Nahm sums, quiver A-polynomials and topological recursion

Abstract

We consider a large class of q-series that have the structure of Nahm sums, or equivalently motivic generating series for quivers. First, we initiate a systematic analysis and classification of classical and quantum A-polynomials associated to such q-series. These quantum quiver A-polynomials encode recursion relations satisfied by the above series, while classical A-polynomials encode asymptotic expansion of those series. Second, we postulate that those series, as well as their quantum quiver A-polynomials, can be reconstructed by means of the topological recursion. There is a large class of interesting quiver A-polynomials of genus zero, and for a number of them we confirm the above conjecture by explicit calculations. In view of recently found dualities, for an appropriate choice of quivers, these results have a direct interpretation in topological string theory, knot theory, counting of lattice paths, and related topics. In particular it follows, that various quantities characterizing those systems, such as motivic Donaldson-Thomas invariants, various knot invariants, etc., have the structure compatible with the topological recursion and can be reconstructed by its means.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    R. Kedem and B.M. McCoy, Construction of modular branching functions from Bethe’s equations in the three state Potts chain, J. Statist. Phys. 71 (1993) 865 [hep-th/9210129] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. [2]

    R. Kedem, T.R. Klassen, B.M. McCoy and E. Melzer, Fermionic quasiparticle representations for characters of G(1)1 × G(1)1/G(1)2 , Phys. Lett. B 304 (1993) 263 [hep-th/9211102] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  3. [3]

    W. Nahm, Conformal field theory and torsion elements of the Bloch group, in Les Houches School of Physics: frontiers in number theory, physics and geometry, Springer, Berlin, Heidelberg, Germany (2007), pg. 67 [hep-th/0404120] [INSPIRE].

  4. [4]

    D. Zagier, The dilogarithm function, in Frontiers in number theory, physics and geometry II, Springer, Berlin, Heidelberg, Germany (2007), pg. 3.

  5. [5]

    S. Garoufalidis and T.T. Lê, Nahm sums, stability and the colored Jones polynomial, Res. Math. Sci. 2 (2015) 1.

    MathSciNet  Article  Google Scholar 

  6. [6]

    S. Garoufalidis and D. Zagier, Asymptotics of Nahm sums at roots of unity, arXiv:1812.07690 [INSPIRE].

  7. [7]

    M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Num. Theor. Phys. 5 (2011) 231 [arXiv:1006.2706] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  8. [8]

    M. Reineke, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants, Compos. Math. 147 (2011) 943.

    MathSciNet  Article  Google Scholar 

  9. [9]

    M. Reineke, Degenerate cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers, Doc. Math. 17 (2012) 1 [arXiv:1102.3978].

    ADS  MathSciNet  MATH  Google Scholar 

  10. [10]

    S. Meinhardt and M. Reineke, Donaldson-Thomas invariants versus intersection cohomology of quiver moduli, arXiv:1411.4062.

  11. [11]

    H. Franzen and M. Reineke, Semi-stable Chow-Hall algebras of quivers and quantized Donaldson-Thomas invariants, arXiv:1512.03748.

  12. [12]

    P. Kucharski, M. Reineke, M. Stosic and P. Sułkowski, BPS states, knots and quivers, Phys. Rev. D 96 (2017) 121902 [arXiv:1707.02991] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  13. [13]

    P. Kucharski, M. Reineke, M. Stosic and P. Sułkowski, Knots-quivers correspondence, Adv. Theor. Math. Phys. 23 (2019) 1849 [arXiv:1707.04017] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  14. [14]

    P. Kucharski and P. Su-lkowski, BPS counting for knots and combinatorics on words, JHEP 11 (2016) 120 [arXiv:1608.06600] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. [15]

    W. Luo and S. Zhu, Integrality structures in topological strings I: framed unknot, arXiv:1611.06506 [INSPIRE].

  16. [16]

    M. Stosic and P. Wedrich, Rational links and DT invariants of quivers, arXiv:1711.03333 [INSPIRE].

  17. [17]

    M.o. Panfil, M. Stošić and P. Sułkowski, Donaldson-Thomas invariants, torus knots and lattice paths, Phys. Rev. D 98 (2018) 026022 [arXiv:1802.04573] [INSPIRE].

  18. [18]

    T. Ekholm, P. Kucharski and P. Longhi, Physics and geometry of knots-quivers correspondence, arXiv:1811.03110 [INSPIRE].

  19. [19]

    M.o. Panfil and P. Sułkowski, Topological strings, strips and quivers, JHEP 01 (2019) 124 [arXiv:1811.03556] [INSPIRE].

  20. [20]

    T. Ekholm, P. Kucharski and P. Longhi, Multi-cover skeins, quivers and 3d N = 2 dualities, JHEP 02 (2020) 018 [arXiv:1910.06193] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  21. [21]

    M. Stosic and P. Wedrich, Tangle addition and the knots-quivers correspondence, arXiv:2004.10837 [INSPIRE].

  22. [22]

    H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  23. [23]

    J.M.F. Labastida and M. Mariño, Polynomial invariants for torus knots and topological strings, Commun. Math. Phys. 217 (2001) 423 [hep-th/0004196] [INSPIRE].

  24. [24]

    J.M.F. Labastida, M. Mariño and C. Vafa, Knots, links and branes at large N, JHEP 11 (2000) 007 [hep-th/0010102] [INSPIRE].

  25. [25]

    J.M.F. Labastida and M. Mariño, A new point of view in the theory of knot and link invariants, math.QA/0104180 [INSPIRE].

  26. [26]

    P. Ramadevi and T. Sarkar, On link invariants and topological string amplitudes, Nucl. Phys. B 600 (2001) 487 [hep-th/0009188] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    A. Mironov, A. Morozov, A. Morozov, P. Ramadevi, V.K. Singh and A. Sleptsov, Checks of integrality properties in topological strings, JHEP 08 (2017) 139 [Addendum ibid. 01 (2018) 143] [arXiv:1702.06316] [INSPIRE].

  28. [28]

    S. Garoufalidis, P. Kucharski and P. Sułkowski, Knots, BPS states and algebraic curves, Commun. Math. Phys. 346 (2016) 75 [arXiv:1504.06327] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  29. [29]

    S. Garoufalidis and T.T.Q. Lê, A survey of q-holonomic functions, arXiv:1601.07487 [INSPIRE].

  30. [30]

    B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Num. Theor. Phys. 1 (2007) 347 [math-ph/0702045] [INSPIRE].

  31. [31]

    B. Eynard, Topological expansion for the 1-Hermitian matrix model correlation functions, JHEP 11 (2004) 031 [hep-th/0407261] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. [32]

    L. Chekhov and B. Eynard, Hermitean matrix model free energy: Feynman graph technique for all genera, JHEP 03 (2006) 014 [hep-th/0504116] [INSPIRE].

    ADS  Article  Google Scholar 

  33. [33]

    L. Chekhov and B. Eynard, Matrix eigenvalue model: Feynman graph technique for all genera, JHEP 12 (2006) 026 [math-ph/0604014] [INSPIRE].

  34. [34]

    V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [INSPIRE].

  35. [35]

    O. Dumitrescu, M. Mulase, B. Safnuk and A. Sorkin, The spectral curve of the Eynard-Orantin recursion via the Laplace transform, Contemp. Math. 593 (2013) 263 [arXiv:1202.1159] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  36. [36]

    R. Dijkgraaf, H. Fuji and M. Manabe, The volume conjecture, perturbative knot invariants and recursion relations for topological strings, Nucl. Phys. B 849 (2011) 166 [arXiv:1010.4542] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  37. [37]

    G. Borot and B. Eynard, All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, arXiv:1205.2261 [INSPIRE].

  38. [38]

    S. Gukov and P. Sulkowski, A-polynomial, B-model and quantization, JHEP 02 (2012) 070 [arXiv:1108.0002] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  39. [39]

    V. Bouchard and B. Eynard, Reconstructing WKB from topological recursion, arXiv:1606.04498 [INSPIRE].

  40. [40]

    L.O. Chekhov, B. Eynard and O. Marchal, Topological expansion of β-ensemble model and quantum algebraic geometry in the sectorwise approach, Theor. Math. Phys. 166 (2011) 141 [arXiv:1009.6007] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  41. [41]

    V. Bouchard and B. Eynard, Think globally, compute locally, JHEP 02 (2013) 143 [arXiv:1211.2302] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  42. [42]

    M. Kontsevich and Y. Soibelman, Airy structures and symplectic geometry of topological recursion, arXiv:1701.09137 [INSPIRE].

  43. [43]

    J.E. Andersen, G. Borot, L.O. Chekhov and N. Orantin, The ABCD of topological recursion, arXiv:1703.03307 [INSPIRE].

  44. [44]

    V. Bouchard, P. Ciosmak, L. Hadasz, K. Osuga, B. Ruba and P. Sułkowski, Super quantum Airy structures, arXiv:1907.08913 [INSPIRE].

  45. [45]

    V. Bouchard and P. Sulkowski, Topological recursion and mirror curves, Adv. Theor. Math. Phys. 16 (2012) 1443 [arXiv:1105.2052] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  46. [46]

    I. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Wiley, U.S.A. (1994).

    Book  Google Scholar 

  47. [47]

    RISC Combinatorics group, A. Riese, qZeil.m webpage, http://www.risc.jku.at/research/combinat/software/qZeil/.

  48. [48]

    B. Eynard and N. Orantin, 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 (2015) 483 [arXiv:1205.1103] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hélder Larraguível.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2005.01776

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Larraguível, H., Noshchenko, D., Panfil, M. et al. Nahm sums, quiver A-polynomials and topological recursion. J. High Energ. Phys. 2020, 151 (2020). https://doi.org/10.1007/JHEP07(2020)151

Download citation

Keywords

  • Matrix Models
  • Topological Strings
  • Differential and Algebraic Geometry