Spectral Theories and Topological Strings on del Pezzo Geometries

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.

63 pages, 22 eps figures; clarifications added and references added

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Cited by in corpus (8)

• Topological strings and Wilson loops
• Non-perturbative Tests of Duality Cascades in Three Dimensional Supersymmetric Gauge Theories
• Duality Cascades and Affine Weyl Groups
• E-strings, $F_4$, and $D_4$ triality
• D-type Minimal Conformal Matter: Quantum Curves, Elliptic Garnier Systems, and the 5d Descendants
• Duality Cascades and Parallelotopes
• 3d dualities with decoupled sectors and brane transitions
• Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case