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 “\( U(1) \) problem.” The BPST instanton, of course, has had a huge importance both for physics and, in mathematics, for four-manifold theory.
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 \( U(1) \) problem were manifestations of the Atiyah–Singer index theorem [2]. Like most physicists of the time, I had never heard of the index theorem, or even the concept of the index, or the names Atiyah and Singer. The role of the index theorem in instanton theory turned out to be the beginning of something big: a new and highly productive interaction of theoretical physics with contemporary developments in geometry. I am pretty sure that this was one of the developments that led Michael Atiyah, Is Singer, Raoul Bott, and other leading geometers of the period to become interested in physics.
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 \( p \)-form
gauge theory, essentially what is now called \( BF \) theory, and showed
that its partition function is the analytic torsion of Ray and Singer.
This result was later important when I was trying to understand the
perturbative expansion of Chern–Simons gauge theory in three
dimensions. It has also been important background for several of my
other papers, especially in computations of volumes of moduli spaces
that I carried out both in the period around 1990 and again just in
the last couple of years.
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 \( BF \) theory and Ray–Singer analytic torsion did not have a major impact at the time, and I was lucky to know about it a decade later when I needed it.
All three papers that I have mentioned so far — on the instanton, the relation to the index theorem, and on \( BF \) theory and the analytic torsion — were important in establishing bridges between topology and quantum field theory. The relationship between these fields has turned out to be very important for condensed matter physics as well as in the understanding of relativistic field theory. Albert Schwarz has certainly been one of the pioneers of this connection.
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 \( B \)-field, open string
theory reduces to a Connes-style version of Yang–Mills theory on a
noncommutative space
[6].
Further developing this subject, he showed that
\( T \)-duality 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.
