Abstract
The β-ensemble with cubic potential can be used to study a quantum particle in a double-well potential with symmetry breaking term. The quantum mechanical perturbative energy arises from the ensemble free energy in a novel large N limit. A relation between the generating functions of the exact non-perturbative energy, similar in spirit to the one of Dunne- Ünsal, is found. The exact quantization condition of Zinn-Justin and Jentschura is equivalent to the Nekrasov-Shatashvili quantization condition on the level of the ensemble. Refined topological string theory in the Nekrasov-Shatashvili limit arises as a large N limit of quantum mechanics.
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References
R. Seznec and J. Zinn-Justin, Summation of divergent series by order dependent mappings: application to the anharmonic oscillator and critical exponents in field theory, J. Math. Phys. 20 (1979) 1398 [INSPIRE].
V. Buslaev and V. Grecchi, Equivalence of unstable anharmonic oscillators and double wells, J. Physics A 26 (1993) 20.
J. Zinn-Justin and U. Jentschura, Multi-instantons and exact results I: conjectures, WKB expansions and instanton interactions, Annals Phys. 313 (2004) 197 [quant-ph/0501136] [INSPIRE].
J. Zinn-Justin and U. Jentschura, Multi-instantons and exact results II: specific cases, higher-order effects and numerical calculations, Annals Phys. 313 (2004) 269 [quant-ph/0501137] [INSPIRE].
J. Zinn-Justin, Multi-instanton contributions in quantum mechanics. 2, Nucl. Phys. B 218 (1983) 333 [INSPIRE].
J. Zinn-Justin, From multi-instantons to exact results, Ann. Inst. Fourier 53 (2003) 1259.
E. Delabaere, H. Dillinger and F. Pham, Exact semiclassical expansions for one-dimensional quantum oscillators, J. Math. Phys. 38 (1997) 6126.
G.V. Dunne and M. Ünsal, Generating energy eigenvalue trans-series from perturbation theory, arXiv:1306.4405 [INSPIRE].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE].
A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].
M. Aganagic, M.C. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum geometry of refined topological strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [INSPIRE].
A. Morozov and S. Shakirov, The matrix model version of AGT conjecture and CIV-DV prepotential, JHEP 08 (2010) 066 [arXiv:1004.2917] [INSPIRE].
D. Krefl and J. Walcher, ABCD of beta ensembles and topological strings, JHEP 11 (2012) 111 [arXiv:1207.1438] [INSPIRE].
M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342 [hep-th/9506102] [INSPIRE].
U.D. Jentschura and J. Zinn-Justin, Instantons in quantum mechanics and resurgent expansions, Phys. Lett. B 596 (2004) 138 [hep-ph/0405279] [INSPIRE].
F. Cachazo, K.A. Intriligator and C. Vafa, A large-N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3 [hep-th/0103067] [INSPIRE].
D. Krefl and J. Walcher, Shift versus extension in refined partition functions, arXiv:1010.2635 [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, On AGT relations with surface operator insertion and stationary limit of Beta-ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin systems via β-deformed matrix models, arXiv:1104.4016 [INSPIRE].
J. Zinn-Justin, Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113 (2002) 1 [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].
J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE].
Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, arXiv:1306.1734 [INSPIRE].
D. Krefl and A. Schwarz, Refined Chern-Simons versus Vogel universality, J. Geom. Phys. 74 (2013) 119 [arXiv:1304.7873] [INSPIRE].
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Krefl, D. Non-perturbative quantum geometry. J. High Energ. Phys. 2014, 84 (2014). https://doi.org/10.1007/JHEP02(2014)084
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DOI: https://doi.org/10.1007/JHEP02(2014)084