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Annales Henri Poincaré

, Volume 20, Issue 12, pp 4055–4162 | Cite as

Exploring 5d BPS Spectra with Exponential Networks

  • Sibasish Banerjee
  • Pietro LonghiEmail author
  • Mauricio Romo
Original Paper

Abstract

We develop geometric techniques for counting BPS states in five-dimensional gauge theories engineered by M theory on a toric Calabi–Yau threefold. The problem is approached by studying framed 3d–5d wall-crossing in the presence of a single M5 brane wrapping a special Lagrangian submanifold L. The spectrum of 3d–5d BPS states is encoded by the geometry of the manifold of vacua of the 3d–5d system, which further coincides with the mirror curve describing moduli of the Lagrangian brane. The information about the BPS spectrum is extracted from the geometry of the mirror curve by construction of a nonabelianization map for the exponential networks. For the simplest Calabi–Yau, \(\mathbb {C}^3\) we reproduce the count of 5d BPS states and match predictions of 3d \(tt^*\) geometry for the count of 3d–5d BPS states. We comment on applications of our construction to the study of enumerative invariants of toric Calabi–Yau threefolds.

Notes

Acknowledgements

We thank Sergei Alexandrov, Murad Alim, Pierrick Bousseau, Richard Eager, Jan Manschot, Joe Minahan, Rahul Pandharipande, Boris Pioline and Johannes Walcher for discussions. We also thank Sergei Alexandrov for related joint work (with two of the authors) which sparked some of the motivation for this project, and the anonymous referee for insightful comments and especially for pointing out a relation between our results and predictions from [7]. SB acknowledges the facilities of Max Planck Institute for Mathematics in Bonn where the majority of the work was completed and Riemann Center for Geometry and Physics in Hannover where the rest was finished, especially the write-up. SB also thanks St. Petersburg state university, Higher School of Economics in Moscow and Uppsala University for providing excellent working conditions. PL and MR thank the Aspen Center for Physics for hospitality during part of this work. PL also thanks Aarhus QGM, ENS Paris, Caltech, The University of California at Berkeley, and Trinity College Dublin for hospitality during completion of this work. MR acknowledges hospitality from Harvard University and Rutgers University. The work of PL is supported by NCCR SwissMAP, funded by the Swiss National Science Foundation. PL also acknowledges support from grants “Geometry and Physics”and “Exact Results in Gauge and String Theories” from the Knut and Alice Wallenberg Foundation during part of this work. MR gratefully acknowledges the support of the Institute for Advanced Study, DOE Grant DE-SC0009988 and the Adler Family Fund.

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Authors and Affiliations

  1. 1.Max Planck Institute for MathematicsBonnGermany
  2. 2.Riemann center of Geometry and PhysicsLeibniz UniversityHannoverGermany
  3. 3.Institute for Theoretical PhysicsETH ZurichZurichSwitzerland
  4. 4.Department of Physics and AstronomyUppsala UniversityUppsalaSweden
  5. 5.School of Natural SciencesInstitute for Advanced StudyPrincetonUSA
  6. 6.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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