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. (more…)