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LettBSL. RBR BSLdefineBSLNPLBRNuclBSL. PhysBSL. RBR BSLdefineBSLLMPLBRLettBSL. MathBSL. PhysBSL. RBR BSLdefineBSLCMPLBRCommBSL. MathBSL. PhysBSL. RBR BSLdefineBSLIzvLBRMathBSL. USSR IzvBSL. RBR % BSLdefineBSLmapleft#1LBRBSLsmashLBRBSLmathopLBR BSLlongleftarrowRBRBSLlimitsBSLsp LBR#1RBRRBRRBR BSLdefineBSLmapright#1LBRBSLsmashLBRBSLmathopLBR BSLlongrightarrowRBRBSLlimitsBSLsp LBR#1RBRRBRRBR BSLdefineBSLmapdown#1LBRBSLBigBSLdownarrowBSLrlapLBR $BSLvcenterLBRBSLhboxLBR$BSLscriptstyle#1$RBRRBR$RBRRBR BSLdefineBSLmapup#1LBRBSLBigBSLuparrowBSLrlapLBR $BSLvcenterLBRBSLhboxLBR$BSLscriptstyle#1$RBRRBR$RBRRBR % % BSLfontBSLssf=cmss10 BSLdefineBSLpfzLBRBSLfrac LBRBSLpartialRBRLBRBSLpartial zRBRRBR BSLdefineBSLordLBRBSLtextLBRordRBRBSL,RBR BSLdefineBSLresLBRBSLtextLBRresRBRBSL,RBR BSLdefineBSLidLBRBSLfrac 1LBR2BSLpi BSLiRBRBSLointBSLsb LBR BSLtsize CBSLsb LBRBSLtauRBRRBRRBR BSLredefineBSLd #1#2LBRBSLdeltaBSLsb LBR#1,#2RBRRBR BSLdefineBSLFunkLBRFunktsionalBSL. AnalBSL. i Prilozhen RBR BSLdefineBSLgleLBRBSLwidehatLBRglRBR(BSLinfty)RBR BSLdefineBSLglbLBRBSLoverlineLBRglRBR(BSLinfty)RBR BSLdefineBSLglLBRgl(BSLinfty)RBR BSLdefineBSLXALBRXBSLsetminus ARBR BSLdefineBSLKNLBRLBRBSLCal KNRBRRBR BSLdefineBSLKELBRBSLwidehatLBRLBRBSLCal KNRBRRBRRBR BSLdefineBSLKNALBRLBRBSLCal KNRBR(A)RBR BSLdefineBSLHlLBRLBRBSLCal HRBRBSLsp BSLlambda(A)RBR BSLdefineBSLFn #1LBRLBRBSLCal FRBRBSLsp LBR#1RBRRBR BSLdefineBSLFlLBRLBRBSLCal FRBRBSLsp BSLlambda(A)RBR BSLdefineBSLFlmLBRLBRBSLCal FRBRBSLsp LBR1-BSLlambdaRBR(A)RBR BSLdefineBSLHELBRBSLCal HRBR BSLdefineBSLfpz#1LBRBSLfrac LBRBSLpartial#1RBRLBRBSLpartial zRBRRBR BSLdefineBSLreLBRBSLtextLBRLBRBSLssf ReBSLkern1.0pt RBRRBRRBR BSLdefineBSLimLBRBSLtextLBRLBRBSLssf ImBSLkern1.0pt RBRRBRRBR % % % Up to now this were only macro definitions (not all of them % are used in the paper) % % BSLtopmatter BSLtitle Algebras of Vector Fields on Riemann SurfacesBSLBSL (Generalized Krichever - Novikov - Algebras) BSLendtitle BSLvskip 1cm BSLauthor Martin Schlichenmaier BSLendauthor BSLaffil University of Mannheim BSLendaffil BSLaddress LBR FakultBSL"at fBSL"ur Mathematik, UniversitBSL"at Mannheim, P.O. Box 103462, D-6800 Mannheim 1, Fed. Rep. of GermanyRBR BSLdate LBR February 91RBR BSLendtopmatter BSLdocument BSLvskip 1cm BSLcenterlineLBRBSLbf 1. Introduction.RBR BSLvskip 0.3cm In the framework of Conformal Field Theory the Virasoro Algebra and its representations have been discovered to be of fundamental importance. Later on mathematicians studied the Virasoro algebra as an example of a simple'' infinite dimensional Lie algebra which does not fit into the classification scheme of Kac -- Moody algebras. By the recent very fruitful contact of quantum field theory and mathematics it plays a more and more important role in mathematics. Just to name one example: the work of Frenkel, Lepowski and Meurman explaining the relation between the largest sporadic simple group (the Monster or Friendly Giant) and the coefficients of of the elliptic modular function $j$. The Virasoro algebra can be described as the universal central extension of the subalgebra $W $ of complex valued analytic vector fields on the the sphere $SBSLsp 1$ which have only a finite number of Fourier modes. For our considerations a different viewpoint is preferable. The above vector field algebra $W$ is isomorphic to the algebra of meromorphic vector fields on $BSLPBSLsp 1$ which are holomorphic on $BSLPBSLsp 1BSLsetminusBSLLBR0,BSLinftyBSLRBRBSL $. It is not only the algebra itself which is of importance, but also its graded structure. To obtain such a graded structure one chooses the special vector fields $$lBSLsb n:=zBSLsp LBRn+1RBRBSLpfzBSL .$$ The set $BSLLBRBSL;lBSLsb nBSLmid nBSLinBSLZBSL,BSLRBR$ is a basis of the vector field algebra $W$. By defining as homogeneous elements of degree $n$ the non-zero elements of $BSL WBSLsb n=BSLCBSLcdot lBSLsb nBSL $ we introduce a grading (with respect to the order of the vector fields at the point $z=0$). We obtain for the commutator $$BSLlbrack lBSLsb n, lBSLsb mBSLrbrack =(m-n)BSL; lBSLsb LBRm+nRBR,BSLqquadBSLtextLBRrespBSL.RBRBSLquad BSLlbrack WBSLsb n,WBSLsb mBSLrbrack BSLsubseteq WBSLsb LBRn+mRBRBSL . $$ By this, we see that the Lie product respects the grading. In other words, $W$ is a graded Lie algebra. By taking the Lie derivative of the forms, the vector fields operate on the forms of weight $BSLlambda$, which are meromorphic on $BSLPBSLsp 1$ and holomorphic on $BSLPBSLsp 1BSLsetminusBSLLBR0,BSLinftyBSLRBRBSL $. To obtain highest weight representations one can now construct semi-infinite wedge products'' of the forms of weight $BSLlambda$. (I will introduce them later on in more detail.) Alas, one does not obtain a (Lie-)representation of the vector field algebra anymore but only a representation of the universal central extension of $W$, the Virasoro algebra $V$: $$BSLCD 0 @>>>BSLC@>>> V@>>> W@>>>0BSLendCDBSL .$$ If $LBSLsb n, BSL nBSLinBSLZ$ denote suitable lifts of the basis elements $lBSLsb n$ and $cBSLne 0$ is a central element then they generate the Virasoro algebra. The structure equations are $$BSLgather BSLlbrack LBSLsb n,LBSLsb mBSLrbrack =(m-n) LBSLsb LBRm+nRBR+BSLfrac 1LBR12RBR(mBSLsp 3-m) BSLdeltaBSLsb LBRm,-nRBRBSLcdot cBSLBSL BSLvphantom LBRhRBRBSLBSL BSLlbrack c,LBSLsb nBSLrbrack =0BSL .BSLendgather$$ The central element $c$ operates as a scalar multiple of the identity on the wedge products. If we normalize the elements $LBSLsb n$ in such a way that the same central extension operates simultanously on every wedge product (i.e. for all weight then $c$ operates as $$-2BSL;(6BSLlambdaBSLsp 2-6BSLlambda+1)BSLcdot idBSL .$$ This expression resembles very much a formula given by Mumford in connection with divisor classes on the moduli space of algebraic curves. Indeed there is a deep relation, as was shown by Arbarello, deConcini, Kac and Procesi BSLciteLBR1RBR. >From the point of view of string theory (where a $SBSLsp 1$ is moving in $D-$dimensional space-time) the above situation describes the twice punctured world sheet of genus zero represented as a cylinder (Fig.BSLtie 1). The punctures representing incoming free and outgoing free string BSLmidspaceLBR3cmRBRBSLcaptionLBRFigure 1RBR Now there are also other world sheets possible, like for example the following (Fig.BSLtie 2): one free incoming string, one free outgoing string and nontrivial self interaction. BSLmidspaceLBR4cmRBRBSLcaptionLBRFigure 2RBR In 1987 Krichever and Novikov LBRBSLbf BSLlbrack 2,3BSLrbrack RBR generalized the above mentioned concept to this situation. In my work LBRBSLbf BSLlbrack 4-7BSLrbrack RBR I generalized this to the case: several incoming and several outgoing strings. The Virasoro case and the Krichever-Novikov case are special cases. I was able to show that after choosing a suitable grading ( this time it is only a generalized one) most of the constructions which could be done in the Virasoro case work here too. In the following, I will give a short review of the results. Let me mention, that the generalized situation was also studied by Rainer Dick LBRBSLbf BSLlbrack 8BSLrbrack RBR and by a chinese group (Guo, Na, Shen, Wang, Yu)BSLciteLBR9RBR. They were working in a different direction and did not introduce such a grading, which is necessary to make the construction with the semi-infinite wedge product. BSLbigbreak BSLvboxLBR RBR BSLbigbreak BSLcenterlineLBRBSLbf 2. The Set-upRBR BSLvskip 0.5cm Let $X$ be a (compact) Riemann surface with a fixed complex structure. We denote by $A$ a set of points, divided into two disjoint, non-empty subset $I$ and $O$ $$A=IBSLcup O,BSLqquad BSL#I=kBSLge 1,BSLquad BSL#O=lBSLge 1,BSLquad N=k+lBSL .$$ The points lying in $I$ I call in-points'', the points lying in $O$ I call out-points''. Let $BSLrho$ be a meromorphic differential with poles of exact order 1 at the points of $A$, positive (real) residues at $I$ and negative residues at $O$. Of course, the sum of the residues has to be zero. To fix $BSLrho $ uniquely I require for $BSLrho$ that it has only purely imaginary periods. Hence (after choosing a base point $BSL QBSLinBSLXABSL $) $$u(P)=BSLreBSLintBSLsb QBSLsp PBSLrho$$ is a well defined harmonic function on $BSLXA$. The level lines BSLquad (for $BSLtauBSLinBSLR$) $$CBSLsb BSLtau=BSLLBRBSL;PBSLinBSLXABSLmid u(P)=BSLtauBSL;BSLRBR$$ of the function $u$ represent in the interpretation of string theory the string configuration at proper time'' $BSLtau$. The set of all level lines $BSL BSLLBRBSL;CBSLsb BSLtauBSLmid BSLtauBSLinBSLRBSL;BSLRBRBSL $ sweeps out the whole of $BSLXA$. For $BSLtauBSLll 0$ the $CBSLsb BSLtau$ are collections of circles around the in-points, for $BSLtauBSLgg 0$ circles around the out-points. Hence the interpretation: $I$ corresponds to free incoming strings, $O$ corresponds to free outgoing strings and the condition $BSLsumBSLsb PBSLresBSLsb P(BSLrho)=0$'' corresponds to the conservation of momentum. Let me show you a picture (Fig.3): BSLmidspaceLBR5cmRBRBSLcaptionLBRFigure 3RBR Let $K$ be the canonical bundle, i.e. its sections are the differentials $f(z)BSL,dz$. Let $LBRBSLCal FRBRBSLsp BSLlambda(A)$ denote the $BSLC-$vector space of meromorphic sections of the tensor power $KBSLsp BSLlambda$, which are holomorphic on $BSLXA$. For the following I assume $BSLlambda$ to be an integer. It is possible to work with more general $BSLlambda$ but technical difficulties are involved. For example, for halfinteger $BSLlambda$ we would have to choose a square root of the canonical bundle, fix it and work with respect to this fixed additional structure on $X$. For $BSLlambda=-1$ we obtain the vector fields. We use the notation $BSLKNA=LBRBSLCal FRBRBSLsp LBR-1RBR(A)$. If $eBSLinBSLKNA$ and $fBSLinBSLFl$ then the Lie derivative $LBSLsb e(f)$ is again in $BSLFl$. This one can see immediately by considering the local representation $$BSLgather e(z)BSLsb BSLvert =BSLalpha(z)BSLpfz,BSLqquad f(z)BSLsb BSLvert =BSLbeta(z)BSL;dzBSLsp BSLlambdaBSLBSL eBSLldot fBSLsb BSLvert :=LBSLsb e(f)BSLsb BSLvert =BSLleft( BSLalpha(z)BSLfpz LBRBSLbetaRBR(z)+BSLlambdaBSLcdotBSLbeta(z) BSLfpz LBRBSLalphaRBR(z)BSLright) BSL;dzBSLsp BSLlambda BSL .BSLendgather$$ For $BSLl=-1$ this is the usual Lie bracket of vector fields, and it defines a Lie algebra structure on $BSLKNA$. By taking the Lie derivative $BSLFl$ becomes a a Lie module over $BSLKNA$. I call $BSLKNA$ the (generalized) Krichever -- Novikov algebra and $BSLFl$ the (generalized) Krichever -- Novikov module of weight $BSLl$. Of course, $BSLFl$ is also a module over the associative algebra $LBRBSLCal FRBRBSLsp 0(A)$ (these are the functions). With the commutator $LBRBSLCal FRBRBSLsp 0(A)$ becomes an abelian Liealgebra $LLBRBSLCal FRBRBSLsp 0(A)$. The algebra of differential operator (of degree $BSLle 1$) is $$LBRBSLCal DRBRBSLsp 1(A)=BSLKNABSLoplus LLBRBSLCal FRBRBSLsp 0(A)$$ with Lie product (the semi-direct product) $$BSLlbrack (e,g),(f,h)BSLrbrack =(BSLlbrack e,fBSLrbrack , LBSLsb eh-LBSLsb fg)BSL .$$ $BSLFl$ gets now a Lie module over $LBRBSLCal DRBRBSLsp 1(A)$. For almost everything what follows there are results for the whole algebra of differential operators. However, due to lack of time I will only discuss the subalgebra $BSLKNA$. Finally, there is a pairing between the spaces $BSLFl$ and $BSLFlm$. This pairing is given by $$(f,h)=BSLid fBSLotimes h BSL .$$ Here $CBSLsb BSLtau$ is a non-singular level line, and $fBSLotimes h$ is a meromorphic 1-differential. The value of the integral does not dependent on the level line chosen. BSLbigbreakBSLvboxLBR RBRBSLbigbreak BSLcenterlineLBRBSLbf 3. The GradingRBR BSLvskip 0.4cm For the definition of $BSLKNA$ and $BSLFl$ the decomposition of $A$ into $I$ and $O$ was of no importance. This will change now. For the following, let the points in $A$ be generic. We denote $$I=BSLLBRPBSLsb 1,PBSLsb 2,BSLldots,PBSLsb kBSLRBR,BSLqquad O=BSLLBRQBSLsb 1,QBSLsb 2,BSLldots,QBSLsb lBSLRBR $$ and choose a local coordinate $zBSLsb i$ at $PBSLsb i,BSL i=1,BSLldots,k$. To start with let $k=l$. We define $$M(BSLl)=(2BSLlambda-1)(g-1)-1BSL .$$ Let $BSLlambdaBSLin BSLZ, BSLlambdaBSLne 0,1$ then there exists for every $p=1,2,BSLldots,k$ and every $nBSLinBSLZ$ exactly one $fBSLsb LBRn,pRBR(BSLl)BSLinBSLFl$ with $$BSLaligned BSLordBSLsb LBRPBSLsb iRBR(fBSLsb LBRn,pRBR(BSLl))&=n-BSLd ip, BSLquad i=1,BSLldots,kBSLBSL BSLordBSLsb LBRQBSLsb iRBR(fBSLsb LBRn,pRBR(BSLl))&=-n,BSLquad i=1, BSLquad i=1,BSLldots,l-1BSLBSL BSLordBSLsb LBRQBSLsb lRBR(fBSLsb LBRn,pRBR(BSLl))&=-n+1+M(BSLl)BSLBSL fBSLsb LBRn,pRBR(BSLl)BSLsb BSLvert &=zBSLsb pBSLsp LBRn-1RBR(1+O(zBSLsb p) )dzBSLsp BSLlambdaBSLendalignedBSLtag *$$ This is proven by using the Riemann Roch theorem together with the fact that for $BSLlambdaBSLne 0,1$ the divisor $KBSLsp BSLlambda $ is non-special and the the points in $A$ are generic. For the exceptional $BSLl$ there are modifications necessary for finitely many values of $n$. These modifactions involve only the behaviour at the Points $QBSLsb i$. For $kBSLne l$ there are also modifications at the points $QBSLsb i $ necessary. Let me mention that there exists explicit representations of the elements $fBSLsb LBRn,pRBR(BSLl)$ in terms of rational function, WeierstraBSLssBSL $BSLsigma-$funktion and theta functions.BSLnp BSLproclaimLBRTheoremRBRBSLquad (a) The elements $$fBSLsb LBRn,pRBR(BSLlambda),BSLquad nBSLinBSLZ,BSLquad p=1,2,BSLldots,k$$ are a basis of $BSLFl$. BSLvskip 0.2cm (b) We have the duality relation $$BSLid fBSLsb LBRn,pRBR(BSLlambda)BSLcdot fBSLsb LBR1-m,rRBR(1-BSLlambda) =BSLd nmBSLcdot BSLd prBSL .$$ BSLvskip 0.2cm (c) There are numbers $CBSLsb LBR...,...RBRBSLsp LBR...RBR(BSLlambda) BSLinBSLC$ and $LBSLin BSLZ$ such that $$LBSLsb LBReBSLsb LBRn,pRBRRBR(fBSLsb LBRm,rRBR(BSLlambda))= LBReBSLsb LBRn,pRBRRBRBSLldot fBSLsb LBRm,rRBR(BSLlambda)= BSLsumBSLsb LBRh=n+m-2RBRBSLsp LBRn+m+LRBRBSLsumBSLsb LBRs=1RBRBSLsp k CBSLsb LBR(n,p),(m,r)RBRBSLsp LBR(h,s)RBR(BSLlambda)fBSLsb LBRh,s RBR(BSLlambda)BSL .$$ $L$ does only depend on $g,k,l,BSLlambda$. BSLendproclaim The main technique to show this is to make local calculations, respBSL. to calculate residues. Please note, that the Virasoro case and the Krichever - Novikov case are contained as special case ($k=l$ and index shift). We can reformulate (c) . For this we define $BSL BSLdeg(fBSLsb LBRn,pRBR(BSLl))=nBSL $ and obtain a grading of $BSLFl$. $$BSLFl=BSLbigoplusBSLsb LBRnBSLinBSLZRBRLBRBSLCal FRBRBSLsb n BSLsp BSLlambda(A),BSLqquad LBRBSLCal FRBRBSLsb nBSLsp BSLlambda(A)=BSLlangleBSL fBSLsb LBRn,pRBR (BSLlambda) BSLmid p=1,BSLldots,kBSL BSLrangle $$ is now the decomposition into (finite dimensional) homogeneous spaces. Now $BSLFl$ is not necessarily a graded module over $BSLKNA$, but by (c) a generalized graded module. If we change the numbering of points in the set $I$ we do not change the grading. If we change the numbering of the elements in $O$ we change the grading. Nevertheless the filtration $$LBRBSLCal FRBRBSLsb LBR(n)RBRBSLsp BSLlambda(A)=BSLbigoplus BSLsb LBRmBSLge nRBR LBRBSLCal FRBRBSLsb nBSLsp BSLlambda(A)BSL $$ induced by the grading will be the same. Hence, the decomposition of $A$ fixes a filtration. A different decomposition yields a non equivalent filtration. Let me add here that it is possible to weight each in or out point in a different manner (for example, one can apply a constant integer shift to the order at certain points or multiply the order at certain points with a positive integer) as long as the overall effect cancels in a suitable manner. The filtration induced by the new grading will be equivalent to the filtration induced by the prescription (*) above. BSLbigbreakBSLvboxLBR RBR BSLbigbreak BSLcenterlineLBRBSLbf 4. Wedge-Product RepresentationsRBR BSLvskip 0.4cm To obtain representations with certain features (in particular, such which are of interests in physics) we want to extend the Lie representation of $BSLKNA$ on $BSLFl$ to the highest exterior power'' of $BSLFl$. Of course, written as this it is nonsense, because of the infinite dimensionality of $BSLFl$. Exploiting the grading, we can make sense out of this and obtain so called highest weight representations'' (for simplicity, I do not distinguish between highest weight and lowest weight). The representation space are the vector space $LBRBSLCal HRBRBSLsp BSLlambda(A) $ of semi-infinite formes of weight $BSLlambda$ with respect to the grading induced by the basis elements. To construct these spaces, we start with the basis $BSL fBSLsb LBRn,pRBR,nBSLinBSLZ,p=1,2,BSLdots,kBSL $ of the space $BSLFl$. I suppress the $BSLl$ in the notation. A basis of $BSLHl$ is given by the elements $$BSLpsi=fBSLsb LBRiBSLsb 1,pBSLsb 1RBRBSLwedge fBSLsb LBRiBSLsb 2,pBSLsb 2RBRBSLwedgeBSLcdotsBSLwedge fBSLsb LBRm,1RBRBSLwedge fBSLsb LBRm,2RBRBSLwedgeBSLcdotsBSLwedge fBSLsb LBRm+1,1RBRBSLwedgeBSLcdotsBSL .$$ Here the indices are in strictly increasing lexikographical order and starting with one index (depending on the base element $BSLpsi$) all following indices will occur. Special base elements are $$BSLPhiBSLsb T=fBSLsb LBRT,1RBRBSLwedge fBSLsb LBRT,2RBRBSLwedgeBSLcdotsBSLwedge fBSLsb LBRT,kRBRBSLwedge fBSLsb LBRT+1,1RBRBSLwedge fBSLsb LBRT+1,2RBRBSLwedgeBSLcdotsBSL .$$ This vectors I call vacuum vector of level $T$. Now the action of $BSLKNA$ on $BSLFl$ should be extended to the highest exterior power'' $BSLHl$. Again guided by the finite dimensional situation we try the naiv Ansatz $$eBSLldot BSLpsi= (eBSLldot fBSLsb LBRiBSLsb 1RBR)BSLwedge fBSLsb LBRiBSLsb 2RBR BSLwedgeBSLcdots +fBSLsb LBRiBSLsb 1RBRBSLwedge (eBSLldot fBSLsb LBRiBSLsb 2RBR) BSLwedgeBSLcdots +BSLcdotsBSL .BSLtag **$$ For simplicity with respect to the notation I restricted myself to the case $k=1$ and suppressed the 2. index. The wedge sign gives the rules how the result has to be written in terms of the bases. The rules are: (1) it is linear in every argument, (2) if 2 factors have to be exchanged, then the expression changes its sign (i.e. it is zero if two factors are identical). Note, this is in fact the definition which works if the vector space one starts with is finite dimensional. Unfortunately, this naiv definition makes only sense if in the sum (**) only finitely many summands will be nonzero. This is always the case if the vector field $e$ is a element of either one of the following subalgebras $$BSLalign BSLKNBSLsp +(A)&=BSLlangle eBSLsb LBRn,pRBRBSLmid nBSLge 3,BSL p=1,BSLldots,kBSLrangleBSLBSL BSLKNBSLsp -(A)&=BSLlangle eBSLsb LBRn,pRBRBSLmid nBSLle -1-L, ,BSL p=1,BSLldots,kBSLrangleBSLBSL BSLendalign$$ This works due to the generalized graded structure. For the (vector space) complement between these two algebras, the action is not well defined. To give an example: we try to calculate $eBSLsb 2BSLldotBSLPhiBSLsb T$. We obtain $$eBSLsb 2BSLldot fBSLsb m=BSLleft((m-1)+BSLlBSLright)fBSLsb m+ BSL BSLtextLBRhigher order termsRBRBSL .$$ Alltogether $$eBSLsb 2BSLldot BSLPhiBSLsb T=''BSLsumBSLsb LBRm=TRBRBSLsp BSLinfty BSLleft((m-1)+BSLlBSLright)BSLPhiBSLsb TBSL .$$ Hence, the action has to be modified. The Physicists call this modification regularization. Unfortunately, the regularized action will only be a projective Lie action, not a linear Lie action any more. This defines a Lie action only for a central extension $BSLKE(A)$ of the vector field algebra. Before I want to say a few words on the mathematical technique of this process, let me give the results. Let $BSL t BSL $ be the central element. If we require the cocycle defining the central extension to be independent of the weight $BSLl$ (i.e. we obtain one central extension operating on all $BSLHl$ for every $BSLl$) then $t$ operates as multiplication with the scalar $$cBSLsb BSLl=-2BSL;(6BSLlBSLsp 2-6BSLl+1)BSL .$$ Starting from the vacuum vector $BSLPhiBSLsb T$ we obtain a submodule $BSL BSLKE(A)BSLldotBSLPhiBSLsb TBSL $ which generalizes the highest weight modules in the Virasoro case. If $EBSLsb LBR2,pRBR$ is a suitable lift of $eBSLsb LBR2,pRBR$ and if we consider $BSLKNBSLsp +(A)$ as submodule of $BSLKE(A)$ we obtain $$BSLalign BSLKNBSLsp +(A)BSLldotBSLPhiBSLsb T&=0BSLBSL EBSLsb LBR2,pRBRBSLldotBSLPhiBSLsb T&=-BSLfrac 12(T-1)(T-2+2BSLl) BSLPhiBSLsb TBSLBSL tBSLldot BSLPhiBSLsb T&=-2BSL;(6BSLlBSLsp 2-6BSLl+1)BSLPhiBSLsb TBSL . BSLendalign $$ There are two different possibilites to obtain this central extension. The first one is a power series method, the second one is a method which works by embedding into the algebra of infinite matrices. By these methods one obtains cohomologous extensions. Here I will concentrate on the 2. method because it is a generalization of the method which was applied in the Virasoro case BSLciteLBR10RBR,BSLciteLBR11RBR. Let $BSL BSLglbBSL $ be the Lie algebra of infinite matrices with only finitely many diagonals. A matrix $BSL A=(aBSLsb LBRijRBR)BSL $ with $(i,j)BSLinLBRBSLZ BSLtimesBSLZRBR$ is a element of this algebra if there is a number $d=d(A)$ with $aBSLsb LBRijRBR=0$ if $BSLvert i-jBSLvert >d$. The action of $BSLKNA$ on $BSLFl$ defines with respect to the basis $fBSLsb LBRn,pRBR$ an embedding of $BSLKNA$ into $BSLglb$. For this, the generalized grading is essential. For $BSLglb$ there exists a well defined modification of the action on $BSLHl$ and a well defined 2-cocycle defining a central extension $BSLgle$ acting on $BSLHl$ (see BSLciteLBR12RBR,BSLciteLBR13RBR,BSLciteLBR14RBR). By pullback one obtains a central extension $BSLKE(A)$ and an action of $BSLKE(A)$ on $BSLHl$. BSLvskip 0.5cm The spaces $BSLHl$ carry a very rich structure. BSLnoindent (1) Of course, there is also a central extension of the algebras of differential operators (of degree $BSLle 1$) acting on $BSLHl$. BSLnoindent (2) It is possible to introduce a natural pairing between right semi-infinite forms of weight $BSLl$ (the forms introduced above) and left semi-infinite forms of weight $(1-BSLl)$ induced by the pairing of $BSLFl$ and $BSLFn LBR1-BSLlRBR(A)$. With respect to this pairing the $EBSLsb LBRn,pRBR$ are selfadjoint operators. BSLnoindent (3) $BSLFl$ operates by wedging'' and $BSLFn LBR1-BSLlRBR$ operates by contracting'' on $BSLHl$. The operators defined by this procedure, generate a Clifford algebra. Physicists call this a $b-c$ system. BSLnoindent (4) There are very interesting compatibility relations of all these structures which deserve further investigation. BSLbigbreak BSLvboxLBR RBR BSLbigbreak BSLcenterlineLBRBSLbf 5. Central ExtensionsRBR BSLbigbreak There is a way to obtain central extensions of the above algebras by purely geometric means. Let me say a few words on central extensions. A central extension $BSLKE(A)$ of the algebra $BSLKNA$ can be given as a short exact sequence of Liealgebras $$BSLCD 0@>>> BSLC@>>>BSLKE(A) @>BSLpsi>>BSLKNA@>>>0BSL .BSLendCDBSLtag ***$$ The central extensions are classified (up to equivalence'') by the elements of $HBSLsp 2(BSLKNA,BSLC)$. These cohomology classes can be represented by 2-cocycles, i.eBSL. by bilinear maps $$BSLchi:BSLKNABSLtimesBSLKNABSLtoBSLC$$ with $$ BSLgather BSLchi(f,g)=-BSLchi(g,f)BSLBSL BSLchi(BSLlbrack f,gBSLrbrack ,h)+ BSLchi(BSLlbrack g,hBSLrbrack ,f)+ BSLchi(BSLlbrack h,fBSLrbrack ,g)=0 BSL . BSLendgather$$ A 2-cocycle $BSLchi$ is a coboundary if it can be written as $$BSLchi(f,g)=BSLkappa(BSLlbrack f,gBSLrbrack ),$$ where $BSLkappa$ is a linear form. The connection with (***) can be obtained by choosing a linear lift $BSLphi$ of the Lie homomorphismus $BSLpsi$. The 2-cocycle $BSLchi$ can be given by $$BSLchi(f,g)BSLcdot t=BSLlbrack BSLphi(f),BSLphi(g)BSLrbrack -BSLphi (BSLlbrack f,gBSLrbrack ),$$ where $t$ is the image of $1$ in $BSLKNA$. In particular, if the elements $eBSLsb LBRn,pRBR$ are a basis of $BSLKNA$, then a basis of $BSLKE(A)$ can be given by $EBSLsb LBRn,pRBR=BSLphi(eBSLsb LBRn,pRBR)$ and the central element $t$. The structure equations for $BSLKE(A)$ read as $$BSLlbrack EBSLsb LBRn,pRBR,EBSLsb LBRm,rRBRBSLrbrack =BSLsumBSLsb LBR h=n+m-2RBRBSLsp LBRn+m+LRBRBSLsumBSLsb LBRs=1RBRBSLsp LBRkRBR CBSLsb LBR(n,p),(m,r)RBRBSLsp LBR(h,s)RBR(-1)EBSLsb LBRh,sRBR+ BSLchi(eBSLsb LBRn,pRBR,eBSLsb LBRm,rRBR)BSLcdot tBSL .$$ To obtain such 2-cocycle I use a method which generalizes what was done by Krichever and Novikov in the 2 point case BSLciteLBR3RBR. If $$eBSLsb BSLvert =f(z)BSLpfz,BSLqquad hBSLsb BSLvert =t(z)BSLpfz $$ are local forms of the vector fields and $R$ is a projective holomorphic connection on $X$ then the following defines a 2-cocycle $$BSLchi(e,h)=BSLfrac cLBR24BSLpi BSLiRBRBSLintBSLsb LBRCBSLsb BSLtauRBR BSLfrac 12BSLleft((f'''t- ft''')-RBSLcdot (f't-ft')BSLright)dz. $$ (1) In the case $g=0$ and 2 points this cocycle reduces to the Virasoro cocycle. BSLnoindent (2) $R$ is necessary to make the definition independent on the coordinates chosen (this is of course necessary for higher genus) BSLnoindent (3) The difference of 2 projective connections is a quadratic differential. If we choose another connection the 2-cocycle will change only by a coboundary. Hence, the cohomology class will not depend on the projective connection chosen. BSLnoindent (4) The cocycle is local (with respect to the introduced grading), i.eBSL. there are constants $A$ and $B$ such that $$BSLchi(eBSLsb LBRn,pRBR,eBSLsb LBRm,rRBR)=0,BSLquad BSLtextLBRifRBRBSLquad n+mBBSL .$$ The defining cocycle for the extension $BSLKE(A)$ operating on $BSLHl$ is also local, and hence there is the question whether it can be given by such a $BSLchi$ as above. By some modification of the arguments of Krichever -- Novikov in BSLciteLBR3RBR it should be possible to prove the following conjecture, which answers the question to the positive. BSLproclaimLBRconjectureRBR Every local cocycle is cohomologous to such a $BSLchi$ with a suitable projective connection. BSLendproclaim The details of the proof'' I have not yet worked out. It is possible to construct such geometric cocycles for $LBSLFn 0(A)$ and $LBRBSLCal DRBRBSLsp 1(A)$. For the former one the cocycle can be given by $$BSLgamma(f,g)=BSLid fdgBSL .$$ I call the algebra $BSLwidehatLBRBSLFn 0RBR(A)$ a generalized Heisenberg algebra. Starting with this algebra I can introduce generalization of the affine Kac-Moody algebras. For $LBRBSLCal DRBRBSLsp 1$ the cocycles $BSLchi$ and $BSLgamma$ also define central extensions. Beside them, there is another (linearly independent) cocycle $$BSLbeta(e,t)=BSLid BSLleft( ft''+TBSLcdot f t'BSLright)dzBSL .$$ Here $T$ is a meromorphic connection, which is allowed to have poles at the points of $A$. In the Virasoro case these 3 cocycles are a basis for $BSL HBSLsp 2(LBRBSLCal DRBRBSLsp 1(A),BSLC)BSL ,$ as was proven in BSLciteLBR1RBR. BSLbigbreak BSLbigbreak BSLcenterlineLBRBSLbf ReferencesRBR BSLvskip 0.4cm BSLparindent=0ptBSLparskip=3pt BSLrefBSLkey LBRBSLbf BSLlbrack 1BSLrbrack RBR BSLby E. Arbarello, C. deConcini, V.G. Kac, C. Procesi BSLpaper Moduli Spaces of Curves and Representation Theory BSLjourBSLCMP BSLvol 117BSLyr 1988BSLpages 1--36 BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 2BSLrbrack RBR BSLby I.M. Krichever, S.P. Novikov BSLpaper Algebras of Virasoro Type, Riemann Surfaces and Structures of the Theory of Solitons BSLjour BSLFunkBSLvol 21BSLissue(2) BSLpages 46BSLyr 1987BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 3BSLrbrack RBR BSLby I.M. Krichever, S.P. Novikov BSLpaper Virasoro Type Algebras, Riemann Surfaces and Strings in Minkowski Space BSLjour BSLFunkBSLvol 21BSLissue(4) BSLpages 47BSLyr 1987BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 4BSLrbrack RBRBSLby M. Schlichenmaier BSLpaper Krichever-Novikov Algebras for More Than Two PointsBSLpaperinfo (preprint April 89)BSLjour BSLLMPBSLvol 19 BSLpages 151--165BSLyr 1990 BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 5BSLrbrack RBRBSLby M. Schlichenmaier BSLpaper Krichever-Novikov Algebras for More Than Two Points: Explicit Generators BSLpaperinfo (preprint Juli 89)BSLjour BSLLMPBSLvol 19BSLpages 327--336 BSLyr 1990 BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 6BSLrbrack RBRBSLby M. Schlichenmaier BSLpaper Central Extensions and Semi-infinite Wedge Representations of Krichever-Novikov Algebras for More Than Two Points BSLpaperinfo (preprint September 89)BSLjour BSLLMPBSLvol 20BSLpages 33--46 BSLyr 1990 BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 7BSLrbrack RBRBSLby M. SchlichenmaierBSLpaper Verallgemeinerte Krichever -- Novikov Algebren und deren Darstellungen BSLpaperinfo PhD. Thesis 1990 UniversitBSL"at Mannheim, Germany BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 8BSLrbrack RBRBSLby R. Dick BSLpaper Krichever-Novikov-like Bases on Punctured Riemann Surfaces BSLjour BSLLMP BSLpaperinfo desy preprint 89-059 (Mai 89) BSLvol 18BSLyr 1989BSLpages 255BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 9BSLrbrack RBRBSLby H-yBSL. Guo, J-sBSL . Na, J-mBSL. Shen, S-kBSL. Wang, Q-hBSL . Yu BSLpaper The Algebras of Meromorphic Vector Fields and Realization on the Space of Meromorphic $BSLl$ - Differentials on Riemann Surfaces (I) BSLpaperinfo preprint AS-ITP-10-89 BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 10BSLrbrack RBR BSLby B.L. Feigin, D.B. Fuks BSLpaper Invariant Skew-Symmetric Differential Operators on the Line and Verma Moduls over the Virasoro Algebra BSLjourBSLFunk BSLvol 16BSLissue(2)BSLyr 1982BSLpages 47--63 BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 11BSLrbrack RBR BSLby V.G. Kac, A.K. Raina BSLbook Highest Weight Representations of Infinite Dimensional Lie Algebras BSLbookinfo AdvBSL. SerBSL. in MathBSL. Physics Vol.2 BSLpubl World Scientific BSLyr 1987 BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 12BSLrbrack RBRBSLby E.BSLtie Date, M.BSLtie Jimbo, T.BSLtie Miwa, M.BSLtie Kashiwara BSLpaper Transformation groups for Soliton Equations BSLjour Publications RIMS Kyoto BSLvol 394BSLyr 1982BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 13BSLrbrack RBRBSLby V.GBSL. Kac, D.HBSL. Peterson BSLpaper Spin and Wedge Representations of Infinite-dimensional Lie Algebras and Groups BSLjour ProcBSL. NatBSL. AcadBSL. SciBSL. USA BSLvol 78BSLyr 1981BSLpages 3308-3312BSLendref BSLrefBSLkey LBRBSLbf BSLlbrack 14BSLrbrack RBRBSLby J.LBSL. VerdierBSLpaper Les representations der algBSLebres de Lie affines: applications BSLa quelques problBSLemes de physique (d'aprBSLes EBSL. Date, MBSL.BSLtie Jimbo, MBSL.BSLtie Kashiwara, TBSL.BSLtie Miwa) BSLpaperinfo SemBSL. Bourbaki, Exp. 596BSLjour Asterisque BSLvol 92-93 BSLendref BSLenddocument ENDBODY