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,and . 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 , . 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, and 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 , 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 . 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 . From a pure physical point of view, the interest in classical solutions with finite energy appeared after the seminal papers of and about monopoles in gauge theories. With Yu. Tyupkin we wrote two papers on monopole solutions in gauge theories , . 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., , , 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
After the pioneering paper of A.S. with , and the paper of G. ’t Hooft about the solution of \( U(1) \) problem in gauge theory , 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 . 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 , 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 , , , . 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 . 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., and Yu. Tyupkin, where an “instanton” solution in Yang–Mills theory was found
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 and about generalizing the Riemann surfaces to the super-Riemann surfaces was very important for constructing the perturbation theory of superstrings (see , ). The beautiful paper with about instantons on a noncommutative space  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.