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.
Highlights
Introduction and motivationIn this paper, we formulate explicitly the correspondence between spectral theories and topological strings on del Pezzo geometries using their group-theoretical structure
These spectral operators fall into the D5 quantum curves, which is consistent with the result that they are described by the free energy of topological strings on the D5 local del Pezzo geometry [29]
The partition functions reduce to matrix models
Summary
We formulate explicitly the correspondence between spectral theories and topological strings on del Pezzo geometries using their group-theoretical structure. Partition functions of the N = 4 supersymmetric Chern-Simons theories with gauge group U(N )k × U(N )0 × U(N )−k × U(N )0 and U(N )k × U(N )−k × U(N )k × U(N )−k were associated to spectral operators H = Q2P2 [28, 29] and H = QPQP [30] respectively These spectral operators fall into the D5 quantum curves, which is consistent with the result that they are described by the free energy of topological strings on the D5 local del Pezzo geometry [29]. It may seem at the first sight that the ST/TS correspondence is only relevant to M2branes at certain parameters. Two appendices are devoted respectively to the classical analysis of the phase space areas for the corresponding Fermi gas systems and a summary of the BPS indices
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