• Corpus ID: 208139465

From topological recursion to wave functions and PDEs quantizing hyperelliptic curves

@article{Eynard2019FromTR,
  title={From topological recursion to wave functions and PDEs quantizing hyperelliptic curves},
  author={Bertrand Eynard and Elba Garcia-Failde},
  journal={arXiv: Mathematical Physics},
  year={2019}
}
Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles. These equations can be used to prove that the WKB solution of many isomonodromic systems… 

Quantum chaos in 2D gravity

We present a quantitative and fully non-perturbative description of the ergodic phase of quantum chaos in the setting of two-dimensional gravity. To this end we describe the doubly non-perturbative

Voros coefficients and the topological recursion for the hypergeometric differential equation of type (1, 4)

  • Yumiko Takei
  • Mathematics
    Integral Transforms and Special Functions
  • 2021
In my joint papers with Iwaki and Koike, we found an intriguing relation between the Voros coefficients in the exact WKB analysis and the free energy in the topological recursion introduced by Eynard

Quantization of Calogero-Painlevé System and Multi-Particle Quantum Painlevé Equations II-VI

We consider the isomonodromic formulation of the Calogero-Painlevé multi-particle systems and proceed to their canonical quantization. We then proceed to the quantum Hamiltonian reduction on a

Witten–Reshetikhin–Turaev Function for a Knot in Seifert Manifolds

In this paper, for a Seifert loop (i.e., a knot in a Seifert three-manifold), first we give a family of an explicit function $\Phi(q; N)$ whose special values at roots of unity are identified with

References

SHOWING 1-10 OF 25 REFERENCES

Geometry of Spectral Curves and All Order Dispersive Integrable System

We propose a definition for a Tau function and a spinor kernel (closely related to Baker{Akhiezer functions), where times parametrize slow (of order 1=N) deformations of an algebraic plane curve.

The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition

In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function,

Invariants of algebraic curves and topological expansion

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties.

All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials

We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity. Our

Quantizing Weierstrass

We study the connection between the Eynard-Orantin topological recursion and quantum curves for the family of genus one spectral curves given by the Weierstrass equation. We construct quantizations

Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP

To any solution of a linear system of differential equations, we associate a matrix kernel, correlators satisfying a set of loop equations, and in the presence of isomonodromic parameters, a Tau

Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: The sl2 case

In this paper, we show that it is always possible to deform a differential equation ∂xΨ(x) = L(x)Ψ(x) with L(x)∈sl2(C)(x) by introducing a small formal parameter ℏ in such a way that it satisfies the

Quantum Curve and the First Painlevé Equation

We show that the topological recursion for the (semi-classical) spectral curve of the first Painlev e equation PI gives a WKB solution for the isomonodromy problem for PI. In other words, the

Reconstructing WKB from topological recursion

We prove that the topological recursion reconstructs the WKB expansion of a quantum curve for all spectral curves whose Newton polygons have no interior point (and that are smooth as affine curves).