by Edward Witten
The first paper by Albert Schwarz that I was aware of was his famous
paper with
A. A. Belavin,
A. M. Polyakov,
and
Yu. S. Tyupkin
introducing the “instanton” solution of Yang–Mills theory
[1].
This paper
appeared while I was a graduate student and had a dramatic impact,
beginning with the discovery by Gerard ’t Hooft that instanton effects
in QCD lead to the solution of the celebrated “
During my postdoctoral years, Albert wrote two papers that had a major
impact on my work. In one, he explained that the fermion zero-modes in
an instanton field that had led to the solution of the
Albert’s second paper from this period that had a major influence on
my work, “Partition function of degenerate quadratic functional and
Ray–Singer invariants”
[3],
was arguably the first paper on topological
quantum field theory. He studied a topologically invariant
gauge theory, essentially what is now called
Both of these papers were pointed out to me by
Sidney Coleman,
who was
one of the senior professors at Harvard, where I was a postdoc. The
paper on the index theorem had a major impact at the time and I am
sure I would have learned about it after a while even if Sidney had
not pointed it out. However, Albert’s paper on
All three papers that I have mentioned so far — on the instanton, the
relation to the index theorem, and on
In the mid-1980s, Albert Schwarz invented the concept of a super Riemann surface (independently of D. Friedan) and did very deep work, with A. A. Rosly and A. A. Voronov, on this subject [4]. Super Riemann surfaces are the natural geometric framework for superstring perturbation theory. When I was working in the last decade on formulating superstring perturbation theory in terms of super Riemann surfaces, I found Schwarz’s work to be the most useful reference, by far.
In the
1990s, Albert Schwarz, with
A. Connes
and
M. R. Douglas,
discovered that in the presence of a strong in this framework becomes the mathematical operation of
Morita equivalence
[7],
an important concept in algebra of which
physicists of the time were quite unaware. To me, though, his most
dramatic paper on gauge theory on a noncommutative space was the paper
with
N. Nekrasov
[8]
showing how noncommutativity resolves the singularity
associated to instanton bubbling. This result fascinated me. What they
said was so beautiful that it was clear that it must be right, but
their claim appeared at first sight to be in conflict with what had
been learned by more pedestrian methods. A desire to understand their
claim better and to resolve the apparent inconsistency was my primary
motivation for work that I did with
Nathan Seiberg
on gauge theory on a noncommutative space.
The Batalin–Vilkovisky framework for quantization is very important and very powerful. Schwarz has made important contributions to understanding it better [5].
Finally, I want to mention his work with S. Gukov and C. Vafa [9], with the first proposal for a physics-based interpretation of Khovanov homology of knots and related “categorified” knot invariants. This work was a main influence for work that I did in this area, initially in the years 2009–11.