Spectral theories and topological strings on del Pezzo geometries
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Springer
Received: August 4, 2020 Accepted: September 14, 2020 Published: October 23, 2020
Sanefumi Moriyama Department of Physics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan Osaka City University Advanced Mathematical Institute (OCAMI), 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan
E-mail: [email protected] 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. Keywords: M-Theory, Matrix Models, Nonperturbative Effects, Topological Strings ArXiv ePrint: 2007.05148
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)154
JHEP10(2020)154
Spectral theories and topological strings on del Pezzo geometries
Contents 1 Introduction and motivation
1 5 6 7 7 9 11 12
3 Reviews: D5 curve
15
4 E6 curve 4.1 Quantum curve 4.2 Weyl group 4.3 Mirror map
19 19 23 25
5 E7 curve 5.1 Quantum curve 5.2 Weyl group 5.3 Mirror map
27 27 30 32
6 E8 curve 6.1 Quantum curve 6.2 Weyl group 6.3 Mirror map
33 33 37 39
7 Observation
40
8 Conclusion and discussions
41
A Phase space area A.1 D5 curve A.2 E6 curve A.3 E7 curve A.4 E8 curve
43 43 45 46 47
B BPS indices
49
–i–
JHEP10(2020)154
2 Summaries: ST/TS correspondence 2.1 Spectral theories 2.2 Topological strings 2.2.1 Perturbations 2.2.2 Mirror map 2.2.3 Non-perturbative effects 2.3 Weyl groups
1
Introduction and motivation
is reminiscent of the inverse transformation from the grand canonical ensemble Z dµ Jk (µ)−N µ Zk (N ) = e , 2πi
(1.2)
the result immediately suggests us to move our analysis to the reduced grand potential Jk (µ) ' Cµ3 /3 [10]. Here we define the grand partition function Ξk (z) and the reduced grand potential Jk (µ) by Ξk (z) =
∞ X
z N Zk (N ),
Ξk (eµ ) =
∞ X n=−∞
N =0
–1–
eJk (µ+2πin) .
(1.3)
JHEP10(2020)154
In this paper, we formulate explicitly the correspondence between spectral theories and topological strings on del Pezzo geometries using their group-theoretical structure. In the introduction, we first explain why this is interesting from
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