# Wavefunctions, integrability, and open strings

## TL;DR

This paper provides evidence supporting a conjecture that quantum mirror curve eigenfunctions can be constructed from WKB expansions combined with open topological string wavefunctions, with explicit solutions in certain geometries.

## Contribution

It offers closed-form wavefunctions in maximally supersymmetric cases and links the conjecture to solutions of quantum Baxter equations in integrable systems.

## Key findings

- Closed expressions for wavefunctions in specific geometries.
- Conjecture connects wavefunctions to quantum Baxter equations.
- Validates the conjecture for various Planck constant values.

## Abstract

It has been recently conjectured that the exact eigenfunctions of quantum mirror curves can be obtained by combining their WKB expansion with the open topological string wavefunction. In this paper we give further evidence for this conjecture. We present closed expressions for the wavefunctions in the so-called maximally supersymmetric case, in various geometries. In the higher genus case, our conjecture provides a solution to the quantum Baxter equation of the corresponding cluster integrable system, and we argue that the quantization conditions of the integrable system follow from imposing appropriate asymptotic conditions on the wavefunction. We also present checks of the conjecture for general values of the Planck constant.