Not Even Wrong

I was down in Princeton last Thursday, and attended a wonderful talk by Witten, which I’ll try and explain a little bit about here. Presumably within a rather short time he’ll have a paper out on the arXiv giving full details.

The talk concerned Chern-Simons theory, the remarkable 3d QFT that was largely responsible for Witten’s Fields medal. Given an SU(2) connection A on a bundle over a 3-manifold M, one can define its Chern-Simons number CS(A). This number is invariant under the identity component of the group of gauge transformations $\mathcal G$, and jumps by 2π times an integer under topologically non-trivial gauge transformations. The QFT is given by taking CS(A) as the action. The path integral

$$Z(M,k)=\int_{\mathcal A/\mathcal G} dA e^{ikCS(A)}$$

is well-defined for k integral and gives an interesting topological invariant of the 3-manifold M. One can also take a knot K in M, choose an irreducible representation R of SU(2) of spin n/2, and then define a knot invariant by

$$Z(M,K,k,n)=\int_{\mathcal A/\mathcal G} dA e^{ikCS(A)}hol_R(K)$$

where $hol_R(K)$ is the trace of the holonomy in the representation R, around the knot K (this is the Wilson loop).

To simplify matters, consider the special case $Z(K,k,n)=Z(S^3,K,k,n)$, which can be used to study knots in $\mathbf R^3$.

These knot invariants can be evaluated for large k by stationary phase approximation (perturbation theory), and for arbitrary k by reformulating the QFT in a Hamiltonian formalism, and using loop group representation theory and the Verlinde (fusion) algebra.

One thing that has always bothered me about this story is that it has never been clear to me whether such a path integral makes sense at all non-perturbatively. At one point I spent a lot of time thinking about how you would do such a calculation in lattice gauge theory. There, one can imagine various (computationally impractical) ways of defining the action, but integrating a phase over an infinite dimensional space always looked problematic: without some other sort of structure, it was hard to see how one could get a well-defined answer in the limit of zero-lattice spacing. In simpler models with similar structure (e.g. loops on a symplectic manifold), similar problems appear, and are resolved by introducing additional terms in the action.

What Witten proposed in his talk was a method for consistently defining such path integrals by analytic continuation. The idea is to complexify, working with SL(2,C) connections and a holomorphic Chern-Simons functional, then exploit the freedom to choose a different contour to integrate over than the contour of SU(2) connections. By choosing a contour that is not invariant under topologically non-trivial gauge transformations, and only modding out by the topologically trivial ones, Witten also managed to define the theory for non-integral k, making contact with a lot of mathematical work on these knot invariants, which treats them a Laurent polynomials in the square root of

$$q=e^{2\pi i/(k+2)}$$

The main new idea that Witten was using was that the contributions of different critical points p (including complex ones), could be calculated by choosing appropriate contours $\mathcal C_p$ using Morse theory for the Chern-Simons functional. This sort of Morse theory involving holomorphic Morse functions gets used in mathematics in Picard-Lefshetz theory. The contour is given by the downward flow from the critical point, and the flow equation turns out to be a variant of the self-duality equation that Witten had previously encountered in his work with Kapustin on geometric Langlands. One tricky aspect of all this is that the contours one needs to integrate over are sums of the $\mathcal C_p$ with integral coefficients and these coefficients jump at “Stokes curves” as one varies the parameter in one’s integral (in this case, x=k/n, k and n are large). In his talk, Witten showed the answer that he gets for the case of the figure-eight knot.

Mathematicians and mathematical physicists have done quite a bit of work on SL(2,C) Chern-Simons, and studying the properties of knot-invariants as analytic functions. I don’t know whether Witten’s new technique solves any of the mathematical problems that have come up there. He mentioned the relation to 3d gravity, where the relationship between Chern-Simons theory and gravity in the Lorentzian and Euclidean signature cases evidently still remains somewhat mysterious. Perhaps his analytic continuation method may provide some new insight there. It also may apply to a much wider range of QFTs where there are imaginary terms in the action, making the path integral problematic. I’d be very curious to understand how this works out in some simpler models, such as the loop space ones. In any case, it appears to be a quite beautiful new idea about how to define certain QFTs via the path integral.

Update: Witten’s slides for the talk are available here, video here. For slides from other talks at the workshop the talk was part of, see here.

Mathematician Jim Simons is retiring from the job of running the hedge fund Renaissance Technologies. Construction of the building for the Simons Center for Geometry and Physics is proceeding, with opening scheduled for next fall.

An Algerian physicist associated with the LHCb experiment at CERN has been arrested on charges of having associations with al-Qaeda. The media freak out and CERN issues a statement.

I. M. Gelfand died on Monday at the age of 96. For more about him, see here, here and here.

The fourth and latest installment of Oswaldo Zapata’s essay on the history of superstring theory is here.

In Geometric Langlands news, Dennis Gaitsgory is running a seminar at Harvard this fall, with notes and other materials on-line here.

Emanuel Kowalski points out that, morally, Princeton’s Peter Sarnak has a blog.

Update: One more.

From a recent blog posting by economist Brad DeLong, entitled The State of Economics in the 2000s Analogized…:

But I think there also has to be an explanation in terms of the sociology of academic disciplines. And in that light, it seems to me that if I were a journalist, I’d consider writing a piece comparing freshwater economics to the other major recent case in which an academic discipline went completely off the rails, namely English departments’ swing into postmodernism in the ’80s and early ’90s. Offhand, there seem to be some real similarities, e.g.:

1. In both cases, the people involved maintained, credibly, that you couldn’t really assess the work in question without putting a lot of effort into understanding it.
2. In both cases, that required mastering difficult stuff. (In econ, all the math and models; in pomo lit stuff, mastering the literally incomprehensible language in which a lot of that stuff was written.)
3. In both cases, that deterred a lot of people on the outside who were generally puzzled and skeptical, but didn’t want to spend years getting into a position in which they could credibly say: yes, this is, in fact, nuts.
4. So in both cases practitioners were largely insulated from criticism they had to take seriously.

Relatedly, in both cases it took shocks from the outside to expose the problems in this (in the case of English, things like the Sokal hoax; in the case of econ, the near-collapse of the global economy.)

Both cases involved a lot of arrogance, and a generally dismissive attitude towards other approaches. Since, in both cases, practitioners were able to seize significant amounts of control over a discipline before their approach crashed and burned, this did real damage to the disciplines in question (leading to, e.g., large chunks of previous disciplinary history being forgotten.)

In the last sentence DeLong identifies clearly what is most sad and disturbing about this kind of story.

Update: As a commenter points out, the text quoted is from DeLong’s blog, but is not his own words, he’s quoting someone else.

I’d recently been wondering whether the archives of the Bourbaki group would be put on-line, and today noticed that there’s a project to do so, with results available here. One can read copies of “La Tribu”, internal reports on the activities of the group, up through 1953. There are a wide variety of interesting mathematical documents, often consisting of attempts to write up one subject or another, efforts that sometimes made it into the published books, often not.

One subject that Bourbaki struggled with over the years was that of how to set out the foundations of differential geometry. My colleague Hervé Jacquet likes to tell about how Chevalley at one point made an effort to do so, with the peculiar starting point of defining things in terms of “cubes”. I wasn’t sure whether to believe him, but here it is. According to Borel, in 1957 Grothendieck presented the group with his own take on the question of manifolds:

Grothendieck lost no time and presented to the next Congress, about three months later, two drafts:

Chap. 0: Preliminaries to the book on manifolds. Categories of manifolds, 98 pages

Chap. I: Differentiable manifolds, The differential formalism, 164 pages

and warned that much more algebra would be needed, e.g., hyperalgebras. As was often the case with Grothendieck’s papers, they were at points discouragingly general, but at others rich in ideas and insights. However, it was rather clear that if we followed that route, we would be bogged down with foundations for many years, with a very uncertain outcome.

I don’t see these documents on the list, perhaps documents from the later years are still to appear.

The documents often start out with some unvarnished comments, here’s an example, from Chevalley’s report on a text about semi-simple Lie algebras:

Au moment d’écrire ces observations, je me demande si ce ramassis des méthodes les plus éculées et les plus pisseuses, ces résultats les moins généraux possibles établis de la manière la plus incompréhensible possible, ne sont pas un canular intrabourbachique monté par le rédacteur. Même s’il en est ainsi, je me laisse prendre au canular et présente les observations suivantes.

Unlike physics, mathematics has managed to remain immune from efforts to promote pseudo-scientific agendas, financed with the goal of mixing up science and religion. I don’t see any reason to believe this is going to change, but I just noticed that the Templeton foundation is funding a program here in New York later this month on the topic of Mathematics and Religion.

The program will take place at the Philoctetes Center, which is run out of a townhouse on the Upper East Side and supports a variety of activities that you can read about here. The organization ran into serious trouble with its funding recently since its investments were managed by Bernard Madoff. A year before the scandal broke, Philoctetes sponsored a panel discussion (accessible here) on The Future of the Stock Market, which featured Madoff as a panelist. Because of these losses, the Center has had to look for funding elsewhere, and has found some from the Templeton Foundation.

One notable thing about the Mathematics and Religion panel is that it doesn’t include much at all in the way of mathematicians. Of the six participants, one is Max Tegmark, a physicist prominently involved in Templeton-funded multiverse studies, but the only mathematician is Edward Nelson. Nelson is quite far from the mainstream of mathematics, with a religion-infused recent paper entitled Warning signs of a possible collapse of contemporary mathematics, available here. Unlike the case of multiverse pseudo-science, which has drawn support from leading figures in the physics community, this sort of point of view about mathematics has attracted zero interest among mathematicians.

The Mathematics and Religion panel isn’t any threat to mathematics, and is part of a larger and much more worthy program about mathematics at Philoctetes funded by Templeon. In November there will be a panel discussion on Mathematics and Beauty that sounds interesting, I might even try and make it over there to see it (last year I did attend a talk at Philoctetes given by Barry Mazur). The Mathematics and Religion panel is associated with something more serious, a talk by Loren Graham on his book Naming Infinity. It’s a book I read earlier this year, but don’t think I ever got around to writing about here on the blog. I wasn’t completely convinced by some of the claims it makes about the relation between religious practices and the work of certain Russian mathematicians. The story it tells about the religious sect of “Name Worshipers” and the history it recounts of one part of the Russian mathematical community are quite fascinating.