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.