A hunka hunka burnin’ knot homology

One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?

This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.

At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong).

Let me give a little background: As many readers will know, there is an invariant of knots attached to every representation of a simple complex Lie group, of which the Jones polynomial is a special case for the defining representation of SL(2,\mathbb{C}). So, as knot homologies have appeared over recent years categorifying the Jones polynomial, and other special cases of these invariants, like the defining representations of SL(n,\mathbb{C}) and SO(2n,\mathbb{C}) (both work of Khovanov and Rozansky), some of us (I know I’m not the only one) have started to wonder if this isn’t a bunch of special cases of a construction for all representations. I believe this is one of things Khovanov and Lauda had in mind when they wrote down a categorification of quantum groups (and is an idea that goes back at least as far as Crane and Frenkel).

For a while, the idea has been floating around that one might be able to do this using the geometry of quiver varieties. I’ve been thinking about this idea for about 3 years, and I’m sure I was not the first (I know Hao Zheng has also had it in mind for while). But geometry is hard, so for a long time various problems and confusions stood in the way of me (or anyone else, it seems) actually carrying it through.

Luckily, other people were around to remove the geometry from the picture and somehow cook things down into some combinatorics that you can actually get a handle on. Specifically, there’s the recent work of Khovanov and Lauda, which is essentially a combinatorial rewrite of Lusztig’s geometric theory of canonical bases without a quiver variety or perverse sheaf in sight. Once I actually understood this fact (it took a while to sink in), I knew exactly what I needed to do: take all the intuitions I had about the geometry of quiver varieties, and convert them into the same kind of combinatorics.

The result is this paper, preliminarily titled “Knot invariants and higher representation theory.” It constructs a bigraded vector space for each oriented link with components labeled by representations of a complex simple Lie group G, for any group and any representations, whose Euler characteristic is the corresponding quantum invariant of that link. Better yet, it proposes a functoriality map for cobordisms between knots (but does not prove that this doesn’t depend on the handle decomposition of the cobordism. That will not be a fun thing to prove). The method is all combinatorics and combinatorial representation theory, with geometry only peeking through at moments; computations will not be easy, but they are finite and only involve a lot of taking projective resolutions and tensoring modules.

I know there are a reasonable number of blog readers who will be interested, so I thought I would share with you early in the process and see what happens; there may still be mistakes in the paper (though I am pretty confident at this point that all mistakes will be minor and fixable). I’d love to hear any comments people have on the paper (even if you don’t get very far, it could help the exposition to know where people get stuck). It may not be as ambitious as polymath, but I like to think of it as a minor step toward a more open, internet-based mathematics.

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19 thoughts on “A hunka hunka burnin’ knot homology

  1. Any chance of explaining what all of this means for some small example? Say for the adjoint representation of sl_3. Or even just for colored Jones polynomials (though my recollection was that those were pretty easy to understand already).

  2. Nice. You need a few \sloppypar s here and there.

    There is the standard reminder: the CS93 movie moves were subsumed by the
    CRS movie moves Adv. Math. V127, No1. In the latter a height function is assumed in the stills. It was the latter that allowed Baez and Langford, HDA IV to fix the gap in Fisher’s thesis. You can get in trouble with only the CS93 moves if you are not careful.

  3. Nice. You need a few \sloppypar s here and there.

    There is the standard reminder: the CS93 movie moves were subsumed by the
    CRS movie moves Adv. Math. V127, No1. In the latter a height function is assumed in the stills. It was the latter that allowed Baez and Langford, HDA IV to fix the gap in Fisher’s thesis. You can get in trouble with only the CS93 moves if you are not careful.

  4. @2: Sadly, no. The Alexander polynomial is a quantum invariant for the super algebra \mathfrak{gl}(1|1), which just doesn’t fit into this picture (yet, I hope). There’s no clear way how to make this construction work for super-algebras.

  5. Noah,

    I plan to write an actual post on this, but it will be a couple of more days (forgive me; I have the swine flu). In the meantime, you can take some solace in a preview version of my slides for a general talk I’m giving in a week. In particular, look at slides 12 and 25.

  6. I’m having a little trouble reconciling your rough statement of the cobordism hypothesis in the slides with what I’ve read. In particular, I don’t think there should be any claims of universality with respect to all invariants of n-manifolds. If you restrict to the class of invariants that come from extended TQFTs, you come closer to the formal statement, which is about \text{Bord}_n^\text{fr} being freely generated by a fully dualizable object.

  7. I agree with Scott. In fact your statement that a TFT is determined by its value on lower levels is (almost always) false if you are not in the framed case. Here is an easy example: 1D unoriented theories.

    The value of a point is then just a vector space, no additional data. However, the rest of the unoriented theory equips this vector space with a non-degenerate symmetric inner product. This is not determined by the vector space alone.

  8. In my understanding Jacob’s cobordism hypothesis theorem DOES say that the invariant attached to the point determines all the higher invariants – however you cannot assign anything you want to the point. For an n-dimensional framed theory you need to assign a fully dualizable object, and the meaning of the term “fully dualizable” depends on n, and gets increasingly hard to satisfy as n grows..

    If you would like an n-dimensional unoriented theory, as in Chris’ example above, the object you assign to the point has to be a fixed point for the O(n)- action on fully dualizable objects that Jacob constructs using the framed case of the theorem. In the 1d case, this O(1) action on dualizable objects takes every object to its dual, and an O(1) fixed point is indeed a vector space with a nondegenerate symmetric inner product. For an oriented theory you need an SO(n)-fixed point, which for n=1 is nothing but for n=2 ends up meaning a Calabi-Yau category (in the case the target 2-category is that of categories).

    In fact Jacob says something much more general (though I think he also said this is a formal consequence of the original framed theorem, given the right machinery): if you want a theory that takes values on manifolds equipped with a G-structure, for G any group mapping to O(n), you need to assign to the point a G-fixed point in dualizable objects in your category (with G acting through O(n)). This beautifully includes all the above plus for example manifolds with maps (up to homotopy) to some auxiliary (connected) space X – here we take G to be the loops in X (mapping trivially to O(n)), so that a reduction of the structure group of the manifold to G involves a map to BG=X. So such theories are classified by X-families of fully dualizable objects! (I may have messed up the details here, but I think that’s the general gist).

  9. Right. David knows this already, but I just wanted to point out a few things, since other people might be listening in.

    There is an important subtlety of Jacob’s theorem in the case of manifolds with G-structure which is easy to confuse. The general version of his theorem about TFTs does not say that they are the G-fixed points for the G-action on fully dualizable objects, but rather they are the HOMOTOPY fixed points. This is very important because a homotopy fixed is not just a property. It is additional structure. Depending on G, this additional structure is often encoded in the higher dimensional portion of the field theory.

    You can see this in the 1D case. There is no property of vector spaces which automatically endows them with an inner product.

    In fact the notion of an “G-action” on the fully dualizable objects is somewhat misleading. It only exists in the homotopy sense. The notion of fixed point in this context then necessarily involves additional (higher) structures.

    This is even more dramatic in the 2D oriented and unoriented cases.

  10. Of course everything I said is meant to be taken in the proper homotopy sense (as am I), and the notion of “property” is always a misleading one there. Anyway the issue here is semantic — I would say all the structure has to do with what you assign the point, which has to be a G-fixed object (and yes that’s a structure not a property), you would say that to even define the action of O(n) – hence of G – on fully dualizable objects one probably needs the full strength of the cobordism hypothesis, and so implicitly one is using higher dimensional manifolds, and in concrete cases you can spell this out concretely as higher dimensional structures.

  11. This has been a very informative discussion, but to clear the record, I should mention that I was mostly just picking nits about Ben’s use of the phrase “all invariants” since (as far as I know) extended TQFTs don’t give you a complete classification of homeomorphism or diffeomorphism classes of manifolds.

  12. By the way, I’ve put a new version on my website, if anyone is still paying attention. It’s currently in the “I kind of feel like it’s ready for the arXiv, but I bet I’ll find a bunch of typos if I wait a week” stage, so comments would really be appreciated.

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