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October 3, 2013

Witten Looking Anew at the Jones Polynomial

Posted by David Corfield

Guest post by Bruce Bartlett

Today at the Clay Research Conference, Edward Witten gave a talk on A new look at the Jones polynomial of a knot. It was an opportune moment, 25 years after his original original paper.

Let me give a quick report-back. Hopefully the video and slides will be available on the Clay website at some point, but that may take some years!

The main result seems to be that, using the gradient flow lines of the analytically-continued Chern-Simons action, the coefficients b nb_n of the coloured Jones polynomial of a knot K 3K \subset \mathbb{R}^3,

(1)Z q(K;R)= nb nq n Z_q(K; R) = \sum_n b_n q^n

can be interpreted as the “number of solutions” to a certain 4-dimensional super Yang-Mills theory with given second Chern class nn. He is giving a second talk tomorrow morning on Khovanov homology and probably this formula forms part of a bigger picture.

This work of Witten can be seen as `part II’ of his 125 page paper on Analytic Continuation of Chern-Simons Theory from 2010 (see here or my own report here, or the video here).

More precisely, I believe Witten’s talk today is the follow-up to the closing paragraph of that paper, entitled “Four or Five-Dimensional Interpretation”. Those ideas seem to have crystallized more clearly now. The upshot is that the Jones polynomial for a knot in 3\mathbb{R}^3 (interpreted as a path integral over all connections on 3\mathbb{R}^3 of the holonomy around the knot) can be computed from a path integral of N=4N = 4 super Yang-Mills theory on X= 3× +X = \mathbb{R}^3 \times \mathbb{R}^+. Of course, to a mathematician this is still problematic since the right hand side is still a nonrigorous path integral! However, this statement can be made more cogent in two ways.

Firstly, the perturbative expansion of both sides can be computed and compared, and apparently they are the same. On the left hand side, this will be the Vassilliev invariants of Chern-Simons theory, while on the right hand side, it will presumably be some other perturbative expansion. I’m not sure how “deep” this particular statement is (that the two perturbative expansions are the same). No doubt though it is pretty cool.

Secondly, one can use “electric-magnetic duality” to compute the right hand side (the path integral of the 4-dimensional supersymmetric theory). This results in the statement that I gave at the top, that the coefficients b nb_n of the Jones polynomial

(2)Z q(K;R)= nb nq n Z_q(K; R) = \sum_n b_n q^n

can be interpreted as the “number of solutions of the 4d super Yang-Mills equations with given second Chern class nn”. In particular, this gives a long-awaited quantum field theory explanation to the question, “Why is the Jones polynomial a polynomial”?

Tomorrow we hear part II of this talk, relating this to Khovanov homology.

Posted at October 3, 2013 8:47 AM UTC

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Re: Witten Looking Anew at the Jones Polynomial

Witten gave his second talk on Khovanov homology and gauge theory. In fact, I understand now that his two talks were really an exposition of his recent past work on a gauge theory explanation for Khovanov homology and the Jones polynomial as contained in Analytic continuation of Chern-Simons theory and Khovanov homology and gauge theory. I just haven’t kept up (I’m still trying to understand his original paper :-)). It also contains elements of this paper on complexifying the path integral and this more technical paper on the details of the gauge theory realization of Khovanov homology.

So, it’s not new to the experts, but it was new and beautiful to me.

Posted by: Bruce Bartlett on October 3, 2013 10:30 AM | Permalink | Reply to this

Re: Witten Looking Anew at the Jones Polynomial

I haven’t kept up either. Thanks for posting this summary! I’m also at Oxford, but the internet connection in my hotel is too lousy for me to post much.

One cool thing is how this work interacts with his work on the Langlands program. For example, thanks to his use of electric-magnetic duality, he winds up relating the knot invariants for a simple Lie group GG to 4d super-Yang–Mills theory with the Langlands dual of GG as gauge group.

This is almost invisible for the Jones polynomial of knots in 3\mathbb{R}^3, since the Langlands dual of SU(2)SU(2) is SO(3)SO(3), and an SO(3)SO(3) connection on the simply-connected space 3× +\mathbb{R}^3 \times \mathbb{R}^+ can be identified with an SU(2)SU(2) connection. But it’s a bigger deal for other groups, whose Langlands duals are more different. More importantly, it hints that Khovanov cohomology and the Langlands program are part of some bigger, scarier, but maybe someday clearer story.

Posted by: John Baez on October 3, 2013 5:32 PM | Permalink | Reply to this

Re: Witten Looking Anew at the Jones Polynomial

I agree it’s completely fascinating! one such unifying structure (perhaps the one you have in mind) in physics that contains both geometric Langlands and Khovanov homology (and just about everything else I know about) is the mysterious but powerful 6-dimensional superconformal field theory “of type (2,0)” attached to an ADE Dynkin diagram, which some of us like to call “Theory X”. By considering reductions on various background geometries one can see a crazy amount of structure (eg the electric-magnetic duality both in Chern-Simons/Khovanov stories and in geometric Langlands amounts simply to switching two circles in an appropriate compactification of Theory X). It’s very science fiction-y and far from well-established physics (let alone rigorous math) but a wonderful guiding principle for those of us looking for unifying structures in geometric representation theory…

Posted by: David Ben-Zvi on October 5, 2013 1:15 AM | Permalink | Reply to this

Re: Witten Looking Anew at the Jones Polynomial

one such unifying structure (perhaps the one you have in mind) in physics that contains both geometric Langlands and Khovanov homology (and just about everything else I know about) is the mysterious but powerful 6-dimensional superconformal field theory “of type (2,0)”

But if all this falls out from just one (albeit a mysteriously powerful one) qft, which is just one element of a larger network of qfts, as in these tables, linked by a cascade of reductions, and if something like Weil’s Rosetta Stone story is correct, what should there be in the arithmetic column surrounding the ordinary Langlands correspondence?

To start with, what is the analogue of a qft in the arithmetic case? And could there be cascades of these analogues?

Posted by: David Corfield on October 6, 2013 10:22 AM | Permalink | Reply to this

Re: Witten Looking Anew at the Jones Polynomial

The slides for the talks are available here.

Posted by: Bruce Bartlett on October 4, 2013 10:02 AM | Permalink | Reply to this

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