25 Years of Topological Quantum Field Theory

It occurred to me today that right about now is the time someone should have chosen as the date for a celebration of the 25th anniversary of the birth of the idea of “Topological Quantum Field Theory”, as well as some much less well-known ideas about the relationship of QFT and mathematics that still await full investigation.

Just about 25 years ago, from May 12-16 1987, there was a remarkable conference that I attended at Duke, to celebrate the “Mathematical Heritage of Hermann Weyl”, two years after the centenary of his birth. The proceedings were published a year or so later. At this conference, Michael Atiyah gave an amazing talk with the title New invariants for manifolds of dimensions 3 and 4. In it he unveiled a vista of new ideas about topology that would dominate the subject for years to come. For symplectic manifolds he described Andreas Floer’s unpublished new ideas about what came to be known as “Floer Homology” and how these gave new invariants of such manifolds and their Lagrangian submanifolds, invariants related to very recent work of Gromov (now known as “Gromov-Witten invariants”). Replacing 1d (Lagrangian paths) and 2d (pseudo-holomorphic curves) objects in a symplectic manifold by 3d (flat connections) and 4d (instantons) objects in a space of connections on a 4d manifold gave yet another whole new world of mathematics. This is the subject of Floer Homology and Donaldson invariants for 4d manifolds, possibly with boundary, (and was based on work of Floer and Donaldson that was still unpublished). Finally, the Euler characteristic of Floer Homology was identified with a new invariant due to Casson (also unpublished, it seemed like nothing Atiyah was talking about was yet written up), which was a 3d invariant that fit beautifully into the whole picture.

There’s a copy of Atiyah’s write-up of the talk online here (perhaps the AMS will ignore any intellectual property issues here for the greater good). I see that, increasingly like everything else in the world, electronic access to the book is controlled by Google, see Google Play, which I didn’t even know existed.

An inspiration for Floer had been Witten’s ground-breaking paper “Supersymmetry and Morse Theory”, which dealt with the relationship between Morse theory and some supersymmetric quantum mechanics models. Atiyah explained some of these ideas in the talk and towards the end conjectured that quantum field theories were part of the story. His conjectural QFTs would have Floer homology as their ground states and would turn out to be the basic examples of TQFTs. After repeated prodding from Atiyah, Witten a year later produced such theories as twisted N=2 supersymmetric QFTs: a sigma model for the symplectic manifold case, and a supersymmetric Yang-Mills theory for the 4d case. In his final remarks, Atiyah raised the issue of knot invariants and the Jones polynomial, suggesting that this too would have a QFT interpretation, something that came to fruition a couple years later with Witten’s Chern-Simons theory that won him a Fields medal. Witten was at the talk, and I recall him coming down to the podium to ask Atiyah some questions about the Jones polynomial immediately after the lecture.

The Duke conference was also significant to me for personal reasons. At the time I was a postdoc at the ITP in Stony Brook, looking for a job and trying to figure out my future. It was becoming clear that physics departments didn’t want to hear from any young theorists interested in mathematics who weren’t doing string theory. I had been spending a lot of time at Stony Brook learning more mathematics and talking to some of the geometers there, who were housed on the floor below the ITP. My trip to the conference at Duke was motivated partly by a desire to visit my grandparents who were in North Carolina for the summer, as well as a plan to investigate prospects for a career change into mathematics. The Atiyah talk bowled me over, convincing me that the intersection of mathematics and QFT had an exciting future. Getting to know a bit more about the mathematical community showed me it could be a great place to work, in many ways much more welcoming and open to new ideas than the physics community. I soon moved up to Cambridge for a year, where the Harvard Physics department let me use a desk, and found a part-time job teaching calculus at Tufts.

What’s remarkable to me now looking at the conference volume is how much exciting material was being discussed, in addition to the fantastic Atiyah talk. Raoul Bott gave a wonderful talk on Borel-Weil-Bott (and its relation to quantization), David Vogan on representation theory in general (and its relation to quantization). Roger Howe has a contribution also about deep connections between quantum mechanics and what he calls the “oscillator representation”. Jim Lepowsky was talking about Kac-Moody Lie algebras, vertex operators and the Monster group, Is Singer about quantizing gauge theory and string theories, and there were a host of other wonderful talks on topology and geometry.

One topic that I didn’t really appreciate at all at the time was that of Langlands theory. Langlands himself was there, talking about Shimura varieties, and James Arthur talked about the Trace formula and its applications to Langlands theory. I think I may have missed Witten’s talk, since I don’t remember it, but his contribution to the conference proceedings is about how to abstractly think about the theory of 2-d free fermions, in a form that makes sense on an arbitrary curve. A few weeks later (June 23), his amazing paper Quantum Field Theory, Grassmanians and Algebraic Curves was submitted to Communications in Mathematical Physics. If I had to point to a paper that truly looks like 21st century work that fell by accident into the 20th century, this would be it. It gives some strikingly different ways of thinking about QFT in 2d, including tantalizing connections to the structures (“automorphic representations”) that show up in Langlands theory, and has provided inspiration to many people over the years, including the geometric Langlands program. Atiyah’s lecture pointed to new ideas relating QFT to cutting edge geometry and topology, ideas that quickly led to lots of progress, while Witten’s ideas related QFT to representation theory and Langlands theory, in ways that we still have yet to fathom.