by Alexander M. Polyakov
In the spring of 1968 my friend Sasha Belavin said that he wanted to introduce me to a very smart mathematician. I was not very enthusiastic since in my experience even the best mathematicians have problems with understanding quantum field theory (QFT). This is curious, since they have deep knowledge of classical dynamics, but when it comes to divergences, renormalization, and gauge fixing, they get stuck. In QFT we change the rules of the game in the middle of the road and a precise mathematical mind can’t tolerate this. Not so in the case of Alik Schwarts! He had his own approach to QFT which allowed him to navigate in this swampy domain. I was impressed by his ability to confront novel things by translating them into his own language (we discussed anomalous dimensions and bootstrap). Several years later, after our joint work on instantons, Alik called me and said that the zero modes of instantons are counted by the famous index theorem (of which I had never heard) and that there is a connection between our instantons and modern work on topology. Several months later the same observation was made by Atiyah and Singer and that started a revolution both in physics and mathematics.
In another episode, about which I feel guilty, Alik told me that the Chern–Simons theory must be related to Jones polynomials. I missed and forgot this profound remark, but a year later E. Witten came to the same conclusion and built a comprehensive theory relating these topics. The list of similar events can be continued — Alik was the first to propose to calculate topological invariants by functional integrals, starting topological QFT.
In 1968 when I met Alik I realized that he is a deep thinker but couldn’t have foreseen that our paths would cross a number of times. I am very glad that they did.
Alexander Polyakov is the Joseph Henry Professor of Physics at Princeton University.
