Khovanov Homology

At his talk last year at the conference in honor of Gelfand’s 90th birthday, Atiyah posed the question of whether there is a quantum field theoretic explanation of why the coefficients of the Jones polynomial are integers. Witten’s Chern-Simons-Witten theory is a 3d QFT that computes the Jones polynomial (a topological invariant of knots or links inside a 3d manifold), but gives no obvious reason the coefficients should be integral.

One thing about the Chern-Simons-Witten story that has always bothered me is that, unlike his other TQFTs, this one is not of a homological nature. In the other TQFTs, the Hilbert space is finite dimensional because there are fermionic variables which cause cancellations such that only the homology of some complex contributes to the observables. To make any real sense of the idea of a path integral whose Lagrangian is the Chern-Simons functional, one has to do something like add a Yang-Mills term, then take a limit. By doing this one can move all but a finite part of the usual gauge theory Hilbert space off to infinite energy. It would be very interesting if there were a version of the theory which instead worked homologically like other TQFTs.

A hot topic in low dimensional topology recently has been the notion of “Khovanov homology”, which associates to a knot a complex whose homology is the Jones polynomial. For an introduction to Khovanov homology, see papers by Dror Bar-Natan (a mathematician who was a student of Witten’s) or Jacob Rasmussen. Bar-Natan has a lot of other material about Khovanov homology on his web-site.

One way of answering Atiyah’s question would be to find a 4d TQFT whose Hilbert space is the Khovanov homology of the boundary. Maybe there is some sort of gauge-theory based QFT which generalizes the Chern-Simons-Witten theory and computes Khovanov homology. But after consulting the local expert on these things (Peter Ozsvath), it seems that no one knows whether it is even possible to reformulate Khovanov homology in any sort of gauge-theoretical terms. The only known definitions of it are kind of like the pre-Witten skein relation definitions of Jones polynomials. They are based on working with a projection of the knot onto two-dimensions.

A couple weeks ago Sergei Gukov gave a talk in the math department at UCSD with the title “Topological Invariants and Khovanov Homology”, and perhaps his work has some relation to the above speculations.

Gukov is also the co-author of a paper that just appeared on the arXiv entitled “Topological M-theory as Unification of Form Theories of Gravity”. Like M-theory itself, it appears that no one knows what “topological M-theory” is, but it is supposed to be some sort of seven-dimensional theory that is related to topological strings on 6d Calabi-Yaus in much the same way M-theory is a conjectural 11d theory related to 10d superstrings. Lubos Motl has even more questions about this than I do.