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Mellin–Barnes Representation of the Topological String

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Abstract

We invoke integrals of Mellin–Barnes type to analytically continue the Gopakumar–Vafa resummation of the topological string free energy in the string coupling constant, leading to additional non-perturbative terms. We also discuss in a similar manner the refined and Nekrasov–Shatashvili limit version thereof. The derivation is straight-forward and essentially boils down to taking residue. This allows us to confirm some related conjectures in the literature at tree-level.

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References

  1. Aniceto, I., Schiappa, R., Vonk, M.: The resurgence of instantons in string theory. Commun. Num. Theor. Phys. 6, 339 (2012). arXiv:1106.5922 [hep-th]

  2. Santamara, R.C., Edelstein, J.D., Schiappa, R., Vonk, M.: Resurgent transseries and the Holomorphic Anomaly. arXiv:1308.1695 [hep-th]

  3. Krefl, D.: Non-perturbative quantum geometry. JHEP 1402, 084 (2014). arXiv:1311.0584 [hep-th]

  4. Krefl, D.: Non-perturbative quantum geometry II. JHEP 1412, 118 (2014). arXiv:1410.7116 [hep-th]

  5. Baar, G., Dunne, G.V.: Resurgence and the Nekrasov–Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lam+¼ systems. JHEP 1502, 160 (2015). arXiv:1501.05671 [hep-th]

  6. Hatsuda, Y., Moriyama, S., Okuyama, K.: Instant on effects in ABJM theory from fermi gas approach. JHEP 1301, 158 (2013). arXiv:1211.1251 [hep-th]

  7. Hatsuda, Y., Marino, M., Moriyama, S., Okuyama, K.: Non-perturbative effects and the refined topological string. JHEP 1409, 168 (2014). arXiv:1306.1734 [hep-th].

  8. Lockhart, G., Vafa, C.: “Superconformal partition functions and non-perturbative topological strings. arXiv:1210.5909 [hep-th]

  9. Krefl, D., Mkrtchyan, R.L.: Exact Chern–Simons/Topological String duality. arXiv:1506.03907 [hep-th]

  10. Gopakumar, R., Vafa, C.: M theory and topological strings, vol. 1. arXiv:hep-th/9809187

  11. Gopakumar, R., Vafa, C.: M theory and topological strings, vol. 2. arXiv:hep-th/9812127

  12. Hatsuda, Y.: Spectral zeta function and non-perturbative effects in ABJM Fermi-gas. arXiv:1503.07883 [hep-th]

  13. Paris, R.B.; Kaminski, D.: Asymptotics and Mellin-Barnes integrals. Encyclopaedia of Mathematics and its Applications, vol. 85. Cambridge University Press, Cambridge (2001)

  14. Bhattacharya, J., Bhattacharyya, S., Minwalla, S., Raju, S.: Indices for Superconformal Field Theories in 3, 5 and 6 Dimensions. JHEP 0802, 064 (2008). arXiv:0801.1435 [hep-th]

  15. Okounkov, A., Reshetikhin, N., Vafa, C.: Quantum Calabi–Yau and classical crystals. Progr. Math. 244, 597 (2006). arXiv:hep-th/0309208

  16. Iqbal, A., Nekrasov, N., Okounkov, A., Vafa, C.: Quantum foam and topological strings. JHEP 0804, 011 (2008). arXiv:hep-th/0312022

  17. Klemm, A., Kreuzer, M., Riegler, E., Scheidegger, E.: Topological string amplitudes, complete intersection Calabi–Yau spaces and threshold corrections. JHEP 0505, 023 (2005). arXiv:hep-th/0410018

  18. Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. JHEP 1111, 129 (2011). arXiv:hep-th/0702146

  19. Pasquetti, S., Schiappa, R.: Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models. Ann. Henri Poincare 11, 351 (2010). arXiv:0907.4082 [hep-th]

  20. Huang, Mx., Klemm, A.: Direct integration for general Ω backgrounds. Adv. Theor. Math. Phys. 16(3), 805 (2012). arXiv:1009.1126 [hep-th]

  21. 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]

  22. Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008). arXiv:hep-th/0310272

  23. Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052 [hep-th]

  24. Marino, M.: Chern–Simons theory and topological strings. Rev. Mod. Phys. 77, 675 (2005). arXiv:hep-th/0406005

  25. Mkrtchyan, R.L.: Nonperturbative universal Chern–Simons theory. JHEP 1309, 054 (2013). arXiv:1302.1507 [hep-th]

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  1. Center for Theoretical Physics, SNU, Seoul, South Korea

    Daniel Krefl

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  1. Daniel Krefl
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Krefl, D. Mellin–Barnes Representation of the Topological String. Lett Math Phys 106, 1561–1574 (2016). https://doi.org/10.1007/s11005-016-0882-2

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  • DOI: https://doi.org/10.1007/s11005-016-0882-2

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