Monthly Archives: July 2013

Follow-up: Physics and Mathematics of Link homology

For the past few weeks, I attended the SMS workshop “Physics and Mathematics of Link homologies” at the University of Montreal.  It was organized by Sergei Gukov, Mikhail Khovanov and Johannes Walcher.  The lectures were videotaped, although they don’t seem to have been posted to the website yet.  The main themes were: Chern-Simons theory, generalizations of the A-polynomial, categorification of homology theories, and the search for a unified picture.  In particular, how the Chern-Simons interpretation of link and 3-manifold invariants can lead to conjectures about the structure of these invariants.

Resources: If you’re interested in learning about the physics, here are some resources that were suggested to me over the course of the workshop

Lectures on knot homology and quantum curves, Sergei Gukov and Ingmar Saberi
Quantum Fields and Strings: A Course for Mathematicians, Deligne, Etinghof, Freed, Jeffrey, Kazhdan, Morgan, Morrison, Witten (collected lecture notes from a year-long series of talks in ’96-’97 at the IAS)
Conformal Field Theory and Topology, Toshitake Kohno
Chern-Simons Theory, Matrix Models, and Topological Strings, Marcos Marino
Mirror symmetry, Clay Mathematics monographs
Quantum Mechanics for Mathematicians, Leon Takhtajan
Supersymmetry and Supergravity, Julius Wess and Jonathan Bagger
Introduction to Superanalysis F.A. Berezin
Lecture notes on global analysis, JD Moore

Classic papers:
Quantum field theory and the Jones polynomial, Witten
Supersymmetry and Morse theory, Witten
The Yang-Mills equations over Riemann surfaces, Atiyah and Bott

(Note for those who attended the workshop: One of our discussion questions after-hours
was to write down a Math-Physics dictionary.  The IAS book has a great glossary at the beginning that does exactly this.

Similarly, another question was: What is the Wess-Zumino-Witten model?  The Kohno book has an involved, explicit description of this conformal field theory model.  It also elaborates on lots of the moments in Witten’s papers where he skips over what seem like crucial parts of the proof by appealing to “the general principles of quantum/conformal field theory”)

Again, if you have any other suggested resources, post a comment and I’ll update my list. I’ve added these to the resources page.

Moving forward,

The workshop had one great feature which I would like to continue here, as our target audience is grad students.

One of the participants pointed out that most of the talks could have been given anywhere from 10-25 years ago.   But one of the best parts of the conference was that the speakers mentioned lots of basic facts that, had you been around for the past two decades, you’d know, but that no one had ever mentioned to me.

For example, Gukov mentioned the hierarchy of U(1)-gauge theory equations:
dim 4: Seiberg-Witten
dim 3: Monopole
dim 2: Vortex
and that these are really just one set of equations (SW), modified by dimensional reduction.  I.e. The monopole equations describe t-invariant solutions to the SW-equations on Y^3 \times \mathbb{R}.  Further, if we assume Y^3 is just \Sigma \times \mathbb{R}, we get the vortex equations.

Presumably, I could have figured this out for myself after awhile, but mentioning it helped dissipate a lot of my confusion.

I suspect one reason for this phenomenon is that if you’ve been around for 20-30 years, then you’ve been saying these basic facts over and over again, and it just gets boring to repeat them.  But if you’re a grad student trying to make sense of the enormous, interconnected framework of mathematics motivated by these ideas, you weren’t around to get bored by hearing it ad nauseum and only showed up after everyone stopped mentioned it because they assumed everyone knows by now.  As a grad student, you may learn those bits integral to your area of specialization, but the big, intuitive picture is lost.

I’d like to return to the vortex equations here in the future, for several reasons (1) to get a feel for gauge theory in a lower-dimensional (n=2) setting, (2) to give intuition in terms of complex geometry for (a) the description the SW moduli on Kahler surfaces and (b) Taubes’ Gromov invariant, and (3) to give intuition for why Heegaard Floer homology, which ostensibly is given by the Lagrangian intersection Floer homology in \text{Sym}^g(\Sigma_g), is really an invariant arising from gauge theory and to motivate its equivalence to Monopole Floer and Embedded Contact Homology.

I’d also like to go back and blog some of the classic papers mentioned above, like Witten’s approach to the Jones polynomial.

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