Advertisement

Topological strings, quiver varieties, and Rogers–Ramanujan identities

  • Shengmao Zhu
Article
  • 30 Downloads

Abstract

Motivated by some recent works on BPS invariants of open strings/knot invariants, we guess there may be a general correspondence between the Ooguri–Vafa invariants of toric Calabi–Yau 3-folds and cohomologies of Nakajima quiver varieties. In this short note, we provide a toy model to explain this correspondence. More precisely, we study the topological open string model of \({\mathbb {C}}^3\) with one Aganagic–Vafa brane \({\mathcal {D}}_\tau \), and we show that, when \(\tau \le 0\), its Ooguri–Vafa invariants are given by the Betti numbers of certain quiver variety. Moreover, the existence of Ooguri–Vafa invariants implies an infinite product formula. In particular, we find that the \(\tau =1\) case of such infinite product formula is closely related to the celebrated Rogers–Ramanujan identities.

Keywords

Topological strings Ooguri–Vafa invariants Quiver varieties Rogers–Ramanujan identities 

Mathematics Subject Classification

14N35 14N10 11P84 05E05 

Notes

Acknowledgements

The author would like to thank Professor Ole Warnaar for useful discussions [71], and showing him some insights about Formula (57).

Funding Funding was provided by NSFC (Grant No. 11201417).

References

  1. 1.
    Andrews, G.E.: Partially ordered sets and the Rogers–Ramanujan identities. Aequat. Math. 12, 94–107 (1975)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254(2), 425–478 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aspinwall, P., Morrison, D.: Topological field theory and rational curves. Commun. Math. Phys. 151, 245–262 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Aganagic, A., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. arXiv:hep-th/0012041
  5. 5.
    Aganagic, A., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57(1–2), 1–28 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Baxter, R.J.: The hard hexagon model and the Rogers–Ramanujan identities. In: Exactly Solved Models in Statistical Mechanics, Chap. 14. Academic Press, London (in press)Google Scholar
  7. 7.
    Behrend, K., Fantechi, B.: The intrinsic normal cone. Invent. Math. 128, 45–88 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bressoud, D.M.: An easy proof of the Rogers–Ramanujan identities. J. Number Theory 16, 235–241 (1983)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bouchard, V.: Lectures on complex geometry, Calabi–Yau manifolds and toric geometry. arXiv:hep-th/0702063
  10. 10.
    Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Remodeling the B-model. Commun. Math. Phys. 287, 117–178 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Crawley-Boevey, W.: Geometry of the moment map for representations of quivers. Compos. Math. 126, 257–293 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Crawley-Boevey, W., Van den Bergh, M.: Absolutely indecomposable representations and Kac–Moody Lie algebras. Invent. Math. 155, 537–559 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Candelas, P., De La Ossa, X.C., Green, P.S., Parkes, L.: Pair of Calabi–Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Chuang, W., Diaconescu, D.-E., Donagi, R., Pantev, T.: Parabolic refined invariants and Macdonald polynomials. arXiv:1311.3624
  15. 15.
    de Cataldo, M.A.A., Hausel, T., Migliorini, L.: Topology of Hitchin systems and Hodge theory of character varieties: the case \(A_1\). Ann. Math. 175(3), 1329–1407 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Diaconescu, D.-E.: Local curves, wild character varieties, and degenerations. arXiv:1705.05707
  17. 17.
    Diaconescu, D.-E., Donagi, R., Pantev, T.: BPS states, torus links and wild character varieties. arXiv:1704.07412
  18. 18.
    Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. arXiv:math-ph/0702045
  19. 19.
    Eynard, E., 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), 483–567 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Fang, B., Liu, C.-C. M., Zong, Z.: On the remodeling conjecture for toric Calabi–Yau 3-orbifolds. arXiv:1604.07123
  21. 21.
    Garsia, A.M., Haiman, M.: A remarkable q; t-Catalan sequence and q-Lagrange inversion. J. Algebr. Combin. 5, 191–244 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Garsia, A., Milne, S.: Method for constructing bijections for classical partition identities. Proc. Nat. Acad. Sci. USA 18, 2026–2028 (1981)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gopakumar,R., Vafa, C.: M-theory and topological strings-II. arXiv:hep-th/9812127
  24. 24.
    Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3(5), 1415–1443 (1999)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Griffin, M.J., Ono, K., Warnaar, S.O.: A framework of Rogers–Ramanujan identities and their arithmetic properties. Duke Math. J. 165(8), 1475–1527 (2016)MathSciNetMATHGoogle Scholar
  26. 26.
    Hardy, G.H.: Ramanujan. Cambridge University Press, London (1940; reprinted by Chelsea, New York, 1959)Google Scholar
  27. 27.
    Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror symmetry. Clay mathematics monographs, vol. 1Google Scholar
  28. 28.
    Hua, J.: Counting representations of quivers over finite fields. J. Algebr. 226, 1011–1033 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hausel, T.: Kac’s conjecture from Nakajima quiver varieties. Invent. Math. 181, 21–37 (2010)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J. 160, 323–400 (2011)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Positivity for Kac polynomials and DT-invariants of quivers. Ann. Math. 177, 1147–1168 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Hausel, T., Mereb, M. Wong, M.L.: Arithmetic and representation theory of wild character varieties. arXiv:1604.03382
  33. 33.
    Hosono, S., Saito, M., Takahashi, A.: Relative Lefschetz actions and BPS state counting. Int. Math. Res. Not. 15, 783–816 (2001)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Ionel, E.N., Parker, T.H.: The Gopakumar–Vafa formula for symplectic manifolds (preprint). arXiv:1306.1516
  35. 35.
    Kac, V.G.: Root systems, representations of quivers and invariant theory. In: Invariant Theory (Montecatini: Lecture Notes in Math. 996), vol. 1983, pp. 74–108. Springer, New York (1982)Google Scholar
  36. 36.
    Kac, V.G.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  37. 37.
    Konishi, Y.: Integrality of Gopakumar–Vafa invariants of toric Calabi–Yau threefolds. Publ. Res. Inst. Math. Sci. 42(2), 605–648 (2006)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Katz, S., Liu, C.-C.M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5(1), 1–49 (2001)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Kiem, Y.H., Li, J.: Categorication of Donaldson–Thomas invariants via perverse sheaves (preprint). arXiv:1212.6444
  40. 40.
    Kirillov, A. Jr.: Quiver representations and quiver varieties. In: Graduate Studies in Mathematics, vol. 174. American Mathematical Society, Providence (2016)Google Scholar
  41. 41.
    Kronheimer, P.B.: The construction of ALE spaces as a hyper-Kahler quotients. J. Differ. Geom. 29, 665–683 (1989)CrossRefMATHGoogle Scholar
  42. 42.
    Kronheimer, P.B., Nakajima, H.: Yang–Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990)Google Scholar
  43. 43.
    Kucharski, P., Sulkowski, P.: BPS counting for knots and combinatorics on words. arXiv:1608.06600
  44. 44.
    Lusztig, G.: On quiver varieties. Adv. Math. 136, 141–182 (1998)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Li, J., Liu, C.-C., Liu, K., Zhou, J.: A mathematical theory of the topological vertex. Geom. Topol. 13, 527–621 (2009)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Labastida, J.M.F., Mariño, M.: Polynomial invariants for torus knots and topological strings. Commun. Math. Phys 217(2), 423 (2001)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Labastida, J.M.F., Mariño, M.: A new point of view in the theory of knot and link invariants. J. Knot Theory Ramif. 11, 173 (2002)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Labastida, J.M.F., Mariño, M., Vafa, C.: Knots, links and branes at large N. J. High Energy Phys. 11, Paper 7 (2000)Google Scholar
  49. 49.
    Lepowsky, J., Wilson, R.L.: The Rogers–Ramanujan identities: Lie theoretic interpretation and proof. Proc. Nat. Acad. Sci. USA 78, 699–701 (1981)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Li, J., Song, Y.: Open string instantons and relative stable morphisms. In: The Interaction of Finite-Type and Gromov–Witten Invariants (BIRS 2003), Volume 8 of Geom. Topol. Monogr., Coventry: Geom. Topol. Publ., 2006, pp. 49–72Google Scholar
  51. 51.
    Li, J., Tian, G.: Virtual moduli cycle and Gromov–Witten invariants of algebraic varieties. J. Am. Math. Soc. 11(1), 119–174 (1998)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Liu, K., Peng, P.: Proof of the Labastida–Mariño–Ooguri–Vafa conjecture. J. Differ. Geom. 85(3), 479–525 (2010)CrossRefMATHGoogle Scholar
  53. 53.
    Liu, C.-C., Liu, K., Zhou, J.: A proof of a conjecture of Mariño–Vafa on Hodge integrals. J. Differ. Geom. 65(2003)Google Scholar
  54. 54.
    Luo, W., Zhu, S.: Integrality structures in topological strings I: framed unknot. arXiv:1611.06506
  55. 55.
    MacDolnald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Charendon Press, Oxford (1995)Google Scholar
  56. 56.
    Mariño, M.: open string amplitudes and large order behavior in topological string theory. arXiv:hep-th/0612127
  57. 57.
    Mariño, M., Vafa, C.: Framed knots at large N. In: Orbifolds Mathematics and Physics, Madison, WI, 2001, in: Contemp. Math., vol. 310, pp. 185–204. American Mathematical Society, Providence (2002)Google Scholar
  58. 58.
    Maulik, D., Toda, Y.: Gopakumar–Vafa invariants via vanishing cycles (preprint). arXiv:1610.07303
  59. 59.
    Mironov, A., Morozov, A., Morozov, A., Sleptsov, A.: Gaussian distribution of LMOV numbers. arXiv:1706.00761
  60. 60.
    Mozgovoy, S.: Motivic Donaldson–Thomas invariants and Kac conjecture (2010). arXiv:1103.2100
  61. 61.
    Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras. Duke Math. J. 76, 365–416 (1994)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Nakajima, H.: Quiver varieties and Kac–Moody algebras. Duke Math. J. 91, 515–560 (1998)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Okounkov, A., Pandharipande, R.: Hodge integrals and invariants of the unknot. Geom. Topol. 8, 675–699 (2004)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Ooguri, H., Vafa, C.: Knot invariants and topological strings. Nucl. Phys. B 577(3), 419–438 (2000)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Peng, P.: A simple proof of Gopakumar–Vafa conjecture for local toric Calabi–Yau manifolds. Commun. Math. Phys. 276, 551–569 (2007)MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. Invent. Math. 178(2), 407–447 (2009)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Pandharipande, R., Solomon, J., Walcher, J.: Disk enumeration on the quintic 3-fold. J. Am. Math. Soc. 21(4), 1169–1209 (2008)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894)MathSciNetGoogle Scholar
  69. 69.
    Schur, J.: Ein Beitrag zur addiven Zahlentheorie. Sitzungsber. Preuss. Akad. Wiss. Phw-Math. Kl., 302–321 (1917)Google Scholar
  70. 70.
    Stembridge, J.R.: Hall–Littlewood functions, plane partitions, and the Rogers–Ramanujan identities. Trans. Am. Math. Soc. 319, 469–498 (1990)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Warnaar, S.O.: Private Communications (2017)Google Scholar
  72. 72.
    Witten, E.: Topological sigma models. Commun. Math. Phys. 118, 411 (1988)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center of Mathematical SciencesZhejiang UniversityHangzhouChina

Personalised recommendations