Topological strings, strips and quivers

Abstract

We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The Topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].

  4. [4]

    E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  5. [5]

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. [7]

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

  8. [8]

    J.M.F. Labastida and M. Marino, A new point of view in the theory of knot and link invariants, J. Knot Theor. Ramifications 11 (2002) 173 [math/0104180].

  9. [9]

    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  MATH  Google Scholar 

  10. [10]

    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 [arXiv:1702.06316] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. [11]

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

    ADS  Google Scholar 

  12. [12]

    P. Kucharski, M. Reineke, M. Stosic and P. Sulkowski, Knots-quivers correspondence, arXiv:1707.04017 [INSPIRE].

  13. [13]

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

  14. [14]

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

  15. [15]

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

  16. [16]

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

    ADS  Google Scholar 

  17. [17]

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

  18. [18]

    M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [INSPIRE].

  19. [19]

    M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A 57 (2002) 1 [hep-th/0105045] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  20. [20]

    J.A. Harvey and G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489 [hep-th/9609017] [INSPIRE].

  21. [21]

    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  MATH  Google Scholar 

  22. [22]

    A. Iqbal and A.-K. Kashani-Poor, The Vertex on a strip, Adv. Theor. Math. Phys. 10 (2006) 317 [hep-th/0410174] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics, Cambridge University Press (2004).

  24. [24]

    G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. [25]

    T. Kimura and Y. Sugimoto, Quantum mirror curve of periodic chain geometry, arXiv:1810.01885 [INSPIRE].

  26. [26]

    T. Mainiero, Algebraicity and Asymptotics: An explosion of BPS indices from algebraic generating series, arXiv:1606.02693 [INSPIRE].

  27. [27]

    P. Smolinski, From topological strings to quantum invariants of knots and quivers, MSc Thesis, University of Warsaw (2017).

  28. [28]

    A.I. Efimov, Cohomological Hall algebra of a symmetric quiver, Compos. Math. 148 (2012) 1133 [arXiv:1103.2736].

    MathSciNet  Article  MATH  Google Scholar 

  29. [29]

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  30. [30]

    E. Basor, B. Conrey and K.E. Morrison, Knots and ones, arXiv:1703.00990 [INSPIRE].

  31. [31]

    M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, BPS Quivers and Spectra of Complete N = 2 Quantum Field Theories, Commun. Math. Phys. 323 (2013) 1185 [arXiv:1109.4941] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  32. [32]

    J. Manschot, B. Pioline and A. Sen, On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants, JHEP 05 (2013) 166 [arXiv:1302.5498] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. [33]

    R. Eager, S.A. Selmani and J. Walcher, Exponential Networks and Representations of Quivers, JHEP 08 (2017) 063 [arXiv:1611.06177] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    M. Gabella, P. Longhi, C.Y. Park and M. Yamazaki, BPS Graphs: From Spectral Networks to BPS Quivers, JHEP 07 (2017) 032 [arXiv:1704.04204] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. [35]

    S. Zhu, Topological strings, quiver varieties and Rogers-Ramanujan identities, arXiv:1707.00831 [INSPIRE].

  36. [36]

    R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].

  37. [37]

    R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].

  38. [38]

    A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Prog. Math. 244 (2006) 597 [hep-th/0309208] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  39. [39]

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  40. [40]

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

    MathSciNet  Article  MATH  Google Scholar 

  41. [41]

    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 

  42. [42]

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

  43. [43]

    H. Franzen and M. Reineke, Semi-Stable Chow-Hall Algebras of Quivers and Quantized Donaldson-Thomas Invariants, [arXiv:1512.03748].

  44. [44]

    W. Nahm, Conformal field theory and torsion elements of the Bloch group, 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. 67-132 (2007) [DOI:https://doi.org/10.1007/978-3-540-30308-4_2] [hep-th/0404120] [INSPIRE].

  45. [45]

    W. Koepf, P.M. Rajković and S.D. Marinković, Properties of q-holonomic functions, J. Differ. Equ. Appl. 13 (2007) 621.

    MathSciNet  Article  MATH  Google Scholar 

  46. [46]

    S. Garoufalidis, A.D. Lauda and T.T.Q. Lê, The colored HOMFLYPT function is q-holonomic, Duke Math. J. 167 (2018) 397 [arXiv:1604.08502] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  47. [47]

    M. Aganagic and C. Vafa, Large N Duality, Mirror Symmetry and a Q-deformed A-polynomial for Knots, arXiv:1204.4709 [INSPIRE].

  48. [48]

    M. Aganagic, T. Ekholm, L. Ng and C. Vafa, Topological Strings, D-Model and Knot Contact Homology, Adv. Theor. Math. Phys. 18 (2014) 827 [arXiv:1304.5778] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  49. [49]

    S. Gukov, S. Nawata, I. Saberi, M. Stosic and P. Sulkowski, Sequencing BPS Spectra, JHEP 03 (2016) 004 [arXiv:1512.07883] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  50. [50]

    A. Schwarz, V. Vologodsky and J. Walcher, Framing the Di-Logarithm (over Z), Proc. Symp. Pure Math. 90 (2015) 113 [arXiv:1306.4298] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  51. [51]

    N. Halmagyi, A. Sinkovics and P. Sulkowski, Knot invariants and Calabi-Yau crystals, JHEP 01 (2006) 040 [hep-th/0506230] [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 Miłosz Panfil.

Additional information

ArXiv ePrint: 1811.03556

Rights and permissions

This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Panfil, M., Sułkowski, P. Topological strings, strips and quivers. J. High Energ. Phys. 2019, 124 (2019). https://doi.org/10.1007/JHEP01(2019)124

Download citation

Keywords

  • M-Theory
  • Topological Field Theories
  • Topological Strings