Advertisement

Annales Henri Poincaré

, Volume 18, Issue 2, pp 559–622 | Cite as

Spectral Theory and Mirror Curves of Higher Genus

  • Santiago Codesido
  • Alba Grassi
  • Marcos MariñoEmail author
Article

Abstract

Recently, a correspondence has been proposed between spectral theory and topological strings on toric Calabi–Yau manifolds. In this paper, we develop in detail this correspondence for mirror curves of higher genus, which display many new features as compared to the genus one case studied so far. Given a curve of genus g, our quantization scheme leads to g different trace class operators. Their spectral properties are encoded in a generalized spectral determinant, which is an entire function on the Calabi–Yau moduli space. We conjecture an exact expression for this spectral determinant in terms of the standard and refined topological string amplitudes. This conjecture provides a non-perturbative definition of the topological string on these geometries, in which the genus expansion emerges in a suitable ’t Hooft limit of the spectral traces of the operators. In contrast to what happens in quantum integrable systems, our quantization scheme leads to a single quantization condition, which is elegantly encoded by the vanishing of a quantum-deformed theta function on the mirror curve. We illustrate our general theory by analyzing in detail the resolved \({\mathbb C}^3/{\mathbb Z}_5\) orbifold, which is the simplest toric Calabi–Yau manifold with a genus two mirror curve. By applying our conjecture to this example, we find new quantization conditions for quantum mechanical operators, in terms of genus two theta functions, as well as new number-theoretic properties for the periods of this Calabi–Yau.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Grassi, A., Hatsuda, Y., Mariño, M.: Topological strings from quantum mechanics. arXiv:1410.3382 [hep-th]
  2. 2.
    Aganagic, M., Dijkgraaf, R., Klemm, A., Mariño, M., Vafa, C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451 (2006). arXiv:hep-th/0312085 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aganagic, M., Cheng, M.C.N., Dijkgraaf, R., Krefl, D., Vafa, C.: Quantum geometry of refined topological strings. JHEP 1211, 019 (2012). arXiv:1105.0630 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052 [hep-th]
  5. 5.
    Drukker, N., Mariño, M., Putrov, P.: From weak to strong coupling in ABJM theory. Commun. Math. Phys. 306, 511 (2011). arXiv:1007.3837 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mariño, M., Putrov, P.: ABJM theory as a Fermi gas. J. Stat. Mech. 1203, P03001 (2012). arXiv:1110.4066 [hep-th]MathSciNetGoogle Scholar
  7. 7.
    Hatsuda, Y., Moriyama, S., Okuyama, K.: Instanton effects in ABJM theory from Fermi gas approach. JHEP 1301, 158 (2013). arXiv:1211.1251 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hatsuda, Y., Moriyama, S., Okuyama, K.: Instanton bound states in ABJM theory. JHEP 1305, 054 (2013). arXiv:1301.5184 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Hatsuda, Y., Mariño, M., Moriyama, S., Okuyama, K.: Non-perturbative effects and the refined topological string. JHEP 1409, 168 (2014). arXiv:1306.1734 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kallen, J., Mariño, M.: Instanton effects and quantum spectral curves. Annales Henri Poincaré 17(5), 1037 (2016). arXiv:1308.6485 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kashaev, R., Mariño, M.: Operators from mirror curves and the quantum dilogarithm. arXiv:1501.01014 [hep-th]
  12. 12.
    Mariño, M., Zakany, S.: Matrix models from operators and topological strings. Annales Henri Poincaré 17(5), 1075 (2016). arXiv:1502.02958 [hep-th]
  13. 13.
    Kashaev, R., Mariño, M., Zakany, S.: Matrix models from operators and topological strings, 2. arXiv:1505.02243 [hep-th]
  14. 14.
    Gu, J., Klemm, A., Mariño, M., Reuter, J.: Exact solutions to quantum spectral curves by topological string theory. JHEP 1510, 025 (2015). arXiv:1506.09176 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Rodriguez Villegas, F.: Modular Mahler measures, I. In: Topics in Number Theory, p. 17. Kluwer Acad. Publ., Dordrecht (1999)Google Scholar
  16. 16.
    Doran, C., Kerr, M.: Algebraic K-theory of toric hypersurfaces. Commun. Number Theory Phys. 5, 397 (2011). arXiv:0809.4669 [math.AG]MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Katz, S.H., Klemm, A., Vafa, C.: Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173 (1997). arXiv:hep-th/9609239 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chiang, T.M., Klemm, A., Yau, S.T., Zaslow, E.: Local mirror symmetry: calculations and interpretations. Adv. Theor. Math. Phys. 3, 495 (1999). arXiv:hep-th/9903053 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Witten, E.: Phases of N = 2 theories in two-dimensions. Nucl. Phys. B 403, 159 (1993). arXiv:hep-th/9301042 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222
  21. 21.
    Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Alg. Geom. 3, 493 (1994). arXiv:alg-geom/9310003 MathSciNetzbMATHGoogle Scholar
  22. 22.
    Aganagic, M., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1 (2002). arXiv:hep-th/0105045 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mariño, M.: Open string amplitudes and large order behavior in topological string theory. JHEP 0803, 060 (2008). arXiv:hep-th/0612127 ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Remodeling the B-model. Commun. Math. Phys. 287, 117 (2009). arXiv:0709.1453 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Eynard, B., Orantin, N.: Computation of open Gromov–Witten invariants for toric Calabi–Yau 3-folds by topological recursion, a proof of the BKMP conjecture. Commun. Math. Phys. 337(2), 483 (2015). arXiv:1205.1103 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Huang, M.X., Klemm, A., Poretschkin, M.: Refined stable pair invariants for E-, M- and \([p, q]\)-strings. JHEP 1311, 112 (2013). arXiv:1308.0619 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Klemm, A., Poretschkin, M., Schimannek, T., Westerholt-Raum, M.: Direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form. arXiv:1502.00557 [hep-th]
  28. 28.
    De la Ossa, X., Florea, B., Skarke, H.: D-branes on noncompact Calabi–Yau manifolds: K theory and monodromy. Nucl. Phys. B 644, 170 (2002). arXiv:hep-th/0104254 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mukhopadhyay, S., Ray, K.: Fractional branes on a noncompact orbifold. JHEP 0107, 007 (2001). arXiv:hep-th/0102146 ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Karp, R.L.: On the \({\mathbb{C}}^n/{\mathbb{Z}}_m\) fractional branes. J. Math. Phys. 50, 022304 (2009). arXiv:hep-th/0602165 ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Coates, T.: Wall-crossings in toric Gromov–Witten theory, II: local examples. arXiv:0804.2592 [math.AG]
  32. 32.
    Simon, B.: Trace Ideals and Their Applications, 2nd edn. American Mathematical Society, Providence (2000)Google Scholar
  33. 33.
    Simon, B.: Notes on infinite determinants of Hilbert space operators. Adv. Math. 24, 244 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Grothendieck, A.: La théorie de Fredholm. Bulletin de la Société Mathématique de France 84, 319 (1956)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Fredholm, I.: Sur une classe d’équations fonctionnelles. Acta Math. 27, 365 (1903)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Stessin, M., Yang, R., Zhu, K.: Analyticity of a joint spectrum and a multivariable analytic Fredhom theorem. N. Y. J. Math. 17, 39 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Chagouel, I., Stessin, M., Zhu, K.: Geometric spectral theory for compact operators. arXiv:1309.4375
  38. 38.
    Babelon, O., Bernard, D., Talon, M.: An Introduction to Classical Integrable Systems. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  39. 39.
    Gutzwiller, M.C.: The quantum mechanical Toda lattice. Ann. Phys. 124, 347 (1980)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Gutzwiller, M.C.: The quantum mechanical Toda lattice: II. Ann. Phys. 133, 304 (1981)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Sklyanin, E.K.: The quantum Toda chain. Lect. Notes Phys. 226, 196 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Gaudin, M., Pasquier, V.: The periodic Toda chain and a matrix generalization of the Bessel function’s recursion relations. J. Phys. A 25, 5243 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kharchev, S., Lebedev, D.: Integral representation for the eigenfunctions of quantum periodic Toda chain. Lett. Math. Phys. 50, 53 (1999). arXiv:hep-th/9910265 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    An, D.: Complete set of eigenfunctions of the quantum Toda chain. Lett. Math. Phys. 87, 209 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Mironov, A., Morozov, A.: Nekrasov functions and exact Bohr-Zommerfeld integrals. JHEP 1004, 040 (2010). arXiv:0910.5670 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  46. 46.
    Mironov, A., Morozov, A.: Nekrasov functions from exact BS periods: the case of SU(N). J. Phys. A 43, 195401 (2010). arXiv:0911.2396 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  47. 47.
    Kozlowski, K.K., Teschner, J.: TBA for the Toda chain. arXiv:1006.2906 [math-ph]
  48. 48.
    Matsuyama, A.: Periodic Toda lattice in quantum mechanics. Ann. Phys. 222, 300 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Balian, R., Parisi, G., Voros, A.: Discrepancies from asymptotic series and their relation to complex classical trajectories. Phys. Rev. Lett. 41, 1141 (1978)ADSCrossRefGoogle Scholar
  50. 50.
    Balian, R., Parisi, G., Voros, A.: Quartic oscillator. In: Feynman Path Integrals. Lecture Notes in Physics, vol. 106, p. 337 (1979)Google Scholar
  51. 51.
    Huang, M.X.: On gauge theory and topological string in Nekrasov–Shatashvili limit. JHEP 1206, 152 (2012). arXiv:1205.3652 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    Huang, M.X., Klemm, A., Reuter, J., Schiereck, M.: Quantum geometry of del Pezzo surfaces in the Nekrasov–Shatashvili limit. JHEP 1502, 031 (2015). arXiv:1401.4723 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994). arXiv:hep-th/9309140 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Gopakumar, R., Vafa, C.: M theory and topological strings. 2. arXiv:hep-th/9812127
  55. 55.
    Iqbal, A., Kozcaz, C., Vafa, C.: The refined topological vertex. JHEP 0910, 069 (2009). arXiv:hep-th/0701156 ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    Choi, J., Katz, S., Klemm, A.: The refined BPS index from stable pair invariants. Commun. Math. Phys. 328, 903 (2014). arXiv:1210.4403 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Nekrasov, N., Okounkov, A.: Membranes and sheaves. arXiv:1404.2323 [math.AG]
  58. 58.
    Huang, M.X., Klemm, A.: Direct integration for general \(\Omega \) backgrounds. Adv. Theor. Math. Phys. 16(3), 805 (2012). arXiv:1009.1126 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Eynard, B., Mariño, M.: A holomorphic and background independent partition function for matrix models and topological strings. J. Geom. Phys. 61, 1181 (2011). arXiv:0810.4273 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Aganagic, M., Bouchard, V., Klemm, A.: Topological strings and (almost) modular forms. Commun. Math. Phys. 277, 771 (2008). arXiv:hep-th/0607100 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Hatsuda, Y.: Spectral zeta function and non-perturbative effects in ABJM Fermi-gas. JHEP 1511, 086 (2015). arXiv:1503.07883 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    Faddeev, L.D., Kashaev, R.M.: Quantum dilogarithm. Mod. Phys. Lett. A 9, 427 (1994). arXiv:hep-th/9310070 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Garoufalidis, S., Kashaev, R.: Evaluation of state integrals at rational points. Commun. Number Theor. Phys. 09(3), 549 (2015). arXiv:1411.6062 [math.GT]MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Faddeev, L.D.: Discrete Heisenberg-Weyl group and modular group. Lett. Math. Phys. 34, 249 (1995). arXiv:hep-th/9504111 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Fuji, H., Hirano, S., Moriyama, S.: Summing up all genus free energy of ABJM matrix model. JHEP 1108, 001 (2011). arXiv:1106.4631 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Alim, M., Yau, S.T., Zhou, J.: Airy equation for the topological string partition function in a scaling limit. Lett. Math. Phys. 106(6), 719 (2016). arXiv:1506.01375 [hep-th]
  67. 67.
    Huang, M.X., Wang, X.F.: Topological strings and quantum spectral problems. JHEP 1409, 150 (2014). arXiv:1406.6178 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Ellegaard Andersen, J., Kashaev, R.: A TQFT from quantum Teichmüller theory. Commun. Math. Phys. 330, 887 (2014). arXiv:1109.6295 [math.QA]
  69. 69.
    Mariño, M., Schiappa, R., Weiss, M.: Multi-instantons and multi-cuts. J. Math. Phys. 50, 052301 (2009). arXiv:0809.2619 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  70. 70.
    Mohri, K., Onjo, Y., Yang, S.K.: Closed submonodromy problems, local mirror symmetry and branes on orbifolds. Rev. Math. Phys. 13, 675 (2001). arXiv:hep-th/0009072 MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Goncharov, A.B., Kenyon, R.: Dimers and cluster integrable systems. arXiv:1107.5588 [math.AG]
  72. 72.
    Fock, V.V., Marshakov, A.: Loop groups, clusters, dimers and integrable systems. arXiv:1401.1606 [math.AG]
  73. 73.
    Eager, R., Franco, S., Schaeffer, K.: Dimer models and integrable systems. JHEP 1206, 106 (2012). arXiv:1107.1244 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  74. 74.
    Moriyama, S., Nosaka, T.: ABJM membrane instanton from pole cancellation mechanism. Phys. Rev. D 92(2), 026003 (2015). arXiv:1410.4918 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  75. 75.
    Moriyama, S., Nosaka, T.: Exact instanton expansion of superconformal Chern–Simons theories from topological strings. JHEP 1505, 022 (2015). arXiv:1412.6243 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    Hatsuda, Y., Honda, M., Okuyama, K.: Large N non-perturbative effects in \({\cal{{N}}}=4\) superconformal Chern–Simons theories. JHEP 1509, 046 (2015). arXiv:1505.07120 [hep-th]
  77. 77.
    Wang, X., Zhang, G., Huang, M.X.: New exact quantization condition for toric Calabi–Yau geometries. Phys. Rev. Lett. 115, 121601 (2015). arXiv:1505.05360 [hep-th]ADSCrossRefGoogle Scholar
  78. 78.
    Hosono, S., Klemm, A., Theisen, S.: Lectures on mirror symmetry. Lect. Notes Phys. 436, 235 (1994). arXiv:hep-th/9403096 ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Santiago Codesido
    • 1
  • Alba Grassi
    • 1
  • Marcos Mariño
    • 1
    Email author
  1. 1.Département de Physique Théorique et Section de MathématiquesUniversité de GenèveGenevaSwitzerland

Personalised recommendations