Table Of Content

FROM THE VIRASORO ALGEBRA TO KRICHEVER–NOVIKOV TYPE ALGEBRAS 3 AND BEYOND 1 0 2 MARTINSCHLICHENMAIER n a ABSTRACT. StartingfromtheVirasoroalgebraanditsrelativesthegeneraliza- J tion to higher genus compact Riemann surfaces was initiated by Krichever and 1 Novikov. The elements of these algebras are meromorphic objects which are 3 holomorphic outside a finiteset of points. A crucial and non-trivial point isto establish an almost-grading replacing the honest grading in the Virasoro case. ] G Such an almost-grading isgiven by splitting the set of points of possible poles intotwonon-emptydisjointsubsets. KricheverandNovikovconsideredthetwo- A point case. Schlichenmaier studied the most general multi-point situation with . h arbitrarysplittings. HerewewillreviewthepathofdevelopmentsfromtheVira- t soroalgebratoitshighergenusandmulti-pointanalogs.Thestartingpointwillbe a aPoissonalgebrastructureonthespaceofmeromorphicformsofallweights.As m sub-structuresthevectorfieldalgebras,functionalgebras,Liesuperalgebrasand [ therelatedcurrentalgebrasshowup.Allthesealgebraswillbealmost-graded.In detailalmost-gradedcentralextensionsareclassified. Inparticular,forthevector 1 v fieldalgebraitisessentiallyunique. Thedefiningcocyclearegiveningeometric 5 terms.Someapplications,includingthesemi-infinitewedgeformrepresentations 2 arerecalled. Finally,someremarksonthebyKricheverandSheinmanrecently 7 introducedLaxoperatoralgebrasaremade. 7 . 1 0 3 1 1. INTRODUCTION : v Lie groups and Lie algebras are related to symmetries of systems. By the use Xi of the symmetry the system can be better understood, maybe it is even possible to solve it in a certain sense. Here we deal with systems which have an infinite r a numberofindependent degreesoffreedom. TheyappearforexampleinConformal Field Theory (CFT),see e.g. [2], [67]. Butalso in the theory of partial differential equationsandatmanyotherplacesin-andoutsideofmathematicstheyplayanim- portant role. The appearing Lie groups and Lie algebras are infinite dimensional. Someofthesimplest nontrivial infinitedimensional Liealgebras aretheWittalge- bra and its central extension the Virasoro algebra. We will recall their definitions in Section 2. In the sense explained (in particular in CFT)they are related to what Partial support by the Internal Research Project GEOMQ11, University of Luxembourg, is acknowledged. 1 2 MARTINSCHLICHENMAIER is called the genus zero situation. For CFT on arbitrary genus Riemann surfaces the Krichever-Novikov (KN) type algebras, to be discussed here, will show up as algebrasofglobalsymmetryoperators. These algebras are defined via meromorphic objects on compact Riemann surfaces S of arbitrary genus with controlled polar behaviour. More precisely, poles areonly allowed atafixed finiteset ofpoints denoted by A. The“classical” exam- ples are the algebras defined by objects on the Riemann sphere (genus zero) with possiblepolesonlyat{0,¥ }. Thisyieldse.g. theWittalgebra, theclassical current algebras, including theircentralextensions theVirasoro,andtheaffineKac-Moody algebras [21]. For higher genus, but still only for two points where poles are allowed,theyweregeneralised byKricheverandNovikov[26],[27],[28]in1987. In 1990 the author [37], [38], [39], [40] extended the approach further to the general multi-point case. This extension was not a straight-forward generalization. The crucial point is to introduce a replacement of the graded algebra structure present in the “classical” case. Krichever and Novikov found that an almost-grading, see Definition 4.1 below, will be enough to do the usual constructions in representation theory, like triangular decompositions, highest weight modules, Verma modules which are de- mandedbytheapplications. In[39],[40]itwasrealizedthatasplittingofAintotwo disjoint non-empty subsets A=I∪O is crucial for introducing an almost-grading and the corresponding almost-grading was given. In the two-point situation there is only one such splitting (up to inversion) hence there is only one almost-grading, which in the classical case is a honest grading. Similar to the classical situation a Krichever-Novikovalgebra,shouldalwaysbeconsideredasanalgebraofmeromor- phicobjectswithanalmost-grading comingfromsuchafixedsplitting. I like to point out that already in the genus zero case (i.e. the Riemann sphere case) with more than two points where poles are allowed the algebras will only be almost-graded. Infact, quite anumberofinteresting newphenomena willshowup alreadythere,see[41],[15],[16],[8]. In this review no proofs are supplied. For them I have to refer to the original articlesand/ortotheforthcomingbook[53]. Forsomeapplicationsjointlyobtained withOleg Sheinman, see also [66]. Formore on the Witt and Virasoro algebra see forexamplethebook[18]. After recalling the definition of the Witt and Virasoro algebra in Section 2 we startwithdescribingthegeometricset-upofKrichever-Novikov(KN)typealgebras inSection3. WeintroduceaPoissonalgebrastructureonthespaceofmeromorphic forms(holomorphic outsideofthefixedsetAofpointswherepolesareallowed)of all weights (integer and half-integer). Special substructures will yield the function algebra, the vector field algebra and more generally the differential operator algebra. Moreover, wediscuss alsotheLiesuperalgebras ofKNtypedefinedviaforms KRICHEVER-NOVIKOVTYPEALGEBRAS 3 of weight -1/2. An important example role also is played by the current algebra (arbitrary genus-multi-point) associated toafinite-dimensional Liealgebra. In Section 4 we introduce the almost-grading induced by the splitting of A into “incoming” and“outgoing” points, A=I∪O. In Section 5 we discuss central extensions for our algebras. Central extensions appear naturally in the context of quantization and regularization of actions. We give for all our algebras geometrically defined central extensions. The defining cocycle for the Virasoro algebra obviously does not make any sense in the higher genusand/or multi-point case. Forthegeometric description weuseprojective and affine connections. In contrast to the classical case there are a many inequivalent cocycles and central extensions. If we restrict our attention to the cases where we canextendthealmost-gradingtothecentralextensionstheauthorobtainedcomplete classification anduniqueness results. Theyaredescribed inSection5.3. In Section 6 we present further results. In particular, we discuss how from the representation of the vector field algebra (or more general of the differential operator algebra) on the forms of weight l one obtains semi-infinite wedge representations (fermionic Fock space representations) of the centrally extended algebras. These representations have ground states (vacua), creation and annihila- tion operators. We add some words about b−c systems, Sugawara construction, Wess-Zumino-Novikov-Witten (WZNW) models, Knizhnik-Zamolodchikov (KZ) connections, anddeformations oftheVirasoroalgebra. Recently, a new class of current type algebras the Lax operator algebras, were introducedbyKricheverandSheinman[25],[29]. IwillreportontheminSection7. In the closing Section 8 some historical remarks (also on related works) on Krichever-Novikov type algebras and some references are given. More references canbefoundin[53]. 2. THE WITT AND VIRASORO ALGEBRA 2.1. The Witt Algebra. The Witt algebra W, also sometimes called Virasoro algebra without central term1, is the complex Lie algebra generated as vector space bytheelements {e |n∈Z}withLiestructure n [e ,e ]=(m−n)e , n,m∈Z. (1) n m n+m Oneofitsrealization isascomplexification oftheLiealgebraofpolynomialvector fieldsVect (S1)onthecircleS1,whichisasubalgebraofVect(S1),dieLiealgebra pol ¥ ofallC vectorfieldsonthecircle. Inthisrealization d e :=−iexpinj , n∈Z. (2) n dj 1Inthebook[18]argumentsaregivenwhyitismoreappropriatejusttouseVirasoroalgebra,as Wittintroduced“his”algebrainacharacteristic pcontext.Nevertheless,Idecidedtostickheretothe mostcommonconvention. 4 MARTINSCHLICHENMAIER TheLieproductistheusualLiebracketofvectorfields. Ifweextendthesegenerators tothewholepunctured complex planeweobtain d e =zn+1 , n∈Z. (3) n dz Thisgives another realization ofthe Witt algebra asthe algebra of those meromor- phicvectorfieldsontheRiemannsphereP1(C)whichareholomorphicoutside {0} and{¥ }. Letzbethe(quasi)globalcoordinate z(quasi,becauseitisnotdefinedat¥ ). Let w=1/zbethelocalcoordinateat¥ . AglobalmeromorphicvectorfieldvonP1(C) willbegivenonthecorresponding subsetswhere zresp. waredefinedas d d v= v (z) , v (w) , v (w)=−v (z(w))w2. (4) 1 2 2 1 (cid:18) dz dw(cid:19) Thefunction v willdetermine the vectorfield v. Hence, wewillusually justwrite 1 v andinfactidentifythevectorfieldvwithitslocalrepresentingfunctionv ,which 1 1 wewilldenotebythesameletter. Forthebracketwecalculate d d d [v,u]= v u−u v . (5) (cid:18) dz dz (cid:19)dz The space of all meromorphic vector fields constitute a Lie algebra. The subspace of those meromorphic vector fields which are holomorphic outside of {0,¥ } is a Liesubalgebra. Itselementscanbegivenas d v(z)= f(z) (6) dz where f isameromorphicfunctiononP1(C),whichisholomorphicoutside{0,¥ }. ThoseareexactlytheLaurentpolynomials C[z,z−1]. Consequently, thissubalgebra has the set {e ,n∈Z} as basis elements. The Lie product is the same and it can n beidentifiedwiththeWittalgebra W . Thesubalgebraofglobalholomorphicvectorfieldsishe ,e ,e i . Itisisomor- −1 0 1 C phictotheLiealgebra sl(2,C). The algebra W is more than just a Lie algebra. It is a graded Lie algebra. If we set for the degree deg(e ):=n then deg([e ,e ])=deg(e )+deg(e ) and we n n m n m obtainthedegreedecomposition W = W , W =he i . (7) n n n C Mn∈Z Notethat[e ,e ]=ne ,whichsaysthatthedegreedecompositionistheeigen-space 0 n n decomposition withrespecttotheadjoint actionofe onW. 0 Algebraically W canalsobegivenasLiealgebraofderivationsofthealgebra of Laurentpolynomials C[z,z−1]. KRICHEVER-NOVIKOVTYPEALGEBRAS 5 2.2. The Virasoro Algebra. In the process of quantizing or regularization one is often forced to modify an action of a Lie algebra. A typical example is given by the product of infinite sums of operators. Quite often they are only well-defined if acertain“normalordering” isintroduced. Inthiswaythemodifiedactionwillonly be a projective action. This can be made to an honest Lie action by passing to a suitablecentralextension oftheLiealgebra. Forthe Wittalgebra the universal one-dimensional central extension istheVira- soro algebra V. As vector space it is the direct sum V =C⊕W. If we set for x∈W, xˆ:=(0,x), and t :=(1,0) then its basis elements are eˆ , n∈Z and t with n theLieproduct 1 [eˆ ,eˆ ]=(m−n)eˆ − (n3−n)d −mt, [eˆ ,t]=[t,t]=0, (8) n m n+m 12 n n for2alln,m∈Z. Ifwesetdeg(eˆ ):=deg(e )=nanddeg(t):=0thenV becomes n n a graded algebra. The algebra W will only be a subspace, not a subalgebra of V. Itwillbeaquotient. Insomeabuse ofnotation weidentify theelementxˆ∈V with x ∈ W. Up to equivalence and rescaling the central element t, this is beside the trivial(splitting) centralextensiontheonlycentralextension. 3. THE KRICHEVER-NOVIKOV TYPE ALGEBRAS 3.1. The Geometric Set-Up. For the whole article let S be a compact Riemann surfacewithoutanyrestrictionforthegenus g=g(S ). Furthermore,letAbeafinite subsetofS . Laterwewillneedasplitting ofAintotwonon-empty disjoint subsets I and O, i.e. A=I∪O. Set N :=#A, K :=#I, M :=#O, with N =K+M. More precisely, let I=(P ,...,P ), and O=(Q ,...,Q ) (9) 1 K 1 M be disjoint ordered tuples of distinct points (“marked points”, “punctures”) on the Riemannsurface. Inparticular, weassume P 6=Q foreverypair (i,j). Thepoints i j inI arecalled the in-points, thepoints inOtheout-points. Sometimesweconsider I andOsimplyassets. Inthearticlewesometimesrefertotheclassicalsituation. Bythisweunderstand S =P1(C)=S2, I={z=0}, O={z=¥ } (10) ThefollowingFigures1,2,3exemplifythedifferent situations: Our objects, algebras, structures, ... will be meromorphic objects defined on S which are holomorphic outside of the points in A. To introduce the objects let K =KS bethecanonicallinebundleofS ,resp. thelocallyfreecanonically sheaf. The local sections of the bundle are the local holomorphic differentials. If P∈S is a point and z a local holomorphic coordinate at P then a local holomorphic differential can be written as f(z)dz with a local holomorphic function f defined in a 2Hered l istheKroneckerdeltawhichisequalto1ifk=l,otherwisezero. k 6 MARTINSCHLICHENMAIER FIGURE 1. Riemannsurfaceofgenuszerowithoneincomingand oneoutgoing point. FIGURE 2. Riemann surface ofgenus twowith one incoming and oneoutgoing point. P1 Q1 P2 FIGURE 3. Riemann surface of genus two with two incoming pointsandoneoutgoing point. neighbourhood of P. A global holomorphic section can be described locally with respect to acovering by coordinate charts (U,z) by asystem of local holomor- i i i∈J phic functions (f) , which are related by the transformation rule induced by the i i∈J coordinate changemapz =z (z)andthecondition fdz = f dz yielding j j i i i j j dz −1 j f = f · . (11) j i (cid:18)dz (cid:19) i Moreover, ameromorphic section of K is given as acollection of local meromorphicfunctions (h) forwhichthetransformation law(11)stillistrue. i i∈J KRICHEVER-NOVIKOVTYPEALGEBRAS 7 Inthefollowingl iseitheranintegerorahalf-integer. If l isanintegerthen (1)K l =K ⊗l forl >0, (2)K 0=O,thetriviallinebundle, and (3)K l =(K ∗)⊗(−l ) forl <0. Here as usual K ∗ denotes the dual line bundle to the canonical line bundle. The dual line bundle is the holomorphic tangent line bundle, whose local sections are the holomorphic tangent vector fields f(z)(d/dz). If l is a half-integer, then we first have to fix a “square root” of the canonical line bundle, sometimes called a theta-characteristics. Thismeanswefixalinebundle LforwhichL⊗2=K . After such a choice of L is done we set K l =K l =L⊗2l . In most cases we L will drop the mentioning of L, but we have to keep the choice in mind. Also the fine-structure ofthealgebras weareabouttodefinewilldepend onthechoice. But themainproperties willremainthesame. Remark3.1. ARiemannsurfaceofgenusghasexactly22g non-isomorphic square roots of K . For g= 0 we have K = O(−2), and L =O(−1), the tautological bundle, is the unique square root. Already for g = 1 we have 4 non-isomorphic ones. As in this case K = O one solution is L = O. But we have also other 0 bundlesL,i=1,2,3. NotethatL hasanon-vanishingglobalholomorphicsection, i 0 whereas this is not the case for L ,L , L . In general, depending on the parity of 1 2 3 dimH(S ,L), one distinguishes even and odd theta characteristics L. For g=1 the bundleO isanodd,theothersareeventhetacharacteristics. Weset l l l F :=F (A):={f isaglobalmeromorphic sectionofK | suchthat f isholomorphic overS \A}. (12) l We will drop the set A in the notation. Obviously, F is an infinite dimensional C-vectorspace. Recallthat inthecase ofhalf-integer l everything depends onthe thetacharacteristic L. The elements of the space Fl we call meromorphic forms of weight l (with respect to the theta characteristic L). In local coordinates z we can write such a i l formas fdz ,with f alocalholomorphic, resp. meromorphicform. i i i Specialimportantcasesoftheweightsarethefunctions(l =0),thespaceisalso denotedbyA,thevectorfields(l =−1),denoted byL,thedifferentials (l =1), andthequadratic differentials (l =2). Nextweintroduce algebraic operations onthespaceofallweights l F := F . (13) lM∈1Z 2 Theseoperations willallowustointroduce thealgebras weareheading for. 8 MARTINSCHLICHENMAIER 3.2. Associative Structure. The natural map of the locally free sheaves of rang one K l ×K n →K l ⊗K n ∼=K l +n , (s,t)7→s⊗t, (14) definesabilinearmap ·:Fl ×Fn →Fl +n . (15) Withrespecttolocaltrivialisationsthiscorrespondstothemultiplicationofthelocal representing meromorphicfunctions (sdzl ,tdzn )7→sdzl ·tdzn =s·t dzl +n . (16) Ifthere isno danger ofconfusion then wewillmostly usethe samesymbol forthe sectionandforthelocalrepresenting function. Thefollowingisobvious Proposition 3.2. The vector space F is an associative and commutative graded (over 1Z)algebras. Moreover,A =F0 isasubalgebra. 2 Definition3.3. Theassociative algebra A istheKrichever-Novikov function algebra(associated to(S ,A)). Ofcourse,itisthealgebraofmeromorphicfunctionsonS whichareholomorphic l outside of A. The spaces F are modules over A. In the classical situation A = C[z,z−1],thealgebraofLaurentpolynomials. 3.3. Lie Algebra Structure. Next we define a Lie algebra structure on the space F. Thestructure isinducedbythemap Fl ×Fn →Fl +n +1, (s,t)7→[s,t], (17) whichisdefinedinlocalrepresentatives ofthesectionsby dt ds (sdzl ,tdzn )7→[sdzl ,tdzn ]:= (−l )s +n t dzl +n +1, (18) (cid:18) dz dz(cid:19) andbilinearly extendedtoF. Proposition3.4. [43],[53] (a)Thebilinear map[.,.]definesaLiealgebra structureonF. (b)ThespaceF withrespectto·and[.,.]isaPoissonalgebra. Nextweconsidercertainimportantsubstructures. 3.4. The Vector Field Algebra and the Lie Derivative. For l =n =−1in (17) weendupinF−1 again. Hence, Proposition 3.5. The subspace L =F−1 is a Lie subalgebra, and the Fl ’s are LiemodulesoverL. KRICHEVER-NOVIKOVTYPEALGEBRAS 9 As forms of weight −1 are vector fields, L could also be defined as the Lie algebra of those meromorphic vector fields on the Riemann surface S which are holomorphic outside of A. The product (18) gives the usual Lie bracket of vector fieldsandtheLiederivative fortheiractions onforms. Duetoitsimportance letus specialize this. Weobtain (naming thelocal functions withthe samesymbol asthe section) d d df de d [e,f](z)=[e(z) ,f(z) ]= e(z) (z)− f(z) (z) , (19) | dz dz (cid:18) dz dz (cid:19)dz dg de (cid:209) (g)(z)=L (g) =e.g = e(z) (z)+l g(z) (z) (dz)l . (20) e | e | | (cid:18) dz dz (cid:19) Definition3.6. ThealgebraL iscalledKrichever-Novikovtypevectorfieldalgebra (associated to(S ,A). IntheclassicalcasethisgivestheWittalgebra. 3.5. The Algebra of Differential Operators. In F, considered as Lie algebra, A =F0 isanabelianLiesubalgebra andthevectorspacesum F0⊕F−1=A ⊕ L isalsoaLiesubalgebra ofF. Inanequivalentwayitcanalsobeconstructed as semi-direct sumofA considered asabelian Liealgebra and L operating onA by takingthederivative. Definition 3.7. This Lie algebra is called the Lie algebra of differential operators ofdegree≤1ofKNtype(associated to(S ,A))andisdenotedbyD1. Inmoredirectterms D1=A ⊕L asvectorspacedirectsumandendowedwith theLieproduct [(g,e),(h,f)]=(e.h− f.g,[e,f]). (21) ThespacesFl willbeLie-modulesoverD1. Itsuniversalenvelopingalgebrawillbethealgebraofalldifferentialoperatorsof arbitrarydegree[40],[42],[46]. 3.6. TheSuperalgebraofHalfForms. Nextweconsidertheassociative product ·F−1/2×F−1/2→F−1=L. (22) Weintroduce thevectorspaceandtheproduct S :=L ⊕F−1/2, [(e,j ),(f,y )]:=([e,f]+j ·y ,e.j − f.y ). (23) Usually we will denote the elements of L by e,f,..., and the elements of F−1/2 byj ,y ,.... Thedefinition(23)canbereformulatedasanextensionof[.,.]onL toa“super- bracket”(denoted bythesamesymbol)onS bysetting dj 1 de [e,j ]:=−[j ,e]:=e.j =(e − j )(dz)−1/2 (24) dz 2 dz 10 MARTINSCHLICHENMAIER and [j ,y ]:=j ·y . (25) We call the elements of L elements of even parity, and the elements of F−1/2 elementsofoddparity. Forsuchelements xwedenotebyx¯∈{0¯,1¯}theirparity. Thesum(23)canalsobedescribed as S =S0¯⊕S1¯, where Si¯isthesubspace ofelementsofparity i¯. Proposition 3.8. [52] The space S with the above introduced parity and product isaLiesuperalgebra. Definition3.9. ThealgebraS istheKrichever-NovikovtypeLiesuperalgebra(as- sociatedto(S ,A)). Classically this Lie superalgebra corresponds to the Neveu-Schwarz superalgebra. Seeinthiscontextalso[10],[3],[5]. 3.7. Jordan Superalgebra. Leidwanger and Morier-Genoux introduced in [30] a Jordansuperalgebra intheKrichever-Novikov setting, i.e. J :=F0⊕F−1/2=J0¯⊕J1¯. (26) Recall that A = F0 is the associative algebra of meromorphic functions. They l definethe(Jordan) product ◦isviathealgebrastructures forthespaces F by f ◦g:= f ·g ∈F0, f ◦j := f ·j ∈F−1/2 (27) j ◦y :=[j ,y ] ∈F0. Byrescaling theseconddefinitionwiththefactor1/2oneobtains aLieantialgebra. See[30]formoredetailsandadditional results onrepresentations. 3.8. Current Algebras. Westart with ga complex finite-dimensional Liealgebra andendowthetensorproduct g=g⊗CA withtheLiebracket [x⊗ f,y⊗g]=[x,y]⊗ f ·g, x,y∈g, f,g∈A. (28) The algebra g is the higher genus current algebra. It is an infinite dimensional Lie algebra and might be considered as the Lie algebra of g-valued meromorphic functions ontheRiemannsurface withpoles onlyoutside of A. Notethat weallow alsothecaseofganabelian Liealgebra. Definition3.10. ThealgebragiscalledcurrentalgebraofKricheverNovikovtype (associated to(S ,A)). Sometimesalsothenameloopalgebraisused. In the classical case the current algebra g is the standard current algebra g = g⊗C[z−1,z]withLiebracket [x⊗zn,y⊗zm]=[x,y]⊗zn+m x,y∈g, n,m∈Z. (29)