Abstract
In the correspondence between spectral problems and topological strings, it is natural to consider complex values for the string theory moduli. In the spectral theory side, this corresponds to non-Hermitian quantum curves with complex spectra and resonances, and in some cases, to PT-symmetric spectral problems. The correspondence leads to precise predictions about the spectral properties of these non-Hermitian operators. In this paper we develop techniques to compute the complex spectra of these quantum curves, providing in this way precision tests of these predictions. In addition, we analyze quantum Seiberg-Witten curves with PT symmetry, which provide interesting and exactly solvable examples of spontaneous PT-symmetry breaking.
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References
A. Grassi, Y. Hatsuda and M. Mariño, Topological strings from quantum mechanics, Annales Henri Poincaré17 (2016) 3177 [arXiv:1410.3382] [INSPIRE].
S. Codesido, A. Grassi and M. Mariño, Spectral theory and mirror curves of higher genus, Annales Henri Poincaré18 (2017) 559 [arXiv:1507.02096] [INSPIRE].
S. Codesido, A. Grassi and M. Mariño, Exact results in \( \mathcal{N} \) = 8 Chern-Simons-matter theories and quantum geometry, JHEP07 (2015) 011 [arXiv:1409.1799] [INSPIRE].
C.M. Bender and T.T. Wu, Anharmonic oscillator, Phys. Rev.184 (1969) 1231 [INSPIRE].
S. Codesido, J. Gu and M. Mariño, Operators and higher genus mirror curves, JHEP02 (2017) 092 [arXiv:1609.00708] [INSPIRE].
A. Grassi and M. Mariño, The complex side of the TS/ST correspondence, J. Phys.A 52 (2019) 055402 [arXiv:1708.08642] [INSPIRE].
R.M. Kashaev and S.M. Sergeev, Spectral equations for the modular oscillator, arXiv:1703.06016 [INSPIRE].
K. Konishi and G. Paffuti, Quantum mechanics: a new introduction, Oxford University Press, Oxford U.K. (2009).
G. Álvarez and C. Casares, Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator, J. Phys.A 33 (2000) 5171.
E. Caliceti, S. Graffi and M. Maioli, Perturbation theory of odd anharmonic oscillators, Commun. Math. Phys.75 (1980) 51.
J. Gu and T. Sulejmanpasic, High order perturbation theory for difference equations and Borel summability of quantum mirror curves, JHEP12 (2017) 014 [arXiv:1709.00854] [INSPIRE].
C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys.70 (2007) 947 [hep-th/0703096] [INSPIRE].
E. Delabaere and D. T. Trinh, Spectral analysis of the complex cubic oscillator, J. Phys.A 33 (2000) 8771.
C.M. Bender et al., Complex WKB analysis of energy-level degeneracies of non-Hermitian Hamiltonians, J. Phys.A34 (2001) L31.
T. Gulden, M. Janas, P. Koroteev and A. Kamenev, Statistical mechanics of Coulomb gases as quantum theory on Riemann surfaces, Zh. Eksp. Teor. Fiz.144 (2013) 574 [arXiv:1303.6386] [INSPIRE].
A. Grassi and M. Mariño, A solvable deformation of quantum mechanics, SIGMA15 (2019) 025 [arXiv:1806.01407] [INSPIRE].
R. De La Madrid and M. Gadella, A pedestrian introduction to Gamow vectors, Amer. J. Phys.70 (2002) 626.
N. Moiseyev, Non-Hermitian quantum mechanics, Cambridge University Press, Cambridge U.K. (2011).
B. Simon and A. Dicke, Coupling constant analyticity for the anharmonic oscillator, Annals Phys.58 (1970) 76 [INSPIRE].
C.M. Bender and A. Turbiner, Analytic continuation of Eigenvalue problems, Phys. Lett.A 173 (1993) 442 [INSPIRE].
B.C. Hall, Quantum theory for mathematicians, Springer, Germany (2013).
G. Álvarez, Bender-Wu branch points in the cubic oscillator, J. Phys.A 28 (1995) 4589.
R. Yaris et al., Resonance calculations for arbitrary potentials, Phys. Rev.A 18 (1978) 1816.
P. Kościk and A. Okopińska, The optimized Rayleigh-Ritz scheme for determining the quantum-mechanical spectrum, J. Phys.A 40 (2007) 10851.
A. Kuroś, P. Kościk and A. Okopińska, Determination of resonances by the optimized spectral approach, J. Phys.A 46 (2013) 085303.
G. Álvarez, Coupling-constant behavior of the resonances of the cubic anharmonic oscillator, Phys. Rev.A 37 (1988) 4079.
T. Sulejmanpasic and M. Ünsal, Aspects of perturbation theory in quantum mechanics: the BenderWu Mathematica package, Comput. Phys. Commun.228 (2018) 273 [arXiv:1608.08256] [INSPIRE].
A. Grassi, M. Mariño and S. Zakany, Resumming the string perturbation series, JHEP05 (2015) 038 [arXiv:1405.4214] [INSPIRE].
K. Ito, M. Mariño and H. Shu, TBA equations and resurgent Quantum Mechanics, JHEP01 (2019) 228 [arXiv:1811.04812] [INSPIRE].
M. Serone, G. Spada and G. Villadoro, The power of perturbation theory, JHEP05 (2017) 056 [arXiv:1702.04148] [INSPIRE].
J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, Annales Henri Poincaré17 (2016) 1037 [arXiv:1308.6485] [INSPIRE].
M.-x. Huang and X.-f. Wang, Topological strings and quantum spectral problems, JHEP09 (2014) 150 [arXiv:1406.6178] [INSPIRE].
M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M – and [p, q]-strings, JHEP11 (2013) 112 [arXiv:1308.0619] [INSPIRE].
M.-x. Huang, A. Klemm, J. Reuter and M. Schiereck, Quantum geometry of del Pezzo surfaces in the Nekrasov-Shatashvili limit, JHEP02 (2015) 031 [arXiv:1401.4723] [INSPIRE].
R. Kashaev and M. Mariño, Operators from mirror curves and the quantum dilogarithm, Commun. Math. Phys.346 (2016) 967 [arXiv:1501.01014] [INSPIRE].
R. Kashaev, M. Mariño and S. Zakany, Matrix models from operators and topological strings, 2, Annales Henri Poincaré17 (2016) 2741 [arXiv:1505.02243] [INSPIRE].
J. Gu, A. Klemm, M. Mariño and J. Reuter, Exact solutions to quantum spectral curves by topological string theory, JHEP10 (2015) 025 [arXiv:1506.09176] [INSPIRE].
X. Wang, G. Zhang and M.-x. Huang, New exact quantization condition for toric Calabi-Yau geometries, Phys. Rev. Lett.115 (2015) 121601 [arXiv:1505.05360] [INSPIRE].
M. Mariño, Spectral theory and mirror symmetry, Proc. Symp. Pure Math.98 (2018) 259 [arXiv:1506.07757] [INSPIRE].
M. Aganagic et al., Quantum geometry of refined topological strings, JHEP11 (2012) 019 [arXiv:1105.0630] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in the proceedings of the 16thInternational Congress on Mathematical Physics (ICMP09), August 3–8, Prague, Czech Republic (2009), arXiv:0908.4052 [INSPIRE].
S. Zakany, Quantized mirror curves and resummed WKB, JHEP05 (2019) 114 [arXiv:1711.01099] [INSPIRE].
P. Dorey, A. Millican-Slater and R. Tateo, Beyond the WKB approximation in PT-symmetric quantum mechanics, J. Phys.A 38 (2005) 1305 [hep-th/0410013] [INSPIRE].
E. Delabaere and F. Pham, Unfolding the quartic oscillator, Ann. Phys.261 (1997) 180.
E. Delabaere, H. Dillinger and F. Pham, Exact semiclassical expansions for one-dimensional quantum oscillators, J. Math. Phys.38 (1997) 6126.
E. Delabaere and F. Pham, Resurgent methods in semi-classical asymptotics, Annales I.H.P.71 (1999) 1.
C.M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett.80 (1998) 5243 [physics/9712001] [INSPIRE].
R. Balian, G. Parisi and A. Voros, Quartic oscillator, in Feynman path integrals, A. Albeverio et al. eds., Springer, Germany (1979).
S. Codesido and M. Mariño, Holomorphic anomaly and quantum mechanics, J. Phys.A 51 (2018) 055402 [arXiv:1612.07687] [INSPIRE].
H. Neuberger, Nonperturbative contributions in models with a nonanalytic behavior at infinite N , Nucl. Phys.B 179 (1981) 253 [INSPIRE].
M. Mariño, Instantons and large N. An introduction to non-perturbative methods in quantum field theory, Cambridge University Press, Cambridge U.K. (2015).
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys.B 426 (1994) 19 [Erratum ibid.B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
P.C. Argyres and A.E. Faraggi, The vacuum structure and spectrum of N = 2 supersymmetric SU(N ) gauge theory, Phys. Rev. Lett.74 (1995) 3931 [hep-th/9411057] [INSPIRE].
A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, Simple singularities and N = 2 supersymmetric Yang-Mills theory, Phys. Lett.B 344 (1995) 169 [hep-th/9411048] [INSPIRE].
A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP04 (2010) 040 [arXiv:0910.5670] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE].
Y. Hatsuda and M. Mariño, Exact quantization conditions for the relativistic Toda lattice, JHEP05 (2016) 133 [arXiv:1511.02860] [INSPIRE].
L. Anderson and M.M. Roberts, Mass deformed ABJM and P T symmetry, JHEP04 (2019) 036 [arXiv:1807.10307] [INSPIRE].
A. Klemm, W. Lerche and S. Theisen, Nonperturbative effective actions of N = 2 supersymmetric gauge theories, Int. J. Mod. Phys.A 11 (1996) 1929 [hep-th/9505150] [INSPIRE].
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Emery, Y., Mariño, M. & Ronzani, M. Resonances and PT symmetry in quantum curves. J. High Energ. Phys. 2020, 150 (2020). https://doi.org/10.1007/JHEP04(2020)150
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DOI: https://doi.org/10.1007/JHEP04(2020)150