A bird’s eye view of topological recursion

Eynard-Orantin Theory (topological recursion) has got to be one of the biggest ideas in quantum topology in recent years (see also HERE). Today I’d like to attempt a bird’s eye explanation of what all the excitement is about from the perspective of low-dimensional topology.

Arithmetic Chern-Simons

The most compelling aspect of Quantum Topology for me is its connection to analytic number theory. Today I’d like to draw your attention to recent work of Minhyong Kim on Arithmetic Chern-Simons Theory (see his paper for more details). I was fortunate to hear him give a talk about this last Wednesday and a colloquium talk on related subjects the day before. People have been talking about such “quantum topological number theory” for a long time- e.g. this 2010 MO question – but we haven’t seen much of an uptake so far. This isn’t an easy direction to pursue because one needs to know both quantum topology and analytic number theory, but I was left with the strong feeling that “There’s gold in them thar hills”, both for topologists and for number theorists.

Heisenberg-picture TQFT

This interesting-looking preprint has just appeared on ArXiv:

Theo Johnson-Freyd, Heisenberg-picture quantum field theory, arXiv:1508.05908

It argues for a different category-theoretical formalism for TQFT than the `Schroedinger-picture‘ Atiyah-Segal-type axiomatization that we are used to. The `Heisenberg-picture‘ functor it proposes has as its target a category whose top level is pointed vector spaces instead of numbers, and whose second to top level is associative algebras instead of vector spaces. The preprint argues that this formalism is better physically motivated, and one might dream that it is better-suited to analyze “semiclassical limit” conjectures such as the AJ conjecture and its variants.

I’m very happy to see this sort of playing-around with the foundations of TQFT, which I am happy to believe are too rigid. I expect there should be a useful Dirac picture also, and that there are other alternative axiomatizations also. Let’s see where this all leads!

Recent coloured HOMFLYPT-related stuff

One of the main ways in which I keep my finger on the pulse of what is hot now in low dimensional topology is to write lots and lots of reviews, both for Zentralblatt MATH and also for MathSciNet. In the last year or so, what has been increasingly coming through the pipe is papers about knot homology and mirror symmetry. There seems to be a lot happening in this field right now.

MOO is classical

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.

Jones’s new polynomial

Check out this exciting new preprint by Vaughan Jones!

V.F.R. Jones, Some Unitary Representations of Thompson’s Groups and , arXiv:1412.7740.

Understanding the anomaly

I’ve recently been looking at the following paper in which -TQFT anomalies are treated carefully and various old constructions of Turaev and Walker are elucidated:

Gilmer, P.M. and Masbaum, G., Maslov Index, Mapping Class Groups, and TQFT, Forum Math. 25 (2013), 1067-1106.

It makes me think a lot about just what the anomaly `actually means’…

A celebration of diagrammatic algebra

Relaxing from my forays into information and computation, I’ve recently been glancing through my mathematical sibling Kenta Okazaki’s thesis, published as:

K. Okazaki, The state sum invariant of 3–manifolds constructed from the linear skein.
Algebraic & Geometric Topology 13 (2013) 3469–3536.

It’s a wonderful piece of diagrammatic algebra, and I’d like to tell you a bit about it!

Tangle Machines- Positioning claim

Avishy Carmi and I are in the process of finalizing a preprint on what we call “tangle machines”, which are knot-like objects which store and process information. Topologically, these roughly correspond to embedded rack-coloured networks of 2-spheres connected by line segments. Tangle machines aren’t classical knots, or 2-knots, or knotted handlebodies, or virtual knots, or even w-knot. They’re a new object of study which I would like to market.

Below is my marketing strategy.

My positioning claim is:

• Tangle machines blaze a trail to information topology.

My three supporting points are:

1. Tangle machines pre-exist in a the sense of Plato. If you look at a knot from the perspective of information theory, you are inevitably led to their definition.
2. Tangle machines are interesting mathematical objects with rich algebraic structure which present a plethora of new and interesting questions with information theoretic content.
3. Tangle machines provide a language in which one might model “real-world” classical and quantum interacting processes in a new and useful way.

Next post, I’ll introduce tangle machines. Right now, I’d like to preface the discussion with a content-free pseudo-philosophical rant, which argues that different approaches to knot theory give rise to different `most natural’ objects of study.

The next big thing in quantum topology?

The place to be in May for a quantum topologist is Vietnam. After some wonderful-sounding mini-courses in Hanoi, the party with move to Nha Trang (dream place to visit) for a quantum topology conference.

I’d like to tell you very briefly about some exciting developments which I expect will be at the centre of the Nha Trang conference, and which I expect may significantly effect the landscape in quantum topology. The preprint in question is -Efficient triangulations and the index of a cusped hyperbolic -manifold by Garoufalidis, Hodgson, Rubinstein, and Segerman (with a list of authors like that, you know it’s got to be good!).