Celebratio Mathematica

This is not a nar­rat­ive of my life. I will not talk about events of my life that were not dir­ectly con­nec­ted to my work. I will talk only briefly about the re­la­tion of my pa­pers to the work of oth­er people.

More in­form­a­tion about my life and my work can be found in my and E. Wit­ten’s in­ter­views for the AIP or­al his­tory pro­ject (www.aip.org/his­tory-pro­grams/niels-bo­hr-lib­rary/or­al-his­tor­ies).1

I star­ted as a to­po­lo­gist, later switched to math­em­at­ic­al phys­ics or, bet­ter to say, to phys­ic­al math­em­at­ics. (The no­tion of math­em­at­ic­al phys­ics is not very well defined, but phys­ic­al math­em­at­ics has the fol­low­ing defin­i­tion giv­en in cor­res­pond­ing sec­tion of Nuc­le­ar Phys­ics B: Math­em­at­ic­al ideas which are of cur­rent in­terest for their po­ten­tial ap­plic­a­tion to the­or­et­ic­al phys­ics. Of course in this defin­i­tion phys­ic­al math­em­at­ics is a time de­pend­ent no­tion. Prof. F. Hirzebruch told me that his work on to­po­lo­gic­al ques­tions of al­geb­ra­ic geo­metry was con­sidered as very ab­stract, very far from any ap­plic­a­tions. Calabi–Yau man­i­folds also were re­garded as a purely math­em­at­ic­al toy.)

I worked in many dir­ec­tions, ap­ply­ing tools from vari­ous branches of math­em­at­ics, chan­ging my field of in­terest every 4–5 years. Let me list some dir­ec­tions of my re­search.

1. Geo­metry of uni­form con­tinu­ity. Volume in­vari­ant (growth) of a group (1952–1955).
2. To­po­lo­gic­al ques­tions of the cal­cu­lus of vari­ations. Space of closed curves. Genus of fiber space (1956–1962).
3. Du­al­ity of func­tors in autonom­ous cat­egor­ies (1962–1968, 1971).
4. Scat­ter­ing mat­rix in quantum field the­ory (adia­bat­ic defin­i­tion, ax­io­mat­ic the­ory) (1971–1974).
5. Particle-like solu­tions to clas­sic­al equa­tions of mo­tion and quantum particles. To­po­lo­gic­al charges. Mag­net­ic mono­poles. Sym­met­ric gauge fields (1975–1978, 1980).
6. In­stan­tons. Quantum fluc­tu­ations of in­stan­tons (1975–1979).
7. To­po­lo­gic­al quantum field the­or­ies (1978–1979, 1987, 1996, 2000).
8. To­po­lo­gic­ally non­trivi­al string-like fields. Alice strings (1981–1982).
9. Geo­metry of su­per­grav­ity. Field-space demo­cracy (1981–1986).
10. Su­per­con­form­al man­i­folds. Su­per­mod­uli spaces. Mul­tiloop con­tri­bu­tion to su­per­string the­ory (1985–1991).
11. Math­em­at­ic­al prob­lems of 2D grav­ity. String equa­tion (1990–1991).
12. Geo­metry of Batal­in–Vilko­visky form­al­ism (1992–1994).
13. Non­com­mut­at­ive al­geb­ra­ic equa­tions (1995, 1997, 2000).
14. Non­com­mut­at­ive geo­metry and its ap­plic­a­tions to string/M-the­ory (1997–).
15. Max­im­ally su­per­sym­met­ric gauge the­or­ies (2003–).
16. Ap­plic­a­tions of arith­met­ic geo­metry to phys­ics (2005–).
17. Ho­mo­logy of mod­uli spaces of al­geb­ra­ic curves and of in­fin­ite-di­men­sion­al Grass­man­ni­an (2011–).
18. Ho­mo­logy of Lie al­geb­ras of su­per­sym­met­ries and su­per-Poin­caré Lie al­gebra, their ap­plic­a­tions (2010–).
19. Quantum curves (1991, 2013 ).
20. In­clus­ive scat­ter­ing mat­rix (2019–).
21. Geo­met­ric ap­proach to quantum the­ory. Jordan al­geb­ras in quantum the­ory. (2019–).

This list does not cov­er all dir­ec­tions of my work (for ex­ample, my pa­pers on com­bin­at­or­i­al in­vari­ance of Pontry­agin classes and on to­po­logy of in­fin­ite-di­men­sion­al spaces are not covered); it in­cludes only the fields that drew my at­ten­tion for sig­ni­fic­ant peri­ods of time.

I nev­er worked in fash­ion­able dir­ec­tions, but some dir­ec­tions of my work be­came fash­ion­able (geo­met­ric group the­ory, to­po­lo­gic­al meth­ods in phys­ics, to­po­lo­gic­al quantum field the­or­ies, non­com­mut­at­ive geo­metry in phys­ics, etc).

My first pa­per (joint with N. Ramm) was writ­ten in 1952 un­der the guid­ance of Pro­fess­or V. A. Efre­movich and pub­lished in 1953 [1]. I was for­tu­nate to have Pro­fess­or V. A. Efre­movich as my ad­visor — he was an out­stand­ing math­em­atician and re­mark­able hu­man be­ing. Most of his work was de­voted to the geo­metry of uni­form con­tinu­ity. In 1935 he proved that a map of met­ric spaces is uni­formly con­tinu­ous if and only if it trans­forms two sets with zero dis­tance between them in­to two sets hav­ing the same prop­erty. This ob­ser­va­tion led him to in­tro­duce prox­im­ity spaces (spaces with a re­la­tion of prox­im­ity between two sets; in met­ric spaces this re­la­tion cor­res­ponds to zero dis­tance.) An­oth­er ax­io­mat­iz­a­tion of the no­tion of uni­form con­tinu­ity was giv­en at the same by A. Weil who in­tro­duced uni­form spaces. My first pa­per was de­voted to the ana­lys­is of the re­la­tion between prox­im­ity spaces and uni­form spaces. Un­for­tu­nately, Efre­movich’s res­ults were not pub­lished on time: he was ar­res­ted dur­ing the purges of 1937 and was freed in 1944 be­cause he was con­sidered ter­min­ally ill. Against the odds he sur­vived and re­turned to math­em­at­ics. However, in 1949 he was fired from his Mo­scow job in a new wave of purges; this was the reas­on why he landed as Pro­fess­or in Ivan­ovo (300 km from Mo­scow where his fam­ily lived).

My story was a little bit sim­il­ar. My par­ents were im­prisoned in 1937. My fath­er was sen­tenced to 10 years without the right to write let­ters. It was dis­covered later that this sen­tence was a eu­phem­ism for death pen­alty. My moth­er was ar­res­ted as a spouse of “an en­emy of the people” and after sev­er­al years in pris­on camp was ex­iled to Kaza­kh­stan. Even my grand­moth­er was ar­res­ted; after a couple of months in jail she found me in an orphan­age un­der a changed name. In 1941 my grand­moth­er and I joined my moth­er and in 1948 we were able to leave Kaza­kh­stan.2

We settled in Ivan­ovo. There I met Pro­fess­or V. A. Efre­movich while still a high school stu­dent — he or­gan­ized “math­em­at­ic­al circles” and olympi­ads. I entered Ivan­ovo Ped­ago­gic­al In­sti­tute in 1951. (My ap­plic­a­tion to Mo­scow Uni­versity was pre­dict­ably un­suc­cess­ful — I was a son of “an en­emy of the people”.)

By the time I gradu­ated from high school I knew already a little bit of math­em­at­ics: cal­cu­lus, ele­ments of Lob­achevsky geo­metry. There­fore in my fresh­man year I was able to study gen­er­al to­po­logy and work in this dir­ec­tion. This was an ideal start­ing point from which Pro­fess­or V. A. Efre­movich led me to more geo­met­ric­al ques­tions. My first ser­i­ous work [2] was in­spired by Efre­movich’s re­mark that the “volume in­vari­ant” of uni­ver­sal cov­er­ing of a com­pact man­i­fold is a to­po­lo­gic­al in­vari­ant of the man­i­fold. (If two com­pact man­i­folds are homeo­morph­ic, then the nat­ur­al homeo­morph­ism between uni­ver­sal cov­er­ings is uni­formly con­tinu­ous. Efre­movich proved that un­der cer­tain con­di­tions the growth of the volume of a ball with ra­di­us tend­ing to in­fin­ity is an in­vari­ant of uni­formly con­tinu­ous homeo­morph­isms.) I proved that the volume in­vari­ant of uni­ver­sal cov­er­ing can be ex­pressed in terms of the fun­da­ment­al group of the ori­gin­al man­i­fold; in mod­ern lan­guage it is de­term­ined by the growth of the fun­da­ment­al group. I also gave es­tim­ates for volume in­vari­ants of man­i­folds with non­posit­ive and with neg­at­ive curvature. Thir­teen years later J. Mil­nor pub­lished a pa­per con­tain­ing the same res­ults with the only dif­fer­ence that Mil­nor was able to use in his proofs some the­or­ems de­rived after the ap­pear­ance of my pa­per. At the mo­ment of writ­ing his first pa­per in this dir­ec­tion Mil­nor did not know about my work, but his second pa­per con­tained cor­res­pond­ing ref­er­ences. The no­tion of growth of a group (volume in­vari­ant of a group in my ter­min­o­logy) was stud­ied later in nu­mer­ous pa­pers (one should men­tion, in par­tic­u­lar, the res­ults by Gro­mov and Grig­orchuk). A new in­ter­est­ing field — geo­met­ric group the­ory — was born from these pa­pers.

I am very grate­ful to P. S. Al­ex­an­drov and V. A. Efre­movich for the pos­sib­il­ity to spend sev­er­al months at Mo­scow Uni­versity, in the best Math­em­at­ics De­part­ment of Rus­sia, re­main­ing a stu­dent at Ivan­ovo. (The status of vis­it­ing stu­dent, not un­com­mon in the US, was un­heard of in Rus­sia. ) In 1955 I gradu­ated from Ivan­ovo Ped­ago­gic­al In­sti­tute and was nom­in­ated for gradu­ate school. (Without such a nom­in­a­tion I would be ob­liged to teach math­em­at­ics and phys­ics in high school.) In the same year I was ad­mit­ted to gradu­ate school at Mo­scow Uni­versity. I was very lucky — in 1954, after Stal­in’s death, my fath­er was ex­on­er­ated. Moreover, 1955 was the best year for a Jew to enter the Uni­versity: the dis­crim­in­a­tion of Stal­in’s time was ab­ol­ished, the new peri­od of dis­crim­in­a­tion star­ted later. My ad­visor at Mo­scow was P. S. Al­ex­an­drov, but at this mo­ment I was already able to work in­de­pend­ently (I wrote 7 pa­pers in my un­der­gradu­ate years) and he gave me com­plete free­dom.

My years in gradu­ate school were quite pro­duct­ive. I would like to men­tion my pa­per with V. Rokh­lin con­tain­ing a proof of com­bin­at­or­i­al in­vari­ance of ra­tion­al Pontry­agin char­ac­ter­ist­ic classes [3]. (At the same time this res­ult was ob­tained by R. Thom.) Most of my pa­pers of this time were de­voted to to­po­lo­gic­al prob­lems that ap­pear in the cal­cu­lus of vari­ations. In one of my pa­pers I stud­ied the to­po­logy of the space of closed curves on a man­i­fold [6] This space is a in­fin­ite-di­men­sion­al or­bi­fold (in mod­ern ter­min­o­logy). The fact that it is not a man­i­fold was not un­der­stood in pre­ced­ing pa­pers by M. Morse and R. Bott; this led to some er­rors. I cor­rec­ted them and used new meth­ods to study the ho­mo­logy of this space. (By the way, this space ap­peared later in pa­pers by Sul­li­van and his col­lab­or­at­ors in re­la­tion to string the­ory; he uses the name string ho­mo­logy for its ho­mo­logy. It was con­sidered also in my pa­per about A-mod­el and gen­er­al­ized Chern–Si­mons the­ory.) In an­oth­er pa­per [4] I in­tro­duced and stud­ied a no­tion of genus of a fiber space; it gen­er­al­ized the no­tions of the Lusternik–Schnirelman cat­egory and of Krasnosel­sky genus of a cov­er­ing. It was ana­lyzed fur­ther in my pa­pers [5], [7] [9] The same no­tion was re­dis­covered (un­der an­oth­er name) 25 years later by S. Smale and used to es­tim­ate to­po­lo­gic­al com­plex­ity of al­gorithms. S. Smale found the simplest es­tim­ates of genus and ap­plied them to es­tim­ate to­po­lo­gic­al com­plex­ity of al­gorithm of solu­tion of al­geb­ra­ic equa­tion hav­ing de­gree \( n \). His es­tim­ate from be­low was \( (\log_2n)^{2/3} \). V. Va­siliev, ap­ply­ing my es­tim­ates of genus, strengthened this res­ult and proved that the com­plex­ity is close to \( n \). (If \( n = 2^k + a \) where \( 0 \leq a < 2^k \) then the com­plex­ity is between \( 2^k - 1 \) and \( n - 1 \)). Later the no­tion of genus was ap­plied to ro­bot mo­tion plan­ning by M. Farber and oth­er schol­ars. My res­ults were based on meth­ods of to­po­logy that were new at that time, es­pe­cially on the use of spec­tral se­quences. When I entered gradu­ate school Mo­scow math­em­aticians did not have a work­ing know­ledge of these meth­ods; I was one of the first Rus­si­an math­em­aticians to suc­cess­fully ap­ply them. I gave nu­mer­ous talks ex­plain­ing these meth­ods, later V. G. Boltyansky, M. M. Post­nikov and I or­gan­ized a sem­in­ar on Geo­met­ric To­po­logy de­voted mostly to ap­plic­a­tions of spec­tral se­quences. The sem­in­ar was very suc­cess­ful; it at­trac­ted many bril­liant young math­em­aticians. (First of all I should men­tion S. Novikov, who after re­ceiv­ing the Fields Medal for his work in to­po­logy, made very im­port­ant con­tri­bu­tions to in­teg­rable sys­tems and math­em­at­ic­al phys­ics.) The sem­in­ar an­nounce­ment con­tained the fol­low­ing de­scrip­tion of geo­met­ric to­po­logy: Gen­er­al to­po­logy stud­ies simple prop­er­ties of com­plic­ated spaces; geo­met­ric to­po­logy stud­ies com­plic­ated prop­er­ties of simple spaces (man­i­folds, poly­hedra). Ex­perts in gen­er­al to­po­logy, in­clud­ing P. S. Al­ex­an­drov, one of cre­at­ors of this field, were in­sul­ted by this de­scrip­tion. It seems that P. Al­ex­an­drov thought that this was my fault (al­though among the spon­sors of the sem­in­ar were two full pro­fess­ors and I was a gradu­ate stu­dent). I don’t re­mem­ber who in­ven­ted the “crim­in­al” sen­tence (the an­nounce­ment was writ­ten by Boltyansky and me), but in any case we did not have any in­ten­tion to in­sult any­body.

In 1958 I de­fen­ded my Can­did­ate of Sci­ences (\( = \) PhD) dis­ser­ta­tion. At this time I was already an au­thor of 15 pa­pers; many of them were well known. (In a re­view book Math­em­at­ics in USSR: 40 years pub­lished in 1957 my res­ults were men­tioned many times.) However, it was not simple for me to find a job after gradu­ation. The main obstacles came from the KGB. It did not give me clear­ance in two places that wanted to hire me (the Labor­at­ory for The­or­et­ic­al Phys­ics in Joint In­sti­tute for Nuc­le­ar Re­search (JINR), Dubna, and the Di­vi­sion of Ap­plied Math­em­at­ics of the Steklov In­sti­tute). I ac­cep­ted an of­fer from Vor­onezh Uni­versity. At this time the De­part­ment of Math­em­at­ics at Vor­onezh Uni­versity was a lively place. It was in­form­ally headed by Pro­fess­or M. Krasnosel­sky, who cre­ated a very act­ive group work­ing in non­lin­ear func­tion­al ana­lys­is and us­ing to­po­lo­gic­al meth­ods in their work. In Vor­onezh I con­tin­ued my work on the genus of fiber space that was close to the in­terests of Vor­onezh group. In 1960 I de­fen­ded my Doc­tor dis­ser­ta­tion in Mo­scow Uni­versity; it was based on my work on genus. (A Doc­tor de­gree in Rus­sia cor­res­ponds to the Ger­man Ha­bil­it­a­tion; a full pro­fess­or as a rule should have this de­gree.) I am very grate­ful to Krasnosel­sky who con­vinced me that my res­ults were suf­fi­cient for the Doc­tor dis­ser­ta­tion and, what was more im­port­ant, ex­plained to me that I should be­come a Doc­tor if I would like to get an apart­ment from the uni­versity. The de­fense of the dis­ser­ta­tion is not a fi­nal step in the Rus­si­an sys­tem; the dis­ser­ta­tion must be cer­ti­fied by a spe­cial or­gan­iz­a­tion, the so-called VAK. In my case everything went smoothly, but sev­er­al years later the VAK be­came a ma­jor tool in the fight against So­viet math­em­aticians of Jew­ish ori­gin; I was very lucky that I did not post­pone sub­mis­sion of my dis­ser­ta­tion.

In Vor­onezh I worked also on to­po­lo­gic­al ques­tions arising in func­tion­al ana­lys­is. I ana­lyzed the to­po­logy of sev­er­al classes of op­er­at­ors. In par­tic­u­lar, I cal­cu­lated ho­mo­topy groups of the space of Fred­holm op­er­at­ors; my res­ults agreed with res­ults of Atiyah and Janich, who stud­ied this space from the view­point of K-the­ory at the same time. I proved that the em­bed­ding of fi­nite-di­men­sion­al unit­ary group \( U (n) \) in­to the in­fin­ite-di­men­sion­al unit­ary group is ho­mo­top­ic to zero and con­jec­tured that the in­fin­ite-di­men­sion­al unit­ary group is as­pher­ic; this con­jec­ture was proven by Kuiper. I con­struc­ted ho­mo­topy in­vari­ants of non­lin­ear Fred­holm op­er­at­ors in Banach spaces and gave a talk about these res­ults in 1961 at the Con­gress of Math­em­aticians of the USSR (Len­in­grad) but nev­er pub­lished these res­ults. I was con­vinced that they were in­com­plete: for op­er­at­ors of in­dex 0 I ob­tained a residue \( \text{mod }2 \) in­stead of an in­teger as an in­vari­ant. Moreover, Pro­fess­or Yu. Bor­iso­vich in­formed me that Cac­ciop­poli stud­ied the de­gree of Fred­holm op­er­at­ors of in­dex zero in the thirties. Later S. Smale in­de­pend­ently found con­struc­tions that were very sim­il­ar to my work. He also did not have a com­plete pic­ture (the con­struc­tion of in­vari­ants in terms of ori­ented cobor­d­ism classes was in­ven­ted later by El­worthy and Tromba), but he did not know about the pa­per of Cac­ciop­poli and he pub­lished his res­ults.

One more dir­ec­tion of my work in Vor­onezh was re­lated to cat­egory the­ory. Be­ing a gradu­ate stu­dent in Mo­scow I ad­vised sev­er­al very gif­ted un­der­gradu­ate stu­dents. A couple of them were will­ing to con­tin­ue to work un­der my guid­ance after my de­par­ture from Mo­scow. I was re­luct­ant to agree — at that time email did not ex­ist, long dis­tance phone calls were ex­pens­ive. However, I agreed to su­per­vise in­form­ally D. Fuchs’ work on his PhD dis­ser­ta­tion. As I un­der­stand now I was fol­low­ing the ad­vice of one tu­tor that pre­pared high school stu­dents to uni­versity en­trance ex­am­in­a­tion: “To be a very suc­cess­ful tu­tor you should work with stu­dents that are able to pass the ex­am­in­a­tion without your help”. D. Fuchs found the top­ic of his dis­ser­ta­tion by him­self. Moreover, for some time I worked in the dir­ec­tion he star­ted. Fuchs’ goal was to un­der­stand Span­i­er du­al­ity in to­po­logy in terms of du­al­ity of func­tors. To­geth­er with D. Fuchs, B. Mityagin and my Vor­onezh stu­dents (V. Kuznet­sov and R. Pokaz­eeva), I stud­ied du­al­ity of func­tors in autonom­ous cat­egor­ies (in cat­egor­ies where the set of morph­isms is equipped with the struc­ture of an ob­ject of the cat­egory — we used the word D-cat­egory for this no­tion) [8], [10], [11], [13], [12], [14], [16].

In 1960, dur­ing the week I spent in Mo­scow after the de­fense of my Doc­tor of Sci­ences dis­ser­ta­tion, I met my fu­ture wife, L. Kiss­ina. We mar­ried sev­er­al months later. My wife was born in Mo­scow and did not want to leave her city. I con­vinced her to give it a try and to live in Vor­onezh for a year. She found that Vor­onezh was great, but it was a great vil­lage and Mo­scow was a great city. As I prom­ised I star­ted to look for a job in Mo­scow. One of our friends learned from phys­i­cists work­ing in MEPhI (Mo­scow En­gin­eer­ing Phys­ics In­sti­tute) that they were not sat­is­fied with the level of math­em­aticians there. This friend men­tioned me as a young Doc­tor of Sci­ences look­ing for a job in Mo­scow. (Prob­ably, I was at this mo­ment the young­est Doc­tor of Sci­ences in the USSR.) In 1964 I was offered a po­s­i­tion as Pro­fess­or of The­or­et­ic­al Phys­ics at MEPhI. The ori­gin­al idea of the phys­i­cists was that at some mo­ment I would be trans­ferred to the De­part­ment of Math­em­at­ics to strengthen it. This nev­er happened. Math­em­aticians were not eager to have me in their de­part­ment. I en­joyed teach­ing vari­ous courses in The­or­et­ic­al Phys­ics and was eager to stay in the Phys­ics De­part­ment and phys­i­cists were not dis­ap­poin­ted with my work.

I was in­ter­ested in phys­ics for a long time. The pos­sib­il­ity to work with phys­i­cists was very ex­cit­ing for me. My first goal was to un­der­stand thor­oughly the main phys­ic­al the­or­ies. I thought a lot about the found­a­tions of quantum mech­an­ics, es­pe­cially about meas­ure­ment the­ory. However, con­ver­sa­tions with phys­i­cists con­vinced me that these ques­tions were not in­ter­est­ing for them and I turned my at­ten­tion to oth­er things. After years of work on the more con­crete prob­lems of the­or­et­ic­al phys­ics I have also lost in­terest in found­a­tions. There was some evol­u­tion in phys­i­cists’ at­ti­tude to meas­ure­ment the­ory; the mod­ern view on these prob­lems does not con­tra­dict my old ideas. (I pub­lished some of my ideas to­geth­er with Yu. Ty­up­kin in 1987 [66]. Very re­cently, in 2019, I used the res­ults of this pa­per in a geo­met­ric ap­proach to quantum the­ory.)

I be­lieve that phys­i­cists were right in avoid­ing thorny ques­tions of meas­ure­ment the­ory. The quantum prob­ab­il­it­ies can be de­rived from de­co­her­ence; the subtle prob­lems of meas­ure­ment the­ory are not rel­ev­ant in this de­riv­a­tion.

The last fifty years were marked by very in­tens­ive and fruit­ful in­ter­ac­tion between math­em­at­ics and the­or­et­ic­al phys­ics. However, in the six­ties this pro­cess had only just star­ted — in Rus­sia slightly earli­er than in the West. A hand­ful of Rus­si­an math­em­aticians star­ted to study ser­i­ously the­or­et­ic­al phys­ics and to work on re­lated math­em­at­ic­al prob­lems; I be­longed to this small group that in­cluded F. Berez­in, R. Dobrush­in, L. Fad­deev, Yu. Man­in, R. Min­los, S. Novikov, and Ya. Sinai.

We fol­lowed the foot­steps of A. Kolmogorov and I. Gel­fand. Later many very tal­en­ted young Rus­si­an math­em­aticians moved in the same dir­ec­tion; I would like to men­tion A. Beil­in­son, I. Bern­stein, V. Drin­feld, B. Fei­gin, D. Fuchs, and D. Kazh­dan.

At the same time many ex­traordin­ary young phys­i­cists moved to­ward math­em­at­ics; first of all Sasha Polyakov and three oth­er Sashas (Be­lav­in, Mig­dal, Zamo­lod­chikov), V. Knizh­nik, A. Moro­zov, M. Shi­f­man, and A. Vain­stein.

To­geth­er with F. Berez­in I or­gan­ized a sem­in­ar de­voted to math­em­at­ic­al prob­lems of phys­ics in Mo­scow Uni­versity. Later R. Dobrush­in, R. Min­los and Ya. Sinai joined us as or­gan­izers.

My at­ten­tion was con­cen­trated mostly on at­tempts to un­der­stand quantum field the­ory. To­geth­er with my stu­dents V. Fateev, V. Likhachev, and Yu. Ty­up­kin I wrote sev­er­al pa­pers on the scat­ter­ing mat­rix in quantum field the­ory. We stud­ied the scat­ter­ing mat­rix in the ax­io­mat­ic ap­proach [18] and the re­la­tion of an adia­bat­ic S-mat­rix to the scat­ter­ing of phys­ic­al particles (see, for ex­ample, [17]). (Much later the ideas of an ax­io­mat­ic ap­proach based on asymp­tot­ic com­mut­ativ­ity were used in my pa­per “Space and time from trans­la­tion sym­metry” [122]. They are rel­ev­ant for string field the­ory.)

I was not sat­is­fied with text­books in quantum field the­ory. Of course, I un­der­stood that there is no reas­on to re­quire math­em­at­ic­al rig­or from such a book. My main con­cern was that the main con­cepts, such as the no­tion of scat­ter­ing mat­rix, were ill-defined, that the rules of the game were changed in the middle of the game. Phys­ics stu­dents were con­vinced that renor­mal­iz­a­tion is ne­ces­sary only to re­move di­ver­gences, and did not un­der­stand that it ap­pears already in stat­ist­ic­al phys­ics where all Feyn­man dia­grams are con­ver­gent. I have writ­ten two books de­voted to the found­a­tions of quantum field the­ory. The ex­pos­i­tion in these books was not rig­or­ous, but it was based on clear defin­i­tions. The second edi­tion of my book Math­em­at­ic­al Found­a­tions of Quantum Field The­ory was pub­lished in 2018. Re­cently, in 2020, the Eng­lish trans­la­tion of the book (with the ad­di­tion of some new res­ults) was pub­lished by World Sci­entif­ic.

At the end of eighties I pub­lished a book Quantum Field The­ory and To­po­logy con­tain­ing a short in­tro­duc­tion to to­po­logy and to its ap­plic­a­tions in quantum field the­ory; the second edi­tion was pub­lished in 2018. An Eng­lish trans­la­tion of this book was pub­lished by Spring­er Ver­lag as two volumes (Quantum Field The­ory and To­po­logy and To­po­logy for Phys­i­cists.)

In the six­ties to­po­logy did not play any sig­ni­fic­ant role in quantum field the­ory. I con­tin­ued to work on some to­po­lo­gic­al and al­geb­ra­ic prob­lems, but this work was not re­lated to phys­ics in any way. The re­la­tions between phys­ics and mod­ern math­em­at­ics changed drastic­ally in the sev­en­ties. The change was promp­ted by dis­cov­ery that soliton solu­tions of clas­sic­al equa­tions of mo­tion are re­lated to quantum particles and that to­po­lo­gic­al con­sid­er­a­tions can be used to guar­an­tee sta­bil­ity of solitons and cor­res­pond­ing quantum particles. I will not dis­cuss in de­tail the his­tory of this dis­cov­ery that star­ted with al­most un­noticed pa­pers by Skyrme. My in­terest in this sub­ject was in­spired by in­flu­en­tial pa­pers by G. ’t Hooft and A. Polyakov who in­de­pend­ently sug­ges­ted in 1974 that clas­sic­al solu­tions in one of the mod­els de­scrib­ing in­ter­ac­tions of gauge fields and scal­ar fields (the Georgi–Glashow mod­el) can be in­ter­preted as mag­net­ic mono­poles. In a pa­per joint with V. Fateev and Yu. Ty­up­kin [21] we proved that the space of fi­nite en­ergy fields in gauge the­or­ies is dis­con­nec­ted in gen­er­al and the com­pon­ents of this space are labeled by num­bers that may be called to­po­lo­gic­al charges or to­po­lo­gic­al in­vari­ants of mo­tion. To­po­lo­gic­al charges can be iden­ti­fied with ele­ments of some ho­mo­topy groups; one can cal­cu­late these groups by means of ho­mo­topy the­ory. If the gauge group is simply con­nec­ted and the group of un­broken gauge sym­met­ries is one-di­men­sion­al then there ex­ists only one to­po­lo­gic­al charge and it can be iden­ti­fied with mag­net­ic charge [28]. This state­ment leads to a pre­dic­tion that mag­net­ic mono­poles ex­ist in any grand uni­fic­a­tion the­ory, i.e., in a gauge the­ory de­scrib­ing strong, weak and elec­tro­mag­net­ic in­ter­ac­tion at the same time.

Mag­net­ic mono­poles and oth­er to­po­lo­gic­ally non­trivi­al fields were con­sidered in many of my pa­pers, for ex­ample in [19], [20], [23], [27], [26], [28], [25], [29], [40] with the help of ho­mo­topy the­ory, ho­mo­logy the­ory, char­ac­ter­ist­ic classes and in­dex the­ory. The joint pa­per of 1975 with A. Be­lav­in, A. Polyakov, and Yu. Ty­up­kin [24] was es­pe­cially in­flu­en­tial. In this pa­per we in­tro­duced and ana­lyzed to­po­lo­gic­ally non­trivi­al ex­trema of eu­c­lidean ac­tion in gauge the­ory (pseudo­particles in the ter­min­o­logy of our pa­per is the same as gauge in­stan­tons in mod­ern ter­min­o­logy). In­stan­tons were stud­ied later in nu­mer­ous pa­pers, in­clud­ing pa­pers writ­ten by me (of­ten in col­lab­or­a­tion with my stu­dents V. Fateev, I. Fro­lov, V. Ro­man­ov, and Yu. Ty­up­kin). In par­tic­u­lar, I cal­cu­lated the di­men­sion of the mod­uli space [30] of in­stan­tons by means of in­dex the­ory and ana­lyzed quantum fluc­tu­ations of in­stan­tons [31], [32], [37], [38], [36], [43]. A slightly dif­fer­ent cal­cu­la­tion of the di­men­sion of the in­stan­ton mod­uli space was giv­en by Atiyah, Hitchin and Sing­er. It seems that it was in­spired by my pre­print, that did not con­tain com­plete res­ults, but their pa­per ap­peared after pub­lic­a­tion of mine (and refers to it). Both of these pa­pers are de­voted to con­sid­er­a­tion of in­stan­tons on the four-di­men­sion­al sphere, however, it was clear from the very be­gin­ning that mod­uli space of in­stan­tons on more gen­er­al four-di­men­sion­al man­i­folds can be stud­ied in a sim­il­ar way. Atiyah, Hitchin, Sing­er worked in this dir­ec­tion and their res­ults formed a basis of fam­ous pa­pers by Don­ald­son on clas­si­fic­a­tion of four-di­men­sion­al man­i­folds.

I also stud­ied one-di­men­sion­al to­po­lo­gic­ally stable con­fig­ur­a­tions in quantum field the­ory (to­po­lo­gic­ally stable string-like ob­jects) [41], [55], [46]. I have shown that such con­fig­ur­a­tions ap­pear in gauge the­or­ies if the gauge group is con­nec­ted but the group of un­broken sym­met­ries is dis­con­nec­ted and that a particle that goes around to­po­lo­gic­ally stable string-like ob­ject can change its type. The last state­ment found an ex­per­i­ment­al con­firm­a­tion in con­densed mat­ter phys­ics. It led also to the state­ment that in grand uni­fic­a­tion the­or­ies de­scrib­ing weakly in­ter­act­ing “real” and “mir­ror” worlds there ex­ist quite un­usu­al string-like ob­jects (Alice strings) hav­ing the prop­erty that a particle go­ing around such a string be­comes a mir­ror particle.3 Elec­tric charge is not loc­al­ized in the pres­ence of Alice strings — people say that it be­comes a Cheshire charge, re­mem­ber­ing the Cheshire cat from L. Car­roll’s book. In prin­ciple Alice strings can be de­tec­ted by means of as­tro­nom­ic­al ob­ser­va­tions; there ex­ist some ob­ser­va­tions that can be ex­plained by means of Alice strings very nat­ur­ally; however, all of them have more com­plic­ated, but less exot­ic ex­plan­a­tions.

When I in­ven­ted Alice strings my first thought was that they give won­der­ful ma­ter­i­al for sci­ence fic­tion. It seems that oth­er people shared this opin­ion. J. Grib­bin wrote a book The Alice En­counter based on this idea. This is an ex­cerpt from the syn­op­sis of this book:

There is about 10 times more dark mat­ter (DM, also known here as Alice mat­ter) than bright stuff in our Galaxy. The DM is spread out in a roughly uni­form sphere (a spher­ic­al dis­tri­bu­tion of Alice stars), with our flattened disk Galaxy em­bed­ded in it. The Alice mat­ter is a kind of mir­ror im­age shad­ow stuff; the term look­ing glass mat­ter has been used by some sci­ent­ists. Alice mat­ter can be turned in­to or­din­ary mat­ter (and vice versa) by send­ing it through a loop of Alice string, a nat­ur­ally oc­cur­ring cos­mic phe­nomen­on.

I en­joyed my work on to­po­lo­gic­al in­teg­rals of mo­tions and to­po­lo­gic­ally non­trivi­al solu­tions. My re­search in to­po­logy gave me an ex­cel­lent back­ground for this work and I was happy to work in a new and fast-mov­ing field. I gave sev­er­al talks with titles like “Ap­plic­a­tions of to­po­logy to phys­ics”, and these talks were in­ter­est­ing both to phys­i­cists, who real­ized there ex­ists a power­ful new tool, and to math­em­aticians, who dis­covered a new rich source of ex­cit­ing math­em­at­ic­al prob­lems.

I was even hap­pi­er when in 1978 I found a way to use ideas from phys­ics in to­po­logy [34], [35]. I real­ized that ac­tion func­tion­als that do not de­pend ex­pli­citly on the met­ric should give rise to to­po­lo­gic­al in­vari­ants. In lec­tures about these new ideas I talked about “ap­plic­a­tions of phys­ics to to­po­logy”. Of course, this was not a pre­cise state­ment; however, the very fact that one could use ideas from phys­ics to ob­tain to­po­lo­gic­al res­ults was fas­cin­at­ing. I have found that Re­idemeister tor­sion (or, more pre­cisely, its dif­fer­en­tial coun­ter­part defined by Ray and Sing­er) can be ob­tained as a par­ti­tion func­tion of some met­ric-in­de­pend­ent ac­tion func­tion­al. The gauge trans­form­a­tions for this func­tion­al were not in­de­pend­ent (in today’s ter­min­o­logy the the­ory was re­du­cible), hence it was im­possible to per­form quant­iz­a­tion us­ing the Fad­deev–Pop­ov ap­proach; I de­veloped tech­niques al­low­ing quant­iz­a­tion of the­or­ies of this kind.

I un­der­stood very well that the idea to con­struct to­po­lo­gic­al in­vari­ants from met­ric-in­de­pend­ent ac­tion func­tion­als (such as the Chern–Si­mons func­tion­al) was quite gen­er­al and it should lead to new in­ter­est­ing in­vari­ants; moreover, it was clear that this is also a way to find in­vari­ants of oth­er struc­tures (say, in­vari­ants of com­plex man­i­folds). However, at that time I was able to ana­lyze non­quad­rat­ic ac­tion func­tion­als only in semi­clas­sic­al ap­prox­im­a­tion or in the frame­work of per­turb­a­tion the­ory. It was easy to write down per­turb­a­tion series for in­vari­ants, but it was very dif­fi­cult to ana­lyze these series. Much later, in 1987 V. Tur­aev told me that V. Jones con­struc­ted an in­vari­ant of knots that can be con­sidered as a non­com­mut­at­ive gen­er­al­iz­a­tion of the Al­ex­an­der poly­no­mi­al. One can ex­press the Al­ex­an­der poly­no­mi­al in terms of Re­idemeister tor­sion. V. Tur­aev asked me wheth­er it is pos­sible to modi­fy the ac­tion func­tion­al I used to ob­tain Re­idemeister tor­sion in a way that leads to the Jones poly­no­mi­al. Pretty soon I came to a con­jec­ture that the role of such a modi­fic­a­tion should be played by the Chern–Si­mons ac­tion func­tion­al; I talked about this con­jec­ture at the to­po­lo­gic­al con­fer­ence in Baku [70]. I was not able to prove this con­jec­ture; the per­turb­at­ive ex­pres­sions I tried to ana­lyze to­geth­er with Yu. Ty­up­kin were too com­plic­ated. The dif­fi­culties arising in this way were over­come only in 1992 by Axel­rod–Sing­er [e4] and Kont­sevich (in an un­pub­lished talk giv­en at Har­vard in the 1991–92 aca­dem­ic year). However, in 1988 E. Wit­ten in­de­pend­ently con­jec­tured that the Jones poly­no­mi­al was re­lated to the Chern–Si­mons ac­tion func­tion­al and gave a (heur­ist­ic) proof of this con­jec­ture [e2]. My 1978 pa­per [34] was a pre­curs­or to a series of re­mark­able pa­pers where the meth­ods of the­or­et­ic­al phys­ics were ap­plied to to­po­logy, al­geb­ra­ic geo­metry, sym­plect­ic geo­metry and oth­er branches of math­em­at­ics. Many of those pa­pers were re­lated to to­po­lo­gic­al quantum field the­or­ies (my pa­per gave the first ex­amples of such the­or­ies). The ax­ioms of to­po­lo­gic­al quantum field the­or­ies were sug­ges­ted by M. Atiyah, who con­jec­tured that Don­ald­son in­vari­ants and Flo­er ho­mo­logy can be ob­tained from to­po­lo­gic­al quantum field the­or­ies. This con­jec­ture was proven by E. Wit­ten, who found a new way to con­struct to­po­lo­gic­al quantum field the­or­ies. (I con­sidered met­ric-in­de­pend­ent ac­tion func­tion­als; E. Wit­ten has shown that many very in­ter­est­ing the­or­ies can be con­struc­ted if we al­low BRST-trivi­al de­pend­ence on the met­ric.)

Today to­po­lo­gic­al quantum field the­or­ies play an im­port­ant role both in math­em­at­ics and phys­ics. I re­turned to them sev­er­al times (for ex­ample, in 2000, when I was in­vited to give a plen­ary lec­ture on TQFT at In­ter­na­tion­al Con­gress for Math­em­at­ic­al Phys­ics). In my pa­per with S. Gukov and C. Vafa [114] I gave an in­ter­pret­a­tion of Khovan­ov and Khovan­ov–Roz­ansky in­vari­ants of knots in terms of to­po­lo­gic­al strings.

In the eighties I stud­ied the geo­metry of su­per­grav­ity (to­geth­er with my stu­dents M. Baran­ov, A. Gay­duk, O. Khu­daver­d­i­an, V. Ro­man­ov, and A. Rosly) [39], [42], [44], [44], [44], [47], [49], [50], [57], [56], [52], [54], [60], [58]. This work led me to an idea that fields should by con­sidered on an equal foot­ing with space-time co­ordin­ates [39], [42]. Fol­low­ing the sug­ges­tion of E. Brez­in I called this idea field-space demo­cracy. Al­though ex­pli­cit ref­er­ences to my pa­pers on field-space demo­cracy are rare the idea is alive. (In par­tic­u­lar, field-space demo­cracy was used by Town­send to con­struct ac­tion func­tion­als of D-branes.) An un­ex­pec­ted con­sequence of my pa­per on field-space demo­cracy was an in­vit­a­tion to at­tend a con­fer­ence on para­psy­cho­logy in Italy. (The or­gan­izers of the con­fer­ence had a goal to present para­psy­cho­logy as a sci­ence, there­fore they in­cluded a sec­tion de­voted to new ideas in the the­ory of space and time. A. Salam was lis­ted as one of the speak­ers; per­haps, the or­gan­izers knew my name from him.) Of course, I did not at­tend the con­fer­ence.

My work in su­per­grav­ity was based on the the­ory of G-struc­tures, in par­tic­u­lar on the the­ory of Cauchy–Riemann struc­tures (CR-struc­tures). CR-struc­tures were used later by A. Rosly to for­mu­late the­or­ies in terms of iso­top­ic ana­logs of twis­tors (isotwis­tors). I also worked in this dir­ec­tion to­geth­er with A. Rosly. Rosly’s idea of isotwis­tors (un­der the name of har­mon­ic space) was used by V. Ogievet­sky and his group (Sokatchev, Ivan­ov, Kal­itzin, and Galper­in) to ob­tain very in­ter­est­ing ex­pli­citly su­per­sym­met­ric for­mu­la­tions of some su­per­sym­met­ric the­or­ies. Later, I came back to this circle of ideas study­ing max­im­ally su­per­sym­met­ric gauge the­or­ies to­geth­er with M. Movshev.

In the second half of eighties I worked on geo­met­ric ques­tions arising in su­per­string the­ory (to­geth­er with M. Baran­ov, I. Fro­lov, A. Rosly, and A. Voro­nov) [53], [59], [63], [61], [64], [62], [65], [67], [71], [69], [68], [74]. I in­tro­duced a no­tion of a su­per­con­form­al man­i­fold (this no­tion was in­tro­duced in­de­pend­ently by D. Friedan and S. Shen­ker un­der the name “su­per Riemann sur­face”). We stud­ied the mod­uli space of su­per­con­form­al man­i­folds (su­per­mod­uli space), that ap­pears in the cal­cu­la­tion of mul­tiloop con­tri­bu­tions for su­per­strings. Re­cently the res­ults of these pa­pers were pop­ular­ized by E. Wit­ten who has writ­ten sev­er­al re­view pa­pers on re­lated top­ics. I would like to men­tion my pa­per about uni­ver­sal mod­uli space [68]; it seems that this space de­serves fur­ther study. I talked about my res­ults in this dir­ec­tion in an in­vited lec­ture at the In­ter­na­tion­al Con­gress of Math­em­aticians (Kyoto, 1990).

In [51] I have a defin­i­tion of su­per­space in terms of func­tors on the cat­egory of Grass­mann al­geb­ras. This defin­i­tion can be con­sidered as a math­em­at­ic­al form­al­iz­a­tion of the stand­ard defin­i­tion used by phys­i­cists; it is es­pe­cially con­veni­ent for in­fin­ite-di­men­sion­al su­per­man­i­folds. Later to­geth­er with A. Konechny I gen­er­al­ized this defin­i­tion [91], [102]. The more gen­er­al defin­i­tion is con­veni­ent when we are deal­ing with two-di­men­sion­al su­per­man­i­folds with in­de­pend­ent left-mov­ing and right-mov­ing modes and with cor­res­pond­ing mod­uli spaces.

At this mo­ment I have to in­ter­rupt the dis­cus­sion of my work to talk about polit­ic­al changes in the So­viet Uni­on and about changes in my life. This was the time of Gorbachev’s peres­troika. All of my col­leagues were fas­cin­ated by new pos­sib­il­it­ies and afraid of im­min­ent dangers. For me, the first sign of new times was an al­most suc­cess­ful at­tempt to go to Swansea. I was in­vited to give a lec­ture at the In­ter­na­tion­al Con­gress for Math­em­at­ic­al Phys­ics and the KGB had no ob­jec­tions. I was ex­cluded from the So­viet del­eg­a­tion by bur­eau­crats in the Academy.

I re­ceived many in­vit­a­tions to spend some time in the West; I could not ac­cept any of them un­til 1989. In early eighties L. Michel told me that I had a stand­ing in­vit­a­tion from IHES, but a new form­al let­ter would be sent only if I have a chance to come. Be­fore “peres­troika” I was al­lowed to go abroad only twice — I vis­ited Bul­garia in 1980 and Czechoslov­akia in 1981. In 1988 I was able to at­tend a con­fer­ence in Po­land. There I met K. Gawedzki and asked him to in­form the ad­min­is­tra­tion of IHES that the situ­ation had changed. I nev­er re­ceived the prom­ised in­vit­a­tion let­ter. Later, in Italy, K. Gawedzki ex­plained to me that usu­ally IHES in­vited about 20 math­em­aticians from Rus­sia every year and only a couple of in­vit­a­tions were ac­cep­ted. That year all twenty in­vited schol­ars came; IHES did not have any pos­sib­il­ity to send an ad­di­tion­al in­vit­a­tion. I came to IHES for the first time in 1995. Later I spent many sum­mers in this won­der­ful in­sti­tute. IHES has ex­traordin­ar­ily strong per­man­ent mem­bers; I was for­tu­nate to col­lab­or­ate with sev­er­al of them (with A. Connes, M. Douglas, M. Kont­sevich, N. Nekra­sov).

In 1989 I was able to ac­cept an in­vit­a­tion from the In­ter­na­tion­al Cen­ter for The­or­et­ic­al Phys­ics (ICTP) to spend 1 1/2 months in Trieste. I was im­me­di­ately in­vited to come back in Ju­ly to par­ti­cip­ate in the Su­per­mem­brane Con­fer­ence. I ex­plained to Prof. Sez­gin that there was no chance for me to be al­lowed to make an­oth­er form­al trip to Italy on such short no­tice, but that I could try to come as a private cit­izen. By my re­quest Prof. Sez­gin form­ally in­vited me and my wife to vis­it him in Italy. My fam­ily con­sidered the at­tempt to come back to Italy in two months as com­pletely hope­less, but things were mov­ing very fast in the So­viet Uni­on. On Ju­ly 17, 1989 I left the So­viet Uni­on — forever.

Very pos­it­ive changes in the So­viet Uni­on went hand in hand with dan­ger­ous de­vel­op­ments. Gorbachev’s eco­nom­ic policy was dis­astrous. In­stead of rad­ic­al changes on a small scale he ac­cep­ted large scale half-baked re­forms. As a res­ult the eco­nomy, that was already in bad shape be­fore Gorbachev, rap­idly de­teri­or­ated. Na­tion­al­ist­ic sen­ti­ments both in So­viet re­pub­lics and Rus­sia, sup­pressed un­der com­mun­ist rulers, flour­ished in more demo­crat­ic times; the gov­ern­ment an­ti­semit­ism de­clined, but the ap­pear­ance of openly an­ti­semit­ic or­gan­iz­a­tions and news­pa­pers was very dis­turb­ing. My chil­dren did not see any fu­ture for them in the So­viet Uni­on. I de­cided to leave Rus­sia fear­ing for my fam­ily.

I would like to ex­press my sin­cere grat­it­ude to all people that helped me to come to the US and settle here, first of all to D. Gross, A. Jaffe, J. Schwarz, I. Sing­er, E. Wit­ten and B. Zu­mino. I spent the aca­dem­ic year 1989–1990 in Prin­ceton (in the In­sti­tute for Ad­vanced Study) and in Cam­bridge (at Har­vard and MIT). Since 1990 I have worked in the De­part­ment of Math­em­at­ics at UC Dav­is. This is now a very strong de­part­ment; it was com­pletely trans­formed due to the ef­forts of its chair­men, first of all A. Kren­er, then C. Tracy, M. Mu­lase, J. Hunter, B. Nachtegaele, J. Hass, D. Romik, and A. Thompson.

When I came to the US I real­ized that it is ne­ces­sary to make a choice: should I look for a po­s­i­tion in a de­part­ment of math­em­at­ics or in phys­ics de­part­ment? I star­ted as a math­em­atician, but I worked in phys­ics de­part­ment in Rus­sia and I al­ways char­ac­ter­ized my­self cit­ing a Rus­si­an pro­verb “Gentle calf sucks milk of two cows”. I de­cided that nobody can say that I am not a math­em­atician, but some people can say that I am not a phys­i­cist. My de­cision to join the De­part­ment of Math­em­at­ics in Dav­is was a good one; now we have sev­er­al ex­cel­lent schol­ars work­ing on the in­ter­face between math­em­at­ics and phys­ics.

Rus­si­an sci­ence was con­cen­trated mostly in Mo­scow and Len­in­grad; Amer­ic­an math­em­aticians and phys­i­cists are dis­persed over the whole coun­try. Hap­pily now, us­ing in­ter­net and tele­phone, schol­ars are able to col­lab­or­ate even across the ocean; Amer­ic­an schol­ars used this pos­sib­il­ity earli­er than schol­ars of oth­er coun­tries. I star­ted such a “long dis­tance col­lab­or­a­tion” im­me­di­ately after com­ing to the West. My first joint pa­per after leav­ing Rus­sia was writ­ten to­geth­er with A. Sen from In­dia (our col­lab­or­a­tion star­ted when we met in Trieste for a couple of days); it was de­voted to bound­ary con­form­al field the­ory [73]. I col­lab­or­ated with V. Kac, H. S. La and P. Nel­son, K. Anagnosto­poulos and M. Bovick, A. Tseyt­lin, L. Fried­lander, M. Kont­sevich, A. Connes, S. Novikov, M. Douglas, M. Rief­fel, N. Nekra­sov, B. Pi­oline, A. Pol­ishchuk, S. Gukov, C. Vafa, V. Volo­god­sky, A. Mikhail­ov, D. Kre­fl, J. Wal­cher, Xiao­jun Liu and with many people from Dav­is: D. Fuchs, M. Pen­kava, O. Za­boron­sky, A. Konechny, M. Al­ex­an­drov, A. As­tashkevich, M. Di­eng, M. Luu, M. Movshev, Yujun Chen, Xi­ang Tang, I. Sha­piro, Jia-Ming Li­ou, and Ren­jun Xu.

I would like to thank all of my coau­thors.

As usu­al I worked in vari­ous dir­ec­tions. One of my first pa­pers in Amer­ica was writ­ten in col­lab­or­a­tion with V. Kac [72]; we stud­ied the par­ti­tion func­tion of 2D Grav­ity from the view­point of geo­metry of in­fin­ite-di­men­sion­al Grass­man­ni­an. More com­plete res­ults in this dir­ec­tion were ob­tained later in my pa­pers and in a joint pa­per with K. Anagnosto­poulos and M. Bovick [75]. In par­tic­u­lar, I have stud­ied in de­tail the so called string equa­tion: \( [A, B] = \text{const} \), when \( A \) and \( B \) are dif­fer­en­tial op­er­at­ors on a line. I came back to this them­at­ics later in [131]. Sev­er­al of my pa­pers were de­voted to the geo­metry of Batal­in–Vilko­visky quant­iz­a­tion [76], [77], [79], [84]. I am con­vinced that the BV-form­al­ism should be used not only as a tool, that al­lows us to quant­ize de­gen­er­ate Lag­rangi­ans, but also a start­ing point in clas­sic­al and quantum field the­ory. This idea to­geth­er with the geo­met­ric ap­proach to BV-form­al­ism was ap­plied in a pa­per writ­ten jointly with M. Kont­sevich and my stu­dents M. Al­ex­an­drov and O. Za­boron­sky [84] to give a geo­met­ric con­struc­tion of a BV sigma-mod­el (now called AK­SZ mod­el) con­tain­ing many im­port­ant the­or­ies as par­tic­u­lar cases. Later M. Kont­sevich used this con­struc­tion in his fam­ous pa­per on quant­iz­a­tion of Pois­son man­i­folds.

The geo­met­ric ap­proach to the BV-form­al­ism is based on the no­tion of a Q-man­i­fold (\( = \) su­per­man­i­fold equipped with an odd vec­tor field with square equal to zero). This no­tion is closely re­lated to the no­tion of an \( L_{\infty} \)-al­gebra that can be re­garded as form­al Q-man­i­fold; \( A_{\infty} \)-al­geb­ras can be char­ac­ter­ized as form­al non­com­mut­at­ive Q-man­i­folds. Hoch­schild co­homo­logy cor­res­ponds to \( A_{\infty} \)-de­form­a­tions and cyc­lic co­homo­logy cor­res­ponds to de­form­a­tions of \( A_{\infty} \)-al­geb­ras with in­vari­ant in­ner product [82]. My work with M. Movshev on su­per­sym­met­ric gauge the­or­ies [125] is based on these re­la­tions.

The no­tion of Q-man­i­fold was one of the pre­curs­ors of de­rived al­geb­ra­ic geo­metry, a new field that is im­port­ant both in math­em­at­ics and phys­ics.

My work on non­com­mut­at­ive al­geb­ra­ic equa­tions star­ted with a proof of “mat­rix Vi­eta the­or­em” in a joint pa­per with D. Fuchs [81]. We proved that for gen­er­ic roots \( x_1 ,\dots, x_n \) of a mat­rix al­geb­ra­ic equa­tion \[ x^n + a_1 x^{n-1} + \dots + a_n = 0 \] one has \[ \operatorname{Tr} x_1 + \dots + \operatorname{Tr} x_n = {}-\operatorname{Tr}a_1 , \quad \det x_1 \cdot\, \cdots \,\cdot \det x_n = \det a_n. \] The coef­fi­cients \( a_1 ,\dots, a_n \) can be ex­pressed in terms of gen­er­ic roots \( x_1 , \dots, x_n \); cor­res­pond­ing ex­pres­sions can be con­sidered as non­com­mut­at­ive ana­logs of ele­ment­ary sym­met­ric func­tions and our res­ults can be for­mu­lated as a state­ment about these func­tions. We did not think that our work was re­lated to phys­ics in any way, but sev­er­al years later phys­i­cists P. Aschieri, D. Brace, B. Mor­ariu and B. Zu­mino came to the non­com­mut­at­ive al­geb­ra­ic equa­tions when study­ing Born–In­feld-type Lag­rangi­ans. My pa­per [99] was in­spired by their work; it was based on an­oth­er proof of the mat­rix Vi­eta the­or­em giv­en by Connes and me [85]. The pa­pers on non­com­mut­at­ive al­geb­ra­ic equa­tions are re­lated to pa­pers by I. Gel­fand, V. Re­takh and their col­lab­or­at­ors; all these pa­pers be­long to an emer­ging field that still does not have a good name.

One of main dir­ec­tions of my work in re­cent years can be de­scribed as the ap­plic­a­tion of Connes’ non­com­mut­at­ive geo­metry to string/M-the­ory.

The “second string re­volu­tion” of the nineties made clear that su­per­string the­ory should be em­bed­ded in­to a more gen­er­al the­ory that was christened M-the­ory. It was con­jec­tured that M-the­ory lives in el­ev­en-di­men­sion­al space; it should give su­per­strings after com­pac­ti­fic­a­tion on a circle and el­ev­en-di­men­sion­al su­per­grav­ity in its low-en­ergy lim­it. All ex­ist­ing su­per­string the­or­ies can be ob­tained from M-the­ory in some lim­its; all of them are re­lated by du­al­it­ies that can trans­form ob­jects that al­low a per­turb­at­ive de­scrip­tion, in­to more exot­ic ob­jects — branes.

A math­em­at­ic­al defin­i­tion of M-the­ory is not known; however in 1996 Banks, Fisc­hler, Shen­ker and Suss­kind sug­ges­ted that M-the­ory can be for­mu­lated in terms of a mat­rix mod­el. This mod­el (the BFSS mod­el or M(at­rix) the­ory) is defined as a re­duc­tion of ten-di­men­sion­al su­per­sym­met­ric Yang–Mills the­ory to \( (1+0) \)-di­men­sion­al space (all spa­tial di­men­sions are re­duced to a point). I was very im­pressed by the BFSS con­jec­ture; it gave me a math­em­at­ic­al frame­work to work with M-the­ory.

It is clear today that in some sense the BFSS con­jec­ture is cor­rect; however, the role of the BFSS mod­el is not quite clear. I be­lieve that the com­plete M-the­ory can be ex­trac­ted from the BFSS mod­el; it seems that my opin­ion is much more op­tim­ist­ic than the view­point of the cre­at­ors of this mod­el. It is pos­sible, however, that the path from M(at­rix) mod­el to M-the­ory is very com­plic­ated and that there ex­ist oth­er, bet­ter ap­proaches. (E. Wit­ten and J. Schwarz in­de­pend­ently ex­pressed in con­ver­sa­tions with me the same opin­ion: that it is clear now that M(at­rix) mod­el is cor­rect, but it is not clear wheth­er it is use­ful.)

When I star­ted think­ing about the BFSS mod­el I real­ized that there ex­ists an­oth­er at­tract­ive mat­rix mod­el that can be used to un­der­stand string/M-the­ory: the re­duc­tion of ten-di­men­sion­al su­per­sym­met­ric Yang–Mills the­ory to a point. I found very soon that I was not the first schol­ar that came to this idea: this mod­el was sug­ges­ted by the Ja­pan­ese phys­i­cists Ishi­bashi, Kawai, Kitaza­wa and Tsuchiya; it is now known un­der the name IKKT mod­el.

The BFSS mod­el can be ob­tained from IKKT by means of com­pac­ti­fic­a­tion on a circle and Wick ro­ta­tion. This re­mark led me to con­sid­er­a­tion of the gen­er­al prob­lem of com­pac­ti­fic­a­tion of the mat­rix mod­el on a circle and, more gen­er­ally, on a tor­us. This prob­lem was con­sidered already in the ori­gin­al BFSS pa­per. However, I found that the lo­gic used in this pa­per and oth­er pa­pers leads to more gen­er­al com­pac­ti­fic­a­tions; they can be de­scribed in terms of Connes’ non­com­mut­at­ive geo­metry. More pre­cisely, these com­pac­ti­fic­a­tions lead to non­com­mut­at­ive tori, which were thor­oughly ana­lyzed by A. Connes, M. Rief­fel and oth­er math­em­aticians. In a joint pa­per with A. Connes and M. Douglas [87] we have shown how non­com­mut­at­ive tori arise very nat­ur­ally in mat­rix mod­els and what is the phys­ic­al mean­ing of com­pac­ti­fic­a­tions on non­com­mut­at­ive tori. This pa­per paved the way to the ap­plic­a­tion of beau­ti­ful the­or­ems of non­com­mut­at­ive geo­metry proven by A. Connes and his fol­low­ers to string/M-the­ory. Among nu­mer­ous pa­pers with ap­plic­a­tions of non­com­mut­at­ive geo­metry to phys­ics, I would like to single out the very in­flu­en­tial Seiberg–Wit­ten pa­per [e5] that gave, in par­tic­u­lar, a clear ex­plan­a­tion of the ap­pear­ance of non­com­mut­at­ive geo­metry in the frame­work of string the­ory, as well as an ana­lys­is of the re­la­tion between gauge fields on non­com­mut­at­ive spaces to or­din­ary gauge fields and the proof of back­ground in­de­pend­ence of non­com­mut­at­ive Born–In­feld the­ory. (A little bit earli­er back­ground in­de­pend­ence was dis­covered in my pa­per with B. Pi­oline [95] at the level of the en­ergy spec­trum.)

I also have writ­ten many pa­pers de­vel­op­ing the ideas of [87]. In [93], [89] it was shown that Mor­ita equi­val­ence of non­com­mut­at­ive tori is re­lated to phys­ic­al du­al­ity of gauge the­or­ies on these tori and to T-du­al­ity in string the­ory. I am con­vinced that the math­em­at­ic­al no­tion of Mor­ita equi­val­ence of as­so­ci­at­ive al­geb­ras and its gen­er­al­iz­a­tion for dif­fer­en­tial as­so­ci­at­ive al­geb­ras should be re­garded as the math­em­at­ic­al found­a­tion of du­al­it­ies in string/M-the­ory. Among oth­er pa­pers I would like to men­tion the pa­per with N. Nekra­sov [90], where we in­tro­duced and stud­ied non­com­mut­at­ive in­stan­tons, and pa­pers with A. Konechny, de­voted mostly to the study of BPS-states on non­com­mut­at­ive tori and their or­bi­folds (see [96], [94], [101], [100]). To­geth­er with A. Konechny I have writ­ten a re­view pa­per “In­tro­duc­tion to mat­rix the­ory and non­com­mut­at­ive geo­metry”, pub­lished in Phys­ics Re­ports (see [107]).

More re­cent pro­jects (to­geth­er with M. Movshev) were de­voted to max­im­ally su­per­sym­met­ric gauge the­or­ies and their de­form­a­tions. These the­or­ies in­clude ten-di­men­sion­al su­per Yang–Mills the­ory and its di­men­sion­al re­duc­tions, in par­tic­u­lar, BFSS and IKKT mat­rix mod­els. Our meth­ods are closely re­lated to non­com­mut­at­ive su­per­geo­metry, more pre­cisely, to the the­ory of \( A_{\infty} \)-al­geb­ras ( \( = \) form­al non­com­mut­at­ive Q-man­i­folds).

Our cal­cu­la­tions led us to the study of ho­mo­logy groups of vari­ous su­per Lie al­geb­ras. We have writ­ten sev­er­al pa­pers (in col­lab­or­a­tion with Ren­jun Xu and A. Mikhail­ov) de­voted to the cal­cu­la­tion of ho­mo­logy of the al­gebra of su­per­sym­met­ries, of su­per Poin­caré al­gebra and of oth­er ho­mo­logy groups that ap­pear in quantum field the­ory and string the­ory, in par­tic­u­lar, in pure spinor form­al­ism.

To­geth­er with Jia-Ming (Frank) Li­ou I have stud­ied the ho­mo­logy groups of mod­uli spaces of al­geb­ra­ic curves and their re­la­tion to the ho­mo­logy of in­fin­ite-di­men­sion­al Grass­man­ni­an [124], [126], [132], [139]. Our pa­pers were in­spired by the old idea that non­per­turb­at­ive string the­ory should be for­mu­lated in terms of in­fin­ite-di­men­sion­al Grass­man­ni­an (see [88] and ref­er­ences therein). They did not find yet any ap­plic­a­tions to phys­ics, but they con­tain some in­ter­est­ing math­em­at­ic­al res­ults.

I have stud­ied two-di­men­sion­al to­po­lo­gic­al quantum field the­or­ies coupled to grav­ity (\( = \)to­po­lo­gic­al string the­or­ies), in par­tic­u­lar, the­or­ies of this kind defined on bordered sur­faces. I have ap­plied meth­ods of arith­met­ic geo­metry in this study. To­geth­er with M. Kont­sevich and V. Volo­god­sky I have ana­lyzed in­teg­ral­ity of in­stan­ton num­bers us­ing \( p \)-ad­ic meth­ods [115]. This pa­per and sub­sequent pa­pers writ­ten in col­lab­or­a­tion with V. Volo­god­sky [120] [117] were based on the ap­plic­a­tion of Frobeni­us map on \( p \)-ad­ic co­homo­logy; in a pa­per writ­ten in col­lab­or­a­tion with I. Sha­piro a con­struc­tion of this map based on the ideas of su­per­geo­metry was giv­en [118]. I am con­vinced that very soon the meth­ods of num­ber the­ory will play an im­port­ant role in quantum field the­ory and string the­ory. In the pa­per en­titled “Twis­ted de Rham co­homo­logy, ho­mo­lo­gic­al defin­i­tion of in­teg­ral and phys­ics over a ring” [119] (with I. Sha­piro) we ar­gue that to­geth­er with con­ven­tion­al phys­ic­al the­or­ies based on real or com­plex num­bers one can con­sider the­or­ies where the role of num­bers is played by ele­ments of an ar­bit­rary ring. The study of such un­con­ven­tion­al the­or­ies can be used to ob­tain in­ter­est­ing res­ults in con­ven­tion­al frame­work, but it is pos­sible that these the­or­ies have dir­ect phys­ic­al mean­ing.

I con­tin­ued the work on ap­plic­a­tions of arith­met­ic geo­metry to­geth­er with V. Volo­god­sky and J. Wal­cher [133].

It was sug­ges­ted many years ago that one can con­sider a pair of or­din­ary dif­fer­en­tial op­er­at­ors \( P, Q \) obey­ing the string equa­tion \( [P, Q] = \hbar \) as a quantum curve. (This sug­ges­tion is based on the fact that com­mut­ing dif­fer­en­tial op­er­at­ors spe­cify a clas­sic­al al­geb­ra­ic curve.) I stud­ied the string equa­tion in the 90s us­ing the tech­niques of in­fin­ite-di­men­sion­al Grass­man­ni­an. More re­cently I have ob­tained new res­ults about string equa­tion re­lat­ing it to new ap­proaches to quantum curves in pa­pers by Eynard–Or­ant­in, Gukov–Sułkowski, Mu­lase, etc. I have in­tro­duced a no­tion of dis­crete quantum curve as a solu­tion to the equa­tion \( KL = \lambda LK \) where \( K, L \) are dif­fer­ence op­er­at­ors and proved sim­il­ar res­ults for this no­tion. Quantum curves are re­lated to many prob­lems of phys­ics, to­geth­er with M. Luu I have ana­lyzed their re­la­tion to \( (p, q) \)-min­im­al mod­els coupled to 2D-grav­ity (see [136]).

To­geth­er with A. Mikhail­ov [138], [127] I ana­lyzed fam­il­ies of gauge con­di­tions in BV form­al­ism and the con­struc­tion of string amp­litudes in this set­ting.

My re­cent pa­pers were de­voted to scat­ter­ing amp­litudes.

Deep con­nec­tions of scat­ter­ing amp­litudes to pos­it­ive Grass­man­ni­an and cluster al­geb­ras were found in [e6]. The pos­it­ive Grass­man­ni­an is closely re­lated to the quantum Grass­man­ni­an. In the pa­per writ­ten in col­lab­or­a­tion with M. Movshev [140] we have shown us­ing the the­ory of quantum cluster al­geb­ras that the con­struc­tion of dif­fer­en­tial forms rep­res­ent­ing scat­ter­ing amp­litudes and defined on cells of the pos­it­ive Grass­man­ni­an can be “\( q \)-de­formed” (there are some ob­jects on quantum Grass­man­ni­an gen­er­al­iz­ing these forms). We con­jec­ture that the \( q \)-de­formed amp­litudes have phys­ic­al mean­ing in the frame­work of gauge the­or­ies on non­com­mut­at­ive spaces.

The pa­per [e6] and fol­low-up pa­pers are based on the idea that quantum the­ory can be for­mu­lated not in terms of fields, but in terms of particles. These pa­pers are deal­ing with four-di­men­sion­al the­or­ies; I am work­ing now on gen­er­al­iz­a­tion of this idea to any di­men­sion.

Re­cently I have in­tro­duced the no­tion of in­clus­ive scat­ter­ing mat­rix closely re­lated to the no­tion of in­clus­ive cross-sec­tion. This no­tion came from the for­mu­la­tion of quantum the­ory in terms of \( L \)-func­tion­als giv­en in my first phys­ic­al pa­per pub­lished in 1967 [15]. To a dens­ity mat­rix \( K \) in the rep­res­ent­a­tion of ca­non­ic­al com­mut­a­tion re­la­tions (CCR) or ca­non­ic­al an­ti­com­mut­a­tion re­la­tions (CAR) one can as­sign a func­tion­al \[ L(\alpha^{\ast} , \alpha) = \operatorname{Tr} e^{-\alpha a^{+}} e^{\alpha^{\ast}a} K, \] where \( \alpha a^{+} = \sum \alpha_k a^{+}_k \),   \( \alpha^{\ast} a = \sum \alpha^{\ast}_k a_k \),  \( a^{+}_k \),  \( a_k \) obey CCR or CAR. This func­tion­al (called \( L \)-func­tion­al) can be con­sidered as a pos­it­ive func­tion­al on the Weyl al­gebra or Clif­ford al­gebra. I have shown that the quantum the­ory can be for­mu­lated in terms of \( L \)-func­tion­als; this al­lows us to avoid prob­lems due to the ex­ist­ence of in­equi­val­ent rep­res­ent­a­tions of CCR and CAR in the case of in­fin­ite num­bers of de­grees of free­dom.

My stu­dents Yu. Ty­up­kin, V. Fateev and I wrote sev­er­al pa­pers ap­ply­ing the for­mu­la­tion of quantum the­ory in terms of \( L \)-func­tion­als to vari­ous prob­lems (see, for ex­ample, [e1], [22]), but we aban­doned this dir­ec­tion in fa­vor of to­po­lo­gic­al meth­ods in phys­ic­al prob­lems. In the 80s Umez­a­wa re­dis­covered this for­mu­la­tion un­der the name of TFD (ther­mofield dy­nam­ics) and ap­plied it to equi­lib­ri­um and nonequi­lib­ri­um stat­ist­ic­al phys­ics.

The in­clus­ive scat­ter­ing mat­rix for ele­ment­ary ex­cit­a­tions of any trans­la­tion-in­vari­ant sta­tion­ary state can be defined also in the frame­work of al­geb­ra­ic quantum field the­ory [141] and ex­pressed in terms of gen­er­al­ized Green func­tions on shell [141], [143]. This fact should be im­port­ant in quantum stat­ist­ic­al phys­ics. (The stand­ard defin­i­tion of scat­ter­ing mat­rix works only for ele­ment­ary ex­cit­a­tions of ground state.) It can be used also to re­late quantum scat­ter­ing to the scat­ter­ing of solitons.

Re­cently I have sug­ges­ted a for­mu­la­tion of quantum the­ory where the con­vex set of states is taken as a start­ing point [144], [145]. I have shown that in this geo­met­ric ap­proach one can prove de­co­her­ence and de­rive the for­mu­las for prob­ab­il­it­ies from the first prin­ciples. The no­tion of in­clus­ive scat­ter­ing mat­rix ap­pears very nat­ur­ally in this ap­proach [147]. The geo­met­ric ap­proach is closely re­lated to the for­mu­la­tion of quantum the­ory in terms of Jordan al­geb­ras [146].