TBA-like integral equations from quantized mirror curves

  • Kazumi Okuyama
  • Szabolcs ZakanyEmail author
Open Access
Regular Article - Theoretical Physics


Quantizing the mirror curve of certain toric Calabi-Yau (CY) three-folds leads to a family of trace class operators. The resolvent function of these operators is known to encode topological data of the CY. In this paper, we show that in certain cases, this resolvent function satisfies a system of non-linear integral equations whose structure is very similar to the Thermodynamic Bethe Ansatz (TBA) systems. This can be used to compute spectral traces, both exactly and as a semiclassical expansion. As a main example, we consider the system related to the quantized mirror curve of local P2. According to a recent proposal, the traces of this operator are determined by the refined BPS indices of the underlying CY. We use our non-linear integral equations to test that proposal.


Topological Strings Bethe Ansatz 


Open Access

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  1. [1]
    S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A new supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    A.B. Zamolodchikov, Painleve III and 2 − D polymers, Nucl. Phys. B 432 (1994) 427 [hep-th/9409108] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    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].
  4. [4]
    P. Dorey, C. Dunning and R. Tateo, The ODE/IM Correspondence, J. Phys. A 40 (2007) R205 [hep-th/0703066] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  5. [5]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Exact Results on the ABJM Fermi Gas, JHEP 10 (2012) 020 [arXiv:1207.4283] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    P. Putrov and M. Yamazaki, Exact ABJM Partition Function from TBA, Mod. Phys. Lett. A 27 (2012) 1250200 [arXiv:1207.5066] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    F. Calvo and M. Mariño, Membrane instantons from a semiclassical TBA, JHEP 05 (2013) 006 [arXiv:1212.5118] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    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].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE].
  10. [10]
    M.-x. Huang and X.-f. Wang, Topological Strings and Quantum Spectral Problems, JHEP 09 (2014) 150 [arXiv:1406.6178] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    A. Grassi, Y. Hatsuda and M. Mariño, Topological Strings from Quantum Mechanics, arXiv:1410.3382 [INSPIRE].
  12. [12]
    R. Kashaev and M. Mariño, Operators from mirror curves and the quantum dilogarithm, arXiv:1501.01014 [INSPIRE].
  13. [13]
    R. Kashaev, M. Mariño and S. Zakany, Matrix models from operators and topological strings, 2, arXiv:1505.02243 [INSPIRE].
  14. [14]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton Effects in ABJM Theory from Fermi Gas Approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Ellegaard Andersen and R. Kashaev, A TQFT from Quantum Teichmüller Theory, Commun. Math. Phys. 330 (2014) 887 [arXiv:1109.6295] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    Y. Hatsuda, Spectral zeta function and non-perturbative effects in ABJM Fermi-gas, JHEP 11 (2015) 086 [arXiv:1503.07883] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M.-x. Huang, On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit, JHEP 06 (2012) 152 [arXiv:1205.3652] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    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].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].MathSciNetGoogle Scholar
  20. [20]
    Y. Hatsuda and K. Okuyama, Probing non-perturbative effects in M-theory, JHEP 10 (2014) 158 [arXiv:1407.3786] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    S. Codesido, A. Grassi and M. Mariño, Spectral Theory and Mirror Curves of Higher Genus, arXiv:1507.02096 [INSPIRE].
  22. [22]
    M. Mariño, Spectral Theory and Mirror Symmetry, arXiv:1506.07757 [INSPIRE].
  23. [23]
    T. Nosaka, Instanton effects in ABJM theory with general R-charge assignments, arXiv:1512.02862 [INSPIRE].

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© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of PhysicsShinshu UniversityMatsumotoJapan
  2. 2.Département de Physique ThéoriqueUniversité de GenèveGenèveSwitzerland

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