vi = �hv0; !(g�n)vi; v 2 V ; v0 2 V 0; g 2 g;where ! is as in (2.1). We denote the resulting dual module by V ?.An important sour e of gk-modules is the indu ed module onstru tion. Any g-module Vmay be regarded as a module for the subalgebra p = Ln�0 g[n℄, with g[n℄ a ting triviallyfor n > 0 and k a ting as the multipli ation by a s alar k 2 C : The indu ed gk-module Vkis de�ned as the spa e Vk = U(g)U(p) V; (2.24)with gk a ting by left multipli ation.For the remainder of this se tion, we will assume that omplex numbers {; k; �k satisfy{ =2 Q; k = { � h_; �k = �{ � h_: (2.25)Theorem 2.6. There exists a �ltration0 � F(0){ � F(1){ � F(2){ = F{ (2.26)of gk � g�k-submodules of F{ su h thatF(2){ =F(1){ �= M�2P+ V���2;k V ?���2;�k; (2.27)F(1){ =F(0){ �= M�2P+ �V�;k V ?���2;�k � V���2;k V ?�;�k� ; (2.28)F(0){ �=M�2P V�;k V ?�;�k: (2.29)Proof. The operator L0 determines a Z-grading deg of F{ , whi h is expli itly des ribed bydeg 1� = 0; degXn = �n for X = a; �a; �; ; ��; � : (2.30)The lowest graded subspa e F{ [0℄ = F (�0; 0)F ( ��0; � 0) C [P℄ of the vertex algebra F{ isidenti�ed with the Fo k spa e F for the �nite-dimensional Lie algebra g. Moreover, sin e {is generi , F{ an be onstru ted as the indu ed gk � g�k-module from the g� g-module F:F{ = U(g� g)U(p�p) F:

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 25We onstru t the �ltration (2.26) by indu ing it from the �nite-dimensional one (1.10):F(0){ = U(g� g)U(p�p) F(0) ; F(1){ = U(g� g)U(p�p) F(1) :It is easy to he k that (1.11),(1.12),(1.13) respe tively imply (2.27),(2.28),(2.29), whi hproves the theorem. �The analogue of the Corollary 1.4, des ribing the realization of the subalgebra R(G) �R(G0), is given below.Theorem 2.7. There exists a subspa e F{ � F{ , satisfying(1) F{ is a vertex operator subalgebra of F{ , and is generated by the quantum �elds(2.18), (2.19) and Y(1; z). In parti ular, F is a gk � g�k-submodule of F{ .(2) As a gk � g�k-module, F{ is generated by the ve tors f1�g�2P+, and we haveF{ �= M�2P+ V�;k V ?�;�k: (2.31)Proof. As before, we identify the lowest graded subspa e F{ [0℄ � F{ with the Fo k spa eF for the �nite-dimensional Lie algebra g. Re all from Corollary 1.4 that the g� g-moduleF ontains the distinguished submodule F. We de�ne the subspa e F{ as the gk � g�k-submodule of F{ , indu ed from F:F{ = U(g� g)U(p�p) F:It immediately follows from Corollary 1.4 that F{ is generated by the ve tors f1�g�2P+, andhas the de omposition (2.31).Next, we need to show that F{ is a vertex subalgebra. Let F0{ denote the spa e, spannedby the Laurent oeÆ ients of F{ -valued �elds Y(a; z)b for all possible a; b 2 F{. We willestablish that F0{ = F{.Indeed, F0{ is a gk � g�k-submodule of F{ , and an be indu ed from its lowest graded omponent F0 = F0{[0℄, whi h is a g� g-submodule of F. It suÆ es to prove that F0 = F.For any a; b 2 F, the lowest graded omponent of Y(a; z)b is equal to the produ t ab inthe algebra F, and sin e F is a subalgebra, we have ab 2 F. (Note that any element a 2 F an be obtained this way, for example, by taking b = 1). Using the ommutation relationswith the two opies of g, we an prove that the lowest graded omponent of Y(a; z)b lies inF for any a; b 2 F{.It follows that F0 = F and hen e F0{ = F{, whi h means that the restri tions of theoperators Y(�; z), orresponding to the subspa e F{, are well-de�ned. Thus F{ is a vertexsubalgebra of F{ . It is lear that as a vertex subalgebra F{ is generated by the quantum�elds (2.18), (2.19) and fY(�; z)g�2P+, and the latter are generated by the single operatorY(1; z).Finally, F{ ontains both LW (z) and �LW (z) - hen e also L(z) - and therefore is a vertexoperator subalgebra of F{ . �Remark 6. The vertex operator algebras F{ and F{ give expli it realizations of the modi�edregular representations R0{(G) and R0{(G0) we dis ussed in the introdu tion. It would beinteresting to onstru t them invariantly by using the orrelation fun tions approa h [FZ℄,interpreting the rational fun tions h10;Y(v1; z1) : : :Y(vn; zn)1i for v1; : : : ; vn 2 F �= F{ [0℄ assolutions of di�erential equations similar to the Knizhnik-Zamolod hikov equations.

26 IGOR B. FRENKEL AND KONSTANTIN STYRKAS2.5. Semi-in�nite ohomology of g. The fa t that the level of the diagonal a tion of g inthe modi�ed regular representations is equal to the spe ial value �2h_ allows us to introdu ethe semi-in�nite ohomology of g with oeÆ ients in F{ and in F{. In this se tion we showthat for generi values of { these ohomologies lead to the same algebras of formal hara tersas in the �nite-dimensional ase.We re all the de�nition of the semi-in�nite ohomology [Fe, FGZ℄. The main new ingredi-ent is the "spa e of semi-in�nite forms" �12 , whi h repla es the �nite-dimensional exterioralgebra �. We summarize its properties in the followingProposition 2.8. Let �12 = V g� V(g0+ � g0). Then(1) The Cli�ord algebra, generated by f�(gn); "(g0n)gg2g;g02g0;n2Zwith relationsf�(xm); �(yn)g = f"(x0m); "(y0n)g = 0; f�(xm); "(y0n)g = Æm;n hy0; xi: (2.32)a ts irredu ibly on �12 , so that for any !� 2 V g�; !+ 2 V(g0+ � g0) we have�(xn)(!� 1) = (0; n � 0(xn ^ !�) 1; n < 0 ; "(x0n)(1 !+) = (1 (x0n ^ !+); n � 00; n < 0 :(2) �12 is a bi-graded vertex superalgebra, with va uum 1 = 1 1, and generated by�(x; z) =Xn2Z�(xn)z�n�1; j�(x; z)j=� 1; deg �(x; z) = 1; x 2 g;"(x0; z) =Xn2Z"(x0�n)z�n; j"(x0; z)j=1; deg "(x0; z) = 0; x0 2 g0:(3) �12 has a g-module stru ture on the level k = 2h_, de�ned by�(xn) =Xm2ZXi : "((g0i)m)�([gi; x℄n+m) :; x 2 g:One an think of V g� as the spa e spanned by formal \semi-in�nite" forms! = �0i1 ^ �0i2 ^ �0i3 ^ : : : ; in+1 = in + 1 for n� 0;where f�jgj2N is a homogeneous basis of g�. A monomial �j1 ^ � � � ^ �jm 2 V g� is identi�edwith the semi-in�nite form with the orresponding fa tors missing:! = � �01 ^ �02 ^ � � � ^ �0j1�1 ^ �0j1 ^ �0j1+1 ^ � � � ^ �jm�1 ^ �jm ^ �jm+1 ^ : : : :In other words, if �j 2 g, then �(�j) operates as usual by eliminating the fa tor �0j .De�nition 3. The BRST omplex, asso iated with a g-module V on the level k = �2h_, isthe omplex C12 +�(g; Ck; V ) = �12 +� V , with the di�erentiald =Xn2ZXi "((g0i)n)�V ((gi)n)� 12 Xm;n2ZXi;j : "((gi)0m)"((gj)0n)�([gi; gj℄m+n) :; (2.33)where fgig is any basis of g, and fg0ig is the dual basis of g0. The orresponding ohomologyis denoted H 12 +�(g; Ck; V ).

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 27The BRST omplex above gives the relative (to the enter) version of the semi-in�nite ohomology. We don't onsider any other type of ohomology, and thus simply drop theword 'relative' everywhere. The ondition k = �2h_ is equivalent to d2 = 0.If V is a vertex algebra, then its semi-in�nite ohomology inherits a vertex superalgebrastru ture [LZ1℄.The following theorem is similar to the redu tion theorem of [FGZ℄ (see also [L℄), andrelates the semi-in�nite ohomology for generi values of { with the lassi al ohomology ofLie algebras.Theorem 2.9. Let V be a g� g-module, and let { 2 C be generi . Set k = { � h_ and�k = �{�h_, and denote V be the indu ed gk� g�k-module. Then with respe t to the diagonalg-a tion V is a level k = �2h_ module, andH 12 +�(g; Ck; V ) �= H�(g; V ):Proof. As a ve tor spa e, the module V has a de ompositionV = U(g�) V U(g+)0;where we identi�ed the fa tor U(g�), oming from the right indu ed a tion of g�k, with U(g+)0using the non-degenerate (sin e { is generi !) ontravariant pairing. Therefore, as ve torspa es C�(g; Ck; V ) = C�� C�0 C�+; (2.34)where C�� =^ g� U(g�); C�0 =^ g0 V; C�+ =^ g0+ U(g+)�We write the di�erential d as d = d� + d0 + d+ + Æ;where d� are the BRST di�erentials for g�,d� =Xn<0Xi "((g0i)n)�l((gi)n)� 12 Xm;n<0Xi;j : "((gi)0m)"((gj)0n)�([gi; gj℄m+n) :;d+ =Xn>0Xi "((g0i)n)�r((gi)n)� 12 Xm;n>0Xi;j : "((gi)0m)"((gj)0n)�([gi; gj℄m+n) :;the di�erential d0 is de�ned as in (1.24) with the g-a tion �V repla ed by�(x) = �V (x) +Xn6=0 : "((gj)0n)�([x; gj℄n) :; x 2 g; (2.35)and Æ in ludes all the remaining terms:Æ =Xn>0Xi "((g0i)n)�l((gi)n) +Xn<0Xi "((g0i)n)�r((gi)n)�� Xm>0;n<0Xi;j : "((gi)0m)"((gj)0n)�([gi; gj℄m+n) : :Following [FGZ℄, we introdu e the skewed degree f deg byf deg(w� w0 w+) = degw+ � degw�; w� 2 C�; w0 2 C0;

28 IGOR B. FRENKEL AND KONSTANTIN STYRKASwhere the 'deg' gradings in the omplexes C� are inherited from C 12 (g;k; V ). We setBp = �v 2 C 12 (g;k; V ) ���� f deg v � p� :One an he k that d� and d0 preserve the �ltered degree, and that Æ(Bp) � Bp+1. Thus,fBpgp2Zis a de reasing �ltration of the omplex C 12 (g;k; V ), and the asso iated graded omplex has the redu ed di�erentialdred = d� + d0 + d+:We now ompute the orresponding redu ed ohomology, whi h will provide a bridge toH12 +�(g; Ck; V ).It is lear that d2� = fd+;d�g = 0, and that the di�erentials d� : C�� ! C�+1� a t in theirrespe tive fa tors of (2.34). One an also he k that (d0)2 = fd0;d�g = 0; moreover,d0(C�� C�0 C�+) � (C�� C�+10 C�+)despite the fa t that d0 does not a t in C�0 .It is a well-known fa t in homologi al algebra thatHn(C+;d+) = Hn(g+;U(g+)0) = Æn;0 C ;with 1 10 2 C+ representing the non-trivial ohomology lass. Similarly, one hasHn(C�;d�) = H�n(g�;U(g�)) = Æn;0 C ;and 1 1 2 C� represents the non-trivial ohomology. Further, one an he k that thesubspa e 1 C�0 1 � C 12 +�(g; Ck; V ) is stabilized by d0, and that the g-a tion (2.35) onthat subspa e redu es to the g a tion 1 �V 1. It follows thatH�red(g; Ck; V ) �= H�(1 C0 1;d0) �= H�(C0;d) �= H�(g; V ):We now return to the ohomology of C 12 +�(g; Ck; V ). Sin e d preserves the 'deg' grading,it an be omputed separately for ea h sub omplexC12 +�(g; Ck; V )[m℄;m 2Z. The �ltrationfBp[m℄gp2Zof this omplex is �nite for ea h m, and leads to a �nitely onverging spe tralsequen e with Ep;q1 [m℄ = Hqred(Bp[m℄=Bp+1[m℄).For m 6= 0 we have Hqred(Bp[m℄=Bp+1[m℄) = 0 for all p, hen e the spe tral sequen e iszero, and H 12 +�(g; Ck; V )[m℄ = 0. For m = 0 we note thatB0[0℄ = C12 +�(g; Ck; V )[0℄; B1[0℄ = 0;whi h means that Ep;q1 [0℄ = 0 unless p = 0, and the ollapsing spe tral sequen e impliesH 12 +�(g; Ck; V )[0℄ �= H�red(B0[0℄=B1[0℄) �= H�(g;V ):This ompletes the proof of the theorem. �Corollary 2.10. The vertex superalgebras H 12 +�(g; Ck; F{);H 12 +�(g; Ck; F{ ) degenerateinto ommutative superalgebras. Moreover, we have ommutative superalgebra isomorphismsH 12 +�(g; Ck; F{ ) �= H�(g;F); H 12 +�(g; Ck; F{ ) �= H�(g;F): (2.36)In parti ular, H 12 +0(g; Ck; F{) �= H 12 +0(g; Ck; F{ ) �= C [P℄W :

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 29Proof. Theorem 2.9 gives us isomorphisms (2.36) on the level of ve tor spa es. It is also learfrom its proof that the semi-in�nite ohomology is on entrated in the subspa e of deg = 0,and thus the operators Y(�; z) on ohomology are redu ed to their onstant terms. In parti -ular, they are independent of z, whi h means that the vertex superalgebra degenerates intoa ommutative superalgebra. The multipli ation is easily tra ed ba k to the multipli ationsin F �= F{[0℄ and in the exterior algebra � = V g0, whi h shows that (2.36) are superalgebraisomorphisms. �3. Modified regular representations of the Virasoro algebra.3.1. Virasoro algebra and the quantum Drinfeld-Sokolov redu tion. In this se tionwe present a onstru tion of the regular representation of the Virasoro algebra, whi h goesin parallel with onstru tions in the previous se tions. However, instead of beginning with aspa e of fun tions on the orresponding group (whi h is, stri tly speaking, a semigroup in the omplex ase), we will use the quantum Drinfeld-Sokolov redu tion [FeFr2℄ (see also [FrB℄and referen es therein), applied to the modi�ed regular representations of g onstru tedin Se tion 1. As a result we obtain ertain bimodules over the Virasoro algebra, whi hhave the stru ture similar to their aÆne ounterparts. The result of the quantum Drinfeld-Sokolov redu tion applied to the a tual regular representation of g should have a standardinterpretation in terms of the spa e of fun tions on the Virasoro semigroup, but we will notneed this fa t for our purposes.Re all that the Virasoro algebra Vir is the in�nite-dimensional omplex Lie algebra, gen-erated by fLngn2Zand a entral element ; subje t to the ommutation relations[Lm; Ln℄ = (m� n)Lm+n + m3 �m12 :The Virasoro algebra has a Z-grading Vir = �n2ZVir[n℄, determined bydegLn = �n; deg = 0:There is a fun torial orresponden e between ertain representations of aÆne Lie algebrasand their W-algebra ounterparts, alled the quantum Drinfeld-Sokolov redu tion [FeFr2℄(see also [FrB℄ and referen es therein). We review this pro edure for the ase g = bsl(2; C ),when the orresponding W-algebra is identi�ed with the Virasoro algebra.De�nition 4. For any gk-module V , the omplex (CDS(V );dDS),CDS(V ) = V �( ; �); dDS =Xn2Z �n�V (en) + �1;is alled the BRST omplex of the quantum Drindeld-Sokolov redu tion. The orresponding ohomology is denoted HDS(V ).The BRST omplex above is very similar to the semi-in�nite ohomology omplex for thenilpotent loop algebra n+ = Ln2ZC en . Indeed, the orresponding spa e of semi-in�niteforms �12 (n+) is identi�ed with �( ; �) by �(en) � n; "(e0n) � ��n, and the only modi�- ation is the additional term �1 in the di�erential.The BRST omplex inherits the gradings j�j and deg from �( ; �) and V . Sin e jdDS j = 1,the grading j � j des ends to the ohomology HDS(V ). However, with respe t to the other

30 IGOR B. FRENKEL AND KONSTANTIN STYRKASgrading, the di�erential dDS is not homogeneous. We introdu e a modi�ed grading deg0 bydeg0 e(z) = 0; deg0 h(z) = 1; deg0 f(z) = 2;deg0 (z) = 0; deg0 �(z) = 1:The di�erential dDS then satis�es deg0 dDS = 0, and the grading deg0 des ends to HDS(V ).The ohomology H0DS(gk) of the va uum module inherits a vertex algebra stru ture. Wehave the following result (details of the proof an be found in [FrB℄).Proposition 3.1. For k 6= �h_ we have H0DS(gk) �= Vir , where = 1� 6k+h_ � 6k.For any gk-module V , the vertex algebra Vir �= H0DS(gk) a ts on H0DS(V ). For { 6= 0, set~F�;{ = H0DS(W�;{�h_): The following identi�es the Vir -module stru ture on ~F�;{.Proposition 3.2. Let { 6= 0, and let = 13 � 6{ � 6{ . Then ~F�;{ �= F{(a) C 1� as ave tor spa e, and the Vir -a tion is given byLF (z) = 14{ : a(z)2 : +{ � 12{ a0(z): (3.1)Proof. In the ve tor spa e fa torization of W�;{�h_ = F (�; ) F{(a) C1� , the di�erentialdDS a ts only in the �rst omponent. Therefore, we must have~F�;{ = HDS(F (�; )) F{(a) C 1� :A spe tral sequen e redu es the ohomology H0DS(F (�; )) to the ohomology of the semi-in�nite Weil omplex F (�; ) �( ; �). The latter splits into an in�nite produ t of �nite-dimensional Weil omplexes, and thus has one-dimensional ohomology, on entrated indegree 0.The in lusion of vertex algebras g{�h_ ,! W0;{�h_ indu es an in lusion Vir ,! ~F0;{, andthe expli it formula (3.1) for L(z) in terms of a(z) is a result of a dire t omputation. �The realization (3.1) of Virasoro modules was known long before the quantum Drinfeld-Sokolov redu tion, and is alled the Feigin-Fuks onstru tion in the literature. We use thesupers ript "F" to distinguish this standard a tion from the modi�ed Virasoro a tions, whi hwe will be onsidering later.3.2. Bosoni realization of the regular representation. The Virasoro analogue of thePeter-Weyl theorem is more subtle than in the ase of lassi al and aÆne Lie algebras. Thereis no lear way to al ulate the two ommuting Vir-a tions in a way similar to Theorem 1.2and Theorem 2.1. However, there exists a Fo k spa e realization analogous to Theorem 2.4,whi h we will all the regular representation of the Virasoro algebra.Theorem 3.3. Let { 6= 0, and let = 13 � 6{ � 6{ and � = 13 + 6{ + 6{ .(1) The spa e ~F{ has a Vir �Vir� -module stru ture, de�ned byL(z) = 14{ : a(z)2 : +{ � 12{ a0(z)� 1{ Y(�2; z); (3.2)�L(z) = � 14{ : �a(z)2 : +{ + 12{ �a0(z) + 1{ Y(�2; z): (3.3)(2) The spa e ~F{ has a ompatible VOA stru ture with rank ~F{ = 26.

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 31Proof. The formulas (3.2),(3.3) are nothing else but the result of the two-sided quantumDrinfeld-Sokolov redu tion, whi h onsists of two redu tions applied separately to the two ommuting g-a tions of Theorem 2.4, f. formulas (2.21),(2.22) and Proposition 3.2.Rather than give detailed proof of this fa t, we hoose to verify the ommutation relationsdire tly. Introdu e notationÆL(z) = L(z)� LF (z) = 1{ Y(�2; z);ÆL(z) = �L(z)� �LF (z) = � 1{ Y(�2; z):Without the additional terms ÆL(z); ÆL(z), both (3.2) and (3.3) give two ommuting opiesof the standard onstru tion (3.1) with the spe i�ed entral harges. Therefore, it suÆ esto show that the presen e of these extra terms does not violate the ommutation relationsfor Vir �Vir� .Straightforward omputations immediately show thatÆL(z) ÆL(w) � ÆL(z) ÆL(w) � ÆL(z) ÆL(w) � 0:LF (z)Y(�2; w) � Y(�2; w)(z � w)2 � 1{ : a(w)Y(�2; w) :z � w ;�LF (z)Y(�2; w)� Y(�2; w)(z � w)2 + 1{ : �a(w)Y(�2; w) :z � w :We now prove the ommutation relations for the a tion (3.2). We haveL(z)L(w)� LF (z)LF (w) = LF (z) ÆL(w) + ÆL(z)LF (w) + ÆL(z) ÆL(w) �� 1{ �Y(�2;w)(z �w)2 � 1{ : a(w)Y(�2; w) :z � w �+ 1{ �Y(�2; z)(z � w)2 + 1{ : a(z)Y(�2; w) :z �w � �� 1{ �2Y(�2; w)(z � w)2 + Y0(�2; w)z � w � � 2 ÆL(w)(z � w)2 + (ÆL)0(w)z � w ;and thereforeL(z)L(w) � LF (z)LF (w) + 2 ÆL(w)(z � w)2 + (ÆL)0(w)z � w � � =2(z � w)4 + 2LF (w)(z � w)2 + (LF )0(w)z �w �++ 2 ÆL(w)(z �w)2 + (ÆL)0(w)(z � w)2 = =2(z � w)4 + 2L(w)(z � w)2 + L0(w)z �w:We have established that adding the extra term ÆL(z) to the a tion (3.1) preserves the ommutation relations for Vir : Similarly, the formula (3.3) gives a representation of Vir� :We now show that the two a tions of Vir and Vir� ommute. Using (2.16), we getÆL(z)�LF (w) � 1{ �Y(�2; z)(z � w)2 � 1{ : �a(z)Y(�2; z) :z � w � �� 1{ �Y(�2; w)(z �w)2 + Y0(�2; w)z � w � 1{ : �a(w)Y(�2; w) :z �w � �� 1{ �Y(�2; w)(z �w)2 � 1{ : a(w)Y(�2; w) :z � w � :

32 IGOR B. FRENKEL AND KONSTANTIN STYRKASNote that (2.16) implies ÆL(z)�LF (w) � �LF (z) ÆL(w), and thusL(z)�L(w) = LF (z)�LF (w) + LF (z) ÆL(w) + ÆL(z)�LF (w) + ÆL(z)ÆL(w) � 0;whi h means that the two Virasoro a tions ommute.It is easy to see that the formula (3.2) an be written asL(z) = Y �(a�1)24{ 10 + { � 12{ a�210 + 1{ 1�2; z� ;and similarly for (3.3), whi h means that the vertex algebra stru ture is ompatible withVir �Vir� -module stru ture on ~F{ .We introdu e the VOA stru ture in ~F{ by setting L(z) = L(z) + �L(z) = LF (z) + �LF (z).One immediately he ks that L(z) is a Virasoro quantum �eld with entral harge 26; andsatis�es L�110 = 0: It suÆ es to he k the remaining relation[L�1;Y(v; z)℄ = ddzY(v; z); v 2 ~F{ ; (3.4)for ea h of the generating quantum �elds, whi h is done by dire t omputations. �3.3. Vir �Vir� -module stru ture of ~F{ for generi {. We now des ribe the so le �l-tration of the Vir �Vir� -module ~F{ for generi {, when it is ompletely analogous to the�nite-dimensional and aÆne ases, given by Theorem 1.3 and Theorem 2.6. In this subse tionwe assume that { =2 Q; = 13 � 6{ � 6{; � = 13 + 6{ + 6{:For a Vir -module ~V , the restri ted dual spa e ~V 0 an be equipped with a Vir -a tion byhLn v0; vi = hv0; L�nvi:We denote the resulting dual module by ~V ?.We denote by ~V�; the irredu ible Vir -module, generated by a highest weight ve tor ~vsatisfying L0 ~v = �~v and Ln~v = 0 for n > 0. For any � 2 h�, set�(�) = �(� + 2)4{ � �2 ; ��(�) = ��(� + 2)4{ � �2 :Theorem 3.4. There exists a �ltration0 � ~F(0){ � ~F(1){ � ~F(2){ = ~F{ (3.5)of Vir �Vir� -submodules of ~F{ su h that~F(2){ =~F(1){ �= M�2P+ ~V�(���2); ~V ?��(���2);� ; (3.6)~F(1){ =~F(0){ �= M�2P+�~V�(�); ~V ?��(���2);� � ~V�(���2); ~V ?��(�);� � ; (3.7)~F(0){ �=M�2P ~V�(�); ~V ?��(�);� : (3.8)

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 33Proof. One an derive from Proposition 3.2 that the for generi { the redu tion sends exa tsequen es of gk-modules to exa t sequen es of Vir -modules, whi h implies in parti ular thatHnDS(V�;k) = ( ~V�(�); ; n = 00; n 6= 0 ;It is then easy to he k that the images ~F(0;1;2){ of the gk � g�k-submodules F(0;1;2){ from Theo-rem 2.6 under the two-sided quantum Drinfeld-Sokolov redu tion satisfy the required prop-erties. �An alternative dire t approa h repeats the steps in the proof of Theorem 1.3. In parti ular,we get a de omposition into blo ks,~F{ = ~F{ (�1)� M�2P+ ~F{ (�): (3.9)We also have the following Virasoro analogue of Corollary 1.4 and Theorem 2.7.Theorem 3.5. There exists a subspa e ~F{ � ~F{ , satisfying(1) ~F{ is a vertex operator subalgebra of ~F{ , and is generated by the quantum �elds(2.18), (2.19) and Y(1; z). In parti ular, ~F{ is a Vir �Vir� -submodule of ~F{ .(2) As a Vir �Vir� -module, ~F{ is generated by the ve tors f1�g�2P+, and we have~F{ �= M�2P+ ~V�(�); ~V ?��(�);� : (3.10)Proof. The desired subspa e ~F{ is the image of the vertex subalgebra F{ under the two-sidedquantum Drinfeld-Sokolov redu tion. We leave te hni al details to the reader. �3.4. Semi-in�nite ohomology of Vir. The entral harge for the diagonal a tion of Virin the modi�ed regular representations is equal to the spe ial value 26. In this se tion westudy the semi-in�nite ohomology of Vir with oeÆ ients in ~F{ and in ~F{.The properties of the appropriate "spa e of semi-in�nite forms" ~�12 for the Virasoroalgebra are summarized in the followingProposition 3.6. Set ~�12 = VVir�VVir0+, where Vir� = Ln��2 CLn and Vir+ =Ln��1 CLn . Then(1) The Cli�ord algebra, generated by fbn; ngn2Zwith relationsfbm; bng = f m; ng = 0; fbm; ng = Æm+n;0: (3.11)a ts irredu ibly on ~�12 , so that for any !� 2 VVir�; !+ 2 VVir0+ we havebn(1 !) = (0; n � �11 (Ln ^ !); n � �2 ; n(! 1) = ((L0�n ^ !) 1; n � 10; n � 2 :(2) ~�12 is a bi-graded vertex superalgebra, with va uum 1 = 1 1, generated byb(z) =Xn2Zbnz�n�2; jb(z)j = �1; deg b(z) = 2; (z) =Xn2Z nz�n+1; j (z)j = 1; deg (z) = �1:

34 IGOR B. FRENKEL AND KONSTANTIN STYRKAS(3) ~�12 has a Vir-module stru ture with entral harge = �26, de�ned by�(Ln) =Xm2Z(m� n) : �mbn+m : :De�nition 5. The BRST omplex, asso iated with a Vir-module ~V with entral harge = 26, is the omplex C 12 +�(Vir; C ; ~V ) = ~�12 +� ~V , with the di�erential~d =Xn2Z �n�~V (Ln) � 12 Xm;n2Z(m� n) : �m �nbm+n : : (3.12)The orresponding ohomology is denoted H 12 +�(Vir; C ; ~V ).As in the aÆne ase, the spe ial value = 26 of the entral harge is required to ensurethat ~d2 = 0.Theorem 3.7. The vertex superalgebras H 12 +�(Vir; C ; ~F{ ) and H 12 +�(Vir; C ; ~F{ ) degen-erate into the ommutative superalgebras, and we have ommutative algebra isomorphismsH 12 +�(Vir; C ; ~F{ ) �= H12 +�(g; Ck; F{); H 12 +�(Vir; C ; ~F{ ) �= H12 +�(g; Ck; F{ ):(3.13)In parti ular, H12 +0(Vir; C ; ~F{ ) �= H 12 +0(Vir; C ; ~F{ ) �= C [P℄W :Proof. The problem of omputing the semi-in�nite ohomology of Vir, as well as its inheritedalgebra stru ture, has been extensively studied by mathemati ians and physi ists working inthe string theory. We take advantage of these results, and onstru t our proof by ombiningentire blo ks from previous papers.We note that for both ~F{ and ~F{ the diagonal a tion of Vir does not ontain additionalvertex operator shifts, and is equal to the sum of two standard Feigin-Fuks a tions.The omprehensive answer for the ohomology of tensor produ ts of Feigin-Fuks and/orirredu ible modules was given in [LZ2℄ for the most diÆ ult ase of the entral harge = p;q, orresponding to { = pq 2 Q. Simpli�ed (for the ase of generi {) version of their omputations, and the spe tral sequen e asso iated with �ltrations of Theorem 3.4, yieldH12 +n(Vir; C ; ~F{ (�)) = (C ; n = 0; 30; otherwise ;H12 +n(Vir; C ; ~F{ (�1)) = (C ; n = 1; 20; otherwise ; H 12 +n(Vir; C ; ~F{ (�)) = 8><>:C ; n = 0; 2C 2 ; n = 10; otherwise ;for ea h � 2 P+, as well as natural isomorphismsH 12 +0(Vir; C ; ~F{ (�)) �= H 12 +0(Vir; C ; ~F{ (�)):The algebra stru ture of H 12 +0(Vir; C ; ~F{ ) is in fa t independent of {, as an be seen fromthe hange of variables pn = an + �an2 ; qn = an � �an2{ :

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 35Indeed, the new ommutation relations be ome [pm; pn℄ = [qm; qn℄ = 0 and [pm; qn℄ = Æm+n;0,and the diagonal Virasoro a tion is given byL(z) =: p(z)q(z) : +p(z)� q(z):For the spe ial ase { = 1, orresponding to the pairing of = 1 and � = 25 modules, the ohomology of a bigger vertex algebra A2D =L�;�2Z~F�;1 ~F�;�1 was identi�ed in [WZ℄ withthe polynomial algebra C [x; y℄ in two variables. The subalgebra H 12 +0(Vir; C ; ~F{ ) is there-fore isomorphi to the polynomial algebra C [�℄, and we an take any nonzero ohomology lass � 2 H 12 +0(Vir; C ; ~F{ (1)) as the generator.The vertex superalgebra stru tures on H 12 +�(Vir; C ; ~F{ ) and H 12 +�(Vir; C ; ~F{ ) degen-erate into ommutative superalgebras. It is lear that both are free C [�℄-modules.It follows immediately that H 12 +�(Vir; C ; ~F{ ) �= C [�℄ V�[�℄, where we an pi k anynon-zero element � 2 H 12 +3(Vir; C ; ~F{ (0)). This settles the ase of ~F{.To get the generators of H 12 +�(Vir; C ; ~F{ ), we pi k non-zero representatives��1 2 H 12 +1(Vir; C ; ~F{ (�1)); �0 2 H 12 +1(Vir; C ; ~F{ (0));su h that �0 is not proportional to � � ��1. One an he k that �0��1 6= 0, and as in Theorem1.12 it follows that H 12 +�(Vir; C ; ~F{ ) �= C [�℄V�[��1; �0℄. The statement now follows fromTheorem 1.12 and Corollary 2.10. �Remark 7. It would be ni e to get a dire t proof of isomorphisms (3.13) by using thete hniques of the quantum Drinfeld-Sokolov redu tion.4. Extensions, generalizations, onje tures4.1. Heterogeneous vertex operator algebra. As we mentioned above, the vertex al-gebra onstru tion for the Virasoro algebra an be obtained from their aÆne analogues byapplying the two-sided quantum Drinfeld-Sokolov redu tion to the left and right g-a tion.One an onsider a similar onstru tion where the redu tion is only applied to the aÆnea tion on one side, thus leading to a vertex operator algebra with two ommuting a tionsof gk and Vir� with appropriate k; � . Indeed, one an see that the following gives a dire trealization of su h a vertex algebra.Theorem 4.1. Let { 6= 0. Set k = { �h_; � = 13 + 6{ + 6{ , and �F{ = F (�; ) ~F{ : Then(1) The spa e �F{ has a gk �Vir� -module stru ture, de�ned bye(z) = (z);h(z) = 2 : �(z) (z) : +a(z);f(z) = � : �(z)2 (z) : ��(z)a(z)� k� 0(z)�Y(�2; z); (4.1)�L(z) = � : �a(z)2 :4{ + { + 12{ �a0(z) + 1{ Y(�2; z) (z): (4.2)(2) The spa e �F{ has a ompatible VOA stru ture with rank �F{ = 28.Proof. The veri� ation of ommutation relations is straightforward. We de�ne the Virasoroquantum �eld byL(z) = 12{ �: h(z)2 :2 + : e(z)f(z) : + : f(z)e(z) :�+ h0(z)2 + �L(z) (4.3)

36 IGOR B. FRENKEL AND KONSTANTIN STYRKASThe entral harge for the Sugawara onstru tion modi�ed by h0(z)2 is equal to 3kk+h_ � 6k,and we omputerank �F{ = �3({ � h_){ � 6({ � h_)�+�13 + 6{ + 6{� = 28: �We will all the vertex operator algebra of Theorem 4.1 the heterogeneous VOA. Note (see[L℄ and referen es therein) that the entral harge = 28 appears as the riti al value in thestudy of 2D gravity in the light- one gauge!The stru ture of the bimodule �F{ in the generi ase is again quite similar to the non-semisimple bimodule R(G0). >From now on we assume that{ =2 Q; k = { � h_; � = 13 + 6{ + 6{:Theorem 4.2. There exists a �ltration0 � �F(0){ � �F(1){ � �F(2){ = �F{of gk �Vir� -submodules of �F{ , su h that�F(2){ =�F(1){ �= M�2P+ V���2;k ~V ?��(���2);� ; (4.4)�F(1){ =�F(0){ �= M�2P+ �V�;k ~V ?��(���2);� � V���2;k ~V ?��(�);� � ; (4.5)�F(0){ �=M�2P V�;k ~V ?��(�);� : (4.6)The heterogeneous VOA ontains a vertex operator subalgebra, analogous to the lassi alPeter-Weyl subalgebra R(G) � R(G0).Theorem 4.3. There exists a subspa e �F{ � �F{ , satisfying(1) �F{ is a vertex operator subalgebra of �F{ , and is generated by the �elds (4.1), (4.2),and Y(1; z). In parti ular, �F{ is a gk �Vir� -submodule of �F{ .(2) As a gk �Vir� -module, �F{ is generated by the ve tors f1�g�2P+, and we have�F{ �= M�2P+ V�;k ~V ��(�);� : (4.7)The proofs of the above theorems are obtained from their aÆne ounterparts by applyingthe quantum Drinfeld-Sokolov redu tion to (right) g�k-a tion, similarly to the Virasoro ase.The fa t that the ranks of VOAs �F{ and �F{ are equal to 28 naturally leads to the on-sideration of the semi-in�nite ohomology of these modules. We note that although thetotal Virasoro quantum �eld L(z) does not ommute with g, the spa es �F{ and �F{ an beregarded as modules over the semi-dire t produ t Virng, su h that[Lm; en℄ = �(n+m+ 1) em+n;[Lm; fn℄ = (m� n+ 1) fm+n;[Lm;hn℄ = �nhm+n +m(m+ 1)Æm+n;0 k: (4.8)

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 37The semi-in�nite ohomology is de�ned by the BRST omplex of the subalgebra Virnn+,where as before n+ =Ln2ZC en is the nilpotent loop subalgebra of g. The ondition = 28ensures that the di�erential squares to zero.De�nition 6. The BRST omplex, asso iated with a Virnn+-module �V with Virasoro entral harge = 28 is the omplex C12 +�(Vir; C ; �V ) = ~�12 +� �( ; �) �V , with thedi�erential �d =Xn2Z �n��V (Ln)� 12 Xm;n2Z(m� n) : �m �nbm+n : ++Xn2Z ��n��V (en)� Xm;n2Z�(m+ n+ 1) �m : ��n m+n : (4.9)The orresponding ohomology is denoted H 12 +�(Virnn+; C ; �V ).Proposition 4.4. The vertex algebras H 12 +�(Virnn+; C ; �F{ ) and H 12 +�(Virnn+; C ; �F{ )degenerate into ommutative algebra stru tures, and we have isomorphismsH12 +�(Virnn+; C ; �F{ ) �= H12 +�(Vir; C ; ~F{ );H12 +�(Virnn+; C ; �F{ ) �= H12 +�(Vir; C ; ~F{ ):In parti ular, H 12 +0(Virnn+; C ; �F{ ) �= H 12 +0(Virnn+; C ; �F{ ) �= C [P℄W :Proof. We use the te hnique from [L℄, where similar isomorphisms were established for rel-ative ohomology spa es. Let dn+ = Pn2Z ��n�(en) be the BRST di�erential for n+; one an show that d2n+ = 0 = fdn+; �dg, whi h leads to the spe tral sequen e asso iated withde omposition �d = dn+ + (�d � dn+). Computing the ohomology with respe t to dn+ �rst,and using Proposition 3.2, we get the desired statement. For full te hni al details (there is aslight di�eren e between BRST redu tion for n+ and quantum Drinfeld-Sokolov redu tion,but it doesn't a�e t the out ome) we refer the reader to [L℄. �4.2. General onstru tion of vertex operator algebras and equivalen e of ate-gories. The vertex operator algebras onstru ted in the previous se tions an be built, like onformal �eld theories, by pairing the left and right modules from ertain equivalent ate-gories of representations of in�nite-dimensional Lie algebras. The operators Y(�; z) are then onstru ted by pairing the left and right intertwining operators. There is a unique hoi e ofthe stru tural oeÆ ients for su h pairing that would ensure the lo ality ondition for thevertex operator algebras; these oeÆ ients are determined by the tensor stru ture on the ategory of representations. Conversely, a natural VOA stru ture on a bimodule an be usedto establish the equivalen e of the left and right tensor ategories.The vertex operator algebra onstru tions in this paper deal with the pairings of di�erent ategories of modules. In the aÆne ase, we pair the g modules on levels k and �k = �2h_�k,symmetri with respe t to the riti al level�h_; the equivalen e of the orresponding tensor ategories was studied in [Fi℄. In the Virasoro ase, we pair the modules with entral harges and � = 26 � .The theorems of Peter-Weyl type an be extended to the quantum group Uq(g), asso iatedwith G. On one hand, the modules from the ategoryO an be q-deformed into modules overUq(g); on the other hand, one an de�ne q-deformations Rq(G) and Rq(G0) of the algebras

38 IGOR B. FRENKEL AND KONSTANTIN STYRKASof regular fun tions, whi h have espe ially simple des ription for g = sl(2; C ). When q isnot a root of unity, we have the quantum analogues of isomorphisms (0.1), (0.5).The Drinfeld-Kohno theorem establishes an isomorphism of tensor ategories of representa-tions of the quantum groups and aÆne Lie algebras when q = exp � �ik+h_ �. This equivalen e,also extended to W-algebras, was made expli it in [S1℄, where intertwining operators forUq(g) were dire tly identi�ed with their VOA ounterparts for gk and W(gk); the key in-gredients were the geometri results in [V℄ on the homology of on�guration spa es. The onstru tion in [S1℄ built onformal �eld theories, asso iated to aÆne Lie algebras and W-algebras based on their quantum group ounterparts, and an be modi�ed to produ e thevertex algebras dis ussed in this paper.The Drinfeld-Kohno equivalen e also allows to ouple ategories of di�erent types, produ -ing in parti ular the heterogeneous VOA of the previous subse tion. Another important aseis the Frenkel-Ka onstru tion, whi h orresponds to the pairing of modules for g andW(g)with entral harge = dimh (see [F℄). However, in general pairings between the quantumgroups and the aÆne Lie or W-algebras lead to the generalized vertex algebra stru tures,satisfying a braided version of the ommutativity axiom. An example of su h stru ture wasproposed in [MR℄.4.3. Integral entral harge, semi-in�nite ohomology and Verlinde algebras. Inthis work we studied the stru ture of the generalized Peter-Weyl bimodules for g only forthe generi values k =2 Q of the entral harge. The stru ture of these bimodules when kis integral is more omplex and undoubtedly even more interesting. In the most spe ial ase when k = �k = �h_, we get a regular representation of the aÆne Lie algebra g at the riti al level, whi h an be viewed as the dire t ounterpart of the �nite-dimensional ase.This spa e admits a realization as a ertain spa e of meromorphi fun tions on the aÆneLie group G, and subspa es of spheri al fun tions with respe t to onjugation give rise tosolutions of the quantum ellipti Calogero-Sutherland system, generalizing the trigonometri analogue in the �nite-dimensional ase [EFK℄.Another spe ial ase is k = �h_+1; �k = �h_�1, when the quantum group degenerates intoits lassi al ounterpart. In this ase the left and right Fo k spa es used in our onstru tionea h have separate vertex algebra stru tures, and the operators Y(�2; w), whi h play animportant role in this paper, are fa tored into produ ts of left and right vertex operatorsused in the basi representations of g. The semi-in�nite ohomology of the orrespondingW-algebras is fundamental to the string theory, and was studied in [WZ℄ for the Virasoroalgebra, and in [BMP℄ for W3.For positive integral k one expe ts the existen e of trun ated versions Fk+h_ and Fk+h_of our vertex operators algebras, similar to the trun ation in the onformal �eld theory,where the positive dominant one P+ is repla ed by the al ove P+k � P+. Then the relativesemi-in�nite ohomology of g with oeÆ ients in Fk+h_ and Fk+h_ with respe t to the entershould be trun ated orrespondingly. The identi� ation of the zero semi-in�nite ohomologygroups with the representation ring of G in Corollary 2.10 leads to the following onje ture.Conje ture 1. For positive integral k, let Vk(g) denote the Verlinde algebra asso iated withintegrable level k representations of g, and let Vk(g) denote its ounterpart asso iated to thebig proje tive modules (see [La℄). Then we have ommutative algebra isomorphismsH 12 +0(g; Ck; Fk+h_ ) �= Vk(g); H 12 +0(g; Ck; Fk+h_ ) �= Vk(g):

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 39In other words, the most essential part of the VOA stru ture, embodied in the 0th oho-mology, is equivalent to the stru ture of the fusion rules of the tensor ategory of g-modules,en oded in the Verlinde algebra.It was also realized re ently (see [FHT℄ and referen es therein) that the Verlinde al-gebra Vk(g) admits an alternative realization in terms of twisted equivariant K-theoryk+h_KdimGG (G) of a ompa t simple Lie group G (whi h, in the notations of [FHT℄, is the ompa t form of the omplex Lie group whi h we denoted by G in this paper).Thanks to the results of [FHT℄, the �rst isomorphism of our onje ture an be restated ina more invariant form.Conje ture 2. We have a natural ommutative algebra isomorphismH 12 +0(g; Ck;R0k+h_ (G)) �= k+h_KdimGG (G):A on eivable dire t geometri proof of the last isomorphism might ombine the realiza-tion of the left hand side using the works [GMS℄ and [AG℄ with the interpretation of theright hand side given in the works [FHT℄ and [AS℄. A similar K-theoreti interpretation ofH12 +0(g; Ck; Fk+h_ ) in our se ond onje ture might add another twirl to the twisted equi-variant K-theory. Referen es[AG℄ S. Arkhipov, D. Gaitsgory, Di�erential operators on the loop group via hiral algebras. Int. Math. Res.Not. 2002, no. 4, 165-210[AS℄ M. Atiyah, G. Segal. Twisted K-theory. math.KT/0407054[BF℄ D. Bernard, G. Felder, Fo k representations and BRST ohomology in SL(2) urrent algebra. Comm.Math. Phys. 127 (1990), no. 1, 145{168.[BGG℄ J. Bernstein, I. Gelfand, S. Gelfand, A ertain ategory of g-modules. (Russian) Funk ional. Anal. iPrilo�zen. 10 (1976), no. 2, 1{8.[BMP℄ P. Bouwknegt, J. M Carthy, K. Pil h, The W3 algebra. Modules, semi-in�nite ohomology and BValgebras. Le ture Notes in Physi s. Monographs, 42. Springer-Verlag, Berlin, 1996.[EFK℄ P. Etingof, I. Frenkel, A. Kirillov Jr., Spheri al fun tions on aÆne Lie groups. Duke Math. J. 80(1995), no. 1, 59{90.[Fe℄ B. Feigin, Semi-in�nite homology of Lie, Ka -Moody and Virasoro algebras. (Russian) Uspekhi Mat.Nauk 39 (1984), no. 2(236), 195{196.[FeFr1℄ B. Feigin, E. Frenkel, Representations of aÆne Ka -Moody algebras and bosonization. Physi s andmathemati s of strings, 271{316, World S i. Publishing, Teane k, NJ, 1990.[FeFr2℄ B. Feigin, E. Frenkel, AÆne Ka -Moody algebras at the riti al level and Gelfand-Dikii algebras.In�nite analysis, Part A, B (Kyoto, 1991), 197{215, Adv. Ser. Math. Phys., 16, World S i. Publishing, RiverEdge, NJ, 1992.[FeFu℄ B. Feigin, D. Fuks, Verma modules over a Virasoro algebra. Funktsional. Anal. i Prilozhen. 17 (1983),no. 3, 91{92.[FeP℄ B. Feigin, S. Parkhomenko, Regular representation of aÆne Ka -Moody algebras. Algebrai and geo-metri methods in mathemati al physi s (Ka iveli, 1993), 415{424, Math. Phys. Stud., 19, Kluwer A ad.Publ., Dordre ht, 1996.[Fi℄ M. Finkelberg, An equivalen e of fusion ategories. Geom. Fun t. Anal. 6 (1996), no. 2, 249{267.[FHT℄ D. Freed, M. Hopkins, C. Teleman, Twisted K-theory and loop group representations I.math.AT/0312155[FrB℄ E. Frenkel, D. Ben-Zvi, Vertex algebras and algebrai urves. Mathemati al Surveys and Monographs,88. Ameri an Mathemati al So iety, Providen e, RI, 2001[F℄ I. Frenkel, Representations of Ka -Moody algebras and dual resonan e models, Appli ations of grouptheory in physi s and mathemati al physi s (Chi ago, 1982), 325{353,

40 IGOR B. FRENKEL AND KONSTANTIN STYRKAS[FGZ℄ I. Frenkel, H. Garland, G. Zu kerman, Semi-in�nite ohomology and string theory. Pro . Nat. A ad.S i. U.S.A. 83 (1986), no. 22, 8442{8446.[FLM℄ I. Frenkel, J. Lepowsky, A. Meurman Vertex operator algebras and the Monster. Pure and AppliedMathemati s, 134. A ademi Press, In ., Boston, MA, 1988.[FZ℄ I. Frenkel, Y. Zhu, Vertex operator algebras asso iated to representations of aÆne and Virasoro algebras.Duke Math. J. 66 (1992), no. 1, 123{168.[GMS℄ V. Gorbounov, F. Malikov, V. S he htman, On hiral di�erential operators over homogeneous spa es.Int. J. Math. Math. S i. 26 (2001), no. 2, 83{106.[HS℄ P. Hilton, U. Stammba h, A ourse in homologi al algebra. Graduate Texts in Mathemati s, Vol. 4.Springer-Verlag, New York-Berlin, 1971.[La℄ A. La howska, A ounterpart of the Verlinde algebra for the small quantum group. Duke Math. J. 118(2003), no. 1, 37{60.[L℄ B. Lian, Semi-in�nite homology and 2D quantum gravity. PhD thesis, Yale University, (1991).[LZ1℄ B. Lian, G. Zu kerman, New perspe tives on the BRST-algebrai stru ture of string theory. Comm.Math. Phys. 154 (1993), no. 3, 613{646.[LZ2℄ B. Lian, G. Zu kerman, Semi-in�nite homology and 2D gravity. I. Comm. Math. Phys. 145 (1992),no. 3, 561{593.[MR℄ G. Moore, N. Reshetikhin, A omment on quantum group symmetry in onformal �eld theory. Nu learPhys. B 328 (1989), no. 3, 557{574.[PS℄ A. Pressley, G. Segal, Loop groups. Oxford Mathemati al Monographs. Oxford S ien e Publi ations.The Clarendon Press, Oxford University Press, New York, 1986.[S1℄ K. Styrkas, Quantum groups, onformal �eld theories, and duality of tensor ategories. PhD thesis, YaleUniversity, (1998).[S2℄ K. Styrkas, Regular representations on the big ells and proje tive modules in the ategory O.math.QA/0410588[V℄ A. Var henko, Multidimensional hypergeometri fun tions and representation theory of Lie algebras andquantum groups. Advan ed Series in Mathemati al Physi s, 21. World S ienti� Publishing Co., In ., RiverEdge, NJ, 1995[W℄ F. Williams, The ohomology of semisimple Lie algebras with oeÆ ients in a Verma module. Trans.Amer. Math. So . 240 (1978), 115{127.[WZ℄ E.Witten, B.Zwieba h, Algebrai stru tures and di�erential geometry in 2d string theory. Nu l. Phys.B377 (1992) , 55{112Department of Mathemati s, Yale University, New Haven, CT 06520, USAMax-Plan k-Institut f�ur Mathematik, D-53111 Bonn, Germany