Quantum curves as quantum distributions

TL;DR

This paper introduces a novel quantum distribution framework for mirror curves in topological string theory, connecting quantum and classical geometries through the Wigner transform of a Fermi gas model.

Contribution

It defines quantum mirror curves as quantum distributions derived from the Fermi gas approach, bridging quantum and classical mirror geometries in topological string theory.

Findings

Quantum mirror curves are modeled as quantum distributions on phase space.

Classical mirror geometry emerges in the large N, strong coupling limit.

Quantum fluctuations are described by an improved universal scaling form.

Abstract

Topological strings on toric Calabi--Yau threefolds can be defined non-perturbatively in terms of a free Fermi gas of N particles. Using this approach, we propose a definition of quantum mirror curves as quantum distributions on phase space. The quantum distribution is obtained as the Wigner transform of the reduced density matrix of the Fermi gas. We show that the classical mirror geometry emerges in the strongly coupled, large N limit in which hbar ~ N. In this limit, the Fermi gas has effectively zero temperature, and the Wigner distribution becomes sharply supported on the interior of the classical mirror curve. The quantum fluctuations around the classical limit turn out to be captured by an improved version of the universal scaling form of Balazs and Zipfel.