Related papers: Quantum curves for Hitchin fibrations and the Eyna…
Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees $\mathcal{D}$-module on a smooth projective…
A geometric quantization using the topological recursion is established for the compactified cotangent bundle of a smooth projective curve of an arbitrary genus. In this quantization, the Hitchin spectral curve of a rank $2$ meromorphic…
The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been…
Prior works relating meromorphic Higgs bundles to topological recursion, in particular those of Dumitrescu-Mulase, have considered non-singular models that allow the recursion to be carried out on a smooth Riemann surface. We start from an…
This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key…
Kontsevich-Soibelman (2017) reformulated Eynard-Orantin topological recursion (2007) in terms of Airy structure which provides some geometrical insights into the relationship between the moduli space of curves and topological recursion. In…
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 of the spectral curve that annihilate the…
We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum…
The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin…
The Eynard-Orantin recursion formula provides an effective tool for certain enumeration problems in geometry. The formula requires a spectral curve and the recursion kernel. We present a uniform construction of the spectral curve and the…
For any (possibly singular) hyperelliptic curve, we give the definition of a hyperelliptic refined spectral curve and the hyperelliptic refined topological recursion, generalising the formulation for a special class of genus-zero curves by…
We formulate geometrically (without reference to physical models) a refined topological recursion applicable to genus zero curves of degree two, inspired by Chekhov-Eynard and Marchal, introducing new degrees of freedom in the process. For…
We give elements towards the classification of quantum Airy structures based on the $W(\mathfrak{gl}_r)$-algebras at self-dual level based on twisted modules of the Heisenberg VOA of $\mathfrak{gl}_r$ for twists by arbitrary elements of the…
We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve.…
We propose a string field Hamiltonian formalism that associates a class of spectral curves and provides their quantization through the Chekhov-Eynard-Orantin topological recursion. As illustrative examples, we present Hamiltonians for the…
We define higher quantum Airy structures as generalizations of the Kontsevich-Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of…
The spectral curve is the key ingredient in the modern theory of classical integrable systems. We develop a construction of the ``quantum spectral curve'' and argue that it takes the analogous structural and unifying role on the quantum…
We investigate the special K\"ahler geometry of the base of the Hitchin integrable system in terms of spectral curves and topological recursion. The Taylor expansion of the special K\"ahler metric about any point in the base may be computed…
The purpose of this paper is to give a twisted version of the Eynard-Orantin topological recursion by a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic definition for a topological recursion to…
We construct a degeneration of the moduli space of Hitchin pairs on smooth projective curves when the curve degenerates to an irreducible curve with a single node. The degeneration constructed here is analogous to the models constructed by…