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According to the Baez-Dolan cobordism hypothesis, an extended TQFT is determined by its value on a single point. This value a fully dualizable object of a symmetric monoidal $n$ category (a fully dualizable object is a higher categorical analogue of a finite dimensional vector space). The Alexander polynomial is a quantum invariant, and comes from a TQFT.

How can an "Alexander polynomial" TQFT be put into an extended TQFT, and what is its value at a single point?

The question I just asked is closely related to this question. I also asked the question on the ldtopology blog here, and Theo Johnson-Freyd suggested that MO might be the place to ask it.


Briefly, I will summarize what an extended TQFT is. A TQFT as a symmetric monoidal functor Z:Cob(n)-> Vect(k) from the tensor category of $n-1$ dimensional manifolds and cobordisms between them to the tensor category of vector spaces over a field k. An extended TQFT is a symmetric monoidal functor Z:Cobk(n)->C from the n-category of cobordisms to a symmetric monoidal n-category C. I vote for the introduction to Lurie's expository account of his work proving the Baez-Dolan cobordism hypothesis as the best place to read about why extended TQFT's are natural objects, to understand their motivation, and to understand why people are so excited about them. An extended TQFT assigns a fully dualizable object to a point, and a higher “trace” on this object to a closed n-dimensional manifolds.

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It's worth sounding a note of caution here. Whenever anyone says that any statement involving the term "n-category" has been "proved", there's a caveat: it's far from clear what exactly "n-category" should mean. Many definitions have been proposed (including the one that Lurie uses), but there are thoroughly nontrivial outstanding questions about the relationships between them. So when a statement is proved using one proposed definition, it doesn't instantly follow that the statement is true for any of the other proposed definitions. –  Tom Leinster Feb 3 '10 at 2:44
    
I think for weak 2-categories we are basically OK though, no? So this question, at least, should be relatively unambiguous (I hope!) although it might have multiple answers... –  Daniel Moskovich Feb 3 '10 at 4:37
    
Yeah, 2's kinda OK, probably. At least, it's not socially controversial. Though if you really want to know whether two notions of 2-category are the same, you have to compare the 3-categories that they form --- and there your headache begins... –  Tom Leinster Feb 4 '10 at 3:44

2 Answers 2

up vote 4 down vote accepted

Understanding the Alexander polynomial can get a bit technical, because the quantum dimension is zero. So I don't think I fully understand how you get a TQFT from the Alexander polynomial.

In a more typical situation, like the Jones polynomial (related to SL(2) instead of the more confusing GL(1|1) in the Alexander setting), you have to be a bit careful about your target 3-category. For example, the usual Reshetikhin-Turaev TQFT doesn't come from a 0123 TQFT valued in tensor categories, because it is not the Drinfeld center of a semisimple category. (In this setting obstructions to extending to points are roughly given by the Witt group of Davydov, Mueger, Nikshych, and Ostrik.)

On the other hand, you can extend Reshetikhin-Turaev down to points if you use a different target category, like conformal nets (see http://arxiv.org/abs/0912.5307).

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Charles Frohman posted this link to my ldtopology blog post on the subject:<br> MR1158467 (93a:57007) Frohman, Charles; Nicas, Andrew The Alexander polynomial via topological quantum field theory. Differential geometry, global analysis, and topology (Halifax, NS, 1990), 27–40, CMS Conf. Proc., 12, Amer. Math. Soc., Providence, RI, 1991. They get an explicit TQFT from the Alexander polynomial, although the approach seems to suffer from various limitations (allow me to be vague because I don't fully understand everything about the construction)... –  Daniel Moskovich Feb 22 '10 at 8:16

I agree with Noah, and I disagree. Recent investigations into the notion of extended TQFT have led to a very rigid notion of what TQFT is. So rigid in fact, that almost nothing is a TQFT anymore, so the answer is as above: There is no answer.

However, maybe you could restate the question so that it does have an answer.

  1. Turaev torsion acts somewhat like a TQFT, and somewhat like the Alexander invariant. There are papers where people work out the Turaev torsion of manifolds with boundary and the associated gluing rules. Can you extend down to get an invariant that looks like a 2-extended TQFT? If so what is the category assigned to a one manifold. Can you build down to an invariant that satisfies some axioms that you could claim correspond to 3-extended TQFT? If so what are you assigning a to a 0-manifold?

  2. Heegaard Floer homology acts like the Alexander invariant, in fact there is a version that categorifies Turaev torsion. There are now bordered versions of Heegaard Floer. Can you keep extending these theories down? If so what do they assign to lower dimensional manifolds?

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