% Needs AMSTEX 2.1 (incl. fonts) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% Uses AMSTEX 2.1 and psbox package %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This collection of files was produced with CERN psbox package % To decompose and tex it: %-save this with a filename CONTAINING ONLY LETTERS and a .TEX % extension (say, JOINTFIL.TEX), in some uncrowded directory; %-make sure you can \input psbox.tex (version>=1.3); % (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get % and tex the file psboxall.tex; more info in psbREAD.ME) %-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by % the first file in the joining (after splitting JOINTFIL.TEX into % it's constituents, TeX will try to process it as it stands). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% use amstex %%%%% %% If it is impossible for you to obtain psbox and/or you %% can not include %% Postscript figures, then you can also split this %% File by hand. Besides the figures it contains %% only a file of private macros called diffmac.tex %% just ignore all error messages related to the fidgures %% Martin Schlichenmaier %% if all fails %% make anonymous ftp to lie.math.uni-mannheim.de %% and obtain a postscript versions as file %% papers/diff.ps or request a hardcopy %% e-mail: schlichenmaier at math.uni-mannheim.de %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BODY \input amstex \documentstyle{amsppt} % \input psbox.tex \splitfile{\jobname} \let\autojoin=\relax %Beginning-Of-File-Named:diff % Differentialoperator article Clausthal talk %\title %Differential Operator Algebras %on compact Riemann Surfaces${ }^1$ %\endtitle %\author Martin Schlichenmaier \endauthor %\address %Department of Mathematics and Computer Science, %University of Mannheim, %D-68131 Mannheim, Germany %\endaddress % % % amstex 2.1 % % % To include encapsulated Postscript images % \input psbox \let\fillinggrid=\relax % \loadbold % % local macros % \input diffmac.tex % \magnification=1200 % % % allgemeine Layoutparameter % \hoffset=0.4cm\voffset=0.4cm \hsize=13.5cm\vsize=18.5cm \baselineskip=13pt plus 0.2pt \parskip=4pt % %\NoBlackBoxes \TagsOnRight % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Numbering of references % \newcount\refCount \def\newref#1 {\advance\refCount by 1 \expandafter\edef\csname#1\endcsname{\the\refCount}} \newref ACKP % Arbarello-Deconcini-Kac-Procesi \newref BREM % Bremner \newref DICK % Dick \newref EGA % EGA IV \newref GUN % Gunning Riemann surfaces \newref GUO % Chinese group \newref HASCH % Hawley/Schiffer proj.connect \newref JKL % Jaffe, Klimek und Lesniewski Heisenberg \newref KACRAI % Kac-Raina booklet \newref KNFA % Krichever-Novikov Funkt.Anal. \newref LI % Li (cocycle) \newref RADUL % cocycle \newref RDS % Ruffing, Deck, Schlichenmaier (Tori) \newref SADOV % Sadov CMPe \newref SCHLL % Schlichenmaier LMP articles \newref SCHLTH % Schlichenmaier thesis \newref SCHLDEG % Schlichenmaier (degenerations) \newref SCHLC % Schlichenmaier (big article) \newref SHEIN % Sheinman (Kac-Moody) % \hfill Mannheimer Manuskripte 164 \hfill hep-th/yymmnnn \hfill October 1993 \vskip 1cm % \topmatter \title Differential Operator Algebras on compact Riemann Surfaces${ }^1$ \endtitle \rightheadtext{Differential Operator Algebras} \leftheadtext{Martin Schlichenmaier} \author Martin Schlichenmaier \endauthor \address Department of Mathematics and Computer Science, University of Mannheim,\nl D-68131 Mannheim, Germany \endaddress \email schlichenmaier\@math.uni-mannheim.de \endemail \keywords Conformal field theory, infinite-dimensional Lie algebras, Riemann surfaces, differential operator algebras, Krichever Novikov algebras \endkeywords \subjclass 17B66, 17B68, 14H99, 30F30, 81T40 \endsubjclass \abstract This talk reviews results on the structure of algebras consisting of meromorphic differential operators which are holomorphic outside a finite set of points on compact Riemann surfaces. For each partition into two disjoint subsets of the set of points where poles are allowed, a grading of the algebra and of the modules of $\l-$forms is introduced. With respect to this grading the Lie structure of the algebra and of the modules are almost graded ones. Central extensions and semi-infinite wedge representations are studied. If one considers only differential operators of degree 1 then these algebras are generalizations of the Virasoro algebra in genus zero, resp\. of Krichever Novikov algebras in higher genus. \endabstract \endtopmatter % % % here the article starts % {\sl ${ }^1$ invited talk at the International Symposium on Generalized Symmetries in Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 -- July 29, 1993} \vskip 1cm \head 1. Introduction \endhead %% It is well-known that the Virasoro algebra together with its representations play a fundamental role in Conformal Field Theory (CFT). Let me recall the realization of the Virasoro algebra which is suitable for the generalization I am considering. The meromorphic vector fields on $\P^1$ (everything is over the complex numbers) which are holomorphic over $\P^1\setminus\{0,\infty\}$ can be given as $p(z)\fpz$, with $p\in\C[z,z^{-1}]$ a Laurent polynomial. The commutator $$[\,p(z)\fpz,r(z)\fpz\,]= \big(p(z)\pfz r(z)-r(z)\pfz p(z)\big) \fpz\tag 1$$ defines a Lie algebra structure. This Lie algebra is called the Witt algebra. Obviously a basis is given by $$ l_n:=z^{n+1}\fpz,\quad n\in\Z\ ,$$ and one calculates immediately $\ [\,l_n,l_m\,]=(m-n)\,l_{n+m}$. The Witt algebra admits a universal central extension $V$ with one-dimensional center $$ 0\ \mapright {}\ \C\ \mapright {} \ V\ \mapright {} \ W\ \mapright {}\ 0 \ .$$ The algebra $V$ is the Virasoro algebra. A basis is given by lifts $L_n\ $ of $\ l_n\ $ and a central element $t$. The well-known structure equations are $$ [\,L_n,L_m\,]=(m-n)L_{n+m}+\frac 1{12}(m^3-m)\delta_{m,-n}\,t, \qquad [t,V]=0\ .\tag 2 $$ It is important to keep in mind that the Virasoro algebra is not only this algebra but this algebra together with the standard grading induced by $\ \deg(L_n)=n$ and $\deg (t)=0 $. >From (2) we see immediately that $V$ is a graded Lie algebra. Such a grading is essential if one wants to construct highest weight representations (e.g\. by using semi-infinite wedge forms). In CFT one is exactly looking for such representations. I do not want to go into the applications to CFT here but let me just say that in string theory the Virasoro algebra (or better the Witt algebra) describes the situation of a world sheet of genus zero with one incoming string (at $P=\{z=0\}$) and one outgoing string (at $Q=\{z=\infty\}$) (see Fig.~1). % % % Figure 1 and Figure 2 (Virasoro and Krichever-Novikov situation) % \centinsert{ \hbox{ \pscaption{\psannotate {\psboxto(4cm;0cm){dcfig1.eps}}{\fillinggrid \at(4\pscm;3\pscm){$P$} \at(9.5\pscm;3\pscm){$Q$} }} {Fig.~1: Virasoro situation ($g=0, \ N = 2$)}} \hskip 1cm \hbox{ \pscaption{\psannotate {\psboxto(6cm;0cm){dcfig2.eps}}{\fillinggrid \at(4\pscm;3\pscm){$P$} \at(20.5\pscm;2.8\pscm){$Q$} }} {Fig.~2: An example of a Krichever - Novikov situation}} } %end of centinsert % % % >From this point of view it is quite natural to generalize the situation to \RSs\ of arbitrary genus and an arbitrary (but fixed) number of incoming strings and an arbitrary (again fixed) number of outgoing string. For one incoming and one outgoing string this was done by Krichever and Novikov (1987/88) \cite\KNFA\ (see Fig.~2) and for arbitrary numbers by myself (1989--) \cite\SCHLL,\cite\SCHLTH \ (see Fig.~3). % % Fig.3 (generalized situation) % \centinsert{ \pscaption{\psannotate {\psboxto(9cm;0cm){dcfig3.eps}}{\fillinggrid \at(5.5\pscm;3\pscm){$P_1$} \at(5.5\pscm;6\pscm){$P_2$} \at(20.5\pscm;5\pscm){$Q_1$} }} {Fig.~3: An example of a generalized situation ($N =3$, 2 in-points, 1 out-point) } } %end of centinsert % I will consider the more general situation of differential operators of arbitrary degree. The degree one case corresponds to the vector field case. In this talk I will review some of the results obtained. For more information I have to refer to \cite\SCHLTH\ and \cite\SCHLC. Let me add that the many-point situation was also studied by Dick \cite\DICK, by Guo, Na, Shen, Wang, Yu, Wu, Chang (see for example \cite\GUO), and by Bremner (3 points and genus 0) \cite\BREM\ but without introducing a grading. Only Sadov \cite\SADOV\ did some work which is related to my approach. % % % \head 2. The set-up and the involved algebras \endhead % For the following let $X$ be a \RS\ (or a nonsingular algebraic curve over an arbitrary algebraically closed field $k$ with char $k=0$), $A$ a finite set of points which is divided into two disjoint subsets $I$ and $O$, $\ A=I\cup O,\ \#I=k\ge 1,\ \#O=l\ge 1$,\nl $N=k+l$. The elements of $I$ are the ``in-points'', the elements of $O$ are the ``out-points''. Let $\rho $ be a meromorphic differential holomorphic outside $A$ with exact pole order 1 at the points of $A$, (given) positive residues at $I$, (given) negative residues at $O$ (of course, obeying $\sum_P\res_P(\rho)=0$), and purely imaginary periods. Fix a point $Q\in \XA$. Then the function $$u(P)=\re \int_Q^P\rho\tag 3$$ is a well-defined harmonic function. The level lines $$C_\t=\{P\in\XA\mid u(P)=\t\}, \quad \tau\in\R\tag 4$$ define a fibering of $\XA$ (see Fig.~4). % % % Fig.4 (level lines) % \centinsert{ \pscaption{\psannotate {\psboxto(9cm;0cm){dcfig4.eps}}{\fillinggrid \at(3.4\pscm;5\pscm){$\boldsymbol\tau\boldsymbol\ll \boldkey0$} \at(17.6\pscm;4.4\pscm){$\boldsymbol\tau\boldsymbol\gg\boldkey 0$} }} {Fig.~4: Fibering by level lines} } %end of centinsert % Every level line separates the in- from the out-points. For $\tau\ll 0$ the level line $C_\tau$ is a disjoint union of deformed circles around the points in $I$. For $\tau\gg 0$ it is a disjoint union of deformed circles around the points in $O$. In the interpretation of string theory this $\tau$ might be interpreted as proper time of the string on the world sheet. Let $K$ be the canonical bundle, i.e\. the bundle whose local sections are the local holomorphic differentials. For every $\l\in\Z$ we consider the bundle $K^\l:=K^{\otimes\l}$, the bundle with local sections the forms of weight $\l$. After fixing a square root of the canonical bundle (a so-called theta characteristic) everything below can even be done for $\l\in \frac 12\Z$. For simplicity we consider here only the case of integer $\l$. We denote by $\Fl$ the vector space of global meromorphic sections of $K^\l$ which are holomorphic on $\XA$. Special cases are the differentials ($\l=1$), the functions ($\l=0$), and the vector fields ($\l=-1$). For the vector fields we also use the special notation $\KN(A)$. The functions (i.e\. the elements in $\Fn 0$) operate by multiplication on $\Fl$. The vector fields (i.e\. the elements in $\KN(A)$) operate by taking the Lie derivative on $\Fl$. In local coordinates the Lie derivative can be described as $$ L_e(g)_|=(e(z)\fpz)\ldot (g(z)\dzl)= \left( e(z)\pfz g(z)+\l\, g(z)\pfz e(z)\right)\dzl \ .\tag 5 $$ Here and in the following I will use the same symbol for the section of the bundle and its local representing function. By (5) $\KN(A)$ becomes a Lie algebra and the vector spaces $\Fl$ become Lie modules over $\KN(A)$ (i.e. $ [L_e,L_f]=L_{[e,f]}$). The algebra $\KN(A)$ I call a generalized Krichever Novikov algebra. For an arbitrary associative algebra $R$ we obtain always a Lie algebra $\ LR\ $ with same underlying vector space and taking the commutator $[a,b]=a\cdot b-b\cdot a$ as Lie product. Obviously, $L\Fn 0$ is an abelian Lie algebra. We take the semi-direct product $$\DO(A)=L\Fn 0\times \KN(A)$$ to obtain the Lie algebra of differential operators of degree $\le 1$ (which obey again the above regularity conditions). Its structure is given by $$ [\,(g,e),(h,f)\,]:=(\,L_eh-L_fg\,,\,[e,f]\,),\tag 6 $$ and we have the exact sequence of Lie algebras $$ 0\ \mapright {}\ L\Fn 0 \ \mapright {}\ \DO(A)\ \mapright {}\ \KN(A)\ \mapright{} \ 0\ .\tag 7$$ Note that $\KN(A)$ is the subalgebra of differential operators of degree 1. The spaces $\Fl$ now become Lie modules over $\DO(A)$. If we want differential operators of arbitrary degree we have to do some universal constructions. I use the following constructions: $$ \DA(A)=U\DO(A)/J,\quad\text{resp.}\quad \DA_\l(A)=U\DO(A)/J_\l, $$ where $U\DO(A)$ is the universal enveloping algebra of $\DO(A)$ (with multiplication $\odot$ and unit $\Eins$), and $J$ resp\. $J_\l$ are the following two-sided ideals $$\gather J:=(\;a\odot b-a\cdot b,\ \Eins-1\mid a,b\in\Fn 0\;),\\ J_\l:=(\;a\odot b-a\cdot b,\ \Eins-1 ,\ a\odot e -a\cdot e +\l L_e(a) \mid a,b\in\Fn 0, e\in \KN(A)\; ). \endgather$$ All $\Fl$ are modules over $\DA(A)$, but only $\Fl$ is a module over $\DA_\l(A)$. It is easy to check that every element $D$ of $\DA(A)$ resp\. of $\DA_\l(A)$ operates as differential operator on $\Fl$. I call the elements of $\DA(A)$ the coherent differential operators and the elements of $\DA_\l(A)$ the differential operators on $\Fl$. Because $\XA$ is affine every algebraic differential operator can be represented by such an element $D$ \cite\EGA. % % \head 3. The grading \endhead It is possible to introduce a grading in $\Fl$ in such a way that the homogeneous subspaces $\Fln n$ of degree $n$ are finite-dimensional and that the Lie module structure is an almost graded one. Of course, this is a crucial step. Nevertheless I want to skip the details here because they can be found in \cite\SCHLL. Essentially the grading is given by the zero order at the points in $I$. To give an example: Let $k=l$, $\l\ne 0,1$, and $\ I=\{P_1,P_2,\ldots,P_k\}\ $, $O=\{Q_1,Q_2,\ldots,Q_k\}\ $ be points in generic positions. Then there is for every $n\in\Z$ and every $p=1,\ldots,k$ up to multiplication with a scalar a unique element $f_{n,p}^\l\in\Fl$ with $$ \aligned \ord_{P_i}(f_{n,p}^\l)&=(n+1-\l)-\d_{i,p},\qquad i=1,\ldots,k,\\ \ord_{Q_i}(f_{n,p}^\l)&=-(n+1-\l),\qquad \qquad i=1,\ldots,k-1,\\ \ord_{Q_k}(f_{n,p}^\l)&=-(n+1-\l)+(2\l-1)(g-1)\ . \endaligned\tag 8 $$ This can be shown either by using Riemann-Roch type arguments or by explicit constructions. After fixing a coordinate at the points $P_i$ the scalar constant can be fixed. The elements $f_{n,p}^\l$ for $n\in\Z$ and $p=1,\ldots,k$ are a basis of $\Fl$. The homogeneous elements are defined to be the elements of the space $$\Fln n:=\langle\; f_{n,p}^\l\mid p=1,\ldots, k\;\rangle, \quad\text{resp.}\quad \Dn n:=\langle\; f_{n,p}^0,f_{n,p}^{-1}\mid p=1,\ldots, k\;\rangle \ .$$ The basis elements obey the important duality relation (after a suitable fixing of the scalar) $$\cintt f_{n,p}^\l\cdot f_{m,r}^{1-\l}=\d_{n,-m}\cdot \d_{p,r},\tag 9$$ where $C_\tau$ is any non-singular level line. In fact, the grading depends on the numbering of the points $Q_i$ but the induced filtration depends only on the partition $A=I\cup O$. Another essentially different partition induces a non-equivalent filtration. Note that the duality (9) is true in all cases not only in the example discussed above. An almost graded structure means that the homogeneous subspaces are finite-dimens\-ion\-al and that there are constants $K$ and $L$ not depending on $n$ and $m$ such that $$[\,\Dn n,\Dn m\,]\subseteq \bigoplus_{h=n+m+K}^{n+m+L}\Dn h,\qquad \Dn n\;.\;\Fln m\subseteq \bigoplus_{h=n+m+K}^{n+m+L}\Fln h\ . \tag 10$$ In our situation $K=0$ and there exist explicit formulas for $L$ which depend on the genus $g$ and on $N=k+l$. In the Virasoro situation ($g=0$, 2 points) $L=0$ and everything reduces to the well-known graded situation. Here I like to stress the fact, that the conditions (10) are necessary to construct semi-infinite wedge representations (see Section~5). % % \head 4. Central extensions \endhead As it is well-known it is not possible to obtain representations of the Witt algebra with certain properties one is interested in (irreducible, unitary highest weight representations). Such representations exist only for the central extension of the Witt algebra, the Virasoro algebra. Hence, it is necessary to study central extensions also for the above algebras. Central extensions can be given by 2-cocycles (with respect to Lie algebra cohomology). Such defining cocycles are known for the Virasoro situation. Unfortunately these cocycles are not defined invariantly. Only after adding some suitable counterterms, we obtain objects which make sense on arbitrary Riemann surfaces. ({\bf 1}) Let $R$ be a holomorphic projective connection \footnote{ see the appendix for the definition} , then $$\chi(e,f)=\cinttt \left( \frac 12(e''f'-e'f'')-R\cdot(e'f-ef')\right) dz \tag 11$$ for $e,f\in\KN(A)\subseteq\DO(A)$, defines a nontrivial 2-cocycle of $\KN(A)$. By pull-back via (7) we obtain a cocycle for $\DO(A)$. Recall, we represent $e$ by $e(z)\fpz$ and the prime means derivative of $e(z)$ with respect to $z$. This cocycle is a generalization to the many-point situation of the cocycle introduced by Krichever and Novikov \cite\KNFA. ({\bf 2}) For the abelian Lie algebra $L\Fn 0$ a non-trivial 2-cocycle is defined by $$\gamma(g,h)=\cintt g\,dh\tag 12$$ for $g,h\in\Fn 0$. The centrally extended Lie algebra is the generalized Heisenberg algebra. For applications see for example \cite\JKL. It is easy to show that $\gamma$ can be extended to a cocycle on $\DO(A)$ via $\gamma((g,e),(h,f)):=\gamma(g,h)$. ({\bf 3}) There is another cocycle of $\DO(A)$ which connects $L\Fn 0$ with $\KN(A)$. Let $T$ be an affine meromorphic connection \footnote{ see the appendix for the definition} with at most one pole at a point of $O$ then $$\beta(e,g)=-\beta(g,e)= \cintt(eg''+T\cdot eg')dz\tag 13 $$ for $e=(0,e)$ and $g=(g,0)\in \DO$ defines a 2-cocycle by obvious extension. In any of the cases above the choice of another connection does not change the cohomology class of the cocycle. Hence, the equivalence class of the central extension does not depend on the chosen connection. Obviously, the cocycle is independent on the value of $\tau$ used to fix the integration curve $C_\tau$. Of course, the above expressions would also define a cocycle if we integrate along any other closed curve on $\XA$. By considering only integration along $C_\tau$ we obtain cocycles which are local in the sense of Krichever and Novikov with respect to the grading of the algebra $\DO(A)$ induced by the partition of $A$. A cocycle is called local if there are constants $M$ and $H$ such that for every homogeneous $e,f\in\DO(A)$ $$\psi(e,f)=0\qquad\text{ if } \deg(e)+deg(f)>H, \quad\text{ or } \deg(e)+deg(f)< M.\tag 14$$ In our situation we have $H=0$, and $M$ depends on the genus and the number of points in $A$. Why is locality important? It allows us to extend the almost grading to the centrally extended algebra by defining the central element $t$ to have degree zero. Recall that the structure equations of the central extension $$0\ \mapright {}\ \C\ \mapright {}\ \DE(A)\ \mapright {r}\ \DO(A)\ \mapright {}\ 0\tag 15$$ defined by the cocycle $\psi$ is given by $$[\Phi(e),\Phi(h)]=\Phi([\,e,h\,])+\psi(e,h)\cdot t,\tag 16$$ where $\Phi$ is a linear splitting map of $r$ (resp\. a lift of the basis elements in $\DO(A)$) and $t$ is the image of $1$ in $\DE(A)$. \remark{Remarks} {\bf 1.} For $g=0,N=2$ everything reduces to the well-known case. In particular, we obtain again the 3 linearly independent cocycles for $\DO(\{0,\infty\})$ studied by Arbarello-DeConcini-Kac-Procesi \cite{\ACKP}. They showed that $H^2(\DO(\{0,\infty\}))$ is 3-dimensional. Hence every one-dimensional central extension is defined up to equivalence by a certain linear combination of the above given ones. {\bf 2.} For the general situation this is not true anymore. Different partitions of the points will yield different level lines and in general non-equivalent central extensions. But the new cocycle is not local anymore with respect to the first partition. Hence, \proclaim{Conjecture} (a) Every local cocycle of $\DO(A)$ is cohomologous to a linear combination of the above 3 cocycles (11), (12), (13). \nl (b) Every local cocycle of $\KN(A)$ is cohomologous to the cocycle (11). \endproclaim {\bf 3.} If we consider $\DA(A)$ or $\DA_\l(A)$ as Lie algebras then $\DO(A)$ is a Lie subalgebra. But it is not clear (and in fact not true) whether any of the above 3 cocycles can be extended to the whole of $\DA_\l(A)$. In the Virasoro situation Radul introduced for $\DA_0(A)$ a ``canonical cocycle'' \cite\RADUL\ (which of course is an extension of a certain linear combination of the above three cocycles). It was shown by Li \cite\LI\ that this is in fact the only one which could be extended. Using semi-infinite wedge representations I was able to show that there is a central extension $\widehat{\DA}_\l(A)$ (as Lie algebra !) of $\DA_\l(A)$. The defining cocycle $\psi_\l$ restricted to $\DO(A)$ is local. If we accept the above conjecture than it has to be cohomologous to a certain linear combination of the above cocycles. In this case it can be calculated as $$ [\psi_\l]=[\chi]+\frac {2\l-1}{2c_\l}[\beta]+\frac {-1}{c_\l}[\gamma], \tag 17$$ where $\ c_\l:=-2(6\l^2-6\l+1)\ $ denotes the famous expression from Mumford's formula and $[..]$ the cohomology class. {\bf 4.} It is possible to define higher genus analogues of untwisted Kac-Moody algebras. For this let $\frak g$ be a Lie algebra with $<.,.>$ an invariant, symmetric, bilinear form on $\frak g$ (i.e\. $<[x,y],z>=$). $\widehat{\frak g}=(\frak g\otimes\Fn 0)\oplus \C\, c$ is now a Lie algebra if we define $$ [x\otimes f,y\otimes g]=[x,y]\otimes f\cdot g - \gamma(f,g)\cdot c,\qquad [c,\widehat{\frak g}]=0 \tag 18 $$ where $\gamma $ is the cocycle introduced in (12). For $g=0,\ A=\{0,\infty\}$ they coincide exactly with the untwisted affine Lie algebras. For $N=2$ the same construction has been given by Krichever and Novikov \cite\KNFA\ and if additionally $g=1$ they were examined in detail by Sheinman \cite \SHEIN. \endremark % % \head 5. Further results \endhead Here space does not permit to explain the semi-infinite wedge representations in detail. Roughly speaking, the semi-infinite wedge spaces $\HE^\l(A)$ are ``restricted $\frac {\infty}2$ external forms'' constructed from the elements of $\Fl$. The elements of $\HE^\l(A)$ are called semi-infinite forms of weight $\l$. To extend the action of $\DA_\l(A)$ (resp\. $\KN(A)$, or $\DO(A)$) on $\Fl$ to $\HW(A)$ one has to ``regularize the action'' and hence has to allow for central extensions of these Lie algebras. The regularization is done by embedding the algebras via their action on $\Fl$ into $\glb$, the algebra of both-sided infinite matrices with only finitely many diagonals. Here the almost graded structure is crucial. For the algebra $\glb$ the regularization and central extension is well-known \cite\KACRAI. Pulling back the action and the cocycle gives a regularized action for a centrally extended algebra of $\DA_\l(A)$ (resp\. of $\KN(A)$, or of $\DO(A)$). The pulled back cocycle is local. Assuming the conjecture to be true it can be written in the form (17). If we consider only the algebra $\KN(A)$ and if we require a coherent action of $\KE(A)$ on every $\HW(A)$ we obtain that the central element operates as $\ (-2)(6\l^2-6\l+1)\cdot identity$. In this manner one obtains highest weight, resp\. Verma module representations of the centrally extended algebras. Let me just give a brief summary what else has been done. It is possible to define an action of $\Fl$ and $\Fn {1-\l}$ on $\HW(A)$ by wedging, resp\. contracting the semi-infinite forms. By this one obtains a Clifford algebra structure (or a $b-c$ system in physicist's language). There are interesting compatibility relations of the $\DE(A)$ action and the $b-c$ action. In addition there is a natural pairing between right semi-infinite forms of weight $\l$ and left semi-infinite forms of weight $1-\l$ induced by the pairing (9). In the case of the torus ($g=1$) explicit calculations have been done in \cite\RDS\ for the 2, 3 and 4 point case. In these cases also degenerations of the Riemann surface have been studied \cite\SCHLDEG. Here it is advantageous to use the language of Algebraic Geometry. It is interesting to note that starting from a two-point situation one is forced to study many-point situations on Riemann surfaces of lower genus. % % \head Appendix: Affine and projective connections \endhead % Let $\ (U_\a,z_\a)_{\a\in J}\ $ be a covering of the Riemann surface by holomorphic coordinates, with transition functions $z_\b=f_{\b\a}(z_\a)=h(z_\a)$. A system of local (holomorphic, meromorphic) functions $\ R=(R_\a(z_\a))\ $ resp\. $\ T=(T_\a(z_\a))\ $ is called a (holomorphic, meromorphic) projective (resp\. affine) connection if it transforms as $$ R_\b(z_b)\cdot (h')^2=R(z_\a)+S(h),\qquad\text{with}\quad S(h)=\frac {h'''}{h'}-\frac 32\left(\frac {h''}{h'}\right)^2\tag 19$$ the Schwartzian derivative, resp\. $$ T_\b(z_b)\cdot h'=T(z_\a)+\frac {h''}{h'}\ .\tag 20$$ Here ${}'$ denotes differentiation with respect to the coordinate $z_\a$. Note that the difference of two affine (projective) connections is always a usual (quadratic) differential. \proclaim{Lemma} (a) There is always a holomorphic projective connection. \nl (b) Given a point $P$ on $X$ there is always a meromorphic affine connection which is holomorphic outside $P$ and has there at most a pole of order 1. \endproclaim \noindent For a proof of (a) see for example \cite{\HASCH} or \cite{\GUN}. A proof of (a) and (b) can also be found in \cite{\SCHLTH}. % % References % \Refs \ref\no\ACKP\by Arbarello, E., DeConcini, C., Kac, V.G., Procesi, C. \paper Moduli Spaces of Curves and Representation Theory \jour\CMP \vol 117\yr 1988\pages 1--36 \endref \ref\no\BREM\by Bremner, M.R. \paper Structure of the Lie algebra of polynomial vector fields on the Riemann sphere with three punctures \jour \JMP \yr 1991 \pages 1607--1608 \endref \ref\no\DICK \by Dick, R. \paper Krichever-Novikov-like bases on punctured Riemann surfaces \jour \LMP\vol 18 \yr 1989 \page 255-265 \moreref \paper Chiral fields on Riemann surfaces and string vertices \paperinfo PhD. Thesis 1990, Universit\"at Hamburg, Germany \endref \ref\no\EGA\by Grothendieck, A.\paper \'El\'ements de g\'eom\'etrie alg\'ebrique IV (3\`eme partie) \jour Publ\. Math\. I.H.E.S. \vol 28\yr 1966 \endref \ref\no\GUN\by Gunning, R.C. \book Lectures on Riemann surfaces \bookinfo Princeton Mathematical Notes Nr.2 \publ Princeton University Press \yr 1966 \endref \ref\no\GUO\by Guo, H-y., Na, J-s., Shen, J-m., Wang, S-k., Yu, Q-h. \paper The algebras of meromorphic vector fields..... \jour Commun. Theor. Phys. \vol 12 \yr 1989 \pages 349--354 \endref \ref\no\HASCH\by Hawley, N.S., Schiffer M. \paper Half order differentials on Riemann surfaces \jour Acta Math. \vol 115 \yr 1966 \pages 199--236 \endref \ref\no\JKL\by Jaffe, A., Klimek, S., Lesniewski, A. \paper Representations of the Heisenberg algebra on a Riemann surface \jour \CMP \vol 126 \yr 1989 \pages 421--431 \endref \ref\no\KACRAI\by Kac, V.G., Raina, A.K. \book Highest Weight Representations of Infinite Dimensional Lie Algebras \bookinfo Adv\. Ser\. in Math\. Physics Vol.2 \publ World Scientific \yr 1987 \endref \ref\no\KNFA\by Krichever, I.M., Novikov, S.P. \paper Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons \jour \FA \vol 21\issue 2\yr 1987\page 46 \moreref \paper Virasoro type algebras, Riemann surfaces and strings in Minkowski space \jour \FA \vol 21\issue 4\yr 1987\page 47 \moreref \paper Algebras of Virasoro type, energy-momentum tensors and decompositions of operators on Riemann surfaces \jour \FA \vol 23\issue 1\yr 1989\pages 19--23 \endref \ref\no\LI\by Li, Wang-Lai \paper 2-cocycles on the algebra of differential operators \jour Journal of Algebra \vol 122 \yr 1989 \pages 64--80 \endref \ref\no\RADUL\by Radul, A.O. \paper A central extension of the Lie algebra of differential operators on the circle and $W$-algebras \jour Pis'ma Zh. \'Eksp. Teor. Fiz. \vol 50\issue 8 \yr 1989 \pages 341--343 \endref \ref\no\RDS\by Ruffing, A., Deck, Th., Schlichenmaier, M. \paper String branchings on complex tori and algebraic representations of generalized Krichever-Novikov algebras \jour \LMP \vol 26 \yr 1992 \pages 23--32 \endref \ref\no\SADOV\by Sadov, V.A. \paper Bases on multipunctured Riemann surfaces and interacting strings amplitudes \jour \CMP \vol 136 \yr 1991 \pages 585--597 \endref \ref\no\SCHLL \by Schlichenmaier, M. \paper Krichever-Novikov algebras for more than two points \jour \LMP \vol 19 \yr 1990 \pages 151 --165 \moreref \paper Krichever-Novikov algebras for more than two points: explicit generators \jour \LMP \vol 19 \yr 1990 \pages 327--336 \moreref \paper Central extensions and semi-infinite wedge representations of Krichever-Novikov algebras for more than two points \jour \LMP \vol 20 \yr 1990 \pages 33--46 \endref \ref\no\SCHLTH\by Schlichenmaier, M. \paper Verallgemeinerte Krichever - Novikov Algebren und deren Darstellungen \paperinfo \nl PhD. Thesis 1990, Universit\"at Mannheim, Germany \endref \ref\no\SCHLDEG\by Schlichenmaier, M. \paper Degenerations of generalized Krichever-Novikov algebras on tori \jour \JMP \vol 34\issue 8 \yr 1993 \pages 3809--3824 \endref \ref\no\SCHLC\by Schlichenmaier, M. \paper On the algebra of differential operators on compact Riemann surfaces with marked points \paperinfo (in preparation) \endref \ref\no\SHEIN\by Sheinman, O.K. \paper Elliptic affine Lie algebras \jour \FA \vol 24\issue 3\yr 1990\page 210--219 \moreref \paper Highest weight modules over certain quasigraded Lie algebras on elliptic curves \jour \FA \vol 26\issue 3\yr 1992\page 203--208 \endref % % \endRefs % \enddocument \autojoin \bye %Beginning-Of-File-Named:diffmac.tex % Macros for Differentialoperatoralgebras % % \font\ssf=cmss10 % Blackboard-Bold % \def\R{\Bbb R } \def\N{\Bbb N } \def\P{\Bbb P } \def\Q{\Bbb Q } \def\Z{\Bbb Z } \def\C{\Bbb C } \def\G{\Bbb G } \def\O{\Bbb O } \def\Eins{\hbox{\ssf 1\kern-1.0pt{\vrule height6.5pt depth0pt}}} % % additional fonts % %\font\bigbf=cmbx10 scaled 1200 %\font\bigbff=cmbx10 scaled 1440 %\font\small=cmr8 % % \define\np{\vfil\eject} \define\nl{\hfil\newline} \define\ldot{\,.\,} % \redefine\l{\lambda} \define\ifft{\qquad\text{if and only if}\qquad} \define\a{\alpha} \redefine\d{\delta} \define\g{\gamma} \redefine\b{\beta} \redefine\t{\tau} \redefine\i{{\,\bold i\,}} % % \define\PL{Phys\. 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