One of the foundational papers in Quantum Topology, and one of the main reasons that the subject is called *Quantum Topology*, is Edward Witten’s landmark paper Quantum field theory and the Jones polynomial. One of the things Witten did in that paper was to define a –manifold invariant as a partition function with action functional proportional to the Chern-Simons –form. A partition function is a path integral, so Witten’s invariant is a physical construction rather than a mathematical one. Quantum topology of –manifolds is, to a large extent, the field whose goal is to mathematically reconstruct, and to understand, Witten’s invariant. Meanwhile, for –manifolds with a metric, Witten defined a –manifold invariant as a partition function in another landmark paper Topological quantum field theory.

I should warn you that I don’t know any physics so some (all?) of what I say below might be rubbish. Still, pressing boldly ahead…

Up until recently, mathematicians only understood tiny corners of Witten’s invariants, or, more broadly, of invariants (topological or otherwise) of manifolds (with or without extra structure) which come from quantum field theory partition functions. But I’ve recently glanced through two papers which seem to finally be going further, seeing more. The tiny corners we have seen already give rise to mathematical invariants of preternatural power (surely that’s the best word to describe it!), such as Ohtsuki series of rational homology –spheres (), Donaldson invariants, and Seiberg–Witten invariants.

The first paper is A unified quantum $SO(3)$ invariant for rational homology –spheres by A. Beliakova, I. Bühler, and T. Le. Beliakova and Le also wrote a survey paper explaining these and related results, called On the unification of quantum –manifold invariants.

Let’s first recall the background to this paper.

The Kontsevich invariant is the universal quantum invariant for links. Quantum link invariants lift to Witten–Reshetikhin–Turaev (WRT) invariants of –manifolds, while the Kontsevich integral lifts to the LMO invariant of –manifolds. It turns out to be a deep and difficult question whether the LMO invariant dominates the WRT invariants (surprisingly so- for links the corresponding fact is easy). For a semisimple Lie algebra, for a **prime** root of unity , and for a rational homology –sphere (), T. Ohtsuki famously showed that this is indeed the case by showing, roughly, that the perturbative expansion of the WRT invariants (the *Ohtsuki series*) coincides with the LMO invariant composed with the weight system, which is determined modulo by the WRT invariant at a th root of unity. As I mentioned in the introductory paragraph, and as shown by L. Rozansky, the Ohtsuki series is the contribution of the flat connection to Witten’s invariant, so this is the part of the story which mathematicians can be said to partially understand in any serious sense (forget that we can’t really calculate it outside the simplest examples- even the Kontsevich invariant for links can’t be calculated for any but the very simplest links). Anything beyond the Ohtsuki series would be new.

Ohtsuki’s techniques relied heavily on being a **prime** root of unity. What happens when is a **non-prime** root of unity? For integral homology –spheres () K. Habiro constructed an *unified WRT invariant* whose evaluation at any root of unity coincides with the value of the WRT invariant at that root, and, roughly, whose Taylor series is the Ohtsuki series. Habiro’s invariant is valued in the Habiro ring , which is a ring with all kinds of nice properties, which is related to all kinds of sexy ideas like the field with one element. The bottom line is that belonging to the Habiro ring implies that the WRT invariants are seen to be not just a random collection of algebraic integers, but rather as coming together to form “an analytic function on roots of unity”. This is a conceptual breakthough, and it solves our problem. One key property of analytic functions is that they are uniquely determined by their values at countably many points; so it is for , so it turns out that knowing that the LMO invariant dominates WRT invariants for prime roots of unity implies the corresponding statement for all roots of unity.

The Beliakova-Bühler-Le paper extends Habiro’s idea to s. Namely, with some effort, the authors construct a unified WRT invariant for a , with , containing a link coloured by odd numbers. This unified invariant dominates the WRT invariants at **all** roots of unity, and is valued in a modified Habiro ring . The case recovers Habiro’s .

Now here’s the point: this paper gives us new and interesting Ohtsuki series. This strongly points to the existence of a refined LMO invariant for s which captures more information from the Chern–Simons theory than just the contribution of the trivial connection, one mathematical step closer to the full preternatural power of the Chern–Simons quantum field theory invariant, as seen by physicists. For s there are non-flat connections which should be contributing, so this actually looks really interesting.

Now to counterbalance the hype, I’m going to start griping… this isn’t really a criticism of this paper at all, but just a personal opinion about this whole branch of mathematics:

- Nothing is calculated explicitly. Maybe nothing can be calculated explicitly except in completely degenerate cases. So (cynically speaking) what good is it?
- There is a lot of sophisticated algebra and number theory in the paper, but almost no topology. Quantum topology of –manifolds pays homage to topology via the Kirby Theorem, which translates everything into a statement about links (or tangles) modulo combinatorial moves, which in turn are viewed as morphisms in some category of representations, which makes everything into algebra, which you then fire super-heavy howitzers of representation theory and of number theory at. Philosophically, I think that quantum –manifold topology ought to be more about… well… topology!

The second paper is Instantons beyond topological theory I by E. Frenkel, A. Losev, and N. Nekrasov. This paper was discussed five years ago (when it was still a preprint) in a post in Not Even Wrong by Peter Woit, who knows immeasurably more about the subject than I do, and commented on by other experts; it was also the topic of an erotic film which won a Grand Festival Award at the Berkeley Film Festival.

This paper is actually the first of a trilogy, which propose a new perspective on the non-perturbative regime of quantum field theory (QFT). It only discusses quantum mechanical aspects (QFT is to be discussed later), which (not knowing any physics at all, really) I interpret to mean “-manifold invariants are to be constructed later”.

The study of correlation functions of BPS (or topological) observables in supersymmetric models of QFT is a traditional bridge between quantum gravity and mathematics in four dimensions. As discussed in the introductory paragraphs, these correlation functions give rise to important topological invariants such as Gromov–Witten and Donaldson invariants. This paper studies models (not supersymmetric per se) which have a topological sector, but they are also interested in the correlation functions of the non-BPS (non-topological, dynamical, or `off-shell’) observables. At a certain limit of the coupling constant (), they can solve the model, and, in two and in four dimensions (I suppose that we only really care about four), the Hamiltonians turn out not always to be diagonalizable and the theories turns out to be logarithmic conformal field theories.

If you didn’t understand the paragraph above then don’t worry- I’m not sure I properly understand the above paragraph myself. The key point for me is just that “quantum topology” might be seeing beyond topology into dynamics. To define a QFT you need a manifold with a metric, and then in topological quantum field theory (TQFT) the partition function is independent of the metric and you get topological invariants of manifolds. BPS observables give rise to powerful **topological** invariants of the underlying manifold (which is equipped with a metric). What we’re now seeing is non-BPS observables which we can “get at” mathematically, and which might allow us to construct powerful **dynamical** invariants of .

Quantum topology beyond topology… not to mention Equation 5.6, the formula for love.

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