Abstract
The exact partition function in ABJM theory on three-sphere can be regarded as a canonical partition function of a non-interacting Fermi-gas with an unconventional Hamiltonian. All the information on the partition function is encoded in the discrete spectrum of this Hamiltonian. We explain how (quantum mechanical) non-perturbative corrections in the Fermi-gas system appear from a spectral consideration. Basic tools in our analysis are a Mellin-Barnes type integral representation and a spectral zeta function. From a consistency with known results, we conjecture that the spectral zeta function in the ABJM Fermi-gas has an infinite number of “non-perturbative” poles, which are invisible in the semi-classical expansion of the Planck constant. We observe that these poles indeed appear after summing up perturbative corrections. As a consequence, the perturbative resummation of the spectral zeta function causes non-perturbative corrections to the grand canonical partition function. We also present another example associated with a spectral problem in topological string theory. A conjectured non-perturbative free energy on the resolved conifold is successfully reproduced in this framework.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
J. Teschner, Exact results on N = 2 supersymmetric gauge theories, arXiv:1412.7145 [INSPIRE].
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].
K. Hosomichi, A review on SUSY gauge theories on S 3, arXiv:1412.7128 [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].
M. Mariño and P. Putrov, Exact Results in ABJM Theory from Topological Strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].
A. Grassi, M. Mariño and S. Zakany, Resumming the string perturbation series, JHEP 05 (2015) 038 [arXiv:1405.4214] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].
J. Ecalle, Les Fonctions Resurgentes, Publ. Math. Orsay I-III (1981).
M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [INSPIRE].
M. Mariño, Open string amplitudes and large order behavior in topological string theory, JHEP 03 (2008) 060 [hep-th/0612127] [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].
S. Garoufalidis, A. Its, A. Kapaev and M. Mariño, Asymptotics of the instantons of Painlevé I, Int. Math. Res. Not. 2012 (2012) 561 [arXiv:1002.3634] [INSPIRE].
I. Aniceto, R. Schiappa and M. Vonk, The Resurgence of Instantons in String Theory, Commun. Num. Theor. Phys. 6 (2012) 339 [arXiv:1106.5922] [INSPIRE].
R. Schiappa and R. Vaz, The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation, Commun. Math. Phys. 330 (2014) 655 [arXiv:1302.5138] [INSPIRE].
R.C. Santamaría, J.D. Edelstein, R. Schiappa and M. Vonk, Resurgent Transseries and the Holomorphic Anomaly, arXiv:1308.1695 [INSPIRE].
I. Aniceto, J.G. Russo and R. Schiappa, Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories, JHEP 03 (2015) 172 [arXiv:1410.5834] [INSPIRE].
C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-Matrix Models and Tri-Sasaki Einstein Spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [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].
Y. Hatsuda and K. Okuyama, Probing non-perturbative effects in M-theory, JHEP 10 (2014) 158 [arXiv:1407.3786] [INSPIRE].
S. Moriyama and T. Nosaka, Partition Functions of Superconformal Chern-Simons Theories from Fermi Gas Approach, JHEP 11 (2014) 164 [arXiv:1407.4268] [INSPIRE].
S. Moriyama and T. Nosaka, ABJM membrane instanton from a pole cancellation mechanism, Phys. Rev. D 92 (2015) 026003 [arXiv:1410.4918] [INSPIRE].
S. Moriyama and T. Nosaka, Exact Instanton Expansion of Superconformal Chern-Simons Theories from Topological Strings, JHEP 05 (2015) 022 [arXiv:1412.6243] [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].
J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE].
A. Grassi, Y. Hatsuda and M. Mariño, Topological Strings from Quantum Mechanics, arXiv:1410.3382 [INSPIRE].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].
S. Matsumoto and S. Moriyama, ABJ Fractional Brane from ABJM Wilson Loop, JHEP 03 (2014) 079 [arXiv:1310.8051] [INSPIRE].
M. Honda and K. Okuyama, Exact results on ABJ theory and the refined topological string, JHEP 08 (2014) 148 [arXiv:1405.3653] [INSPIRE].
H. Awata, S. Hirano and M. Shigemori, The Partition Function of ABJ Theory, PTEP 2013 (2013) 053B04 [arXiv:1212.2966] [INSPIRE].
M. Honda, Direct derivation of “mirror” ABJ partition function, JHEP 12 (2013) 046 [arXiv:1310.3126] [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].
N. Drukker and J. Felix, 3d mirror symmetry as a canonical transformation, JHEP 05 (2015) 004 [arXiv:1501.02268] [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].
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].
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].
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].
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].
R. Couso-Santamaría, R. Schiappa and R. Vaz, Finite N from Resurgent Large-N, Annals Phys. 356 (2015) 1 [arXiv:1501.01007] [INSPIRE].
A. Voros, Spectral zeta functions, Adv. Stud. Pure Math. 21 (1992) 327.
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].
M.-x. Huang and X.-f. Wang, Topological Strings and Quantum Spectral Problems, JHEP 09 (2014) 150 [arXiv:1406.6178] [INSPIRE].
M. Mariño, Localization at large N in Chern-Simons-matter theories, to appear.
S. Hirano, K. Nii and M. Shigemori, ABJ Wilson loops and Seiberg duality, PTEP 2014 (2014) 113B04 [arXiv:1406.4141] [INSPIRE].
M. Mezei and S.S. Pufu, Three-sphere free energy for classical gauge groups, JHEP 02 (2014) 037 [arXiv:1312.0920] [INSPIRE].
A. Grassi and M. Mariño, M-theoretic matrix models, JHEP 02 (2015) 115 [arXiv:1403.4276] [INSPIRE].
J. Kallen, The spectral problem of the ABJ Fermi gas, JHEP 10 (2015) 029 [arXiv:1407.0625] [INSPIRE].
X.-f. Wang, X. Wang and M.-x. Huang, A Note on Instanton Effects in ABJM Theory, JHEP 11 (2014) 100 [arXiv:1409.4967] [INSPIRE].
Y. Hatsuda, M. Honda and K. Okuyama, Large-N non-perturbative effects in \( \mathcal{N}=4 \) superconformal Chern-Simons theories, JHEP 09 (2015) 046 [arXiv:1505.07120] [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.-x. Huang, A. Klemm, J. Reuter and M. Schiereck, Quantum geometry of del Pezzo surfaces in the Nekrasov-Shatashvili limit, JHEP 02 (2015) 031 [arXiv:1401.4723] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [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].
D. Dorigoni, An Introduction to Resurgence, Trans-Series and Alien Calculus, arXiv:1411.3585 [INSPIRE].
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
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1503.07883
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hatsuda, Y. Spectral zeta function and non-perturbative effects in ABJM Fermi-gas. J. High Energ. Phys. 2015, 86 (2015). https://doi.org/10.1007/JHEP11(2015)086
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2015)086