Celebratio Mathematica

I saw Fuchs for the first time in 1969. I liked math­em­at­ics; I wanted to study it. At least in my case, the choice had already been made, even if un­con­sciously. In oth­er words, “math­em­at­ics is for me, it is that world where I would like to be.” The prob­lem was that it was ab­so­lutely un­clear what it meant to do math­em­at­ics. There were many re­search sem­inars at Mekh­mat and ap­par­ently they were the gates to this “brave new world”. Fuchs was an act­ive par­ti­cipant in at least two of them, Gel­fand’s and Arnold’s. I star­ted com­ing to both of them when I was a first-year stu­dent. Fuchs is a big, phys­ic­ally strong man; people told stor­ies about his strength. I heard from An­dry­usha Zele­v­in­sky that he could shift a row of desks with one move­ment of his arm. At these sem­inars he was the ex­pert in to­po­logy and Gel­fand, for ex­ample, al­ways asked him whenev­er a prob­lem in al­geb­ra­ic to­po­logy arose — and such prob­lems did arise be­cause dis­cussed at the sem­in­ar were vari­ous forms of the in­dex the­or­em, works of Novikov, and much more, for ex­ample, al­geb­ra­ic K-the­ory, al­though this was a little later.

In or­der to be­gin do­ing math­em­at­ics, one has to jostle around, try this and that, make sense of books and pa­pers. All of this is im­port­ant, but in do­ing this it is very easy to get lost and des­pair. It is as if you have found your­self in­side a forest with a myri­ad of en­tangled paths that lead no one knows where. This is why it is so im­port­ant to have some­body to talk to. Some­body must show by ex­ample how to walk these paths and what it means to do math­em­at­ics.

Among par­ti­cipants of the Gel­fand and Arnold sem­inars were many re­mark­able math­em­aticians of vari­ous ages; suf­fice it to men­tion such names as Dima Kazh­dan, Seryozha Gel­fand, Gab­ri­elov, Gu­sein-Zade… They were young back then, but some­what older math­em­aticians were there, too — Arnold, Novikov, Kir­illov, Fuchs — as well as more seni­or people: Shilov, Gindikin, Karpele­vich, Min­los, Kostyuchen­ko. My prob­lem was that I was afraid of these people; in my eyes they were (to some ex­tent) Olympi­ans. It was hard even to come closer to them, much less talk to them. In my case the prob­lem was solved by Gel­fand. Fuchs had a repu­ta­tion — quite de­servedly — of be­ing the kind­est per­son at the sem­in­ar. Gel­fand clearly dis­tin­guished Fuchs from the oth­ers. Izrail Moiseevich’s way of deal­ing with speak­ers was un­usu­al: he could in­ter­rupt, he could be un­pleas­ant. But Dmitry Bor­iso­vich was ex­cep­ted; every­body loved him, Gel­fand loved him, too, and it was Gel­fand who asked Fuchs to be­come my ad­viser.

This idea of Gel­fand’s proved to be very suc­cess­ful. Fuchs helped me (not im­me­di­ately of course) to over­come my lack of con­fid­ence, and this, I think, is the ad­viser’s main task. We worked to­geth­er for a long time, and this was very im­port­ant for me and, I be­lieve, for him, too.

Our work had to do with vari­ous top­ics, but mostly we stud­ied rep­res­ent­a­tion the­ory of the Vi­ra­s­oro al­gebra.

Here I will talk about our work, but this won’t be a re­view of pub­lished pa­pers; rather this is an es­say on how Gel­fand–Fuchs the­ory of co­homo­logy of Lie al­geb­ras of vec­tor fields on man­i­folds nat­ur­ally de­veloped in­to what later be­came known as con­form­al field the­ory. Dmitry Bor­iso­vich worked a lot with Gel­fand and later with me, but this was not a change of sub­ject. It turned out (al­though it was not clear at the time) that rep­res­ent­a­tion the­ory of the Vi­ra­s­oro al­gebra lo­gic­ally fol­lows Gel­fand–Fuchs the­ory. Of course the story I’ll tell will be very in­com­plete.

I would also like to write about how we worked to­geth­er. I think we were friends, as far as pos­sible (Dmitry Bor­iso­vich is 14 years my seni­or). I went to his apart­ment on 13th Parkovaya Street once a week. We worked in a one-room apart­ment on the first floor, and in the apart­ment next door lived his moth­er; there we had lunch. The meal usu­ally con­sisted of a cut­let with a side of buck­wheat kasha in­vari­ably fol­lowed by cold cof­fee. Sum­mers I spent in a dacha with my fam­ily. The Fuchs dacha was nearby and I cycled there, just as Fuchs some­times cycled to my place when in Mo­scow. It so happened that while we were stay­ing at the dacha, my second son Zhenya was born in the ma­ter­nity hos­pit­al of the nearby town of Zhukovskiǐ. My wife was still there when Fuchs came, and along with Serezha Tabach­nikov, who was vis­it­ing us, we went to the ma­ter­nity hos­pit­al and stood out­side. Vis­its were not al­lowed, nor would they take any­thing from the vis­it­ors, but the moms would drop a rope from the win­dow, the par­cel would be fastened to the end of the rope and then lif­ted back in­to the room. We did this, too.

I re­mem­ber how it oc­curred to me that one could use the mod­ules of semi-in­fin­ite forms in or­der to prove the Kac de­term­in­ant for­mula (that is, to find the val­ues of the para­met­ers for which the Verma mod­ule over the Vi­ra­s­oro al­gebra is re­du­cible). The very idea that the mod­ules of semi-in­fin­ite forms are at all rel­ev­ant nat­ur­ally arose from think­ing over the res­ults by Fuchs and my­self about an­ti­sym­met­ric in­vari­ant dif­fer­en­tial op­er­at­ors on the line. I was trav­el­ing to Cher­no­go­lovka,1 but I was so ex­cited that I stopped by the Fuchs’ place. The idea was raw but it was clear it was bound to work. When I vis­ited him next, in his mind was the un­der­stand­ing of what I had said along with a whole host of oth­er things to do. Thus our work on the Vi­ra­s­oro al­gebra began.

I learnt a lot from Fuchs. He is very mod­est, kindly, gen­er­ous. He shared everything he knew, con­ceived, un­der­stood. Now I real­ize that he did have dif­fi­culties, per­son­al and math­em­at­ic­al, but to me he seemed to be the ideal hu­man be­ing. So I thought back then and all in all I still think so.

The con­tinu­ous co­homo­logy of Lie al­geb­ras of vec­tor fields on man­i­folds (Gel­fand–Fuchs the­ory) is what formed me as a math­em­atician, made an ex­traordin­ar­ily strong emo­tion­al im­pact on me. I think every­body ex­per­i­ences such a wa­ter­shed mo­ment: you pass it, and now you are a math­em­atician. I stud­ied dif­fer­ent things, I marveled at some of them. I was struck by the main idea of ho­mo­lo­gic­al al­gebra, by much from al­geb­ra­ic geo­metry. I liked very much class field the­ory. However, com­ing to grips with things new to me (and es­pe­cially with class field the­ory) was pain­ful and dif­fi­cult. I did not des­pair, but everything was covered in fog and, what is more im­port­ant, dif­fer­ent things re­mark­able in their own right would not form a whole.

Everything changed when I tried to delve in­to con­tinu­ous co­homo­logy the­ory. For me it is (to this day) an ex­ample of an “ideal” math­em­at­ic­al the­ory. On the one hand, there is its re­mark­able beauty, clar­ity, and lo­gic. On the oth­er, there is its “min­im­al­ism”, the ab­sence of un­ne­ces­sary en­tit­ies.2 In oth­er words, an ideal the­ory is a com­pact world; it lives and de­vel­ops ac­cord­ing to its own laws, but it is “open”. Its open­ness means that such a the­ory is like that droplet which re­flects the en­tire world. This is the point of good min­im­al­ist mu­sic. And an­oth­er thing: an ideal the­ory, per­haps be­cause of its com­pact­ness, car­ries a charge, an en­ergy bomb as it were, a power­ful source of mo­tion.

And so I was shaken, and what happened to me can be con­sidered an epi­phany. This was the res­ult of my at­tempts to un­der­stand this the­ory, but also — and of course to a much great­er de­gree — be­cause of talk­ing with Dmitry Bor­iso­vich.

I was lucky, of course. It is good to enter math­em­at­ics through an area which is “com­pact”; this way it is easi­er to find a prob­lem, to see what you like. Al­geb­ra­ic geo­metry is fright­en­ingly large; it is so much harder. However, as I shall try to ex­plain, the com­pact­ness of Gel­fand–Fuchs the­ory is il­lus­ory, and in fact it ex­tends to in­fin­ity.

The con­tinu­ous co­homo­logy of Lie al­geb­ras of vec­tor fields on man­i­folds im­pli­citly (some­times ex­pli­citly, too) con­tains the rudi­ments of many oth­er math­em­at­ic­al the­or­ies, and I shall try to tell you about a part of all of this. What I shall tell is my own point of view; some of it I un­der­stood on my own, some be­came clear as a res­ult of joint work with Dmitry Bor­iso­vich.

Vec­tor fields are in­fin­ites­im­al sym­met­ries of man­i­folds, and an ana­logue that im­me­di­ately comes to mind is a cur­rent al­gebra. These are al­geb­ras of in­fin­ites­im­al sym­met­ries of bundles. Let $\mathfrak{g}$ be a Lie al­gebra, let $G$ be the cor­res­pond­ing group, and con­sider a trivi­al $G$-bundle over a man­i­fold $M$. In this case the cur­rent group con­sists of maps $M\to G$ with point­wise mul­ti­plic­a­tion; the cur­rent Lie al­gebra ${\mathfrak{g}}^M$ is sim­il­arly defined. Fur­ther­more, the sym­metry group of a pair, a man­i­fold and a bundle over it is the semi­direct product of the dif­feo­morph­isms group and the cur­rent al­gebra. The meth­ods of con­tinu­ous co­homo­logy the­ory can be ap­plied to this cur­rent al­gebra if we al­low all $C^{\mathrm{\infty }}$-maps. Ac­cord­ing to the Gel­fand–Fuchs ideo­logy, one be­gins by solv­ing the cor­res­pond­ing loc­al prob­lem: find the con­tinu­ous co­homo­logy of the Lie al­gebra $$\mathfrak{g}\otimes \mathbb{C}[[x_1,x_2,\dots ,x_n]],$$ equipped with the ad­ic-to­po­logy. This co­homo­logy is dual to the ho­mo­logy of the Lie al­gebra $$\mathfrak{g}\otimes \mathbb{C}[x_1,x_2,\dots ,x_n].$$ Gen­er­al­iz­ing this we ar­rive at the fol­low­ing al­geb­ra­ic ques­tion.

Let $A$ be a com­mut­at­ive al­gebra. What is the ho­mo­logy of the Lie al­gebra $\mathfrak{g}\otimes A$? If $A$ is non­com­mut­at­ive, then $\mathfrak{g}\otimes A$ does not carry a nat­ur­al Lie al­gebra struc­ture, but one can con­sider the Lie al­gebra of $n\times n$ matrices with coef­fi­cients in $A$, ${\mathrm{gl}}_n(A)$.

The loc­al prob­lem in Gel­fand–Fuchs the­ory is to com­pute the con­tinu­ous co­homo­logy of the Lie al­gebra of form­al vec­tor fields on ${\mathbb{R}}^n$, de­noted by $W_n$. Gel­fand and Fuchs solved this prob­lem by us­ing in­vari­ant the­ory. The al­gebra $W_n$ con­tains the Lie sub­al­gebra of lin­ear vec­tor fields — it is iso­morph­ic to ${\mathrm{gl}}_n(\mathbb{R})$ — and one can write down the Serre–Hoch­schild spec­tral se­quence. Its second term is de­scribed in terms of ${\mathrm{gl}}_n$-in­vari­ants in the stand­ard com­plex of $W_n$, and in­vari­ants are labeled by some graphs. Thus one ob­tains a com­plex with a basis whose ele­ments are labeled by graphs of spe­cial type: the $r$-th such graph is an $r$-cycle with an ex­tra edge stick­ing out from each of the ver­tices; Gel­fand and Fuchs called it the “$r$-th hedge­hog”. In the $W_n$-case this com­plex is re­l­at­ively simple.

People have man­aged to ap­ply these ideas to cur­rent al­geb­ras in two cases. The first is the one where $\mathfrak{g}={\mathrm{gl}}_{\mathrm{\infty }}$, the Lie al­gebra of in­fin­ite but fi­nit­ary matrices. The ho­mo­logy of ${\mathrm{gl}}_{\mathrm{\infty }}(A)$ can be found once we know the Hoch­schild ho­mo­logy of $A$. This is how the Connes–Tsy­gan cyc­lic ho­mo­logy made its ap­pear­ance. Connes’ mo­tiv­a­tions were dif­fer­ent, but Tsy­gan dir­ectly fol­lowed the ideas of Gel­fand and Fuchs. Us­ing in­vari­ant the­ory, he found a basis of the com­plex with basis ele­ments also labeled by graphs — uni­ons of cycles with edges dec­or­ated by ele­ments of $A$. These graphs are spe­cial and un­soph­ist­ic­ated, but some­what later there ap­peared a gen­er­al­iz­a­tion, a so-called graph com­plex, where at­tached to a basis ele­ment is a graph of a rather ar­bit­rary form. Such graph-com­plexes arose in knot the­ory, in 3-man­i­fold in­vari­ants, in Kont­sevich’s proof of the form­al­ity con­jec­ture, and in many oth­er places. Of course, graphs were not in­ven­ted by Gel­fand and Fuchs. They had been ex­tens­ively used in many oth­er places, and first of all in quantum field the­ory, where they ap­peared in the form of Feyn­man dia­grams. Gel­fand and Fuchs were strongly mo­tiv­ated by this cir­cum­stance. It tran­spired later that the ap­pear­ance of graphs in quantum field the­ory and in Gel­fand–Fuchs the­ory is a mani­fest­a­tion of a more gen­er­al mech­an­ism.

Gel­fand and Fuchs tried to com­pute the co­homo­logy of the Lie al­gebra of Hamilto­ni­an vec­tor fields. The loc­al prob­lem also leads to a graph-com­plex, and to be looked after are the sym­plect­ic Lie al­gebra in­vari­ants. The com­plex they ob­tained was very in­volved, and no won­der, be­cause if the di­men­sion tends to in­fin­ity, then it be­comes the fam­ous graph-com­plex. The reas­on for this phe­nomen­on re­mains mys­ter­i­ous even now, al­though a par­tial ex­plan­a­tion can be found in Roz­ansky–Wit­ten the­ory. (A re­la­tion to Gel­fand–Fuchs the­ory and char­ac­ter­ist­ic classes of fo­li­ations was re­cog­nized by Kapran­ov and Kont­sevich soon after the Roz­ansky–Wit­ten pa­per ap­peared.)

The second case where the co­homo­logy of $\mathfrak{g}\otimes A$ proved to be com­put­able is where $A=\mathbb{C}[x]$; in oth­er words, the al­gebra is the ring of reg­u­lar func­tions on the line with val­ues in a fi­nite-di­men­sion­al semisimple Lie al­gebra. The meth­ods used are quite dif­fer­ent. This al­gebra is a Lie sub­al­gebra of $\mathfrak{g}\otimes \mathbb{C}[x,x^{-1}]$. At first glance the lat­ter al­gebra is more com­plic­ated, but it is a Kac–Moody Lie al­gebra and is, there­fore, an ob­ject of a very de­veloped the­ory. The an­swer is that $$H^{\ast }(\mathfrak{g}\otimes \mathbb{C}[x])=H^{\ast }(\mathfrak{g}).$$ The al­gebra $\mathfrak{g}$ is, there­fore, ri­gid. It is sur­pris­ing that $$H^{\ast }(\mathfrak{g}\otimes \mathbb{C}[x_1,x_2])\ne H^{\ast }(\mathfrak{g}).$$

The case of ${\mathfrak{g}}^{S^1}$ is in­deed ana­log­ous to that of the Lie al­gebra of vec­tor fields on $S^1$, the start­ing point for Gel­fand and Fuchs. The loc­al com­pu­ta­tion, as we have already poin­ted out, shows that the co­homo­logy of the Lie al­gebra of form­al cur­rents in a neigh­bor­hood of a point equals the co­homo­logy of the cor­res­pond­ing fi­nite-di­men­sion­al $\mathfrak{g}$, which is well known to be equal to the sin­gu­lar co­homo­logy of the cor­res­pond­ing group, con­sidered as a to­po­lo­gic­al space. In the Gel­fand–Fuchs situ­ation, the loc­al res­ult amounts to the fact that $$H^{\ast }(W_1)=H^{\ast }(S^3)=H^{\ast }({\mathrm{SL}}_2(\mathbb{C})).$$ The next step in the Gel­fand–Fuchs ap­proach is the “glu­ing” of loc­al res­ults in­to the glob­al one. In or­der to ac­com­plish this, they con­struc­ted a fil­tra­tion on the stand­ard co­homo­lo­gic­al com­plex of the Lie al­gebra of vec­tor fields on $M$; they called the fil­tra­tion’s “bot­tom” the di­ag­on­al sub­com­plex; it con­sists of co­chains sup­por­ted on the di­ag­on­al. Fur­ther terms of the fil­tra­tion are sup­por­ted on the com­pon­ents of the dis­crim­in­ant sub­man­i­fold of $M\times M\times \cdots \times M$. The res­ult of glu­ing in the case of $M=S^1$ as­serts $H^{\ast }(\mathrm{Lie}(S^1))$ is iso­morph­ic to the co­homo­logy of the space of con­tinu­ous maps $S^1\to S^3$. The idea of the proof is this: the de Rham com­plex that com­putes the co­homo­logy of the in­fin­ite-di­men­sion­al space $\mathrm{Map}(S^1,S^3)$ also car­ries a fil­tra­tion by sup­port, and this al­lows one to prove that the cor­res­pond­ing spec­tral se­quence is the same.

All of this can be ap­plied to ${\mathfrak{g}}^{S^1}$ and gives the fol­low­ing res­ult: $H^{\ast }({\mathfrak{g}}^{S^1})$ is iso­morph­ic to the sin­gu­lar co­homo­logy $H^{\ast }(G^{S^1})$. In terms of 1-al­geb­ras, this res­ult can stated as fol­lows: the group $G$ and the cur­rent al­gebra $\mathfrak{g}\otimes \mathbb{C}[x]$ each define a 1-al­gebra; it turns out that these two 1-al­geb­ras are equi­val­ent. It fol­lows, there­fore, that each co­homo­logy class of $G^{S^1}$ is rep­res­en­ted by a left-in­vari­ant form. This is a well-known phe­nomen­on in the case of a com­pact group, where one uses the in­vari­ant in­teg­ra­tion for the proof. This ar­gu­ment does not work for the group $G^{S^1}$, and for $G^M$, $\dim M>1$, the as­ser­tion is al­to­geth­er wrong. The con­tinu­ous co­homo­logy of a cur­rent al­gebra on a man­i­fold with val­ues in a semisimple Lie al­gebra (for ex­ample, ${\mathrm{sl}}_2(\mathbb{C})$) are still un­known if the di­men­sion of the man­i­fold is great­er than 1.

Gel­fand and Fuchs worked on a rather con­crete prob­lem: they ana­lyzed the stand­ard com­plex of the Lie al­gebra of vec­tor fields on a man­i­fold. (They star­ted with the case of trivi­al coef­fi­cients, but later con­sidered mod­ules of tensor fields.) They used meth­ods of func­tion­al ana­lys­is, spe­cific­ally the the­ory of dis­tri­bu­tions, an ob­ject of Gel­fand’s fas­cin­a­tion in those times. But fun­da­ment­ally, the es­sence of their work was purely al­geb­ra­ic. Among oth­er things, Gel­fand and Fuchs con­struc­ted a sys­tem of sheaves with con­nec­tion on the con­fig­ur­a­tion space of col­lec­tions of points on a man­i­fold. This sys­tem of sheaves en­joys a fac­tor­iz­a­tion prop­erty. The fiber over a point is a stand­ard com­plex of the Lie al­gebra of vec­tor fields over the form­al disc centered at this point; over a fi­nite col­lec­tion of points, the fiber is the tensor product of fibers over each point. In mod­ern lan­guage, this means Gel­fand and Fuchs con­struc­ted an $n$-al­gebra, and the con­tinu­ous co­homo­logy of the Lie al­gebra of vec­tor fields is its fac­tor­iz­a­tion co­homo­logy. The gen­es­is of $n$-al­gebra the­ory is to­po­lo­gic­al, but for a long time it was per­ceived as something exot­ic. Gel­fand and Fuchs con­vinced every­body that this is not the case. These days this sub­ject is very pop­u­lar, be­cause a lot in mod­ern math­em­at­ics and math­em­at­ic­al phys­ics can­not even be stated without us­ing the $n$-al­gebra lan­guage. Un­like many oth­ers, I found this lan­guage rather clear as Gel­fand–Fuchs the­ory con­tained its main in­gredi­ents.

The Vi­ra­s­oro al­gebra is the (uni­ver­sal) cent­ral ex­ten­sion of the Lie al­gebra of vec­tor fields on the circle. The Gel­fand–Fuchs com­pu­ta­tion showed that the co­homo­logy of this Lie al­gebra is the su­per­poly­no­mi­al ring on two gen­er­at­ors, one in di­men­sion 2, an­oth­er in di­men­sion 3; thus the Vi­ra­s­oro al­gebra was born. An­oth­er source is string the­ory, where the Vi­ra­s­oro al­gebra is the main act­or.

It was clear from the be­gin­ning that the Vi­ra­s­oro al­gebra is sim­il­ar to the cent­ral ex­ten­sion of ${\mathfrak{g}}^{S^1}$. The lat­ter’s cent­ral ex­ten­sion is a Kac–Moody Lie al­gebra, and is, there­fore, a sub­ject with a well-de­veloped the­ory. There are a lot of tools to study rep­res­ent­a­tions of Kac–Moody Lie al­geb­ras, al­geb­ra­ic and geo­met­ric. The Vi­ra­s­oro al­gebra also has a cat­egory of highest-weight mod­ules, there are Verma mod­ules, and all the usu­al ques­tions can be asked. The very first ques­tion is this: The Verma mod­ules $V(h,c)$ de­pend on a pair of para­met­ers $(h,c)$. If $(h,c)$ is gen­er­ic, then $V(h,c)$ is ir­re­du­cible. It be­comes re­du­cible if $(h,c)$ lies on a uni­on of cer­tain curves. The equa­tions de­fin­ing these curves were found by Kac, and they are called Kac hy­per­bolas. Kac did not have a proof, and to make a guess he used ex­pli­cit com­pu­ta­tions, some of which were car­ried out by string the­or­ists.

The meth­ods of Kac–Moody al­gebra the­ory, however, do not dir­ectly work in the Vi­ra­s­oro al­gebra case. There are many reas­ons for this, e.g., there is no Casimir ele­ment. Ad­di­tion­ally, there is no ob­vi­ous way to define in­teg­rable rep­res­ent­a­tions. Such rep­res­ent­a­tions are well known in the Kac–Moody case; they are ana­logues of fi­nite-di­men­sion­al mod­ules over a semisimple Lie al­gebra. The ex­ist­ence of fi­nite-di­men­sion­al mod­ules greatly fa­cil­it­ates the study of the cat­egory of highest-weight mod­ules.

Fuchs and I did a lot of work on rep­res­ent­a­tions of the Vi­ra­s­oro al­gebra. We man­aged to prove the Kac con­jec­ture, us­ing the re­la­tion with in­vari­ant dif­fer­en­tial op­er­at­ors on the line. We ana­lyzed the struc­ture of Verma mod­ules, ob­tained a lot of in­form­a­tion on highest-weight mod­ules. The cre­at­ors of con­form­al field the­ory, Be­lav­in, Polyakov, and Zamo­lod­chikov, in­tro­duced the concept of a min­im­al mod­el. We shall say more about these mod­els a little later, but their es­sen­tial part is a spe­cial class of mod­ules over the Vi­ra­s­oro al­gebra. It turned out that these mod­ules are a prop­er ana­logue of in­teg­rable rep­res­ent­a­tions. Fuchs and I con­struc­ted the Bern­stein–Gel­fand–Gel­fand-type res­ol­u­tions of these mod­ules, and in par­tic­u­lar ob­tained the Vi­ra­s­oro al­gebra ver­sions of the Weyl char­ac­ter for­mula. I shall talk more about the meth­ods used a little later, as these meth­ods be­long in con­form­al field the­ory. The Vi­ra­s­oro al­gebra and loop al­geb­ras are par­tic­u­lar ex­amples of what is known as a ver­tex al­gebra. These are ob­jects of a the­ory that has been rap­idly de­vel­op­ing for over 30 years with no end in sight. Ver­tex al­geb­ras also en­joy the ex­ist­ence of min­im­al mod­els, in­teg­rable rep­res­ent­a­tions, res­ol­u­tions, char­ac­ter for­mu­las, etc. One can­not say that this activ­ity grew out of the Gel­fand–Fuchs co­homo­logy; nev­er­the­less there are close con­nec­tions.

Ver­tex al­geb­ras are a piece (an es­sen­tial one) of the math­em­at­ic­al ma­chinery of con­form­al field the­ory. Let us point out that the Vi­ra­s­oro al­gebra is the al­gebra of con­form­al sym­met­ries (loc­al and in­fin­ites­im­al ones), and there­fore it is a key.

Con­form­al field the­ory ori­gin­ated in works of Be­lav­in, Polyakov and Zamo­lod­chikov, as well as Friedan and Shen­ker, and its con­sequences are in­nu­mer­able and hard to pro­cess. This the­ory can­not be called “ideal”; it does not sat­is­fy the “min­im­al­ist­ic” prin­ciples, and yet it has some of this fla­vor too. Con­form­al field the­ory arose in­side quantum field the­ory as its spe­cial case. Quantum field the­ory per se is the very neg­a­tion of the prin­ciple of min­im­iz­ing the num­ber of en­tit­ies. Be­fore us is a world without bound­ary, a tower of Ba­bel, to which tur­rets are ad­ded all the time, parts of the con­struc­tion get aban­doned, oth­er parts col­lapse or sit in neg­lect; mean­while the main aim of reach­ing heav­en is not com­ing much closer. Con­form­al field the­ory is an out­growth of quantum field the­ory ob­tained by means of “cut­ting off the ends”; that is to say, they ex­trac­ted that part which is sup­por­ted by a hard math­em­at­ic­al frame. It turned out that this piece had ac­quired (or per­haps ac­cu­mu­lated) en­ergy that led to an ex­plos­ive growth of a con­sid­er­able chunk of math­em­at­ics, and maybe phys­ics, too. Fur­ther­more, parts of phys­ics and math­em­at­ics began to fuse in­to a whole, and all this was hap­pen­ing right in front of us.

Very much a bird’s eye view on the con­form­al field the­ory struc­ture is as fol­lows. To be­gin with, the the­ory is 2-di­men­sion­al; in oth­er words, it lives on a Riemann sur­face, a 2-di­men­sion­al man­i­fold with met­ric. It pos­sesses fields, which are at­tached to points. Fur­ther­more, as­signed to a point is in fact an en­tire space of states. For ex­ample, a quantum-mech­an­ic­al de­scrip­tion of an elec­tron in­volves a 2-di­men­sion­al space of states, which car­ries an ac­tion of the or­tho­gon­al group; the lat­ter is un­der­stood as the sym­metry group. In con­form­al field the­ory, the sym­metry group is big­ger; it com­prises the set of all (in­fin­ites­im­al) trans­form­a­tions of a punc­tured disc — or rather the cent­ral ex­ten­sion of this Lie al­gebra, the Vi­ra­s­oro al­gebra. There­fore, the space of states $V(p)$ that “sits” in point $p$ is a rep­res­ent­a­tion of the Vi­ra­s­oro al­gebra. The spaces $V(p_1)$, $V(p_2)$, …, $V(p_n)$ “in­ter­act” by means of the Rieman­ni­an sur­face on which they live. This in­ter­ac­tion mani­fests it­self in the cor­rel­a­tion func­tion, which is a func­tion­al $$V(p_1)\otimes V(p_2)\otimes \cdots \otimes V(p_n)\to \mathbb{C}.$$ Let us point out that this sort of struc­ture has also aris­en in math­em­at­ics, in class field the­ory to be pre­cise, as fol­lows. If $K$ is a di­men­sion-1 “glob­al field”, then it has points. The com­ple­tion at a point is a loc­al field $K_p$, and one at­taches to this point a rep­res­ent­a­tion of the group $G(K_p)$, where $G$ is a semisimple Lie group. The func­tion­al $$V(p_1)\otimes V(p_2)\otimes \cdots \otimes V(p_n)\to \mathbb{C}$$ is a map in­vari­ant with re­spect to the group of cur­rents “reg­u­lar” on the com­ple­ment to the col­lec­tion $\{p_1,p_2,\dots ,p_n\}$, and it is in­volved in the defin­i­tion of a mod­u­lar form. The ap­proach of con­form­al field the­ory is more gen­er­al: placed in a neigh­bor­hood of a point is a cer­tain al­gebra, and the ex­ist­ence of a func­tion­al is a part of the struc­ture.

The main at­trib­ute of such func­tion­als is the op­er­a­tion of “fu­sion”: it de­scribes what hap­pens to the spaces $V(p_i)$ and $V(p_j)$ when the points $p_i$ and $p_j$ come to­geth­er (a new space is cre­ated) and as a res­ult what hap­pens with the cor­rel­a­tion func­tion. Let us note that this nicely cor­res­ponds to Gel­fand–Fuchs the­ory. There the ob­ject of study is the stand­ard co­chain com­plex and as­signed to a point is the sub­com­plex of co­chains sup­por­ted at this point. The op­er­a­tion of fu­sion also ex­ists in Gel­fand–Fuchs the­ory; it de­scribes what hap­pens with loc­al co­chains when points col­lide. The concept of a cor­rel­a­tion func­tion is not quite defined in this the­ory, but it is clear what it could be. We want to as­sign a num­ber to a col­lec­tion of co­chains at points $p_1,\dots ,p_n$, and in or­der to do this we need a “cycle” on which we can eval­u­ate the value of the product of co­chains. This sort of thing arises in ad­vanced ver­sions of Gel­fand–Fuchs the­ory, in the study of $n$-al­geb­ras.

Fuchs and I spent many years work­ing on con­form­al field the­ory. Nev­er­the­less, it was very dif­fi­cult to un­der­stand what was hap­pen­ing there; we were hampered by the phys­ics lan­guage, com­pletely un­fa­mil­i­ar and fright­en­ing. Things be­came much easi­er when I un­der­stood that many struc­ture ele­ments of con­form­al field the­ory are not new to me: they had already ap­peared in con­tinu­ous co­homo­logy the­ory (and the the­ory of mod­u­lar forms.)

In ad­di­tion to the concept of a cor­rel­a­tion func­tion there is a re­lated concept of a mod­u­lar func­tor. As an ex­ample, let us con­sider what the founders called a min­im­al mod­el or (more tech­nic­ally) a ra­tion­al con­form­al the­ory. Ac­cord­ing to Segal and Atiyah, a mod­u­lar func­tor is a ver­sion of co­homo­logy the­ory, or something that is called a 3-di­men­sion­al field the­ory. Namely, there is a rule that to each closed 2-di­men­sion­al man­i­fold $\mathcal{E}$ with ori­ent­a­tion at­taches a vec­tor space $V(\mathcal{E})$ so that the change of ori­ent­a­tion re­places the space with its dual. In ad­di­tion, $V(\mathcal{E})$ car­ries a Her­mitian form. Next, if $\mathcal{E}={\mathcal{E}}_1\bigsqcup {\mathcal{E}}_2$, then $$V(\mathcal{E})=V({\mathcal{E}}_1)\otimes V({\mathcal{E}}_2).$$ The most im­port­ant prop­er­ties are as fol­lows. A choice of a com­plex struc­ture on $\mathcal{E}$ defines a vec­tor in $V(\mathcal{E})$. Ana­log­ously, a choice of a 3-di­men­sion­al $M$ such that $\mathcal{E}=\partial M$ also defines a vec­tor in $V(\mathcal{E})$. Note that a very sim­il­ar struc­ture ap­pears in the the­ory of mod­u­lar forms. A mod­u­lar func­tor is something like the space of mod­u­lar forms, hence the name.

Fuchs and I, un­know­ingly, stud­ied the mod­u­lar func­tor that arises via min­im­al mod­els. We worked on the fol­low­ing prob­lem. Let $$\{L_i=z^{i+1}\partial /\partial z,C\}$$ be a basis of the Vi­ra­s­oro al­gebra, $V_{h,c}$ the Verma mod­ule, $(h,c)$ its highest weight, and $L_i$, $i>0$, the an­ni­hil­a­tion op­er­at­ors. We stud­ied the quo­tient space (coin­vari­ants) $$R_{h,c}/\{L_i,L_{i-1},\dots \}{R}_{h,c},$$ where $R_{h,c}$ is a quo­tient of $V_{h,c}$. It turned out that there is a class of rep­res­ent­a­tions for which this quo­tient is fi­nite-di­men­sion­al for any $i<0$. This class co­in­cides with the class of rep­res­ent­a­tions that ap­pear in the min­im­al mod­els. This as­ser­tion was stated by us as a con­jec­ture; it was proved in the pa­per by Beil­in­son, Mazur, and my­self. The mod­u­lar func­tor that arises in a min­im­al mod­el is also a cer­tain space of coin­vari­ants. If $R_{h,c}$ is a rep­res­ent­a­tion from a min­im­al mod­el, then $$c=c_{p,q}=1-6(p-q{)}^2/pq,$$ where $p,q$ is a pair of re­l­at­ively prime pos­it­ive in­tegers. Let ${\mathrm{Vac}}_{p,q}$ be the ir­re­du­cible quo­tient of $V_{0,c_{p,q}}$. The Rieman­ni­an sur­face defines a sub­al­gebra of the Vi­ra­s­oro al­gebra, and the mod­u­lar func­tor is the space of coin­vari­ants with re­spect to this sub­al­gebra. An al­geb­ra­ic curve defines a sub­al­gebra of the Vi­ra­s­oro al­gebra as fol­lows. Let $\mathcal{E}$ be a curve, $p\in \mathcal{E}$ a point, $z$ a loc­al co­ordin­ate around $p$, $f(z)\in \mathbb{C}[[z]]\partial /\partial z$, and $\mathrm{Lie}(\mathrm{Out})$ the Lie al­gebra of vec­tor fields on $\mathcal{E}\setminus \{p\}$. It is clear that there is a nat­ur­al Lie al­gebra em­bed­ding $\mathrm{Lie}(\mathrm{Out})\hookrightarrow \mathbb{C}((z))\partial /\partial z$, the Laurent ex­pan­sion around $p$. In fact, al­though this is less ob­vi­ous, this em­bed­ding can be lif­ted to one in­to the cent­ral ex­ten­sion of $\mathbb{C}((z))\partial /\partial z$, the Vi­ra­s­oro al­geb­ras at­tached to $p$. The above-men­tioned sub­al­gebra spanned by the vec­tor fields $\{L_i,L_{i-1},L_{i-2},\dots \}$ cor­res­ponds to a de­gen­er­a­tion of the curve.

As I have already poin­ted out, the Vi­ra­s­oro al­gebra and the af­fine Lie al­geb­ras are close re­l­at­ives and are the main ex­amples of ver­tex op­er­at­or al­geb­ras. Fuchs and I stud­ied the Vi­ra­s­oro al­gebra rep­res­ent­a­tions us­ing its em­bed­ding in­to sim­pler ver­tex al­geb­ras, Heis­en­berg and Clif­ford. We con­ceived of this meth­od by ourselves, but from the con­form­al field the­ory per­spect­ive this is an in­fin­itely nat­ur­al thing to do. Con­form­al field the­or­ies per se are hard to ana­lyze, people try to em­bed them in­to sim­pler ones, those we know enough about, and es­pe­cially those where we can find cor­rel­a­tion func­tions. This ap­proach is called bo­son­iz­a­tion. Since all of this had worked in the case of the Vi­ra­s­oro al­gebra, it seemed nat­ur­al to try and do the same thing for the af­fine Lie al­geb­ras. Bo­son­iz­a­tion of ${\widehat{\mathrm{sl}}}_2$ was ob­tained by Wakimoto, and then his con­struc­tion was car­ried over to the gen­er­al case. The situ­ation with bo­son­iz­a­tion can be briefly de­scribed as fol­lows. Let $\mathfrak{g}$ be a simple fi­nite-di­men­sion­al Lie al­gebra. “Bo­son­iz­a­tion” of $\mathfrak{g}$ is a ho­mo­morph­ism $U(\mathfrak{g})\to D$, where $D$ is the al­gebra of dif­fer­en­tial op­er­at­ors on ${\mathbb{C}}^N$ or, more gen­er­ally, an al­gebra of dif­fer­en­tial op­er­at­ors on an al­geb­ra­ic vari­ety. The most nat­ur­al such ho­mo­morph­ism $U(\mathfrak{g})\to \mathrm{Dif}(M)$, where $M$ is a vari­ety that car­ries an ac­tion by $\mathfrak{g}$, for ex­ample, a ho­mo­gen­eous space. An al­gebra of dif­fer­en­tial op­er­at­ors can be twis­ted, mean­ing that it acts not on func­tions but on sec­tions of a $\mathfrak{g}$-equivari­ant line bundle over $M$. The most pop­u­lar choice of $M$ is the flag man­i­fold. A ho­mo­morph­ism $U(\mathfrak{g})\to \mathrm{Dif}(M)$ defines, via in­duc­tion, a func­tor from the cat­egory of $\mathfrak{g}$-mod­ules to the cat­egory of $D$-mod­ules. On the oth­er hand, there is a func­tor from the cat­egory of $D$-mod­ules to the cat­egory of con­struct­ible sheaves. This geo­met­ric ap­proach is the most power­ful tool in the study of $\mathfrak{g}$-mod­ules.

In the af­fine Lie al­gebra case such ap­proach works too, be­cause an af­fine Lie al­gebra has its flag man­i­fold and can, there­fore, be mapped in­to the cor­res­pond­ing al­gebra of dif­fer­en­tial op­er­at­ors. Fur­ther­more, in this case there are sev­er­al in­equi­val­ent flag man­i­folds, and hence sev­er­al bo­son­iz­a­tions. In fact, in the af­fine Lie al­gebra case there are flag man­i­folds of three types: thick, thin, and semi-in­fin­ite. The thin one re­sembles the fi­nite-di­men­sion­al flag man­i­fold the most, and is in fact an in­duct­ive lim­it of fi­nite-di­men­sion­al ones. The thick one is more of a pro­ject­ive lim­it, and is built of in­fin­ite-di­men­sion­al cells of fi­nite codi­men­sion, $\mathrm{\infty }-n$, $n\in \mathbb{N}N$. Semi-in­fin­ite flag man­i­folds are a mix­ture of an in­duct­ive and pro­ject­ive lim­it. They also have a cell de­com­pos­i­tion, but now di­men­sion is rather $\mathrm{\infty }+m$, $m\in \mathbb{Z}$. Wakimoto mod­ules are re­lated to the lat­ter class of flag man­i­folds.

For the Vi­ra­s­oro al­gebra, as well as for most ver­tex al­geb­ras, a con­struc­tion re­sem­bling thick and thin flags is yet to be found, but the known bo­son­iz­a­tions are semi-in­fin­ite con­struc­tions in spir­it.

There­fore, a con­sid­er­able chunk of mod­ern the­ory ori­gin­ated (par­tially, of course) in our work on the Vi­ra­s­oro al­gebra rep­res­ent­a­tion the­ory. In a sense, we ex­trac­ted the sub­cat­egory gen­er­ated by the Verma mod­ules, which is akin to the cat­egory of $D$-mod­ules on the thick flag man­i­fold, the sub­cat­egory gen­er­ated by the con­tra­gredi­ent Verma mod­ules, this is the “thin” part, and the Fei­gin–Fuchs rep­res­ent­a­tions; this part is semi-in­fin­ite.

But this is not it, the situ­ation turned out to be even more in­ter­est­ing. There is no thick flag man­i­fold for the Vi­ra­s­oro al­gebra, but there is an im­port­ant ana­logue. Namely, the “thick flag man­i­fold” for the af­fine Lie al­gebra $\hat{\mathfrak{g}}$ is the mod­uli space of the fol­low­ing pairs: a $G$-bundle on $\mathbb{C}{\mathbb{P}}^1$, its trivi­al­iz­a­tion at a fixed point. More gen­er­ally, one can re­place $\mathbb{C}{\mathbb{P}}^1$ with a fixed point with an ar­bit­rary smooth al­geb­ra­ic curve with a few fixed points. An ana­logue of this for the Vi­ra­s­oro al­gebra is the mod­uli of al­geb­ra­ic curves with fixed points and form­al co­ordin­ates around them. Such an ob­ject is ex­actly what ap­pears in Gel­fand–Fuchs the­ory and plays there a key role. The con­struc­tion of loc­al­iz­a­tion can be ap­plied and pro­duces a func­tor that makes a Vi­ra­s­oro al­gebra mod­ule in­to a $D$-mod­ule on such a mod­uli space. The space of coin­vari­ants that we dis­cussed above is the fiber of this $D$-mod­ule at a fixed point. Let us also point out that such a loc­al­iz­a­tion func­tor can be defined for any ver­tex al­gebra. Un­like the af­fine Lie al­gebra case, this loc­al­iz­a­tion is not easy to use for the study of rep­res­ent­a­tion the­ory, but it is very im­port­ant all the same and is used in the study of mod­uli spaces.

In my talk at ICM-90 in Kyoto I pro­posed an in­ter­pret­a­tion of the mod­u­lar func­tor in the lan­guage of Gel­fand–Fuchs the­ory. Namely, let $\mathcal{E}$ be a smooth com­plex al­geb­ra­ic curve. First, one can define $\mathrm{Lie}(\mathcal{E})$, an al­gebra of holo­morph­ic vec­tor fields on $\mathcal{E}$. This is a Lie dg-al­gebra (one pos­sible real­iz­a­tion is the Dol­beau­lt res­ol­u­tion of the sheaf of vec­tor fields) or, equi­val­ently, a sim­pli­cial ob­ject in the cat­egory of Lie al­geb­ras. Fol­low­ing Gel­fand and Fuchs, one defines and com­putes the con­tinu­ous co­homo­logy of $\mathrm{Lie}(\mathcal{E})$. The an­swer is this: the co­homo­logy of the space of con­tinu­ous maps $\mathcal{E}\to S^3$; here $\mathcal{E}$ is re­garded as a to­po­lo­gic­al space. In oth­er words, everything is ex­actly the same as in the case of smooth man­i­folds. In par­tic­u­lar, $H^1(\mathrm{Lie}(\mathcal{E}))$ is 1-di­men­sion­al. This means that $\mathrm{Lie}(\mathcal{E})$ has a non­trivi­al char­ac­ter, and the space of iso­morph­ism classes of such char­ac­ters is iden­ti­fied with $H^2(\mathrm{Lie}(S^1))$ or $H^3(\mathrm{Lie}(\mathbb{R}))$. One spe­cif­ic fea­ture of this, dif­fer­en­tial-graded, situ­ation is that the ho­mo­logy of $\mathrm{Lie}(\mathcal{E})$ is not at all dual to the co­homo­logy, even though there is a pair­ing between them.

The ho­mo­logy of $\mathrm{Lie}(\mathcal{E})$ nat­ur­ally arises in de­form­a­tion the­ory. Namely, $H_0(\mathrm{Lie}(\mathcal{E}))$ is iso­morph­ic to the space of dis­tri­bu­tions on the mod­uli space of curves sup­por­ted at $\mathcal{E}$. If one com­putes the ho­mo­logy with coef­fi­cients in a 1-di­men­sion­al rep­res­ent­a­tion, then one ob­tains dis­tri­bu­tions with val­ues in the cor­res­pond­ing power of the de­term­in­ant line bundle.

If the 1-di­men­sion­al rep­res­ent­a­tion cor­res­ponds to $$c_{p,q}=1-6(p-q{)}^2/pq,$$ then it has a quo­tient, in the de­rived cat­egory sense. The ho­mo­logy with coef­fi­cients in this quo­tient is pre­cisely the mod­u­lar func­tor. All of this sup­ports the idea that the ori­gins of the mod­u­lar func­tor, in fact all of 2-di­men­sion­al field the­ory (to­po­lo­gic­al field the­ory in­cluded), lie in Gel­fand–Fuchs the­ory.

In­teg­ral rep­res­ent­a­tions of cor­rel­a­tion func­tions nat­ur­ally arise in our works. We were try­ing to find ex­pli­cit for­mu­las for sin­gu­lar vec­tors in Verma mod­ules over the Vi­ra­s­oro al­gebra or, equi­val­ently, de­scribe ho­mo­morph­isms between Verma mod­ules. This is an im­port­ant prob­lem and, un­for­tu­nately, there are no “good” for­mu­las in the case of Kac–Moody Lie al­geb­ras. In the Vi­ra­s­oro case what helps is bo­son­iz­a­tion. Namely, the Vi­ra­s­oro al­gebra em­beds in­to the uni­ver­sal en­vel­op­ing al­gebra of the Heis­en­berg Lie al­gebra: $\{h_i,i\in \mathbb{Z}\}$, $[h_i,h_j]=i{\delta }_{i+j,0}$. This em­bed­ding de­pends on two para­met­ers and is called the Fei­gin–Fuchs real­iz­a­tion. As the Heis­en­berg al­gebra has an ir­re­du­cible rep­res­ent­a­tion, we ob­tain a two-para­met­er fam­ily of the Vi­ra­s­oro al­gebra mod­ules. If para­met­ers are gen­er­ic, the mod­ule is iso­morph­ic to a Verma mod­ule. For spe­cial val­ues of the para­met­ers — those that be­long to a uni­on of planar quad­rics — these mod­ules are dif­fer­ent. Of spe­cial in­terest are isol­ated points, those at the in­ter­sec­tion of the curves. We made a de­tailed study of such (Fo­ck) mod­ules. It turned out later that sim­il­ar mod­ules are defined for Kac–Moody, in par­tic­u­lar, for af­fine Lie al­geb­ras. In the cat­egory of mod­ules over Kac–Moody Lie al­geb­ras, there are re­flec­tion func­tors, labeled by ele­ments of the Weyl group. Ap­ply­ing such func­tors to Verma mod­ules, one ob­tains the so-called twis­ted Verma mod­ules. Their com­pos­i­tion series co­in­cide with those of Verma mod­ules, but they are dif­fer­ently as­sembled. In the case of a Kac–Moody Lie al­gebra, the Weyl group is usu­ally in­fin­ite, and there­fore one can “in­fin­itely twist”. For ex­ample, the Weyl group for ${\widehat{\mathrm{sl}}}_2$ is a group freely gen­er­ated by two re­flec­tions, ${\sigma }_1$, ${\sigma }_2$. It turns out that the in­fin­ite product ${\sigma }_1{\sigma }_2{\sigma }_1{\sigma }_2\cdots$ also defines a func­tor on the mod­ule cat­egory, and ap­plied to the Verma mod­ule gives what is known as the Wakimoto mod­ule. The Vi­ra­s­oro al­gebra has no Weyl group; nev­er­the­less this sort of func­tor can be defined. Thus the in­fin­itely twis­ted Verma mod­ule is ex­actly the Fo­ck mod­ule men­tioned above.

In­ter­twin­ing op­er­at­ors between Fo­ck mod­ules are con­struc­ted in the form of in­teg­rals of ver­tex op­er­at­ors and are called a screen­ing. The ver­tex op­er­at­or $B_{\alpha }(z)$, $z\in \mathbb{C}$, also acts between Fo­ck mod­ules and is uniquely de­term­ined by the brack­ets with the Heis­en­berg al­gebra gen­er­at­ors: $$[h_i,B_{\alpha }(z)]=\alpha z^i{B}_{\alpha }(z).$$ It also has nice com­mut­a­tion re­la­tions with the ele­ments of the Vi­ra­s­oro al­gebra. Screen­ing op­er­at­ors can be com­posed, and such com­pos­i­tions are also in­ter­twin­ing op­er­at­ors. This im­plies an ex­pli­cit, al­beit not very simple, for­mula for sin­gu­lar vec­tors in Fo­ck mod­ules. Let us point out that an ex­pli­cit for­mula for mat­rix ele­ments of in­ter­twin­ing op­er­at­ors is ob­tained via con­tour in­teg­ra­tion over a cycle, and the choice of the cycle is an in­ter­est­ing and non­trivi­al ques­tion.

Screen­ing op­er­at­ors and the re­lated in­teg­ral rep­res­ent­a­tions find nu­mer­ous ap­plic­a­tions in con­form­al field the­ory. One of the im­port­ant such ap­plic­a­tions is known as a “to­po­lo­gic­al” real­iz­a­tion of the mod­u­lar func­tor. As we have seen, the mod­u­lar func­tor that comes from the min­im­al mod­els as­signs to an al­geb­ra­ic curve the space of coin­vari­ants of the va­cu­um mod­ule ${\mathrm{Vac}}_{p,q}$. The to­po­lo­gic­al real­iz­a­tion is the iso­morph­ism of this space of coin­vari­ants and the co­homo­logy group of a cer­tain con­struct­ible sheaf on the con­fig­ur­a­tion space $\mathcal{E}\times \mathcal{E}\times \cdots \times \mathcal{E}$. (The num­ber of factors de­pends on the genus of $\mathcal{E}$ and on the min­im­al mod­el.) The de­tails can be found in the book by Bezrukavnikov, Finkel­berg, and Schecht­man.

In the series of works by Schecht­man and Varchen­ko in­teg­ral rep­res­ent­a­tions of solu­tions for Knizh­nik–Zamo­lod­chikov equa­tions were found. Their res­ults can also be ob­tained us­ing the tech­nique of screen­ing op­er­at­ors.

It is also im­port­ant to note the fol­low­ing. The Vi­ra­s­oro al­gebra has a gen­er­al­iz­a­tion, the $W$-al­geb­ras. These are defined as the cent­ral­izer of an ap­pro­pri­ate col­lec­tion of screen­ing op­er­at­ors, in­side a cer­tain Heis­en­berg al­gebra. (There­fore, just as the Vi­ra­s­oro al­gebra, each $W$-al­gebra is a sub­al­gebra of some Heis­en­berg al­gebra.) Hence again screen­ing op­er­at­ors are in­ter­twiners of Fo­ck mod­ules over $W$-al­geb­ras, which al­lows one to re­late the rep­res­ent­a­tion the­ory of $W$-al­geb­ras with quantum groups.

I would like to write about the work on the al­gebra ${\mathbb{L}}^1$ be­cause this sub­ject was very im­port­ant for me. At the be­gin­ning of our col­lab­or­a­tion with Fuchs I thought a lot about this al­gebra and un­der­stood a num­ber of key things, which to a large ex­tent de­term­ined what I later worked on, with Fuchs and on my own.

${\mathbb{L}}^1$ is the al­gebra of vec­tor fields on the line that van­ish at the ori­gin to or­der 2. Fuchs asked the ques­tion about its co­homo­logy when he, with Gel­fand, stud­ied the Lie al­gebra of vec­tor fields on the circle. I be­lieve Fuchs’s train of thought was as fol­lows (this is my re­con­struc­tion.) The ap­proach of Gel­fand and Fuchs uses the fil­tra­tion of the con­tinu­ous co­chain com­plex by sup­port and the com­pu­ta­tion of the cor­res­pond­ing spec­tral se­quence. It is also nat­ur­al to ask about the ho­mo­logy of the al­gebra of vec­tor fields on the circle, and when so do­ing it is un­reas­on­able to con­sider all smooth vec­tor fields; it is more nat­ur­al to con­sider fields that have fi­nite Four­i­er series ex­pan­sions. This way one ar­rives at the Lie al­gebra with basis $\{L_i:i\in \mathbb{Z}\}$ and com­mut­a­tion re­la­tions $$[L_i,L_j]=(j-i)L_{i+j},$$ in oth­er words, at the Lie al­gebra of poly­no­mi­al vec­tor fields on the af­fine curve ${\mathbb{C}}^{\ast }$. De­note this al­gebra by $\mathrm{Lie}({\mathbb{C}}^{\ast })$. There is a pair­ing between the ho­mo­logy of $\mathrm{Lie}({\mathbb{C}}^{\ast })$ and the con­tinu­ous co­homo­logy of $\mathrm{Lie}(S^1)$; the con­jec­ture that this pair­ing is nonde­gen­er­ate is very nat­ur­al, but how can one prove it? There is no doubt this as­ser­tion is val­id for any smooth (per­haps even sin­gu­lar) af­fine vari­ety, but this con­jec­ture turned out to be very dif­fi­cult and it has been proved only re­cently by Hen­nion and Kapran­ov [e3]. In the case of the circle (or ${\mathbb{C}}^{\ast }$) one can pro­ceed as fol­lows. The vari­ety ${\mathbb{C}}^{\ast }$ is a quad­ric; it can be de­gen­er­ated in­to a sin­gu­lar quad­ric, the co­ordin­ate cross on the plane. $\mathrm{Lie}({\mathbb{C}}^{\ast })$ will thus de­gen­er­ate in­to the Lie al­gebra of vec­tor fields in the co­ordin­ate cross, that is, in­to the Lie al­gebra with basis $\{{\overline{L}}_i:i\in \mathbb{Z}\}$ and com­mut­a­tion re­la­tions $$[{\overline{L}}_i,{\overline{L}}_j]=\{\begin{array}{ll}(j-i){\overline{L}}_{i+j}& \text{if }i,j\ge 0,\\ (j-i){\overline{L}}_{i+j}& \text{if }i,j\le 0,\\ 0& \text{otherwise}.\end{array}$$ This de­gen­er­a­tion defines a spec­tral se­quence that con­verges to $H_{\bullet }(\mathrm{Lie}({\mathbb{C}}^{\ast }))$, and its first term is $$(H_{\bullet }({\mathbb{L}}_{-}^1)\otimes H_{\bullet }({\mathbb{L}}_{+}^1){)}^{L_0}.$$

Here ${\mathbb{L}}_{+}^1$ is the Lie al­gebra with basis $\{{\overline{L}}_i:i>0\}$ and ${\mathbb{L}}_{-}^1$ is the Lie al­gebra with basis $\{{\overline{L}}_i:i<0\}$; the ele­ment $L_0$ op­er­ates on both ${\mathbb{L}}_{\pm }^1$. This is known as the Gon­char­ova spec­tral se­quence.

Lida Gon­char­ova was a stu­dent of Dmitry Bor­iso­vich; she was an ex­traordin­ary hu­man be­ing and a gif­ted math­em­atician. She mar­ried Sasha Ge­r­on­im­us, a good math­em­atician, who later be­came a very well known Rus­si­an Or­tho­dox priest. Lida be­came a priest’s wife and quit math­em­at­ics.3

In or­der to use the Gon­char­ova spec­tral se­quence, one needs to know $H_{\bullet }({\mathbb{L}}_{-}^1)$ and $H_{\bullet }({\mathbb{L}}_{+}^1)$. The al­geb­ras ${\mathbb{L}}_{\pm }^1$ are iso­morph­ic to each oth­er and to the al­gebra of vec­tor fields on the line of the form $z^{2}p(z)\partial /\partial z$. De­note this al­gebra by ${\mathbb{L}}^1$. It is graded by the ac­tion of $L_0$, namely, $\mathrm{deg}{L}_i=i$.

The stand­ard chain com­plex of ${\mathbb{L}}^1$ is graded, and its Euler char­ac­ter­ist­ic is $$\prod _{i>0}(1-q^i).$$ The Euler iden­tity $$\prod _{i>0}(1-q^i)=1+\sum _{i>0}(-1{)}^i(q^{\frac{3i^2+i}{2}}+q^{\frac{3i^2-i}{2}})$$ sug­gests mak­ing a nat­ur­al con­jec­ture (and this is what Fuchs did): $\dim H_i({\mathbb{L}}^1)=2$ so that the de­grees of two basis ele­ments are equal to $(3i^2+i)/2$ and $(3i^2-i)/2$. This is the prob­lem Fuchs gave Lida, who was a gradu­ate stu­dent at that time. The prob­lem proved to be un­ex­pec­tedly hard. Lida man­aged to solve it, but her meth­od was very soph­ist­ic­ated and tech­nic­al. Her meth­od also al­lowed to find the ho­mo­logy of the al­geb­ras $${\mathbb{L}}^k=\{z^{k}p(z)\partial /\partial z\},$$ also con­jec­tured by Fuchs, but the at­tempts to ap­ply it to oth­er Lie al­geb­ras failed.

The Euler iden­tity re­lated to ${\mathbb{L}}^1$, as well as its gen­er­al­iz­a­tion, the Mac­don­ald iden­tity, are ana­log­ously re­lated to the max­im­al nil­po­tent sub­al­geb­ras of Kac–Moody Lie al­geb­ras. In the the­ory of Kac–Moody Lie al­geb­ras the Mac­don­ald iden­tit­ies arise in some­what dif­fer­ent — but not al­to­geth­er dif­fer­ent — ways. One such way is where the Mac­don­ald iden­tity be­comes the Weyl char­ac­ter for­mula for the trivi­al mod­ule. The trivi­al rep­res­ent­a­tion en­joys the Bern­stein–Gel­fand–Gel­fand res­ol­u­tion, which con­sists of Verma mod­ules. Verma mod­ules are free over the uni­ver­sal en­vel­op­ing al­gebra of the max­im­al nil­po­tent sub­al­gebra, which im­plies that there is a basis of the ho­mo­logy of the max­im­al nil­po­tent sub­al­gebra whose ele­ments are labeled by the Weyl group ele­ments. Ana­logues of these as­ser­tions are val­id for the Vi­ra­s­oro al­gebra, and this gives us a trans­par­ent proof of the Gon­char­ova the­or­em.

A some­what dif­fer­ent meth­od of deal­ing with the stand­ard com­plex uses a nat­ur­al pos­it­ive def­in­ite form on the max­im­al nil­po­tent sub­al­gebra. This form ex­tends to the com­plex, and al­lows one to define the Laplace op­er­at­or. Its ker­nel is the ho­mo­logy. The spec­tral prob­lem for the Laplace op­er­at­or is easy to solve for Kac–Moody Lie al­geb­ras, but in the Vi­ra­s­oro case it is non­trivi­al, and was solved by Gel­fand, Fuchs and my­self [10]. This pa­per con­tains an in­ac­cur­acy, which was cor­rec­ted by Fe­lix Vain­stein.

I would like to note that the in­form­a­tion on the ho­mo­logy of ${\mathbb{L}}^1$ is used in many com­pu­ta­tions, for ex­ample, in the prob­lem on ho­mo­logy of the Lie al­gebra of vec­tor fields on the line with coef­fi­cients in the mod­ules of tensor fields and their tensor products. In fact, it is this prob­lem that our col­lab­or­a­tion with Fuchs began with. In­form­a­tion about sin­gu­lar vec­tors in Verma mod­ules was im­port­ant for us, be­cause the dif­fer­en­tials in the com­plexes of in­terest were ex­pressed via these vec­tors.

I would like to add a few words to what I wrote at the be­gin­ning. Fuchs played a de­fin­ing role in my de­vel­op­ment as a math­em­atician; his in­flu­ence was more im­port­ant for me than that of Gel­fand’s school, to which we both be­long. Yes, it is dif­fi­cult to be­come a math­em­atician; it is es­pe­cially dif­fi­cult to real­ize what the sub­ject of math­em­at­ics is, what is it in math­em­at­ics that is yours, that you like. Gel­fand used to say that a math­em­atician al­ways works on the same sub­ject. This means that a hu­man be­ing likes some things, and there can­not be too many such things, be­cause the ca­pa­city of a hu­man mind is lim­ited. When ap­plied to Gel­fand him­self, these words sound para­dox­ic­al as he seemed to change his re­search area many times. Nev­er­the­less, he thought he had al­ways worked on the same sub­ject.

Very well, a math­em­atician has de­cided what they like, has de­veloped a de­gree of con­fid­ence. In oth­er words, a trans­ition from child­hood to adult­hood has taken place, even if in­com­pletely. But adult­hood cre­ates new prob­lems. I will write about some of them, those that were es­pe­cially im­port­ant for me.

I gradu­ated from Mo­scow State Uni­versity in 1974. My un­der­gradu­ate thes­is was a piece of re­l­at­ively ori­gin­al re­search, a con­tinu­ation of a joint work by Gel­fand, Fuchs and my­self on de­form­a­tions of char­ac­ter­ist­ic classes of fo­li­ations. Go­ing to gradu­ate school, in my case, was not an op­tion (be­cause Jews were not ad­mit­ted), and I had to do with my life something else. I found a job in a com­puter cen­ter and had to fig­ure out how to com­bine my job with math­em­at­ic­al re­search. I must ad­mit I was quite lost. It was a dif­fi­cult situ­ation where many people would quit math­em­at­ics. I did not want this at all, but life had changed, I had to spend days at work. Abandon­ing math­em­at­ics was for me un­ac­cept­able, but it was un­clear how to or­gan­ize my life. Fuchs told me that my situ­ation was not un­usu­al, that many people were in the same po­s­i­tion. I un­der­stood that it was up to me to solve my prob­lems; that is, I had to find a way to work, think, talk with oth­er math­em­aticians. Fuchs told me that it would be dif­fi­cult and offered to con­tin­ue stay­ing in touch. This meant com­ing to his place once a week and not be­ing dis­tressed if no new thoughts have oc­curred to me since the pre­vi­ous vis­it. He thought it was es­sen­tial to hold out for the time be­ing and then the life would settle down and I would re­gain the abil­ity to think and work. Fuchs had ex­per­i­ence, he knew how it was with oth­er people, and he knew something about me. His sup­port and help dur­ing this dif­fi­cult time was not merely im­port­ant, without it I would have been un­able to over­come all of this. To say that I am grate­ful to Fuchs is to say noth­ing.

Life went on; after the com­puter cen­ter I found my­self at the gradu­ate school of the Yaroslavl Uni­versity; later I de­fen­ded my Ph.D. thes­is in Len­in­grad. Thes­is de­fense was a com­plic­ated pro­cess in those days and without Fuchs’s help it was un­likely to work out. I mar­ried, which of course did not make things any easi­er, but the very idea — keep on do­ing math­em­at­ics no mat­ter what — re­mained, and without Fuchs this would have been im­possible.

All right then, I be­came a math­em­atician, I had writ­ten pa­pers, I had ob­tained some new res­ults. One com­mon prob­lem is this: how do you treat the fact that oth­er people work on the same stuff, use what you have done. This prob­lem is mul­ti­fa­ceted; here we will not be able to dis­cuss it in earn­est. It is clear, however, that every­one com­petes with oth­er people (as in sports). It is in­ev­it­able that you have to de­cide what to do with this cir­cum­stance. As with many oth­er things, there is no sat­is­fact­ory solu­tion here. One ex­treme is when you ig­nore the pub­lic opin­ion, write very few pa­pers, stick to your course. Some people do that, but it is very dif­fi­cult and of­ten coun­ter­pro­duct­ive. Math­em­at­ics is a col­lect­ive en­deavor and it is very im­port­ant to un­der­stand that you are a part of a com­munity that is in­volved in a cer­tain pro­cess (the pro­cess of gain­ing know­ledge, un­der­stand­ing, it is not very im­port­ant what you call this pro­cess.) Prac­tice shows that an­oth­er ex­treme — the com­plete ad­her­ence to con­ven­tions, be­com­ing an act­ive part of the sys­tem — is also coun­ter­pro­duct­ive. One must learn how to re­main true to one­self, and I have learnt this from Fuchs, if at all. True, it is im­possible not to get per­turbed be­cause some people are bet­ter than you at some things, that your work is not men­tioned; fur­ther­more, it is hu­man nature that you can­not help envy­ing those who are smarter, more suc­cess­ful than you. Nev­er­the­less, it is im­port­ant to de­vel­op an abil­ity to live with all of this — yes, those feel­ings are there, but they are at the back­ground, as it were. The main thing — the rules ac­cord­ing to which you work and live — they are a step above. In oth­er words, tor­ments and tribu­la­tions are un­avoid­able, but you have to learn how to deal with them. My solu­tion to these prob­lems is not ideal but, es­pe­cially now, I real­ize that I largely fol­lowed Fuchs. We are of course very dif­fer­ent; nev­er­the­less it was his way of solv­ing these ques­tions that I ap­plied to my life. Note that at no time did Fuchs ex­pli­citly state any kind of mor­al prin­ciple, but as a little child I formed my be­ha­vi­or, my prin­ciples im­it­at­ing the grown-ups.

I have already said what was im­port­ant to say. Nev­er­the­less, I would like to touch upon one more ques­tion. Math­em­at­ics is not only a way to ex­press your­self and real­ize your abil­it­ies. Psy­cho­lo­gic­ally, math­em­at­ics is per­ceived as a pro­cess of gain­ing in­sight in­to a re­mark­ably beau­ti­ful en­tity, and this pro­cess is not in­stant­an­eous, rather it is ex­ten­ded in time. In oth­er words, some­body already did this be­fore us — ad­mired this en­tity and tried to un­der­stand it — and some­body will carry on after us. It is dif­fi­cult, if not out­right im­possible, to ex­plain this with at least some semb­lance of clar­ity, but prac­tic­ally any math­em­atician has a hunch that this must be so. As a prac­tic­al mat­ter, this means that you re­ceived something from your teach­ers, and your task is to pass this along. To do so is some­times an ob­vi­ous ne­ces­sity. I know people who can prop­erly func­tion only when sur­roun­ded by pu­pils. There­fore, teach­ing and work­ing with stu­dents is an es­sen­tial part of math­em­at­ic­al life. I have had quite a few stu­dents; my re­la­tion­ships with them evolved in many dif­fer­ent ways, they were some­times dif­fi­cult, but here I also learnt a lot from Fuchs. He spent much of his time on me, and to a large ex­tent I pro­ject his way of in­ter­ac­tion. I re­ceived from Fuchs a cer­tain “know­ledge” about math­em­at­ics and to the ex­tent I can I pass it along.

The short note [1] is the first Gel­fand–Fuchs pa­per on con­tinu­ous co­homo­logy. This is not their first joint pa­per; the pre­vi­ous ones are pre­par­at­ory, and among oth­er things they con­tain a dis­cus­sion of con­tinu­ous group co­homo­logy. The pa­per [2] con­tains all the main ideas; its second part is [3]. The pa­per [5], joint with D. I. Ka­lin­in, is de­voted to the co­homo­logy of the Lie al­gebra of Hamilto­ni­an vec­tor fields.

The very im­port­ant pa­per [4], joint with D. A. Kazh­dan, ap­peared in 1972. In­tro­duced in it is the key (to me) concept of a prin­cip­al ho­mo­gen­eous space of an in­fin­ite-di­men­sion­al Lie al­gebra, such as the Lie al­gebra of form­al vec­tor fields. They in­tro­duced tech­niques for deal­ing with form­al geo­metry, and these found nu­mer­ous ap­plic­a­tions; in par­tic­u­lar, they make much clear­er “loc­al” proofs of the in­dex the­or­em and its ap­plic­a­tions.

Fuchs’ 1973 study [6] is an in­ter­est­ing re­view pa­per on char­ac­ter­ist­ic classes of fo­li­ations; [9] is a more ex­ten­ded re­view of the same sub­ject.

My first pa­per with Gel­fand and Fuchs [8] deals with vari­ations of char­ac­ter­ist­ic classes of fo­li­ations. Around that time I was greatly im­pressed by the Fuchs pa­per “Quil­len­iz­a­tion and bor­d­isms” [7], an ex­cel­lent in­tro­duc­tion to K-the­ory (at least for me.)

The work by Fuchs and my­self on the Vi­ra­s­oro al­gebra is put to­geth­er (in­com­pletely) in the big re­view pa­per [15]. This pa­per ap­peared in 1990, but it had been writ­ten a year or two earli­er. The So­viet Uni­on was com­ing to an end, and un­for­tu­nately at about the same time our col­lab­or­a­tion ended too. Times changed, and many people, in­clud­ing Gel­fand and Fuchs, moved to the United States. The be­gin­ning of our joint work is the pa­per [11] de­voted to in­vari­ant dif­fer­en­tial op­er­at­ors on the line. There was no Vi­ra­s­oro al­gebra there yet, but it made its ap­pear­ance in [12]. The very fact that the prob­lem on in­vari­ant dif­fer­en­tial op­er­at­ors is re­lated to the semi-in­fin­ite con­struc­tion of Vi­ra­s­oro al­gebra mod­ules al­lowed us both to prove our con­jec­tures on the clas­si­fic­a­tion of in­vari­ant skew-sym­met­ric dif­fer­en­tial op­er­at­ors and to prove the Kac con­jec­ture about the Vi­ra­s­oro al­gebra Verma mod­ules.

The pa­per [13] stud­ies the struc­ture of the Vi­ra­s­oro al­gebra Verma mod­ules and con­structs the Bern­stein–Gel­fand–Gel­fand-type res­ol­u­tions, which im­ply the char­ac­ter for­mu­las of those mod­ules that arise in min­im­al mod­els. All of this is de­scribed in great­er de­tail in the afore­men­tioned [15].

This list of our joint works is by no means com­plete; we also wrote two pa­pers joint with Gel­fand.

I would like to add a few words about “the sources and com­pon­ent parts”4 of our work. I have already writ­ten about one source, co­homo­logy the­ory of Lie al­geb­ras. This is what we worked on at the be­gin­ning. We mostly re­stric­ted ourselves to the 1-di­men­sion­al case, be­cause much more can be un­der­stood there. Fur­ther­more, we in­vest­ig­ated the zeroth (co-)ho­mo­logy with val­ues in tensor products of mod­ules of tensor fields on the line, which is very close to the prob­lem about clas­si­fic­a­tion of in­vari­ant dif­fer­en­tial op­er­at­ors. The second “source” of our work is rep­res­ent­a­tion the­ory of fi­nite-di­men­sion­al and af­fine Lie al­geb­ras, which we tried to emu­late by study­ing, in as much de­tail as we could, the Verma mod­ule struc­ture, ex­pli­cit for­mu­las for in­ter­twin­ing op­er­at­ors and sin­gu­lar vec­tors, as well as Bern­stein–Gel­fand–Gel­fand-type res­ol­u­tions. The prob­lem about coin­vari­ants arose as an at­tempt to un­der­stand an ana­logue of the loc­al­iz­a­tion func­tor for Kac–Moody Lie al­geb­ras. But we began not even here, we wanted to un­der­stand what hap­pens to sin­gu­lar vec­tors un­der quo­tient­ing out by a sub­mod­ule. For ex­ample, a Verma mod­ule can be quo­tien­ted out by the sub­al­gebra gen­er­ated by $\{L_{-3},L_{-4},\dots \}$, that is, by the com­mut­at­or of the max­im­al nil­po­tent sub­al­gebra of the Vi­ra­s­oro al­gebra. One of the re­mark­able fea­tures of Fuchs is his fond­ness for ana­lyz­ing con­crete, seem­ingly very par­tic­u­lar ques­tions. He was good at com­pu­ta­tions, had a sense for de­tail, and very highly val­ued beau­ti­ful and seem­ingly par­tic­u­lar res­ults, which later proved to be very im­port­ant in more ways than one. The third source and com­pon­ent part of our activ­ity is con­form­al field the­ory. Here is how this came about. Be­lav­in, Polyakov, and Zamo­lod­chikov worked on 2-di­men­sion­al con­form­al field the­ory, and from the out­set it was clear that the Vi­ra­s­oro al­gebra is its main act­or. It had been used in string the­ory, a close re­l­at­ive and fore­fath­er of con­form­al field the­ory. Sasha Be­lav­in asked Volodya Drin­feld what math­em­aticians knew about the Vi­ra­s­oro al­gebra, and Volodya replied that there was a man, namely me, in Cher­no­go­lovka (where Be­lav­in, Polyakov, and Zamo­lod­chikov worked) who knew something about it. Sasha Be­lav­in found me, and this was the be­gin­ning of our friend­ship. I also talked with Sasha Zamo­lod­chikov and was struck by his abil­it­ies es­pe­cially as a math­em­atician. His in­tu­ition and com­pu­ta­tion­al abil­it­ies seemed to me bound­less. It soon tran­spired that all Fuchs and I had worked on was a part of con­form­al field the­ory, its math­em­at­ic­al com­pon­ent. It turned out that “we had been speak­ing in prose without know­ing it.” Fuchs and I un­der­stood in­teg­ral rep­res­ent­a­tions in our lan­guage, but ul­ti­mately they were con­struc­ted by Dot­sen­ko and Fateev. The idea of bo­son­iz­a­tion and in­teg­ral rep­res­ent­a­tions gave a huge im­petus to the de­vel­op­ment of both con­form­al field the­ory and pure math­em­at­ics. Once again, Fuchs and I did not un­der­stand that we dealt with ver­tex al­geb­ras and their re­la­tion to quantum groups. That this was the case be­came clear only when we made con­tact with the Cher­no­go­lovka phys­i­cists.

I would like to con­clude by say­ing a few words about Fuchs the teach­er. I was an ob­ject of Fuchs’s ef­forts as a ped­agogue. These ef­forts were very in­dir­ect, he did not teach me any­thing, as it were, but I fol­lowed him around like a puppy, and noth­ing could be bet­ter for me than this. I could not at­tend his fam­ous lec­tures on ho­mo­top­ic­al to­po­logy, but his text­book on this sub­ject (with Fo­men­ko and Gut­en­mach­er) was per­haps the only text­book I ever read al­most to the end. It is ex­tremely well writ­ten, and al­geb­ra­ic to­po­logy is the first math­em­at­ic­al the­ory I learned. All the oth­ers were much easi­er as I had already got­ten one. Fuchs is the au­thor of a num­ber of text­books; I like the one on co­homo­logy of Lie al­geb­ras [14], a re­mark­able, clearly writ­ten book.