Quantum Topology in Hanoi

I recently got back from an interesting trip to Vietnam, where I attended “Quantum Topology in Hanoi“. This conference was held at “VAST”, the Vietnamese Academy of Science and Technology, and organised by Thang Le and Stavros Garoufalidis of Georgia Tech.

Being in Vietnam was great fun, and most participants (including at least one of our readers) enjoyed the crazy bustle of life in Hanoi. Even the afternoon when the power went out, the backup generator failed, the airconditioning was off, and Dylan Thurston talked about the combinatorial model for knot Floer homology in 34°C wasn’t so bad. :-) After the conference, I went to Halong Bay with Dylan and Jana for the weekend, and then up to Sapa for two more days.

My talk was about the “lasagna operad” acting on Khovanov homology, and I decided to try Dror’s ‘single page handout’ approach, and even more adventurously, his freshly made ‘javascript handout browser’. You can see my slide, and Dror’s programming, here.

But what I’d like to talk about now is Nathan Geer’s talk, about “fake quantum dimensions”. I’ll start somewhere near the beginning, reminding you how to build tangle invariants out of braided tensor categories, then explain what goes wrong when quantum dimensions become zero, and finally what Nathan and co. propose to do about it.

Given a braided tensor category with duals \mathcal{C}, we can take a tangle diagram, label its strands by objects in the category, and produce a morphism in the category. Even better, the morphism doesn’t really depend on the tangle diagram, just the tangle up to isotopy. We can express this as “there’s a functor from the category of \mathcal{C}-labelled braids to \mathcal{C}“. This functor, usually called the Reshetikhin-Turaev functor, is defined, of course, using the structural morphisms in \mathcal{C} coming from the definitions of “braided” and “with duals”. The idea is to slice up ‘time’ in the tangle into short enough periods that all we see are caps, cups, and crossings, and we define the functor on these as the appropriate pairing, copairing and braiding morphisms in \mathcal{C}.

This is the way we produce many quantum knot invariants — a knot labelled by some object of the category becomes an element of the ring K=\text{End}_{\mathcal{C}}(e, e), where e is the tensor identity. (Actually, why is it a ring? Perhaps I should have asked that \mathcal{C} have abelian groups for Hom spaces.) If we start with a knot, we have to label each component by some object, but we still get the same sort of answer. In many cases, like the categories coming from quantum groups, we can identify this ring K as something nice, like \mathbb{Z}[q,q^{-1}], in which case our knot invariants are familiar polynomials. The Jones polynomial arises in this way from the category of representations of U_q(\mathfrak{sl}_2), labelling the knot by the standard representation. (The coloured Jones polynomials just use a different label — one of the \mathbb{N} worth of irreps of \mathfrak{sl}_2.)

Let’s now think about a slightly different way to calculate the same number. Suppose we cut open our link into a tangle with two endpoints. Our functor into the braided monoidal category turns this into an endomorphism of the label object, V, and, supposing that object is simple, we get some multiple of the identity morphism. (What sort of multiple? The endomorphism ring \text{End}_{\mathcal{C}}(V, V) of any object V is certainly a K-module, and asking that the object be simple says the endomorphism ring is exactly K, and the multiple is just some element of K.) How is this number related to the result we got for the closed link? It’s not hard to see (and you should go and see it!) that the closed invariant is the quantum dimension of V times the open invariant. (The quantum dimension of V is the value, in K, of an unknot labelled by V.)

Why go to all this effort, thinking about both the closed and open invariants of a link? As soon as we leave the familiar (?) world of quantum groups at generic q, crossing the wall to the mysterious realm of roots of unity, we start to come across objects with quantum dimension zero. As explained above, if we label (a component of) a link by an object with zero quantum dimension, the closed quantum invariant is automatically zero. On the other hand, cutting the link at some point and looking at the open invariant, we often find that this invariant is nontrivial!

We’re immediately faced with the problem, however, of relating the different open invariants we get when cutting the link at different places. When all the quantum dimensions are nonzero, this is easy: two open invariants are proportional, with a factor given by the ratio of the quantum dimensions of the labels at the points we’re cutting. Nathan’s idea is to define a “fake quantum dimension” for each object, d(V), in such a way that we can define a “fake quantum invariant” by cutting at some point, taking the open invariant, and multiplying by the fake quantum dimension. Not only will this give us something well defined, but apparently it reproduces various previously known invariants.

To this end, Nathan Geer (with coauthors Bertrand Patureau-Mirand and V. Turaev) define a K valued function S'(V,W) on pairs of objects in \mathcal{C}, given by the evaluation of (yay fixed-width art!)

      |  V
      |
    --|--
   /  |  %
  /   |   %  W
  |   |   |
  %   |   /
   %  |  /
    -----
      |
      |

(i.e., which multiple of the identity on V is this?). It might be useful for some readers to point out that they’re calling this S', not S, because S is already taken for the closely related S-matrix! The S-matrix is the evaluation of the Hopf link labelled by V and W — namely the closure of the diagram above. They then fix an object V_0, with a special property called “ambidexterity” which I’ll explain below, and define the “fake quantum dimension” of an object V by the formula

d(V) = \frac{S'(V_0, V)}{S'(V, V_0)}.

(Actually, we have to be a little careful here. Let’s just define \mathcal{A} to be the set of objects V for which this makes sense, i.e. S'(V, V_0)\neq 0, and only define the fake quantum dimension for those objects. In the final examples, however, we’ll see that \mathcal{A} is big enough to be interesting.)

If both V and V_0 have non-zero quantum dimension, then each factor of S' above is just the evaluation of the $V, V_0$ Hopf-link divided by either \text{qdim}(V_0) or \text{qdim}(V), and the fake quantum dimension is d(V) = \frac{\text{qdim}(V)}{\text{qdim}(V_0)}. Thus it’s just the usual quantum dimension, scaled by some constant associated with the V_0 we’ve fixed. (Oh, and notice that the fake quantum dimension now lives in the field of fractions of K, not just K.)

What’s this ambidexterity property? Suppose we have some endomorphism f:V \otimes V \rightarrow V \otimes V. There are two ways we can make an endomorphism of V out of this — ‘closing up’ a pair of endpoints on the left, or ‘closing up’ a pair of endpoint on the right. (You could think of these as ‘partial right trace’ and ‘partial left trace’, or ‘partial expectations’, if any of those words sound familiar.) An object V is ‘ambidexterous’ if for every such endomorphism f of V \otimes V, the two resulting endomorphisms of V are equal. That is,

       |  ___         ___  |
       | /   %       /   % |
       | |   |       |   | |
     +-----+ |       | +-----+
     |  f  | |   =   | |  f  |
     +-----+ |       | +-----+
       | |   |       |   | |
       | %   /       %   / |
       |  ---         ---  |

This property isn’t as unlikely as it might first seem. In fact, if when the quantum dimension of V is nonzero, it’s automatic! Moreover, if every f:V \otimes V \rightarrow V \otimes V commutes with the braiding map for V, it’s ambidexterous too.

Finally now, we can define their new invariant. Instead of the usual Reshetikhin-Turaev invariant, define \mathcal{F}', on links with at least one component coloured by an object in \mathcal{A}, by cutting open anywhere coloured from $\mathcal{A}$, seeing what multiple of the identity we have, then multiplying by the fake quantum dimension of the label where we cut.

It might seem like not much has changed, but in fact it has. First of all, their theorem is that this is well-defined; it only depends on the link up to isotopy, and doesn’t depend on where you cut. (The proof of this theorem is really easy. It’s just a single line of calculations with diagrams, but unless anyone is actually still reading, and wants to know about it, I might not embarrass myself with further ASCII art.) Further, it seems that it’s actually interesting, unlike what you get if you use the normal quantum dimension instead.

To back this up, I’ll just describe two of the examples Nathan gave. In both, we see previously known invariants, but appearing as special cases of this unified construction. I don’t know enough about the examples to do more than quote my notes from his talk — but perhaps Nathan will come by Santa Barbara someday and explain the details to me.

1) In the representation category of U_q(\mathfrak{g}), where \mathfrak{g} is a Lie superalgebra of type I, you can take any ‘typical’ object as your ambidextrous V_0, and then \mathcal{A} = \{ \text{all typicals} \}. From this you can recover the multivariable Alexander polynomial of a link.

2) In U_q(\mathfrak{sl}_2), with q a root of unity, you can take V_0 a nilpotent representation, in which case \mathcal{A} = \{ \text{all nilpotents} \}, and the “fake quantum invariant” contains the ADO (Akulsu-Deguchi-Ohtsuki) invariant, which in turn contains the Kashaev invariant.

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