Celebratio Mathematica

Albert S. Schwarz

My life in science

by Albert Schwarz

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 (­tory-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 EynardOr­ant­in, 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].


[1] N. S. Ramm and A. S. Švarc: “Geo­met­riq blizosti, uni­form­naq geo­met­riq i to­po­lo­giq” [Geo­metry of prox­im­ity, uni­form geo­metry and to­po­logy], Mat. Sb., Nov. Ser. 33(75) : 1 (1953), pp. 157–​180. MR 61368 Zbl 0050.​39101 article

[2] A. S. Schwarz: “A volume in­vari­ant of cov­er­ings,” Dok­lady Akad. Nauk SSSR 105 (1955), pp. 32–​34. In Rus­si­an. Zbl 0066.​15903 article

[3] V. A. Rohlin and A. S. Švarc: “O kom­bin­at­or­noy in­vari­ant­nosti klas­sov Pon­trq­gina” [On the com­bin­at­or­i­al in­vari­ance of the Pontry­agin classes], Dokl. Akad. Nauk SSSR 114 : 3 (1957), pp. 490–​493. MR 102070 Zbl 0078.​36803 article

[4] A. S. Švarc: “Rod rassloen­nogo pros­tran­stva” [The genus of a fiber space], Dokl. Akad. Nauk SSSR 119 : 2 (1958), pp. 219–​222. MR 102812 Zbl 0085.​37703 article

[5] A. S. Schwarz: “On the genus of a fiber space,” Dok­lady Akad. Nauk SSSR 126 (1959), pp. 719–​722. In Rus­si­an. Zbl 0087.​38204 article

[6] A. S. Švarc: “Go­mo­lo­gii pros­tran­stv zamknutjx krivjx” [Ho­mo­logy of spaces of closed curves], Tr. Mosk. Mat. O.-va 9 (1960), pp. 3–​44. MR 133824 Zbl 0115.​17001 article

[7] A. S. Švarc: “Rod rassloen­nogo pros­tran­stva” [The genus of a fibre space], Tr. Mosk. Mat. O.-va 10 (1961), pp. 217–​272. An Eng­lish trans­la­tion was pub­lished in Am. Math. Soc. Transl., Ser. 2 55 (1966),. MR 154284 article

[8] D. B. Fuks and A. S. Švarc: “K go­mo­topicheskoy teorii funk­t­orov v kat­egorii to­po­lo­gicheskix pros­tran­stv” [On the ho­mo­topy the­ory of func­tors in the cat­egory of to­po­lo­gic­al spaces], Dokl. Akad. Nauk SSSR 143 : 3 (1962), pp. 543–​546. An Eng­lish trans­la­tion was pub­lished in Sov. Math., Dokl. 3 (1962). MR 137116 article

[9] A. S. Švarc: “Rod rassloen­nogo pros­tran­stva” [The genus of a fibre space], Tr. Mosk. Mat. O.-va 11 (1962), pp. 99–​126. An Eng­lish trans­la­tion was pub­lished in Am. Math. Soc. Transl., Ser. 2 55 (1966),. MR 151982 article

[10] A. S. Švarc: “Dvoystven­nost’funk­t­orov” [Du­al­ity of func­tors], Dokl. Akad. Nauk SSSR 148 : 2 (1963), pp. 288–​291. An Eng­lish trans­la­tion was pub­lished in Sov. Math., Dokl. 4 (1963). MR 168629 article

[11] B. S. Mityagin and A. S. Shvarts: “Func­tors in cat­egor­ies of Banach spaces” [Funk­t­orj v kat­egoriqx banax­ovjx pros­tran­stv], Us­pekhi Mat. Nauk 19 : 2(116) (1964), pp. 65–​130. An Eng­lish trans­la­tion was pub­lished in Russ. Math. Surv. 19:2 (1964). MR 166593 article

[12] L. M. Kiss­ina and A. S. Švarc: “K vo­prosu ob opis­anii dvoystven­nogo funk­t­ora” [On the ques­tion of de­scrip­tion of the du­al­ity func­tor], Dokl. Akad. Nauk SSSR 167 : 2 (1966), pp. 282–​285. MR 219065 article

[13] R. S. Pokaz­eeva and A. S. Švarc: “Dvoystven­nost’funk­t­orov” [Du­al­ity of func­tors], Mat. Sb., Nov. Ser. 71(113) : 3 (1966), pp. 357–​385. An Eng­lish trans­la­tion was pub­lished Am. Math. Soc., Trans­lat., II. Ser. 73 (1968). MR 220650 article

[14] V. V. Kuznecov and A. S. Švarc: “Dvoystven­nost’funk­t­orov i dvoystven­nost’kat­egor­iy” [Du­al­ity of func­tors and du­al­ity of cat­egor­ies], Us­pekhi Mat. Nauk 22 : 1(133) (1967), pp. 168–​170. MR 204491 article

[15] A. S. Schwarz: “A new for­mu­la­tion of the quantum the­ory,” Dokl. Akad. Nauk SSSR 173 : 4 (1967), pp. 793–​796. In Rus­si­an. article

[16] V. V. Kuznecov and A. S. Švarc: “Dvoystven­nost’funk­t­orov i dvoystven­nost’kat­egor­iy” [Du­al­ity of func­tors and du­al­ity of cat­egor­ies], Sib. Mat. Zh. 9 : 4 (1968), pp. 840–​856. An Eng­lish trans­la­tion was pub­lished in Siberi­an Math. J. 9:4 (1968). MR 236241 Zbl 0186.​03002 article

[17] V. N. Likhachev, Yu. S. Ty­up­kin, and A. S. Shvarts: “Adia­baticheskaq teor­ema v kvanto­voy teorii polq” [Adia­bat­ic the­or­em in quantum field the­ory], Teor. Mat. Fiz. 10 : 1 (1972), pp. 63–​84. An Eng­lish trans­la­tion was pub­lished in The­or­et. and Math. Phys. 10:1 (1972). MR 475468 article

[18] V. A. Fateev and A. S. Shvarts: “On ax­io­mat­ic scat­ter­ing the­ory,” The­or­et. and Math. Phys. 14 : 2 (February 1973), pp. 112–​124. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Teor. Mat. Fiz. 14:2 (1973). Zbl 0274.​47006 article

[19] A. Schwarz, V. Fateev, and Yu. Ty­up­kin: To­po­lo­gic­ally non-trivi­al particles in quantum field the­ory. Tech­nic­al re­port 157, Lebedev In­sti­tute, 1975. In Rus­si­an. techreport

[20] A. S. Schwarz, Yu. Ty­up­kin, and V. Fateev: “On to­po­lo­gic­ally non-trivi­al particles in quantum field the­ory,” Pisma Zh. Èksper. Teor­et. Fiz. 22 (1975), pp. 192–​194. In Rus­si­an; trans­lated in­to Eng­lish in JETP Let­ters 22:3 (1975), 88–89. article

[21] A. S. Schwarz, Yu. Ty­up­kin, and V. Fateev: “On the ex­ist­ence of heavy particles in gauge fields the­or­ies,” Pisma Zh. Èksper. Teor­et. Fiz. 21 (1975), pp. 91–​93. In Rus­si­an. article

[22] J. S. Tjup­kin, V. A. Fateev, and A. S. Švarc: “O klas­sicheskom predele mat­ricj rasseq­niq v kvanto­voy teorii polq” [The clas­sic­al lim­it of the scat­ter­ing mat­rix in quantum field the­ory], Dokl. Akad. Nauk SSSR 221 : 1 (1975), pp. 70–​73. MR 378641 article

[23] Yu. S. Ty­up­kin, V. A. Fateev, and A. S. Shvarts: “Con­nec­tion between particle-like solu­tions of clas­sic­al equa­tions and quantum particles,” So­viet J. Nuc­le­ar Phys. 22 : 3 (1975), pp. 321–​325. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Jadernaja Fiz. 22:3 (1975). MR 406170 article

[24] A. A. Be­lav­in, A. M. Polyakov, A. S. Schwartz, and Yu. S. Ty­up­kin: “Pseudo­particle solu­tions of the Yang–Mills equa­tions,” Phys. Lett. B 59 : 1 (1975), pp. 85–​87. MR 434183 article

[25] A. Schwarz, V. Fateev, and Yu. Ty­up­kin: On the particle-like solu­tions in the pres­ence of fer­mi­ons. Tech­nic­al re­port 155, Lebedev In­sti­tute, 1976. In Rus­si­an. techreport

[26] A. S. Schwarz: “Mag­net­ic mono­poles in gauge field the­or­ies,” Us­pekhi Matem­aticheskikh Nauk (Rus­si­an Math­em­at­ic­al Sur­veys) 31 : 3 (1976), pp. 248–​258. In Rus­si­an. article

[27] A. S. Schwarz: “To­po­lo­gic­ally non­trivi­al solu­tions of clas­sic­al equa­tions and their role in quantum field the­ory,” pp. 224–​240 in Non­loc­al, non­lin­ear and non­renor­mal­iz­able field the­or­ies (Proc. Fourth In­ter­nat. Sym­pos. Non­loc­al Field The­or­ies) (Alushta, 1976). Edi­ted by D. I. Blokh­int­sev. 1976. In Rus­si­an. MR 456147 inproceedings

[28] A. S. Schwarz: “Mag­net­ic mono­poles in gauge the­or­ies,” Nuc­le­ar Phys. B 112 : 2 (September 1976), pp. 358–​364. MR 416313 article

[29] J. S. Tjup­kin, V. A. Fateev, and A. S. Švarc: “CHasti­ce­podob­n­je reshe­niq uravn­en­iy kal­ib­ro­voch­njx teor­iy polq” [Particle-like solu­tions of the equa­tions of gauge the­or­ies], Teor. Mat. Fiz. 26 : 3 (1976), pp. 397–​402. An Eng­lish trans­la­tion was pub­lished in The­or­et. and Math. Phys. 26:3 (1976). MR 449236 article

[30] A. S. Schwarz: “On reg­u­lar solu­tions of Eu­c­lidean Yang–Mills equa­tions,” Phys. Lett. B 67 : 2 (1977), pp. 172–​174. MR 443673 article

[31] A. S. Schwarz and I. Fro­lov: “Con­tri­bu­tion of in­stan­tons to the cor­rel­a­tion func­tions of a Heis­en­berg fer­ro­mag­net,” Pisma Zh. Èksper. Teor­et. Fiz. 28 (1978), pp. 274–​276. In Rus­si­an; trans­lated in JETP Let­ters 28, p. 249. article

[32] A. S. Schwarz: “On quantum fluc­tu­ations of in­stan­tons,” Lett. Math. Phys. 2 : 3 (1978), pp. 201–​205. MR 495474 Zbl 0383.​70016 article

[33] V. N. Ro­man­ov, I. V. Fro­lov, and A. S. Švarc: “O sfericheski-sim­met­rich­njx soliton­ax” [Spher­ic­ally sym­met­ric solitons], Teor. Mat. Fiz. 37 : 3 (1978), pp. 305–​318. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 37:3 (1978). MR 524695 article

[34] A. S. Schwarz: “The par­ti­tion func­tion of de­gen­er­ate quad­rat­ic func­tion­al and Ray–Sing­er in­vari­ants,” Lett. Math. Phys. 2 : 3 (1978), pp. 247–​252. MR 676337 Zbl 0383.​70017 article

[35] A. S. Schwarz: “The par­ti­tion func­tion of a de­gen­er­ate func­tion­al,” Comm. Math. Phys. 67 : 1 (1979), pp. 1–​16. MR 535228 Zbl 0429.​58015 article

[36] A. S. Schwarz, A. Be­lav­in, V. Fateev, and Yu. Ty­up­kin: “Quantum fluc­tu­ations of multi-in­stan­tons solu­tions,” Phys­ics Let­ters B 83 : 3–​4 (1979), pp. 317–​320. article

[37] A. S. Schwarz, V. Fateev, and I. Fro­lov: “Quantum fluc­tu­ations of in­stan­tons in the non-lin­ear \( \sigma \)-mod­el,” Nuc­le­ar Phys­ics B 154 (1979), pp. 1–​20. article

[38] A. S. Schwarz and I. Fro­lov: “On the in­stan­ton con­tri­bu­tion in Eu­c­lidean Green func­tions,” Phys­ics Let­ters B 80 : 4–​5 (1979), pp. 406–​409. article

[39] A. S. Schwarz: “Are the field and space vari­ables on an equal foot­ing?,” Nuc­le­ar Phys. B 171 : 1–​2 (1980), pp. 154–​166. MR 582926 article

[40] V. N. Ro­man­ov, V. A. Fateev, and A. S. Schwarz: “Mag­nit­nye mono­pol­ii v ed­inykh teor­iyakh ehlektro­mag­nit­nogo, slabogo i sil’nogo vzaimod­ejstviya” [Mag­net­ic mono­poles in the uni­fied the­or­ies of elec­tro­mag­net­ic, weak, and strong in­ter­ac­tions], Yadernaya Fiz. 32 : 4 (1980), pp. 1138–​1141. An Eng­lish trans­la­tion was pub­lished in Sov. J. Nucl. Phys. 32:4 (1980). MR 624047 article

[41] A. S. Schwarz: “The­or­ies with non­loc­al elec­tric charge con­ser­va­tion,” Pisma Zh. Èksper. Teor­et. Fiz. 34 (1981), pp. 555–​558. In Rus­si­an; trans­lated in JETP Let­ters 34: 10 (1981), 532–534. article

[42] A. V. Gay­duk, V. N. Ro­man­ov, and A. S. Schwarz: “Su­per­grav­ity and field space demo­cracy,” Comm. Math. Phys. 79 : 4 (1981), pp. 507–​528. MR 623965 article

[43] V. A. Fateev, I. V. Fro­lov, A. S. Schwarz, and Yu. S. Ty­up­kin: “Quantum fluc­tu­ations of in­stan­tons,” Sov. Sci. Rev., Sect. C 2 (1981), pp. 1–​51. MR 659266 Zbl 0535.​58026 article

[44] A. S. Schwarz: “Su­per­grav­ity and com­plex geo­metry,” Yadernaya Fiz. 34 : 4 (1981), pp. 1144–​1149. An Eng­lish trans­la­tion was pub­lished in So­viet J. Nuc­le­ar Phys. 34:4 (1981). MR 660226 article

[45] A. S. Schwarz: “Su­per­grav­ity, com­plex geo­metry and \( G \)-struc­tures,” Comm. Math. Phys. 87 : 1 (1982), pp. 37–​63. MR 680647 Zbl 0503.​53048 article

[46] A. S. Schwarz: “Field the­or­ies with no loc­al con­ser­va­tion of the elec­tric charge,” Nuc­le­ar Phys. B 208 : 1 (1982), pp. 141–​158. MR 682897 article

[47] A. A. Roslyĭ and A. S. Schwarz: “Geo­met­riya nemin­im­al’noj i al’ternat­ivnoj min­im­al’noj su­per­ggrav­it­at­sii” [Geo­metry of non­min­im­al and al­tern­at­ive min­im­al su­per­grav­ity], Yadernaya Fiz. 37 : 3 (1983), pp. 786–​794. An Eng­lish trans­la­tion was pub­lished in So­viet J. Nuc­le­ar Phys. 37:3 (1983). MR 719875 article

[48] O. M. Khu­daverdy­an and A. S. Shvarts: “Nor­mal’naq kal­ib­rovka v su­per­grav­itacii” [Nor­mal gauge in su­per­grav­ity], Teor. Mat. Fiz. 57 : 3 (1983), pp. 354–​362. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 57:3 (1983). MR 735394 article

[49] A. A. Rosly and A. S. Schwarz: “Geo­metry of \( N=1 \) su­per­grav­ity,” Comm. Math. Phys. 95 : 2 (1984), pp. 161–​184. MR 760330 Zbl 0644.​53077 article

[50] A. A. Rosly and A. S. Schwarz: “Geo­metry of \( N=1 \) su­per­grav­ity, II,” Comm. Math. Phys. 96 : 3 (1984), pp. 285–​309. MR 769349 Zbl 0644.​53078 article

[51] A. S. Shvarts: “On the defin­i­tion of su­per­space” [K opredel­eniü su­per­pros­tran­stva], Teor. Mat. Fiz. 60 : 1 (1984), pp. 37–​42. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 60:1 (1984). MR 760438 article

[52] A. A. Rosly and A. S. Schwarz: “Geo­met­ric­al ori­gin of new un­con­strained su­per­fields,” pp. 308–​324 in Pro­ceed­ings of the third sem­in­ar on quantum grav­ity (Mo­scow, 23–25 Oc­to­ber 1984). Edi­ted by M. A. Markov, V. A. Berez­in, and V. P. Fro­lov. World Sci­entif­ic (Singa­pore), 1985. MR 829729 incollection

[53] M. A. Baran­ov and A. S. Shvarts: “O mno­go­petle­vom vk­lade v teor­iyu struny” [Mul­tiloop con­tri­bu­tion to string the­ory], Pis’ma Zh. Ek­sper. Teor­et. Fiz. 42 : 8 (1985), pp. 340–​342. An Eng­lish trans­la­tion was pub­lished in JETP Lett. 42:8 (1985). MR 875755 article

[54] M. A. Baran­ov, A. A. Roslyĭ, and A. S. Shvarts: “Geo­met­ric as­pects of su­per­grav­ity,” Prob­lemy Yadern. Fiz. i Kosm. Lucheĭ 24 (1985), pp. 25–​33. MR 887675 article

[55] A. S. Schwarz and Yu. S. Ty­up­kin: “Grand uni­fic­a­tion, strings, and mir­ror particles,” pp. 279–​289 in Group the­or­et­ic­al meth­ods in phys­ics (Zvenig­orod, USSR, 24–26 Novem­ber 1982), vol. 2. Edi­ted by M. A. Markov, V. I. Man’ko, and A. E. Shabad. Har­wood Aca­dem­ic (Chur, Switzer­land), 1985. MR 989351 incollection

[56] M. A. Baran­ov, A. A. Roslyĭ, and A. S. Shvarts: “Su­per­light­like geodesics in su­per­grav­ity,” Yadernaya Fiz. 41 : 1 (1985), pp. 285–​287. An Eng­lish trans­la­tion was pub­lished in So­viet J. Nuc­le­ar Phys. 41:1 (1985). MR 804665 article

[57] M. A. Baran­ov, A. A. Roslyĭ, and A. S. Shvarts: “O svqzi grav­itacii i su­per­grav­itacii” [On the con­nec­tion between grav­ity and su­per­grav­ity], Teor­et. Mat. Fiz. 64 : 1 (1985), pp. 7–​16. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 64:1 (1985). MR 815093 article

[58] A. A. Rosly and A. S. Schwarz: “Su­per­sym­metry in a space with aux­il­i­ary di­men­sions,” Comm. Math. Phys. 105 : 4 (1986), pp. 645–​668. MR 852094 article

[59] M. A. Baran­ov, Yu. I. Man­in, I. V. Fro­lov, and A. S. Shvarts: “The mul­tiloop con­tri­bu­tion in the fer­mi­on string,” So­viet J. Nuc­le­ar Phys. 43 : 4 (1986), pp. 670–​671. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Yadernaya Fiz. 43:4 (1986). MR 857425 article

[60] A. A. Roslyĭ, O. M. Khu­daverdy­an, and A. S. Shvarts: “Su­per­sim­met­riq i kom­pleksnaq geo­met­riq” [Su­per­sym­metry and com­plex geo­metry], pp. 247–​284 in Com­plex ana­lys­is: Sev­er­al vari­ables, III. Edi­ted by G. M. Khen­kin and R. V. Gamkrelidze. Sovre­mennye Prob­lemy Matem­atiki. Fun­da­ment­al’nye Napravlen­iya 9. VIN­ITI (Mo­scow), 1986. An Eng­lish trans­la­tion was pub­lished in 1989. MR 860614 incollection

[61] A. Schwarz and M. Baran­ov: Su­per­con­form­al geo­metry and mul­tiloop con­tri­bu­tion to the string the­ory. Pre­print, MI­FI (Mo­scow), 1987. techreport

[62] M. A. Baran­ov, I. V. Fro­lov, and A. S. Shvarts: “Geo­met­riya dvumernykh su­per­kon­formnykh teor­ij polya” [Geo­metry of two-di­men­sion­al su­per­con­form­al field the­or­ies], Teor. Mat. Fiz. 70 : 1 (1987), pp. 92–​103. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 70:1 (1987). MR 883786 Zbl 0624.​58002 article

[63] A. M. Baran­ov, Yu. I. Man­in, I. V. Fro­lov, and A. S. Schwarz: “A su­per­ana­log of the Sel­berg trace for­mula and mul­tiloop con­tri­bu­tions for fer­mi­on­ic strings,” Comm. Math. Phys. 111 : 3 (1987), pp. 373–​392. MR 900500 Zbl 0624.​58033 article

[64] M. A. Baran­ov and A. S. Schwarz: “On the mul­tiloop con­tri­bu­tion to the string the­ory,” In­ter­nat. J. Mod­ern Phys. A 2 : 6 (1987), pp. 1773–​1796. MR 913613 article

[65] A. S. Shvarts: “The fer­mi­on string and a uni­ver­sal mod­u­lus space,” JETP Lett. 46 : 9 (1987), pp. 428–​431. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Pis’ma Zh. Èksper. Teor­et. Fiz. 46:9 (1987). MR 940601 article

[66] A. S. Schwarz and Yu. S. Ty­up­kin: “Meas­ure­ment the­ory and the Schrödinger equa­tion,” pp. 667–​675 in Quantum field the­ory and quantum stat­ist­ics: Es­says in hon­our of the six­tieth birth­day of E. S. Fradkin, vol. 1. Edi­ted by I. A. Batal­in, C. J. Isham, and G. A. Vilko­visky. Hil­ger (Bris­tol, UK), 1987. MR 943165 incollection

[67] A. A. Rosly, A. S. Schwarz, and A. A. Voro­nov: “Geo­metry of su­per­con­form­al man­i­folds,” Comm. Math. Phys. 119 : 1 (1988), pp. 129–​152. MR 968484 Zbl 0675.​58010 article

[68] A. S. Schwarz: “Fer­mi­on­ic string and uni­ver­sal mod­uli space,” Nuc­le­ar Phys. B 317 : 2 (1989), pp. 323–​343. MR 1001895 article

[69] M. A. Baran­ov, I. V. Fro­lov, and A. S. Schwarz: “Geo­met­riq su­per­kon­form­nogo pros­tran­stva mod­u­ley” [Geo­metry of the su­per­con­form­al mod­uli space], Teor. Mat. Fiz. 79 : 2 (1989), pp. 241–​252. An Eng­lish trans­la­tion was pub­lished in The­or­et. and Math. Phys. 79:2 (1989). MR 1007798 Zbl 0679.​53058 article

[70] A. S. Schwarz: “New to­po­lo­gic­al in­vari­ants arising in the the­ory of quant­ized fields” in Baku In­ter­na­tion­al To­po­lo­gic­al Con­fer­ence, ab­stracts (Baku, Oc­to­ber 3–9, 1987). Edi­ted by B. R. Nur­iev. Elm (Baku, Azerbaijan), 1989. MR 1347199 inproceedings

[71] A. A. Rosly, A. S. Schwarz, and A. A. Voro­nov: “Su­per­con­form­al geo­metry and string the­ory,” Comm. Math. Phys. 120 : 3 (1989), pp. 437–​450. MR 981212 Zbl 0667.​58008 article

[72] V. Kac and A. Schwarz: “Geo­met­ric in­ter­pret­a­tion of the par­ti­tion func­tion of 2D grav­ity,” Phys. Lett. B 257 : 3–​4 (1991), pp. 329–​334. MR 1100639 article

[73] A. S. Schwarz and A. Sen: “Glu­ing the­or­em, star product and in­teg­ra­tion in open string field the­ory in ar­bit­rary back­ground fields,” Int. J. Mod. Phys. A 6 : 30 (1991), pp. 5387–​5407. MR 1137572 Zbl 0802.​53032 article

[74] A. Schwarz: “Geo­metry of fer­mi­on­ic string,” pp. 1377–​1386 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­aticians (Kyoto, Ja­pan, 21–29 Au­gust 1990), vol. 2. Edi­ted by I. Satake. Spring­er (Tokyo), 1991. MR 1159322 Zbl 0751.​53024 incollection

[75] K. N. Anagnosto­poulos, M. J. Bowick, and A. Schwarz: “The solu­tion space of the unit­ary mat­rix mod­el string equa­tion and the Sato Grass­man­ni­an,” Com­mun. Math. Phys. 148 : 3 (1992), pp. 469–​485. MR 1181066 Zbl 0753.​35073 ArXiv hep-​th/​9112066 article

[76] A. Schwarz: “Geo­metry of Batal­in–Vilko­visky quant­iz­a­tion,” Com­mun. Math. Phys. 155 : 2 (1993), pp. 249–​260. MR 1230027 Zbl 0786.​58017 ArXiv hep-​th/​9205088 article

[77] A. Schwarz: “Semi­clas­sic­al ap­prox­im­a­tion in Batal­in–Vilko­visky form­al­ism,” Com­mun. Math. Phys. 158 : 2 (1993), pp. 373–​396. MR 1249600 Zbl 0855.​58005 ArXiv hep-​th/​9210115 article

[78] M. Atiyah, A. Borel, G. J. Chaitin, D. Friedan, J. Glimm, J. J. Gray, M. W. Hirsch, S. MacLane, B. B. Man­del­brot, D. Ruelle, A. Schwarz, K. Uh­len­beck, R. Thom, E. Wit­ten, and C. Zee­man: “Re­sponses to ‘The­or­et­ic­al math­em­at­ics: To­ward a cul­tur­al syn­thes­is of math­em­at­ics and the­or­et­ic­al phys­ics’, by A. Jaffe and F. Quinn,” Bull. Am. Math. Soc., New Ser. 30 : 2 (April 1994), pp. 178–​207. Zbl 0803.​01014 ArXiv math/​9404229 article

[79] A. Schwarz: “Sym­metry trans­form­a­tions in Batal­in–Vilko­visky form­al­ism,” Lett. Math. Phys. 31 : 4 (1994), pp. 299–​301. MR 1293493 Zbl 0802.​58027 ArXiv hep-​th/​9310124 article

[80] M. Pen­kava and A. Schwarz: “On some al­geb­ra­ic struc­tures arising in string the­ory,” pp. 219–​227 in Per­spect­ives in math­em­at­ic­al phys­ics (Taiwan, sum­mer 1992 and Los Angeles, winter 1992). Edi­ted by R. C. Pen­ner and S.-T. Yau. Con­fer­ence Pro­ceed­ings and Lec­ture Notes in Geo­metry and To­po­logy 3. In­ter­na­tion­al Press (Bo­ston), 1994. MR 1314668 Zbl 0871.​17021 ArXiv hep-​th/​9212072 incollection

[81] D. Fuchs and A. Schwarz: “Mat­rix Vi­eta the­or­em,” pp. 15–​22 in Lie groups and Lie al­geb­ras: E. B. Dynkin’s sem­in­ar. Edi­ted by S. G. Gindikin and E. B. Vin­berg. Amer­ic­an Math­em­at­ic­al So­ci­ety Trans­la­tions. Series 2 169. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1995. This book is also no. 26 in the Ad­vances in Math­em­at­ics series. MR 1364450 Zbl 0837.​15011 ArXiv math/​9410207 incollection

[82] M. Pen­kava and A. Schwarz: “\( A_{\infty} \) al­geb­ras and the co­homo­logy of mod­uli spaces,” pp. 91–​107 in Lie groups and Lie al­geb­ras: E. B. Dynkin’s sem­in­ar. Edi­ted by S. G. Gindikin and E. B. Vin­berg. Amer­ic­an Math­em­at­ic­al So­ci­ety Trans­la­tions. Series 2 169. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1995. This book is also no. 26 in the Ad­vances in Math­em­at­ics series. MR 1364455 Zbl 0863.​17017 ArXiv hep-​th/​9408064 incollection

[83] A. Schwarz: “Sigma-mod­els hav­ing su­per­man­i­folds as tar­get spaces,” Lett. Math. Phys. 38 : 1 (1996), pp. 91–​96. MR 1401058 Zbl 0859.​58004 ArXiv hep-​th/​9506070 article

[84] M. Al­ex­an­drov, A. Schwarz, O. Za­boron­sky, and M. Kont­sevich: “The geo­metry of the mas­ter equa­tion and to­po­lo­gic­al quantum field the­ory,” Int. J. Mod. Phys. A 12 : 7 (1997), pp. 1405–​1429. MR 1432574 Zbl 1073.​81655 ArXiv hep-​th/​9502010 article

[85] A. Connes and A. Schwarz: “Mat­rix Vi­eta the­or­em re­vis­ited,” Lett. Math. Phys. 39 : 4 (1997), pp. 349–​353. MR 1449580 Zbl 0874.​15010 article

[86] A. Schwarz and O. Za­boron­sky: “Su­per­sym­metry and loc­al­iz­a­tion,” Com­mun. Math. Phys. 183 : 2 (1997), pp. 463–​476. MR 1461967 Zbl 0873.​58003 ArXiv hep-​th/​9511112 article

[87] A. Connes, M. R. Douglas, and A. Schwarz: “Non­com­mut­at­ive geo­metry and mat­rix the­ory: Com­pac­ti­fic­a­tion on tori,” J. High En­ergy Phys. 1998 : 2 (1998). art­icle no. 3, 35 pages. MR 1613978 Zbl 1018.​81052 ArXiv hep-​th/​9711162 article

[88] A. Schwarz: “Grass­mann and string the­ory,” Com­mun. Math. Phys. 199 : 1 (1998), pp. 1–​24. MR 1660223 Zbl 0921.​58078 ArXiv hep-​th/​9610122 article

[89] A. Schwarz: “Mor­ita equi­val­ence and du­al­ity,” Nucl. Phys., B 534 : 3 (1998), pp. 720–​738. MR 1663471 Zbl 1079.​81066 ArXiv hep-​th/​9805034 article

[90] N. Nekra­sov and A. Schwarz: “In­stan­tons on non­com­mut­at­ive \( \mathbb{R}^4 \), and \( (2,0) \) su­per­con­form­al six di­men­sion­al the­ory,” Com­mun. Math. Phys. 198 : 3 (1998), pp. 689–​703. MR 1670037 Zbl 0923.​58062 ArXiv hep-​th/​9802068 article

[91]A. Konechny and A. Schwarz: “On \( (k\oplus l|g) \)-di­men­sion­al su­per­man­i­folds,” pp. 201–​206 in Su­per­sym­metry and quantum field the­ory (Kharkov, Ukraine, 5–7 Janu­ary 1997). Edi­ted by J. Wess and V. P. Ak­u­lov. Lec­ture Notes in Phys­ics 509. 1998. Pro­ceed­ings of the D. Volkov Me­mori­al Sem­in­ar. MR 1677320 Zbl 0937.​58001 incollection

[92] L. Fried­lander and A. Schwarz: “Grass­man­ni­an and el­lipt­ic op­er­at­ors,” pp. 79–​88 in High­er ho­mo­topy struc­tures in to­po­logy and math­em­at­ic­al phys­ics (Pough­keep­sie, NY, 13–15 June 1996). Edi­ted by J. Mc­Cle­ary. Con­tem­por­ary Math­em­at­ics 227. 1999. Con­fer­ence to hon­or the 60th birth­day of Jim Stasheff. MR 1665462 Zbl 0923.​58053 ArXiv funct-​an/​9704003 incollection

[93] M. A. Rief­fel and A. Schwarz: “Mor­ita equi­val­ence of mul­ti­di­men­sion­al non­com­mut­at­ive tori,” Int. J. Math. 10 : 2 (1999), pp. 289–​299. MR 1687145 Zbl 0968.​46060 ArXiv math/​9803057 article

[94] A. Konechny and A. Schwarz: “BPS states on non-com­mut­at­ive tori and du­al­ity,” Nucl. Phys., B 550 : 3 (1999), pp. 561–​584. MR 1693262 Zbl 0949.​58004 ArXiv hep-​th/​9811159 article

[95] B. Pi­oline and A. Schwarz: “Mor­ita equi­val­ence and \( T \)-du­al­ity (or \( B \) versus \( \Theta \)),” J. High En­ergy Phys. 1999 : 8 (1999). art­icle no. 21, 16 pages. MR 1715542 Zbl 1060.​81612 ArXiv hep-​th/​9908019 article

[96] A. Konechny and A. Schwarz: “\( 1/4 \)-BPS states on non­com­mut­at­ive tori,” J. High En­ergy Phys. 1999 : 9 (1999). art­icle no. 30, 15 pages. MR 1720696 Zbl 0957.​81086 ArXiv hep-​th/​9907008 article

[97] A. Schwarz: “Quantum ob­serv­ables, Lie al­gebra ho­mo­logy and TQFT,” Lett. Math. Phys. 49 : 2 (1999), pp. 115–​122. MR 1728307 Zbl 1029.​81064 ArXiv hep-​th/​9904168 article

[98] A. Schwarz: “Non­com­mut­at­ive su­per­geo­metry and du­al­ity,” Lett. Math. Phys. 50 : 4 (1999), pp. 309–​321. MR 1768707 Zbl 0967.​58004 ArXiv hep-​th/​9912212 article

[99] A. Schwarz: “Non­com­mut­at­ive al­geb­ra­ic equa­tions and the non­com­mut­at­ive ei­gen­value prob­lem,” Lett. Math. Phys. 52 : 2 (2000), pp. 177–​184. MR 1786861 Zbl 0972.​15001 ArXiv hep-​th/​0004088 article

[100] A. Konechny and A. Schwarz: “Mod­uli spaces of max­im­ally su­per­sym­met­ric solu­tions on non-com­mut­at­ive tori and non-com­mut­at­ive or­bi­folds,” J. High En­ergy Phys. 2000 : 9 (2000), pp. article no. 005, 24 pages. MR 1789106 Zbl 0989.​81623 ArXiv hep-​th/​0005174 article

[101] A. Konechny and A. Schwarz: “Com­pac­ti­fic­a­tion of M(at­rix) the­ory on non­com­mut­at­ive tor­oid­al or­bi­folds,” Nuc­le­ar Phys. B 591 : 3 (2000), pp. 667–​684. MR 1797572 Zbl 1042.​81580 ArXiv hep-​th/​9912185 article

[102] A. Konechny and A. Schwarz: “The­ory of \( (k\oplus l|q) \)-di­men­sion­al su­per­man­i­folds,” Se­lecta Math. (N.S.) 6 : 4 (2000), pp. 471–​486. MR 1847384 Zbl 1001.​58001 article

[103] A. Schwarz: “Non­com­mut­at­ive in­stan­tons: A new ap­proach,” Comm. Math. Phys. 221 : 2 (2001), pp. 433–​450. MR 1845331 Zbl 0989.​46040 ArXiv hep-​th/​0102182 article

[104] A. Schwarz: “Theta func­tions on non­com­mut­at­ive tori,” Lett. Math. Phys. 58 : 1 (2001), pp. 81–​90. MR 1865115 Zbl 1032.​53082 ArXiv math/​0107186 article

[105] A. Schwarz: “To­po­lo­gic­al quantum field the­or­ies,” pp. 123–​142 in XIIIth In­ter­na­tion­al Con­gress on Math­em­at­ic­al Phys­ics (Lon­don, 17–22 Ju­ly 2000). Edi­ted by A. Grigory­an, A. Fo­kas, T. Kibble, and B. Zegar­l­in­ski. In­ter­na­tion­al Press (Bo­ston), 2001. MR 1883299 Zbl 1043.​81066 ArXiv hep-​th/​0011260 incollection

[106] A. As­tashkevich and A. Schwarz: “Pro­ject­ive mod­ules over non-com­mut­at­ive tori: Clas­si­fic­a­tion of mod­ules with con­stant curvature con­nec­tion,” J. Op­er. The­ory 46 : 3 (supplement) (2001), pp. 619–​634. Ded­ic­ated to Pro­fess­or D. B. Fuchs on his 60th birth­day. MR 1897158 Zbl 0996.​46031 ArXiv math/​9904139 article

[107] A. Konechny and A. Schwarz: “In­tro­duc­tion to M(at­rix) the­ory and non­com­mut­at­ive geo­metry,” Phys. Rep. 360 : 5–​6 (2002), pp. 353–​465. MR 1892926 Zbl 0985.​81126 ArXiv hep-​th/​0012145 article

[108] M. Di­eng and A. Schwarz: “Dif­fer­en­tial and com­plex geo­metry of two-di­men­sion­al non­com­mut­at­ive tori,” Lett. Math. Phys. 61 : 3 (2002), pp. 263–​270. MR 1942364 Zbl 1019.​58004 ArXiv math/​0203160 article

[109] A. Schwarz: “Gauge the­or­ies on non­com­mut­at­ive Eu­c­lidean spaces,” pp. 794–​803 in Mul­tiple fa­cets of quant­iz­a­tion and su­per­sym­metry: Mi­chael Marinov me­mori­al volume. Edi­ted by M. Olshanet­sky and A. Vain­shtein. World Sci­entif­ic (River Edge, NJ), 2002. MR 1964925 Zbl 1026.​81062 ArXiv hep-​th/​0111174 incollection

[110] A. Schwarz: “Non­com­mut­at­ive su­per­geo­metry, du­al­ity and de­form­a­tions,” Nuc­le­ar Phys. B 650 : 3 (2003), pp. 475–​496. To Alain Connes on his 55th birth­day. MR 1952770 Zbl 1006.​81083 ArXiv hep-​th/​0210271 article

[111] A. Pol­ishchuk and A. Schwarz: “Cat­egor­ies of holo­morph­ic vec­tor bundles on non­com­mut­at­ive two-tori,” Comm. Math. Phys. 236 : 1 (2003), pp. 135–​159. MR 1977884 Zbl 1033.​58009 ArXiv math/​0211262 article

[112] A. Schwarz: “A-mod­el and gen­er­al­ized Chern–Si­mons the­ory,” Phys. Lett. B 620 : 3–​4 (2005), pp. 180–​186. MR 2149784 Zbl 1247.​81441 ArXiv hep-​th/​0501119 article

[113] Y. Chen, M. Kont­sevich, and A. Schwarz: “Sym­met­ries of WDVV equa­tions,” Nuc­le­ar Phys. B 730 : 3 (2005), pp. 352–​363. MR 2180009 Zbl 1276.​81096 ArXiv hep-​th/​0508221 article

[114] S. Gukov, A. Schwarz, and C. Vafa: “Khovan­ov–Roz­ansky ho­mo­logy and to­po­lo­gic­al strings,” Lett. Math. Phys. 74 : 1 (2005), pp. 53–​74. MR 2193547 Zbl 1105.​57011 ArXiv hep-​th/​0412243 article

[115] M. Kont­sevich, A. Schwarz, and V. Volo­god­sky: “In­teg­ral­ity of in­stan­ton num­bers and \( p \)-ad­ic B-mod­el,” Phys. Lett. B 637 : 1–​2 (2006), pp. 97–​101. MR 2230876 Zbl 1247.​14058 ArXiv hep-​th/​0603106 article

[116] A. Schwarz and X. Tang: “Quant­iz­a­tion and holo­morph­ic an­om­aly,” J. High En­ergy Phys. 2007 : 3 (2007), pp. article no. 062, 18 pages. MR 2313939 ArXiv hep-​th/​0611281 article

[117] A. Schwarz and V. Volo­god­sky: “Frobeni­us trans­form­a­tion, mir­ror map and in­stan­ton num­bers,” Phys. Lett. B 660 : 4 (2008), pp. 422–​427. MR 2396265 Zbl 1246.​14053 ArXiv hep-​th/​0606151 article

[118] A. Schwarz and I. Sha­piro: “\( p \)-ad­ic su­per­spaces and Frobeni­us,” Comm. Math. Phys. 282 : 1 (2008), pp. 87–​113. MR 2415474 Zbl 1202.​14017 ArXiv math/​0605310 article

[119] A. Schwarz and I. Sha­piro: “Twis­ted de Rham co­homo­logy, ho­mo­lo­gic­al defin­i­tion of the in­teg­ral and ‘phys­ics over a ring’,” Nuc­le­ar Phys. B 809 : 3 (2009), pp. 547–​560. MR 2478121 Zbl 1192.​81311 ArXiv 0809.​0086 article

[120] A. Schwarz and V. Volo­god­sky: “In­teg­ral­ity the­or­ems in the the­ory of to­po­lo­gic­al strings,” Nuc­le­ar Phys. B 821 : 3 (2009), pp. 506–​534. MR 2547214 Zbl 1203.​81155 ArXiv 0807.​1714 article

[121] M. Movshev, A. Schwarz, and R. Xu: Ho­mo­logy of Lie al­gebra of su­per­sym­met­ries. Pre­print, 2010. ArXiv 1011.​4731 techreport

[122] A. Schwarz: “Space and time from trans­la­tion sym­metry,” J. Math. Phys. 51 : 1 (2010), pp. article no. 015201, 7 pages. MR 2605834 Zbl 1245.​81070 ArXiv hep-​th/​0601035 article

[123] M. V. Movshev, A. Schwarz, and R. Xu: “Ho­mo­logy of Lie al­gebra of su­per­sym­met­ries and of su­per Poin­caré Lie al­gebra,” Nuc­le­ar Phys. B 854 : 2 (2012), pp. 483–​503. MR 2844329 Zbl 1229.​81117 ArXiv 1106.​0335 article

[124] J.-M. Li­ou and A. Schwarz: “Equivari­ant co­homo­logy of in­fin­ite-di­men­sion­al Grass­man­ni­an and shif­ted Schur func­tions,” Math. Res. Lett. 19 : 4 (2012), pp. 775–​784. MR 3008414 Zbl 1300.​14025 ArXiv 1201.​2554 article

[125] M. V. Movshev and A. Schwarz: “Su­per­sym­met­ric de­form­a­tions of max­im­ally su­per­sym­met­ric gauge the­or­ies,” J. High En­ergy Phys. 2012 : 9 (2012), pp. article no. 136, 77 pages. MR 3044913 Zbl 1397.​81390 ArXiv 0910.​0620 article

[126] J.-M. Li­ou and A. Schwarz: “Mod­uli spaces and Grass­man­ni­an,” Lett. Math. Phys. 103 : 6 (2013), pp. 585–​603. MR 3054647 Zbl 1276.​14016 ArXiv 1111.​1649 article

[127] A. Mikhail­ov, A. Schwarz, and R. Xu: “Co­homo­logy ring of the BRST op­er­at­or as­so­ci­ated to the sum of two pure spinors,” Mod­ern Phys. Lett. A 28 : 23 (2013), pp. article no. 1350107, 6 pages. MR 3085958 Zbl 1279.​81052 ArXiv 1305.​0071 article

[128] D. Kre­fl and A. Schwarz: “Re­fined Chern–Si­mons versus Vo­gel uni­ver­sal­ity,” J. Geom. Phys. 74 (2013), pp. 119–​129. MR 3118578 Zbl 1283.​58018 ArXiv 1304.​7873 article

[129] X. Liu and A. Schwarz: Quant­iz­a­tion of clas­sic­al curves. Pre­print, March 2014. ArXiv 1403.​1000 techreport

[130] R. Xu, M. Movshev, and A. Schwarz: “In­teg­ral in­vari­ants in flat su­per­space,” Nuc­le­ar Phys. B 884 (2014), pp. 28–​43. MR 3214872 Zbl 1323.​81093 ArXiv 1403.​1997 article

[131] A. Schwarz: “Quantum curves,” Comm. Math. Phys. 338 : 1 (2015), pp. 483–​500. MR 3345383 Zbl 1318.​81042 ArXiv 1401.​1574 article

[132] J.-M. Li­ou, A. Schwarz, and R. Xu: “Wei­er­strass cycles and tau­to­lo­gic­al rings in vari­ous mod­uli spaces of al­geb­ra­ic curves,” J. Fixed Point The­ory Ap­pl. 17 : 1 (2015), pp. 209–​219. To Pro­fess­or An­drzej Granas. MR 3392990 Zbl 1345.​14042 ArXiv 1308.​6374 article

[133] A. Schwarz, V. Volo­god­sky, and J. Wal­cher: “Fram­ing the di-log­ar­ithm (over \( \mathbb{Z} \)),” pp. 113–​128 in String-Math 2012 (Bonn, Ger­many, 16–21 Ju­ly 2012). Edi­ted by R. Don­agi, S. Katz, A. Klemm, and D. R. Mor­ris­on. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 90. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 2015. MR 3409790 Zbl 1356.​81196 ArXiv 1306.​4298 incollection

[134] M. V. Movshev and A. Schwarz: “Gen­er­al­ized Chern–Si­mons ac­tion and max­im­ally su­per­sym­met­ric gauge the­or­ies,” pp. 327–​340 in String-Math 2012 (Bonn, Ger­many, 16–21 Ju­ly 2012). Edi­ted by R. Don­agi, S. Katz, A. Klemm, and D. R. Mor­ris­on. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 90. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 2015. MR 3409803 Zbl 1356.​81200 ArXiv 1304.​7500 incollection

[135] A. Schwarz: “Ax­io­mat­ic con­form­al the­ory in di­men­sions \( > 2 \) and AdS/CT cor­res­pond­ence,” Lett. Math. Phys. 106 : 9 (2016), pp. 1181–​1197. MR 3533564 Zbl 1347.​81058 ArXiv 1509.​08064 article

[136] M. T. Luu and A. Schwarz: “Four­i­er du­al­ity of quantum curves,” Math. Res. Lett. 23 : 4 (2016), pp. 1111–​1137. MR 3554503 Zbl 1354.​81040 ArXiv 1504.​01582 article

[137] A. Schwarz, V. Volo­god­sky, and J. Wal­cher: In­teg­ral­ity of fram­ing and geo­met­ric ori­gin of 2-func­tions. Pre­print, February 2017. ArXiv 1702.​07135 techreport

[138] A. Mikhail­ov and A. Schwarz: “Fam­il­ies of gauge con­di­tions in BV form­al­ism,” J. High En­ergy Phys. 2017 : 7 (2017), pp. article no. 063, 25 pages. MR 3686735 Zbl 1380.​81371 ArXiv 1610.​02996 article

[139] J.-M. Li­ou and A. Schwarz: “Wei­er­strass cycles in mod­uli spaces and the Krichever map,” Math. Res. Lett. 24 : 6 (2017), pp. 1739–​1758. MR 3762693 Zbl 1393.​14022 ArXiv 1207.​0530 article

[140] M. Movshev and A. Schwarz: “Quantum de­form­a­tion of planar amp­litudes,” J. High En­ergy Phys. 2018 : 4 (2018). art­icle no. 121, 20 pages. MR 3801153 Zbl 1390.​81613 ArXiv 1711.​10053 article

[141] A. Schwarz: Scat­ter­ing mat­rix and in­clus­ive scat­ter­ing mat­rix in al­geb­ra­ic quantum field the­ory. Pre­print, August 2019. To Max­im Kont­sevich on the oc­ca­sion of his 55th birth­day with love and ad­mir­a­tion. ArXiv 1908.​09388 techreport

[142] M. Luu and A. Schwarz: It­er­ated in­teg­rals on af­fine curves. Pre­print, December 2019. ArXiv 1912.​09506 techreport

[143] A. Schwarz: “In­clus­ive scat­ter­ing mat­rix and scat­ter­ing of qua­si­particles,” Nuc­le­ar Phys. B 950 (2020). art­icle no. 114869, 14 pages. MR 4039472 ArXiv 1904.​04050 article

[144] A. Schwarz: “Geo­met­ric ap­proach to quantum the­ory” in Spe­cial is­sue on al­gebra, to­po­logy, and dy­nam­ics in in­ter­ac­tion in hon­or of Dmitry Fuchs, published as SIGMA, Sym­metry In­teg­rabil­ity Geom. Meth­ods Ap­pl. 16. Issue edi­ted by B. Khes­in, F. Ma­likov, V. Ovsi­en­ko, and S. Tabach­nikov. Na­tion­al Academy of Sci­ences of Ukraine (Kiev), 2020. Pa­per no. 020, 3 pages. MR 4080799 Zbl 1436.​81018 ArXiv 1906.​04939 incollection

[145] A. Schwarz: Geo­met­ric and al­geb­ra­ic ap­proaches to quantum the­ory. Pre­print, February 2021. ArXiv 2102.​09176 techreport

[146] A. Schwarz: Scat­ter­ing the­ory in al­geb­ra­ic ap­proach: Jordan al­geb­ras, 2021. In pre­par­a­tion. misc

[147] A. S. Schwarz: Scat­ter­ing in geo­met­ric ap­proach to quantum the­ory. Pre­print, 2021. ArXiv 2107.​08557v1 techreport