Spectral Theories and Topological Strings on del Pezzo Geometries

TL;DR

This paper explores the connection between spectral theories and topological strings on del Pezzo geometries by constructing quantum curves and analyzing their properties, revealing group symmetries and clarifying previous ambiguities.

Contribution

It explicitly constructs quantum curves for del Pezzo geometries and demonstrates how group symmetries clarify spectral and topological string correspondences.

Findings

Quantum curves are explicitly constructed for del Pezzo geometries.

Group symmetries clarify the shift of chemical potential and spectral operator factors.

Decoupling relations for BPS indices extend to quantum mirror maps.

Abstract

Motivated by understanding M2-branes, we propose to reformulate partition functions of M2-branes by quantum curves. Especially, we focus on the backgrounds of del Pezzo geometries, which enjoy Weyl group symmetries of exceptional algebras. We construct quantum curves explicitly and turn to the analysis of classical phase space areas and quantum mirror maps. We find that the group structure helps in clarifying previous subtleties, such as the shift of the chemical potential in the area and the identification of the overall factor of the spectral operator in the mirror map. We list the multiplicities characterizing the quantum mirror maps and find that the decoupling relation known for the BPS indices works for the mirror maps. As a result, with the group structure we can present explicitly the statement for the correspondence between spectral theories and topological strings on del Pezzo geometries.