The Laplace Transform and Quantum Curves

TL;DR

This paper introduces a Laplace transform linking topological recursion wavefunctions to their duals, enabling the derivation of quantum curves for complex spectral curves related to mirror symmetry and knot theory.

Contribution

It defines a novel Laplace transform that facilitates the construction of quantum curves for a broad class of spectral curves, including those previously inaccessible by TR methods.

Findings

Derived a formula for quantising spectral curves of a specific form

Reproduced known quantum curves from literature

Derived new quantum curves, including for Gromov-Witten theory of weighted projective spaces

Abstract

A Laplace transform that maps the topological recursion (TR) wavefunction to its $x$-$y$ swap dual is defined. This transform is then applied to the construction of quantum curves. General results are obtained, including a formula for the quantisation of many spectral curves of the form $e^xP_2(e^y) - P_1(e^y) = 0$ where $P_1$ and $P_2$ are coprime polynomials; an important class that contains interesting spectral curves related to mirror symmetry and knot theory that have, heretofore, evaded the general TR-based methods previously used to derive quantum curves. Quantum curves known in the literature are reproduced, and new quantum curves are derived. In particular, the quantum curve for the $T$-equivariant Gromov-Witten theory of $\mathbb{P}(a,b)$ is obtained.