Abstract
We consider a generalization of the Borel resummation, which turns out to be equivalent to the standard Borel resummation. We apply it to the simplest large N duality between the pure Chern-Simons theory and the topological string on the resolved conifold, and obtain a simple integral formula for the free energy. Expanding this integral representation around the large radius point at finite string coupling g s , we find that it includes not only the M-theoretic resummation à la Gopakumar and Vafa, but also a non-perturbative correction in g s . Remarkably, the obtained non-perturbative correction is in perfect agreement with a proposal for membrane instanton corrections in arXiv:1306.1734. Various other examples are also presented.
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O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
A. Grassi, M. Mariño and S. Zakany, Resumming the string perturbation series, JHEP 05 (2015) 038 [arXiv:1405.4214] [INSPIRE].
R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 1, hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafaa, M theory and topological strings. 2, hep-th/9812127 [INSPIRE].
Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, JHEP 09 (2014) 168 [arXiv:1306.1734] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton effects in ABJM theory from Fermi gas approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].
N.A. Nekrasov and S. L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052.
M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum geometry of refined topological strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 1 (2011) 141 [arXiv:1103.4844] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
M. Hanada, M. Honda, Y. Honma, J. Nishimura, S. Shiba and Y. Yoshida, Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant, JHEP 05 (2012) 121 [arXiv:1202.5300] [INSPIRE].
Y. Hatsuda and K. Okuyama, Probing non-perturbative effects in M-theory, JHEP 10 (2014) 158 [arXiv:1407.3786] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Tests of Seiberg-like duality in three dimensions, arXiv:1012.4021 [INSPIRE].
S. Sinha and C. Vafa, SO and Sp Chern-Simons at large-N, hep-th/0012136 [INSPIRE].
Y. Hatsuda, Spectral zeta function and non-perturbative effects in ABJM Fermi-gas, arXiv:1503.07883 [INSPIRE].
S. Codesido, A. Grassi and M. Mariño, Exact results in \( \mathcal{N}=8 \) Chern-Simons-matter theories and quantum geometry, JHEP 07 (2015) 011 [arXiv:1409.1799] [INSPIRE].
A. Grassi, Y. Hatsuda and M. Mariño, Quantization conditions and functional equations in ABJ(M) theories, arXiv:1410.7658 [INSPIRE].
S. Pasquetti and R. Schiappa, Borel and stokes nonperturbative phenomena in topological string theory and c = 1 matrix models, Annales Henri Poincaré 11 (2010) 351 [arXiv:0907.4082] [INSPIRE].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. (2012) P03001 [arXiv:1110.4066] [INSPIRE].
F. Calvo and M. Mariño, Membrane instantons from a semiclassical TBA, JHEP 05 (2013) 006 [arXiv:1212.5118] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton bound states in ABJM theory, JHEP 05 (2013) 054 [arXiv:1301.5184] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
A. Grassi, Y. Hatsuda and M. Mariño, Topological strings from quantum mechanics, arXiv:1410.3382 [INSPIRE].
M.-x. Huang and X.-f. Wang, Topological strings and quantum spectral problems, JHEP 09 (2014) 150 [arXiv:1406.6178] [INSPIRE].
R. Kashaev and M. Mariño, Operators from mirror curves and the quantum dilogarithm, arXiv:1501.01014 [INSPIRE].
M. Mariño and S. Zakany, Matrix models from operators and topological strings, arXiv:1502.02958 [INSPIRE].
R. Kashaev, M. Mariño and S. Zakany, Matrix models from operators and topological strings, 2, arXiv:1505.02243 [INSPIRE].
X. Wang, G. Zhang and M.-x. Huang, A new exact quantization condition for toric Calabi-Yau geometries, arXiv:1505.05360 [INSPIRE].
M. Mariño, R. Schiappa and M. Weiss, Nonperturbative effects and the large-order behavior of matrix models and topological strings, Commun. Num. Theor. Phys. 2 (2008) 349 [arXiv:0711.1954] [INSPIRE].
M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [INSPIRE].
R.C. Santamaría, J.D. Edelstein, R. Schiappa and M. Vonk, Resurgent transseries and the holomorphic anomaly, arXiv:1308.1695 [INSPIRE].
R. Couso-Santamaría, J.D. Edelstein, R. Schiappa and M. Vonk, Resurgent transseries and the holomorphic anomaly: nonperturbative closed strings in local CP 2, Commun. Math. Phys. 338 (2015) 285 [arXiv:1407.4821] [INSPIRE].
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Hatsuda, Y., Okuyama, K. Resummations and non-perturbative corrections. J. High Energ. Phys. 2015, 51 (2015). https://doi.org/10.1007/JHEP09(2015)051
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DOI: https://doi.org/10.1007/JHEP09(2015)051