## Non-perturbative quantum geometry.(English)Zbl 1333.81346

Summary: The $$\beta$$-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.

### MSC:

 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory

### Keywords:

matrix models; topological strings
Full Text:

### References:

 [1] Seznec, R.; Zinn-Justin, J., Summation of divergent series by order dependent mappings: application to the anharmonic oscillator and critical exponents in field theory, J. Math. Phys., 20, 1398, (1979) · Zbl 0495.65002 [2] Buslaev, V.; Grecchi, V., Equivalence of unstable anharmonic oscillators and double wells, J. Physics, A 26, 20, (1993) · Zbl 0817.47077 [3] Zinn-Justin, J.; Jentschura, U., Multi-instantons and exact results I: conjectures, WKB expansions and instanton interactions, Annals Phys., 313, 197, (2004) · Zbl 1054.81020 [4] Zinn-Justin, J.; Jentschura, U., Multi-instantons and exact results II: specific cases, higher-order effects and numerical calculations, Annals Phys., 313, 269, (2004) · Zbl 1054.81022 [5] Zinn-Justin, J., Multi-instanton contributions in quantum mechanics. 2, Nucl. Phys., B 218, 333, (1983) [6] Zinn-Justin, J., From multi-instantons to exact results, Ann. Inst. Fourier, 53, 1259, (2003) · Zbl 1073.81043 [7] Delabaere, E.; Dillinger, H.; Pham, F., Exact semiclassical expansions for one-dimensional quantum oscillators, J. Math. Phys., 38, 6126, (1997) · Zbl 0896.34051 [8] G.V. Dunne and M. Ünsal, Generating energy eigenvalue trans-series from perturbation theory, arXiv:1306.4405 [INSPIRE]. [9] R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE]. · Zbl 0999.81068 [10] Mironov, A.; Morozov, A., Nekrasov functions and exact Bohr-zommerfeld integrals, JHEP, 04, 040, (2010) · Zbl 1272.81180 [11] Aganagic, M.; Cheng, MC; Dijkgraaf, R.; Krefl, D.; Vafa, C., Quantum geometry of refined topological strings, JHEP, 11, 019, (2012) [12] N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE]. · Zbl 1214.83049 [13] Klemm, A.; Mariño, M.; Theisen, S., Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP, 03, 051, (2003) [14] Morozov, A.; Shakirov, S., The matrix model version of AGT conjecture and CIV-DV prepotential, JHEP, 08, 066, (2010) · Zbl 1291.81263 [15] Krefl, D.; Walcher, J., ABCD of beta ensembles and topological strings, JHEP, 11, 111, (2012) [16] Matone, M., Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett., B 357, 342, (1995) [17] Jentschura, UD; Zinn-Justin, J., Instantons in quantum mechanics and resurgent expansions, Phys. Lett., B 596, 138, (2004) · Zbl 1247.81135 [18] Cachazo, F.; Intriligator, KA; Vafa, C., A large-N duality via a geometric transition, Nucl. Phys., B 603, 3, (2001) · Zbl 0983.81050 [19] D. Krefl and J. Walcher, Shift versus extension in refined partition functions, arXiv:1010.2635 [INSPIRE]. [20] Marshakov, A.; Mironov, A.; Morozov, A., On AGT relations with surface operator insertion and stationary limit of beta-ensembles, J. Geom. Phys., 61, 1203, (2011) · Zbl 1215.81092 [21] G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin systems via β-deformed matrix models, arXiv:1104.4016 [INSPIRE]. · Zbl 1386.81140 [22] Zinn-Justin, J., Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys., 113, 1, (2002) [23] Mariño, M.; Schiappa, R.; Weiss, M., Nonperturbative effects and the large-order behavior of matrix models and topological strings, Commun. Num. Theor. Phys., 2, 349, (2008) · Zbl 1153.81526 [24] J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE]. [25] Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, arXiv:1306.1734 [INSPIRE]. · Zbl 1333.81336 [26] Krefl, D.; Schwarz, A., Refined Chern-Simons versus vogel universality, J. Geom. Phys., 74, 119, (2013) · Zbl 1283.58018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.