Low Dimensional Topology

January 20, 2011

Newsflash: Witten’s new preprint

Filed under: Knot theory,Quantum topology — dmoskovich @ 12:03 pm

A couple of days ago, Edward Witten uploaded a preprint titled Fivebranes and Knots. Based on Witten’s record on such topics, and on a preliminary visual scan of the introduction, it would not be unreasonable to surmise that this preprint could change history. Khovanov homology will never look the same again.

In 1989 Witten published a paper titled Quantum Field Theory and the Jones polynomial. In this paper, the Jones polynomial was put into a physics context of Chern-Simons theory on the three-sphere with gauge group SU(2). The physical perspective is conceptually clearer than the mathematical one, if one ignores the fact that the analytic objects under discussion (path integrals) don’t actually exist; and physics makes the Jones polynomial come to seem a natural mathematical object, whose topological invariance is manifestly obvious. Quantum topology has ever since been dominated by the quest to somehow approximate the generality and physical perfection of Witten’s Theory. And yet, all we really know how to rigourize mathematically is the perturbative expansion of Witten’s invariant around the trivial flat connection; so the only quantum invariants we really understand are quantum invariants of homology 3-spheres.
Now Witten has done the same thing to Khovanov Homology. Khovanov homology now has a conceptually clear physics explanation; and the physics will make predictions about it which are far ahead of anything that we can handle with today’s mathematical tools.
Hear Witten explain his ideas at a UCSB talk in Summer of 2010. A new goal has been set in Khovanov Homology; and I think the Khovanov homologists henceforth will find themselves fruitfully employed unravelling Witten’s more recent mathematical mysteries, and gradually rigourizing and mathematically understanding various segments in Witten’s new construction.
This is trully a momentous occasion for knot theory!

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  1. It should be noted that there is an important conceptual difference between this paper and the original 1989 Jones polynomial paper. Whereas the “non-rigorous” definition in the original ’89 paper relied on a Feynman path integral, and was therefore non-rigorous in some (apparently) fundamentally difficult way, the definition from the new paper is much more like a Floer theory (“count solutions to a certain, possibly well-behave differential equation,”) and therefore might be much easier to put on firm, rigorous ground (at the very least, superficially similar set-ups, such as instanton Floer homology, are rigorous, although the analysis is subtle and difficult). I hope some of the experts in gauge theory and Floer theory take note, and give us their first impressions!

    Comment by Sam Lewallen — January 20, 2011 @ 4:10 pm | Reply

  2. I bet Kronheimer and Mrowka are already putting the finishing touches on a paper making this stuff rigorous :)

    Comment by Ian Agol — February 4, 2011 @ 8:56 pm | Reply

  3. Have there been any expository accounts of the mathematics in Witten’s paper yet?

    Comment by Tom Leness — February 9, 2011 @ 12:27 pm | Reply

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