Abstract
The partition function on the three-sphere of ABJM theory contains non-perturbative corrections which correspond to membrane instantons in M-theory. These corrections can be studied in the Fermi gas approach to the partition function, and they are encoded in a system of integral equations of the TBA type. We study a semiclassical or WKB expansion of this TBA system in the ABJM coupling k, which corresponds to the strong coupling expansion of the type IIA string. This allows us to study membrane instanton corrections in M-theory at high order in the WKB expansion. Using these WKB results, we verify the conjectures for the form of the one-instanton correction at finite k proposed recently by Hatsuda, Moriyama and Okuyama (HMO), which are in turn based on a conjectural cancellation of divergences between worldsheet instantons and membrane instantons. The HMO cancellation mechanism is important since it shows in a precise, quantitative way, that the perturbative genus expansion is radically insufficient at strong coupling, and that non-perturbative membrane effects are essential to make sense of the theory. We propose analytic expressions in k for the full two-membrane instanton correction and for higher-order non-perturbative terms, which pass many consistency checks and provide further evidence for the HMO mechanism.
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References
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [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].
O. Aharony, O. Bergman, D.L. Jafferis and J.M. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
M. Mariño, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, J. Phys. A 44 (2011) 463001 [arXiv:1104.0783] [INSPIRE].
S. Bhattacharyya, A. Grassi, M. Mariño and A. Sen, A one-loop test of quantum supergravity, arXiv:1210.6057 [INSPIRE].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. (2012) P03001 [arXiv:1110.4066] [INSPIRE].
H. Fuji, S. Hirano and S. Moriyama, Summing up all genus free energy of ABJM matrix model, JHEP 08 (2011) 001 [arXiv:1106.4631] [INSPIRE].
K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [INSPIRE].
A. Cagnazzo, D. Sorokin and L. Wulff, String instanton in AdS 4 × CP 3, JHEP 05 (2010) 009 [arXiv:0911.5228] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [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].
S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A new supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [INSPIRE].
P. Fendley and H. Saleur, N = 2 supersymmetry, Painlevé III and exact scaling functions in 2D polymers, Nucl. Phys. B 388 (1992) 609 [hep-th/9204094] [INSPIRE].
A.B. Zamolodchikov, Painlevé III and 2D polymers, Nucl. Phys. B 432 (1994) 427 [hep-th/9409108] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Exact results on the ABJM Fermi gas, JHEP 10 (2012) 020 [arXiv:1207.4283] [INSPIRE].
P. Putrov and M. Yamazaki, Exact ABJM partition function from TBA, Mod. Phys. Lett. A 27 (2012) 1250200 [arXiv:1207.5066] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Nonperturbative tests of three-dimensional dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].
K. Okuyama, A note on the partition function of ABJM theory on S 3, Prog. Theor. Phys. 127 (2012) 229 [arXiv:1110.3555] [INSPIRE].
M. Hanada et al., Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant, JHEP 05 (2012) 121 [arXiv:1202.5300] [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 2, hep-th/9812127 [INSPIRE].
M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].
C.A. Tracy and H. Widom, Proofs of two conjectures related to the thermodynamic Bethe ansatz, Commun. Math. Phys. 179 (1996) 667 [solv-int/9509003] [INSPIRE].
P. Fendley, Airy functions in the thermodynamic Bethe ansatz, Lett. Math. Phys. 49 (1999) 229 [hep-th/9906114] [INSPIRE].
J.A. Harvey and G.W. Moore, Superpotentials and membrane instantons, hep-th/9907026 [INSPIRE].
M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [INSPIRE].
M. Mariño, Topological strings at strong string coupling, talk given at the Banff Workshop. New Recursion Formulae and Integrability for Calabi-Yau Spaces, Banff Canada (2011), http://www.birs.ca/events/2011/5-day-workshops/11w5114/videos.
N.N. Lebedev, Special functions and their applications, Dover Publications (1962).
A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and series. Vol. 3: More special functions, CRC (1990).
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ArXiv ePrint: 1212.5118
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Calvo, F., Mariño, M. Membrane instantons from a semiclassical TBA. J. High Energ. Phys. 2013, 6 (2013). https://doi.org/10.1007/JHEP05(2013)006
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DOI: https://doi.org/10.1007/JHEP05(2013)006