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


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.



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.


  1. 1.
    Gopakumar, R., Vafa, C.: M theory and topological strings. 1. arXiv:hep-th/9809187
  2. 2.
    Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415–1443 (1999). arXiv:hep-th/9811131 MathSciNetzbMATHGoogle Scholar
  3. 3.
    Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. AMS/IP Stud. Adv. Math. 23, 45 (2001)zbMATHGoogle Scholar
  4. 4.
    Dijkgraaf, R., Vafa, C., Verlinde, E.: M-theory and a topological string duality. arXiv:hep-th/0602087
  5. 5.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory, I. ArXiv Mathematics e-prints. arXiv:math/0312059 (2003)
  6. 6.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory, II. ArXiv Mathematics e-prints. arXiv:math/0406092 (2004)
  7. 7.
    Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations. arXiv:0811.2435
  8. 8.
    Joyce, D., Song, Y.: A theory of generalized Donaldson–Thomas invariants. arXiv:0810.5645
  9. 9.
    Gaiotto, D., Moore, G.W., Neitzke, A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163–224 (2010). arXiv:0807.4723 ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013). arXiv:0907.3987 MathSciNetzbMATHGoogle Scholar
  11. 11.
    Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. JHEP 11, 129 (2011). arXiv:hep-th/0702146 ADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: BPS quivers and spectra of complete \({\cal{N}}=2\) quantum field theories. Commun. Math. Phys. 323, 1185–1227 (2013). arXiv:1109.4941 ADSMathSciNetzbMATHGoogle Scholar
  13. 13.
    Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: \({\cal{N}}=2\) quantum field theories and their BPS quivers. Adv. Theor. Math. Phys. 18(1), 27–127 (2014). arXiv:1112.3984 MathSciNetzbMATHGoogle Scholar
  14. 14.
    Manschot, J., Pioline, B., Sen, A.: Wall crossing from Boltzmann black hole halos. JHEP 07, 059 (2011). arXiv:1011.1258 ADSMathSciNetzbMATHGoogle Scholar
  15. 15.
    Alexandrov, S., Pioline, B., Saueressig, F., Vandoren, S.: D-instantons and twistors. JHEP 03, 044 (2009). arXiv:0812.4219 ADSMathSciNetGoogle Scholar
  16. 16.
    Alexandrov, S., Persson, D., Pioline, B.: Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence. JHEP 12, 027 (2011). arXiv:1110.0466 ADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    Manschot, J., Pioline, B., Sen, A.: From black holes to quivers. JHEP 11, 023 (2012). arXiv:1207.2230 ADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    Pioline, B.: Four ways across the wall. J. Phys. Conf. Ser. 346, 012017 (2012). arXiv:1103.0261 Google Scholar
  19. 19.
    Alexandrov, S., Pioline, B.: Attractor flow trees, BPS indices and quivers. arXiv:1804.06928
  20. 20.
    Dimofte, T., Gukov, S., Hollands, L.: Vortex counting and Lagrangian 3-manifolds. Lett. Math. Phys. 98, 225–287 (2011). arXiv:1006.0977 ADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing in coupled 2d–4d systems. JHEP 12, 082 (2012). arXiv:1103.2598 ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Mclean, R.C.: Deformations of calibrated submanifolds. Commun. Anal. Geom. 6, 705–747 (1996)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. arXiv:hep-th/0012041
  24. 24.
    Aganagic, M., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1–28 (2002). arXiv:hep-th/0105045 ADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222
  26. 26.
    Gaiotto, D., Gukov, S., Seiberg, N.: Surface defects and resolvents. JHEP 09, 070 (2013). arXiv:1307.2578 ADSGoogle Scholar
  27. 27.
    Ashok, S.K., Billo, M., Dell’Aquila, E., Frau, M., Gupta, V., John, R.R., Lerda, A.: Surface operators, chiral rings and localization in \({\cal{N}}=2\) gauge theories. JHEP 11, 137 (2017). arXiv:1707.08922 ADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Ashok, S.K., Billo, M., Dell’Aquila, E., Frau, M., Gupta, V., John, R.R., Lerda, A.: Surface operators in 5d gauge theories and duality relations. JHEP 05, 046 (2018). arXiv:1712.06946 ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Cecotti, S., Fendley, P., Intriligator, K.A., Vafa, C.: A new supersymmetric index. Nucl. Phys. B 386, 405–452 (1992). arXiv:hep-th/9204102 ADSMathSciNetGoogle Scholar
  30. 30.
    Cecotti, S., Vafa, C.: On classification of \({\cal{N}}=2\) supersymmetric theories. Commun. Math. Phys. 158, 569–644 (1993). arXiv:hep-th/9211097 ADSMathSciNetzbMATHGoogle Scholar
  31. 31.
    Cecotti, S., Vafa, C.: Topological antitopological fusion. Nucl. Phys. B 367, 359–461 (1991)ADSzbMATHGoogle Scholar
  32. 32.
    Gaiotto, D., Moore, G.W., Neitzke, A.: Spectral networks. Ann. Henri Poincare 14, 1643–1731 (2013). arXiv:1204.4824 ADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Cecotti, S., Gaiotto, D., Vafa, C.: \(tt^*\) geometry in 3 and 4 dimensions. JHEP 05, 055 (2014). arXiv:1312.1008 ADSMathSciNetzbMATHGoogle Scholar
  34. 34.
    Eager, R., Selmani, S.A., Walcher, J.: Exponential networks and representations of quivers. JHEP 08, 063 (2017). arXiv:1611.06177 ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Gaiotto, D.: Surface operators in \({\cal{N}}=2\) 4d gauge theories. JHEP 11, 090 (2012). arXiv:0911.1316 ADSMathSciNetGoogle Scholar
  36. 36.
    Cecotti, S., Neitzke, A., Vafa, C.: R-twisting and 4d/2d correspondences. arXiv:1006.3435
  37. 37.
    Cecotti, S., Vafa, C.: 2d wall-crossing, R-twisting, and a supersymmetric index. arXiv:1002.3638
  38. 38.
    Kachru, S., Zimet, M.: A comment on 4d and 5d BPS states. arXiv:1808.01529
  39. 39.
    Aganagic, M., Ooguri, H., Vafa, C., Yamazaki, M.: Wall crossing and M-theory. Publ. Res. Inst. Math. Sci. Kyoto 47, 569 (2011). arXiv:0908.1194 MathSciNetzbMATHGoogle Scholar
  40. 40.
    Gaiotto, D., Strominger, A., Yin, X.: New connections between 4-D and 5-D black holes. JHEP 02, 024 (2006). arXiv:hep-th/0503217 ADSMathSciNetGoogle Scholar
  41. 41.
    Jafferis, D.L., Moore, G.W.: Wall crossing in local Calabi Yau manifolds. arXiv:0810.4909
  42. 42.
    Maulik, D., Oblomkov, A., Okounkov, A., Pandharipande, R.: Gromov–Witten/Donaldson–Thomas correspondence for toric 3-folds. ArXiv e-prints (2008) arXiv:0809.3976
  43. 43.
    Morrison, A., Mozgovoy, S., Nagao, K., Szendroi, B.: Motivic Donaldson–Thomas invariants of the conifold and the refined topological vertex. arXiv:1107.5017
  44. 44.
    Gopakumar, R., Vafa, C.: M theory and topological strings. 2. arXiv:hep-th/9812127
  45. 45.
    Ooguri, H., Vafa, C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419–438 (2000). arXiv:hep-th/9912123 ADSMathSciNetzbMATHGoogle Scholar
  46. 46.
    Walcher, J.: Opening mirror symmetry on the quintic. Commun. Math. Phys. 276, 671–689 (2007). arXiv:hep-th/0605162 ADSMathSciNetzbMATHGoogle Scholar
  47. 47.
    Klemm, A., Lerche, W., Mayr, P., Vafa, C., Warner, N.P.: Selfdual strings and \({\cal{N}}=2\) supersymmetric field theory. Nucl. Phys. B 477, 746–766 (1996). arXiv:hep-th/9604034 ADSzbMATHGoogle Scholar
  48. 48.
    Dubrovin, B.: Geometry and integrability of topological–antitopological fusion. Commun. Math. Phys. 152, 539–564 (1993). arXiv:hep-th/9206037 ADSMathSciNetzbMATHGoogle Scholar
  49. 49.
    Iritani, H.: \(tt^{*}\)-geometry in quantum cohomology. ArXiv e-prints (2009) arXiv:0906.1307
  50. 50.
    Fan, H.: Schrodinger equations, deformation theory and \(tt^{*}\)-geometry. ArXiv e-prints (2011) arXiv:1107.1290
  51. 51.
    Chuang, W.Y., Diaconescu, D.E., Manschot, J., Moore, G.W., Soibelman, Y.: Geometric engineering of (framed) BPS states. Adv. Theor. Math. Phys. 18(5), 1063 (2014). arXiv:1301.3065 [hep-th]MathSciNetzbMATHGoogle Scholar
  52. 52.
    Coman, I., Pomoni, E., Teschner, J.: From quantum curves to topological string partition functions. arXiv:1811.01978
  53. 53.
    Longhi, P., Park, C.Y.: ADE spectral networks. JHEP 08, 087 (2016). arXiv:1601.02633 ADSMathSciNetzbMATHGoogle Scholar
  54. 54.
    Galakhov, D., Longhi, P., Moore, G.W.: Spectral networks with spin. Commun. Math. Phys. 340(1), 171–232 (2015). arXiv:1408.0207 ADSMathSciNetzbMATHGoogle Scholar
  55. 55.
    Galakhov, D., Longhi, P., Mainiero, T., Moore, G.W., Neitzke, A.: Wild wall crossing and BPS giants. JHEP 11, 046 (2013). arXiv:1305.5454 ADSGoogle Scholar
  56. 56.
    Hollands, L., Neitzke, A.: BPS states in the Minahan-Nemeschansky \(E_{6}\) theory. Commun. Math. Phys. 353(1), 317–351 (2017). arXiv:1607.01743 ADSzbMATHGoogle Scholar
  57. 57.
    Longhi, P.: Wall-crossing invariants from spectral networks. Ann. Henri Poincare 19(3), 775–842 (2018). arXiv:1611.00150 MathSciNetzbMATHGoogle Scholar
  58. 58.
    Gabella, M.: BPS spectra from BPS graphs. arXiv:1710.08449
  59. 59.
    Gukov, S., Witten, E.: Gauge theory, ramification, and the geometric langlands program. arXiv:hep-th/0612073
  60. 60.
    Hanany, A., Hori, K.: Branes and \({\cal{N}}=2\) theories in two-dimensions. Nucl. Phys. B 513, 119–174 (1998). arXiv:hep-th/9707192 ADSMathSciNetzbMATHGoogle Scholar
  61. 61.
    Witten, E.: Phases of \({\cal{N}}=2\) theories in two-dimensions. Nucl. Phys. B 403, 159–222 (1993). arXiv:hep-th/9301042 ADSMathSciNetzbMATHGoogle Scholar
  62. 62.
    Witten, E.: Phases of \({\cal{N}}=2\) theories in two-dimensions. AMS/IP Stud. Adv. Math. 1, 143 (1996)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Dorey, N., Tong, D.: Mirror symmetry and toric geometry in three-dimensional gauge theories. JHEP 05, 018 (2000). arXiv:hep-th/9911094 ADSMathSciNetzbMATHGoogle Scholar
  64. 64.
    Periwal, V., Shevitz, D.: Exactly solvable unitary matrix models: multicritical potentials and correlations. Nucl. Phys. B 344, 731–746 (1990)ADSMathSciNetGoogle Scholar
  65. 65.
    Klemm, A., Sulkowski, P.: Seiberg–Witten theory and matrix models. Nucl. Phys. B 819, 400–430 (2009). arXiv:0810.4944 ADSMathSciNetzbMATHGoogle Scholar
  66. 66.
    Dimofte, T., Gukov, S.: Chern–Simons theory and S-duality. JHEP 05, 109 (2013). arXiv:1106.4550 ADSMathSciNetzbMATHGoogle Scholar
  67. 67.
    Nekrasov, N.: Five dimensional gauge theories and relativistic integrable systems. Nucl. Phys. B 531, 323–344 (1998). arXiv:hep-th/9609219 ADSMathSciNetzbMATHGoogle Scholar
  68. 68.
    Cherkis, S.A., Kapustin, A.: Nahm transform for periodic monopoles and \({\cal{N}}=2\) super Yang–Mills theory. Commun. Math. Phys. 218, 333–371 (2001). arXiv:hep-th/0006050 ADSMathSciNetzbMATHGoogle Scholar
  69. 69.
    Witten, E.: Solutions of four-dimensional field theories via M theory. Nucl. Phys. B 500, 3–42 (1997). arXiv:hep-th/9703166. [452 (1997)]ADSMathSciNetzbMATHGoogle Scholar
  70. 70.
    Gaiotto, D.: \({\cal{N}}=2\) dualities. JHEP 1208, 034 (2012). arXiv:0904.2715 ADSGoogle Scholar
  71. 71.
    Seiberg, N., Witten, E.: Electric–magnetic duality, monopole condensation, and confinement in \({\cal{N}}=2\) supersymmetric Yang–Mills theory. Nucl. Phys. B 426, 19–52 (1994). arXiv:hep-th/9407087. [Erratum: Nucl. Phys. B 430, 485 (1994)]
  72. 72.
    Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in \({\cal{N}}=2\) supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994). arXiv:hep-th/9408099 ADSMathSciNetzbMATHGoogle Scholar
  73. 73.
    Mikhailov, A.: BPS states and minimal surfaces. Nucl. Phys. B 533, 243–274 (1998). arXiv:hep-th/9708068 ADSMathSciNetzbMATHGoogle Scholar
  74. 74.
    Forbes, B., Jinzenji, M.: Extending the Picard–Fuchs system of local mirror symmetry. J. Math. Phys. 46, 082302 (2005). arXiv:hep-th/0503098 ADSMathSciNetzbMATHGoogle Scholar
  75. 75.
    Aganagic, M., Hori, K., Karch, A., Tong, D.: Mirror symmetry in \((2+1)\)-dimensions and \((1+1)\)-dimensions. JHEP 07, 022 (2001). arXiv:hep-th/0105075 ADSMathSciNetGoogle Scholar
  76. 76.
    Aharony, O., Razamat, S.S., Willett, B.: From 3d duality to 2d duality. JHEP 11, 090 (2017). arXiv:1710.00926 ADSMathSciNetzbMATHGoogle Scholar
  77. 77.
    Zagier, D.: The dilogarithm function. In: Proceedings, Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization: Les Houches, France, March 9–21, 2003, pp. 3–65 (2007)Google Scholar
  78. 78.
    Seiberg, N.: Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics. Phys. Lett. B 388, 753–760 (1996). arXiv:hep-th/9608111 ADSMathSciNetGoogle Scholar
  79. 79.
    Intriligator, K.A., Morrison, D.R., Seiberg, N.: Five-dimensional supersymmetric gauge theories and degenerations of Calabi–Yau spaces. Nucl. Phys. B 497, 56–100 (1997). arXiv:hep-th/9702198 ADSMathSciNetzbMATHGoogle Scholar
  80. 80.
    Douglas, M.R., Katz, S.H., Vafa, C.: Small instantons, Del Pezzo surfaces and type I-prime theory. Nucl. Phys. B 497, 155–172 (1997). arXiv:hep-th/9609071 ADSzbMATHGoogle Scholar
  81. 81.
    Haghighat, B., Vandoren, S.: Five-dimensional gauge theory and compactification on a torus. JHEP 09, 060 (2011). arXiv:1107.2847 ADSMathSciNetzbMATHGoogle Scholar
  82. 82.
    Haghighat, B., Manschot, J., Vandoren, S.: A 5d/2d/4d correspondence. JHEP 03, 157 (2013). arXiv:1211.0513 ADSMathSciNetzbMATHGoogle Scholar
  83. 83.
    Alexandrov, S., Banerjee, S., Longhi, P.: Rigid limit for hypermultiplets and five-dimensional gauge theories. JHEP 01, 156 (2018). arXiv:1710.10665 ADSMathSciNetzbMATHGoogle Scholar
  84. 84.
    Gukov, S., Marino, M., Putrov, P.: Resurgence in complex Chern–Simons theory. arXiv:1605.07615
  85. 85.
    Aganagic, M., Neitzke, A., Vafa, C.: BPS microstates and the open topological string wave function. Adv. Theor. Math. Phys. 10(5), 603–656 (2006). arXiv:hep-th/0504054 MathSciNetzbMATHGoogle Scholar
  86. 86.
    Gabella, M., Longhi, P., Park, C.Y., Yamazaki, M.: BPS graphs: from spectral networks to BPS quivers. arXiv:1704.04204
  87. 87.
    Fuji, H., Gukov, S., Sulkowski, P.: Super-A-polynomial for knots and BPS states. Nucl. Phys. B 867, 506–546 (2013). arXiv:1205.1515 ADSMathSciNetzbMATHGoogle Scholar
  88. 88.
    Ekholm, T., Kucharski, P., Longhi, P.: Physics and geometry of knots-quivers correspondence. arXiv:1811.03110
  89. 89.
    Aganagic, M., Ekholm, T., Ng, L., Vafa, C.: Topological strings, D-Model, and knot contact homology. Adv. Theor. Math. Phys. 18(4), 827–956 (2014). arXiv:1304.5778 MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

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

Personalised recommendations