Celebratio Mathematica

I’ve known Al­bert So­lomonovich Schwarz (A.S.) since 1968. At that time I was a gradu­ate stu­dent in the in­sti­tute where he was a pro­fess­or. We star­ted work­ing to­geth­er and later I be­came his PhD stu­dent. I knew his fam­ily very well be­cause A.S. pre­ferred to work with stu­dents at home. As I un­der­stood very soon, his fam­ily ex­per­i­enced tra­gic events dur­ing the peri­od of ter­ror in the So­viet Uni­on. (Un­for­tu­nately, the same is true for many fam­il­ies, in­clud­ing mine.) This was the reas­on why A.S. stud­ied in a pro­vin­cial ped­ago­gic­al in­sti­tute (in Ivan­ovo) at the math­em­at­ic­al de­part­ment where at that time, and for the same reas­on, there were two top So­viet math­em­aticians in to­po­logy, namely, Rohlin and Efre­movich. After fin­ish­ing at the in­sti­tute and de­fend­ing his PhD at Mo­scow Uni­versity, A.S. be­came one of the young­est Doc­tors of Sci­ence in the USSR. At the de­part­ment of The­or­et­ic­al Phys­ics, A.S. was the only pro­fess­or who did not have a back­ground in phys­ics. But this was not the main reas­on why his po­s­i­tion was not com­fort­able. Every­body who knows the sorts of prob­lems that could arise for people of his na­tion­al­ity in the USSR at that time will un­der­stand me. This is why I was so glad when A.S. moved from the So­viet Uni­on to the United States at the end of the 80s.

My work with A.S. played a cru­cial role in my sci­entif­ic life. Be­cause of him, at a rather young age I met such bril­liant sci­ent­ists as F. A. Berez­in, S. P. Novikov, I. M. Gel­fand, L. D. Fad­deev and oth­ers. This was very im­port­ant for me in form­ing an un­der­stand­ing of the level and stand­ards of real sci­entif­ic re­search. In the be­gin­ning of my col­lab­or­a­tion with A.S., we stud­ied and wrote some pa­pers on purely math­em­at­ic­al as­pects of quantum field the­ory [1], [2]. They con­cerned ax­io­mat­ic and con­struct­ive ap­proaches. Even if I was not very ex­cited by this do­main of quantum field the­ory, it was use­ful for me to read pa­pers of Haag, Glimm and Jaffe on this sub­ject. The situ­ation changed after the dis­cov­ery of in­teg­rable non­lin­ear equa­tions and solitons. The re­la­tion of solitons to quantum particles turned our in­terest to math­em­at­ic­al phys­ics. To­geth­er with Yu. Ty­up­kin, we have shown an ex­act re­la­tion between solitons and quantum particles and their form-factors in the ex­ample of clas­sic­al and quantum non­lin­ear Schrödinger equa­tions in the at­tract­ive and re­puls­ive re­gime [5]. Later, us­ing the res­ults of Glimm and Jaffe, we showed that in the two-di­men­sion­al case, quantum particles (kinks) can ap­pear in the strong coup­ling re­gime [4]. From a pure phys­ic­al point of view, the in­terest in clas­sic­al solu­tions with fi­nite en­ergy ap­peared after the sem­in­al pa­pers of G. ’t Hooft and A. Polyakov about mono­poles in gauge the­or­ies. With Yu. Ty­up­kin we wrote two pa­pers on mono­pole solu­tions in gauge the­or­ies [3], [7]. In the first pa­per, a ne­ces­sary con­di­tion in the to­po­logy of the space of clas­sic­al vacua for the ex­ist­ence of such solu­tions was giv­en, while in the second we provided a rig­or­ous proof of the ex­ist­ence of such solu­tions.

After the pi­on­eer­ing pa­per of A.S. with A. Be­lav­in, A. Polyakov and Yu. Ty­up­kin, where an “in­stan­ton” solu­tion in Yang–Mills the­ory was found [6], and the pa­per of G. ’t Hooft about the solu­tion of \( U(1) \) prob­lem in gauge the­ory [14], it be­came clear that in­stan­tons play a very im­port­ant role bey­ond very small dis­tances where we can use per­turb­a­tion the­ory. Us­ing the Atiyah–Sing­er in­dex for el­lipt­ic op­er­at­ors, A.S. found the di­men­sion of the para­met­er space of multi-in­stan­ton solu­tions [8]. For phys­ic­al ap­plic­a­tions it be­came im­port­ant to find in­stan­ton con­tri­bu­tions to vari­ous quant­it­ies in quantum field the­ory. At that time people used the di­lute in­stan­ton gas ap­prox­im­a­tion in gauge the­or­ies. However, in sev­er­al pa­pers writ­ten in col­lab­or­a­tion with I. Fro­lov, we have shown, based on the ex­ample of two-di­men­sion­al Kähler \( \sigma \)-mod­els, where in­stan­ton con­tri­bu­tions have been cal­cu­lated in a closed form, that this ap­prox­im­a­tion does not really work [10], [11], [12], [13]. Namely, it turned out that the in­stan­tons form not a di­lute gas but rather a dense plasma, and their con­tri­bu­tions lead to the ap­pear­ance of a cor­rel­a­tion length in the sys­tem. After the nice geo­met­ric­al ADHM con­struc­tion of multi-in­stan­ton solu­tions, to­geth­er with A. Be­lav­in and Yu. Ty­up­kin we tried to cal­cu­late the in­stan­ton con­tri­bu­tions in gauge the­or­ies. We were able to ex­press quantum fluc­tu­ations around the in­stan­tons in terms of ADHM con­struc­tion, but a closed form (in terms of in­stan­ton para­met­ers) of the meas­ure on the in­stan­ton man­i­fold re­mained un­known (and still is) due to al­geb­ra­ic dif­fi­culties [9]. Nev­er­the­less, in this pa­per we gave some ar­gu­ments that in gauge the­or­ies the in­stan­tons pro­duce the same phe­nom­ena as in two-di­men­sion­al Kähler \( \sigma \)-mod­els.

After study­ing to­geth­er the in­stan­ton con­tri­bu­tions, we worked in dif­fer­ent do­mains of quantum field the­ory and did not have com­mon pa­pers. However, we al­ways re­mained very good friends. I fol­lowed his sci­entif­ic activ­ity and he was in­ter­ested in mine. Here I want to men­tion just a few of his im­port­ant works of that peri­od which were in­ter­est­ing to me and played an im­port­ant role for fu­ture de­vel­op­ment of quantum field the­ory and string the­ory. The pa­per “The par­ti­tion func­tion of de­gen­er­ate quad­rat­ic func­tion­al and Ray–Sing­er in­vari­ants” be­came the pi­on­eer­ing work in the con­struc­tion of to­po­lo­gic­al quantum field the­or­ies and strings. His work with A. Rosly and A. Voro­nov about gen­er­al­iz­ing the Riemann sur­faces to the su­per-Riemann sur­faces was very im­port­ant for con­struct­ing the per­turb­a­tion the­ory of su­per­strings (see [15], [16]). The beau­ti­ful pa­per with N. Nekra­sov about in­stan­tons on a non­com­mut­at­ive space [17] fas­cin­ated many sci­ent­ists and gen­er­ated an in­terest in the ana­lys­is of non­com­mut­at­ive gauge the­or­ies.

I am sure that A.S. will write new im­port­ant pa­pers. But at the end of my notes I want to say that, be­sides the sci­entif­ic cre­ativ­ity and nice hu­man qual­it­ies, A.S. has a huge eru­di­tion in dif­fer­ent fields of know­ledge. This be­came clear to me after a long time spent in dis­cus­sions with him. Fur­ther­more, in my life I met not so many people with such a per­son­al­ity. And I am very happy that Al­bert So­lomonovich Schwarz was my teach­er.