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 For symplectic manifolds he described Andreas Floer’s unpublished new ideas about what came to be known as “Floer Homology” and how these gave new invariants of such manifolds and their Lagrangian submanifolds, invariants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invariants for 4d manifolds, possibly with boundary, (and was based on work of Floer and Donaldson that was still unpublished). Finally, the Euler characteristic of Floer Homology was identified with a new invariant due to Casson (also unpublished, it seemed like nothing Atiyah was talking about was yet written up), which was a 3d invariant that fit beautifully into the whole picture.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.

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14 Responses to 25 Years of Topological Quantum Field Theory

  1. Jeff says:

    Memories. Wasn’t at the conference, I was a brand new grad student, but a few years later my orals were to present that Witten paper. Took some work :-). There’s a great book by John Roe which helped a lot. Of course, I was only interested in the math side…

  2. theoreticalminimum says:

    Thank you for this informative post, Peter.

    It’s sad to realise that Andreas Floer would tragically end his life 4 years after that Duke conference, at the age of 35 and at a time when he was at the peak of his creativity.

    I would recommend the Floer memorial volume (Hofer et al.), which is the best collection of articles I can think of that captures the effervescence of the developments at that time in symplectic geometry and quantum field theories.

  3. Florin says:

    Hello,

    I’ve been quietly following with great interest your website for a couple of years now and although I don’t have a formal training in mathematics I grew very interested in topics such as Langlands program or any other attempt to reveal stronger connections between mathematics and reality/physics. And while I somehow have some grasp on Noether’s theorems and I have some kind of intuitive understanding of the importance of representation theory for say the quark model, the mathematical apparatus for even having a hint about the Langlands program seems to me very abstract and inaccessible.

    So I was wondering, what would be the mathematical requirements for starting to approach this subject and also could you please indicate to a less mathematical introduction to the Langlands program?

    Thank you, your answer would be much appreciated!

    Regards,
    Florin

  4. Peter Woit says:

    Florin,

    The Langlands program brings together several fundamental areas of mathematics, so to really understand and appreciate it, you need some serious background in modern mathematics. Edward Frenkel’s lectures this past semester at Columbia provide a good introduction and explain the links to affine Lie algebra representations and conformal field theory.

    You could also take a look at this survey of the subject from Langlands himself

    http://publications.ias.edu/sites/default/files/gibbs-ps.pdf

    A good place to start might be the popular book by Ash and Gross, called “Fearful Symmetry”.

    [Oops, the correct title is Fearless Symmetry, thanks to all for the correction]

  5. Ossicle says:

    I believe these are the Frenkel lectures Peter refers to:

    http://www.youtube.com/playlist?list=PLA391DE72F863E030&feature=plpp

  6. MathPhys says:

    Jeff,

    Which of John Roe’s books were you refering to? The CBMS write-up? or the longer version? Which of these two in more readable?

  7. Jeff M says:

    MathPhys,

    It’s “Elliptic Operators, Topology, and Asymptotic Methods” from the Pitman Research Notes, 1988, No. 179. My orals predate his CBMS lectures 🙂 It’s basically his lecture notes from a course he gave at Oxford.

  8. paddy says:

    Though I am also mystified about the connection between mathematics and physics:
    (a)It is arguably a trusism that mathematics provides a framework for physics;
    (b)it is certainly arguable that this framework has always seem to come after the fact;
    (c)what delimits and leads physics forward is observation and experiment;
    and (d) “shut up and calculate”.

  9. Pingback: 87th Carnival of Mathematics | Random Walks

  10. SA says:

    This is off-topic but I just thought it was pretty funny: http://arxiv.org/pdf/hep-th/0503249.pdf

  11. Peter Woit says:

    SA,

    What’s funny is that many theorists don’t seem to think this is a joke, see

    http://inspirehep.net/search?ln=en&p=refersto%3Arecid%3A679450

    More comments maybe in an imminent posting…

  12. Florin says:

    Hello Mr. Woit,

    Just wanted to say thank you for all the resources you posted in your comment, much appreciated. Also Ossicle, thanks for the lectures, watching the first videos really rendered me the scale of the program.

    Not sure if I’ll ever make it to approach and actually understand all the topics involved here, but I plan to go on with my calculus manual so that I can access general relativity and quantum field theory and in parallel maybe start learning some group theory, Lie algebra… Too bad I started this late..

  13. SA says:

    Peter,

    Indeed that would be funny (and sad). However, if you take a look at those citing papers, you will see that most are citing tongue-in-cheek and the 1 or 2 other papers are citing it obviously based on a superficial keyword search.

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