# Low Dimensional Topology

## June 7, 2015

### Eynard-Orantin Theory enters Quantum Topology

Filed under: Uncategorized — dmoskovich @ 9:18 am

I’m now reading the following paper:

G. Borot and B. Eynard, All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, Quantum Topol. 6 (2015), 39-138.

In it, the authors apply the Eynard-Orantin topological recursion to conjecture an all-order asymptotic expansion of the coloured Jones polynomial of the complement of a hyperbolic knot, extending the volume conjecture.

To get an overview of Eynard-Orantin Theory, I’m looking at:

1. The original paper.
2. Eynard’s own overview– an expanded version of an ICM talk.
3. Some superb slides on the topic by Mulase.

Eynard-Orantin Theory studies a certain universal recursion formula based on a plane analytic curve (a Riemann surface) which Eynard calls the spectral curve’, the Cauchy differentiation kernel, and the residue calculus on it. Like quantum topology itself, the main question about Eynard-Orantin Theory is what it is exactly that it is computing.

The recursion associates to a Riemann surface with extra structure $\mathcal{S}$ a family $\omega_{n,g}$ of differential forms called invariants’. The initial terms $\omega_{0,1}$ and $\omega_{0,2}$ are canonically given, and remaining terms are defined by the universal recursive formula by `removing pairs of pants’.

Examples of such recursions had occured previously e.g. in work of Mirzakhani.

In low dimensional topology, the curve $\mathcal{S}$ is canonically associated with the A-polynomial. The form $\omega_{0,1}$ essentially determines the hyperbolic volume of the knot complement, while $\omega_{0,2}$ corresponds to the Reidemeister torsion.

Mirror symmetry relates these quantities with quantum invariants such as coloured Jones polynomials. Dijkgraaf, Fuji, and Manabe pointed out that mirror symmetry relates not only $\omega_{0,1}$ with the coloured Jones polynomial, but also the higher terms. They required them to add some ad-hoc terms which Borot-Eynard eliminate by chosing a different and perhaps more suitable wave function.

The topological recursion lives on the B-model side of mirror symmetry. To trully understand it, however, and to prove it in the cases of interest in low dimensional topology, the goal is to find an A-model proof of it.

More and better conjectures are a sure indicator for the vibrancy of the topic. Eynard-Orantin Theory is coming into play in many places in mathematics, and now it is coming to topology. Have the famous conjectures concerning asymptotics of quantum invariants at last met their match?