The simplest quantum 3-manifold invariant is the Murakami-Ohtsuki-Okada (MOO) invariant. It comes from Chern-Simons theory in the way that the Reshetikhin-Turaev invariant comes from Chern-Simons Theory. It has a closed formula in terms of the order of the first cohomology class of the -manifold and an eighth root of unity. Witten’s Chern-Simons theory for gauge group shows that the MOO invariant can be reformulated in terms of classical Riemann theta functions with characteristic, but the relationship is by way of quantum field theory.

A recently published paper by Gelca and Uribe, which is also the topic of a book by Gelca and some nice slides, constructs the MOO invariant from theta functions completely classically essentially without using anything quantum at all (although the representation theory behind it was originally developed for quantum mechanical purposes). Thus, like the Alexander polynomial and the linking number, MOO is seen to be quantum but also classical.

There is also a more analytic, heat-equation-based way of seeing the same thing due to Andersen, but I haven’t read Andersen’s paper and therefore I can’t say anything about that.

Amongst the most useful functions in mathematics are the trigonometric functions. The function arises as a cross-section of the complex exponential function. It is periodic with period (I’ve become a proponent of tau) and it allows you to parametrize the points on a circle.

The formula for the theta function, as formulated by Jacobi, is

Theta functions can be thought of as sort of complex analogues of trigonometric functions. They are not doubly periodic (otherwise they would have to be constant by Liouville’s Theorem), but they are as close to being doubly periodic as possible, satisfying

where and are integers, and has positive imaginary part. Note that adding an integer to the -input of a theta function leaves it unchanged. Riemann taught us that we should think of such a function as a multi-valued function from the torus, that is the complex plane quotiented by .

Personally, I really like theta functions. In an alternative life I’d like to study theta functions. In another alternate life, I’d like to study K-theory. These are all things I think are really beautiful, but I never really sank myself properly into.

Anyway, the theta function is genuinely periodic in the integer direction. If we’re adding to the -argument, which is times the meridian of the torus and times its longitude, then we can throw away, contracting the meridian to a point. Geometrically, this is like viewing the torus as the boundary of a solid torus, and this is where handlebodies and later Heegaard splittings begin to enter the picture.

Riemann generalized theta functions to higher genera, and somebody (Mumford?) generalized them to depend on a natural number (in Riemann or Jacobi’s case ) to yield *theta functions with characteristic* so that adding times the meridian to gives identity. It is these Reimann theta functions with characteristic which are related to the MOO invariant.

Gelca and Uribe’s basic idea is to think of theta functions as oriented framed multicurves in handlebodies. These basically encode the curves around which the theta function is periodic. Thus, the torus meridian is the Jacobi theta function, the Jacobi theta function in characteristic is times the meridian, and in higher genus, a theta function is just an oriented multicurve which represents some element in a lagrangian subspace of the first homology of the surface.

Theta functions are naturally acted on by two groups, the Heisenberg group and the modular group (AKA the mapping class group of the surface).

Acting by the Heisenberg group multiplies the theta function by some exponential function, which can in turn be represented as an oriented multicurve representing an element of the **other** lagrangian. This is where the framing comes in- the theta function is periodic in the contractible direction, and the multiplicative factor is captured by its framing.

Because we only really care about the homology class of the oriented framed multicurve we can quotient by a load of relations and what we really have here is a skein module called the *reduced linking number skein module*.

This whole construction is wonderfully elucidated by Example 5.6 on Pages 240-241 of Gelca’s book, but all sources of this I can find are behind paywalls, I can’t be bothered to scan and then cut and paste, and I can’t be bothered to redraw it… I apologise.

The action of the mapping class group turns out to be more or less what you’d expect, as long as you remember that the it is the class of the multicurve in the skein module, not the multicurve itself, which represents the theta function. Namely, push the framed multicurve representing the theta function to the boundary of the handlebody in all possible ways, Dehn-twist the picture, and push the result back into the handlebody, and you get the correct theta function (times whatever multiplicative factor).

Now things get a bit more interesting. There’s an identity in the theory of theta functions called the exact Egorov identity, which turns out to say that handlesliding the multicurve representing the theta function (or rather a certain theta series) over the curve representing an element of the mapping class group gives the identity. In other words, invariance under handleslides.

And now for MOO. The reduced linking number skein module modulo the action of the mapping class group turns out to be isomorphic to the field of complex numbers. Viewing the theta function multicurve as a surgery link, such that surgery on by this link gives our manifold, the image of this link is the MOO invariant. It is invariant under handleslides by the exact Egorov identity. We have thus constructed the MOO invariant without a shred of quantum field theory.

I’ll conclude with some random thoughts:

- Yoshida has a paper in Annals proposing to reformulate the Reshetikhin-Turaev invariants in terms of “higher level theta functions”. Hansen and I wrote some notes on this paper… Teleman pointed out a gap. I’m currently doubtful that that such a simple formula has any chance to hold.
- I have wondered for a long time whether and in what sense -manifolds “exist”. Surfaces certainly “exist” because they arise inevitably from other objects which “exist”, for example as closed Riemann surfaces. But it’s never been clear to me that -manifolds “exist” in the same way, and reading Poincaré’s original papers, it wasn’t clear to Poincaré either.
At first I was excited about the Gelca-Uribe paper, because it seemed they were pulling –manifolds “out of a hat”, and that their Heegaard splittings were an inevitable consequence of the theory of theta functions. I no longer feel that this is the case. Their whole theory is really about skein modules, and can be reformulated in diagrammatic algebraic terms without reference to topology.

- Continuing on from that last thought, somebody ought to reformulate their paper completely diagrammatically! Doing so might provide new ways of thinking about aspects of the theory of theta functions, and so might be useful in a wider sense (I dream). For example, the exact Egorov identity corresponds to invariance under handleslides. But we know that handleslides are unzips of embedded theta graphs. So if embedded curves correspond to theta functions, what to embedded theta graphs correspond to? What are embedded trivalent graphs in analytic language? What are unzips?
- It would be cool to explain the whole idea in 2-3 pages with no reference to fancy concepts and formulae, and with nothing quantum in sight. The core idea (which I interpret as sort of a diagrammatic calculus for theta functions) is so simple and elegant that I believe it deserves to be better known.

Regarding your comment about whether or not 3-manifolds “exist”, I think I would phrase it a different way but I think there is much to say on this topic. When you look at geometrization of 3-manifolds I find the striking thing is it’s such a peculiar collection of mathematical objects. High-dimensional manifolds can be thought of as “fattenings” of CW-complexes. So they’re close to a generality. 1 and 2-manifolds are so specific that their “purpose” in mathematics is fairly fixed. But 3-manifolds are where the terrain is varied enough that one needs many perspectives to “see” them, and at least as of yet there’s no uniform perspective that allows you to see everything you might like to see in them.

But I think this is exactly the kind of thing that makes mathematics entertaining. One studies PDEs not because one hopes to have a single unifying perspective on the subject that answers every question efficiently. You tend to study particular families that are useful. 3-manifolds are of that variety of question. They’re a useful family, rather than a generality. They’re also not so small that one can quickly bludgeon the subject to death.

Comment by Ryan Budney — March 30, 2015 @ 9:59 am |

This is a great comment!

For me, what would “convince” me that 3-manifolds “exist” in some philosophical sense would be if one could find a strong tie-in with the main body of mathematics whose “existence” is somehow well-established. For links we sort-of have this via quantum invariants, which provide insights into number theory (multiple zetas etc), field with one element, Lie theory etc. and also somehow reflect quite concrete physical objects which one could almost hold in one’s hand. I don’t know of a parallel sense in which 3-manifolds tie in with the rest of mathematics except as examples; for example, 3-manifold quantum invariants tend to come from link invariants, and the tie-ins with the rest of math happen already on the level of links (e.g. the MOO invariant, as discussed in the post).

Comment by dmoskovich — March 30, 2015 @ 10:12 am |

It’s a little bit like the classification of finite simple groups. There are these nice “uniform” families and then there’s the sporadic groups that appear to come out of nowhere. You could view that as a defect in the families you generate finite groups with, or you could perhaps just be content that not every answer needs to be a polished algebraic machine. At least at present, 3-manifolds primarily arise naturally in fairly specific situations like embeddings of other manifolds in the 3-sphere.

Comment by Ryan Budney — March 30, 2015 @ 4:20 pm |

I think you’re saying that 3-manifolds might in some sense not “exist as a mathematical primitive” because there are several classes of objects which behave quite differently which come under the same roof, such as Seifert fibered spaces vs. hyperbolic manifolds. For me this is less of a concern, basically for the reason you state- that not every answer needs to be a polished algebraic machine.

I think that my objection might be disjoint: My main problem is that 3-manifolds aren’t popping up all over mathematics and in the sciences, in diverse contexts. That is what I would expect of a “mathematical primitive”- I would expect it to make veiled appearances all over the place. Perhaps Monsterous Moonshine and its cousins “justify” sporatic simple groups. They are clearly more than just artifacts of suboptimal definitions. But I don’t see a parallel for 3-manifolds- the only place they seem to appear is inside geometric topology.

To sharpen the philosophical objection further, there are several related objects which DO appear in other contexts, but they’re no longer “just” 3-manifolds. For example, there is a whole industry devoted to the analogy between 3-manifolds and number fields, which would remove my objection, except that the 3-manifolds are equipped with a flow. I found the answers here, and linked references, quite illuminating in this respect.

So my personal speculation would be that 3-manifolds could perhaps be thought of as a partial manifestation of some more “fundamental” structure, such as some sort of spacial algebra of the sort that Habiro and Asaeda have looked at… or maybe something completely different. I really don’t know… Dror Bar-Natan may have vaguely similar doubts when during talks he says phrases like, “I don’t know what a 3-manifold is”.

Comment by dmoskovich — April 12, 2015 @ 7:15 am |