- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Thermodynamic Bethe ansatz equations for SU(r+1) quantum Seiberg-Witten curve: Higher-order Mathieu equation

Feiyu Peng1 and Hongfei Shu1,2,3,*

  • *Contact author: shuphy124@gmail.com, shu@zzu.edu.cn
PDF

Phys. Rev. D 114, 066020 – Published 28 September, 2026

DOI: https://doi.org/10.1103/rf3w-jp2k

Abstract

We develop the ordinary differential equation–integrable model correspondence for the higher-order Mathieu equation arising from the quantum Seiberg-Witten curve of the pure SU(r+1) N=2 supersymmetric Yang-Mills theory. From the subdominant solutions, we construct the Q and Y systems and derive the corresponding thermodynamic Bethe ansatz (TBA) equations. The dependence of the moduli parameters is found to be encoded in the boundary conditions of the Y functions at θ→−∞. From these boundary data, we derive an analytic expression for the effective central charge, which also governs the subleading contribution in the large-θ expansion of the TBA equations. Finally, we compare the large-θ expansion of the Q function derived from the TBA equations with that obtained from the WKB method, which yields analytic agreement at subleading order and precise numerical agreement at the higher-order corrections.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (53)

  1. N. Seiberg and E. Witten, Electric—magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B426, 19 (1994).
  2. N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B431, 484 (1994).
  3. A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355, 466 (1995).
  4. E. J. Martinec and N. P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B459, 97 (1996).
  5. R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B460, 299 (1996).
  6. N. A. Nekrasov and S. L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in 16th International Congress on Mathematical Physics (2010), pp. 265–289, .
  7. N. Nekrasov, V. Pestun, and S. Shatashvili, Quantum geometry and quiver gauge theories, Commun. Math. Phys. 357, 519 (2018).
  8. P. Dorey and R. Tateo, Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations, J. Phys. A 32, L419 (1999).
  9. V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Spectral determinants for Schrodinger equation and Q operators of conformal field theory, J. Stat. Phys. 102, 567 (2001).
  10. P. C. Argyres and M. R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B448, 93 (1995).
  11. P. C. Argyres, M. R. Plesser, N. Seiberg, and E. Witten, New N=2 superconformal field theories in four-dimensions, Nucl. Phys. B461, 71 (1996).
  12. D. Gaiotto, Opers and TBA, arXiv:1403.6137.
  13. K. Ito and H. Shu, ODE/IM correspondence and the Argyres-Douglas theory, J. High Energy Phys. 08 (2017) 071.
  14. K. Ito, M. Mariño, and H. Shu, TBA equations and resurgent quantum mechanics, J. High Energy Phys. 01 (2019) 228.
  15. A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. l’I. H. P. Phys. Théor. 39, 211 (1983), https://numdam.org/item/AIHPA_1983__39_3_211_0/.
  16. D. Gaiotto, G. W. Moore, and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299, 163 (2010).
  17. D. Gaiotto, G. W. Moore, and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234, 239 (2013).
  18. S. Cecotti and M. Del Zotto, Y systems, Q systems, and 4D N=2 supersymmetric QFT, J. Phys. A 47, 474001 (2014).
  19. L. F. Alday, J. Maldacena, A. Sever, and P. Vieira, Y-system for Scattering Amplitudes, J. Phys. A 43, 485401 (2010).
  20. Y. Hatsuda, K. Ito, K. Sakai, and Y. Satoh, Thermodynamic Bethe Ansatz equations for minimal surfaces in AdS3, J. High Energy Phys. 04 (2010) 108.
  21. J. Caetano and J. Toledo, χ-systems for correlation functions, J. High Energy Phys. 01 (2019) 050.
  22. H. Ouyang and H. Shu, TBA-like equations for non-planar scattering amplitude/Wilson lines duality at strong coupling, J. High Energy Phys. 05 (2022) 099.
  23. K. Ito and H. Shu, TBA equations for the Schrödinger equation with a regular singularity, J. Phys. A 53, 335201 (2020).
  24. K. Ito and J. Yang, Exact WKB analysis and TBA equations for the stark effect, Prog. Theor. Exp. Phys. 2024, 013A02 (2024).
  25. K. Ito and J. Yang, TBA equations and quantum periods for D-type Argyres-Douglas theories, J. High Energy Phys. 01 (2025) 047.
  26. L. Hollands and A. Neitzke, Exact WKB and abelianization for the T3 equation, Commun. Math. Phys. 380, 131 (2020).
  27. K. Ito, T. Kondo, K. Kuroda, and H. Shu, WKB periods for higher order ODE and TBA equations, J. High Energy Phys. 10 (2021) 167.
  28. K. Ito, T. Kondo, and H. Shu, Wall-crossing of TBA equations and WKB periods for the third order ODE, Nucl. Phys. B979, 115788 (2022).
  29. K. Ito and H. Shu, TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics, J. High Energy Phys. 03 (2024) 122.
  30. K. Ito, H. Shu, and J. Yang, Thermodynamic Bethe ansatz and wall crossing for deformed supersymmetric quantum mechanics, Phys. Rev. D 113, 046002 (2026).
  31. K. Ito and H. Shu, ODE/IM Correspondence and Quantum Periods, Vol. 51 of SpringerBriefs in Mathematical Physics (Springer, New York, 2025), 10.1007/978-981-96-0499-9.
  32. A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, J. High Energy Phys. 04 (2010) 040.
  33. Y. Zenkevich, Nekrasov prepotential with fundamental matter from the quantum spin chain, Phys. Lett. B 701, 630 (2011).
  34. A. Grassi, J. Gu, and M. Mariño, Non-perturbative approaches to the quantum Seiberg-Witten curve, J. High Energy Phys. 07 (2020) 106.
  35. Q. Hao, Exact WKB of solutions by Borel summation and open TBA, arXiv:2507.06922.
  36. A. B. Zamolodchikov, Generalized Mathieu equations and Liouville TBA, in Quantum Field Theories in Two Dimensions (World Scientific, Singapore, 2012), Vol. 2.
  37. D. Fioravanti and D. Gregori, Integrability and cycles of deformed N=2 gauge theory, Phys. Lett. B 804, 135376 (2020).
  38. A. Grassi, Q. Hao, and A. Neitzke, Exact WKB methods in SU(2) Nf=1, J. High Energy Phys. 01 (2022) 046.
  39. K. Imaizumi, Quantum periods and TBA equations for N=2SU(2)Nf=2 SQCD with flavor symmetry, Phys. Lett. B 816, 136270 (2021).
  40. D. Fioravanti, D. Gregori, and H. Shu, Integrability, susy SU(2) matter gauge theories and black holes, Nucl. Phys. B1021, 117200 (2025).
  41. A. Mironov and A. Morozov, Nekrasov functions from exact BS periods: The case of SU(N), J. Phys. A 43, 195401 (2010).
  42. A. Popolitov, On relation between Nekrasov functions and BS periods in pure SU(N) case, arXiv:1001.1407.
  43. D. Fioravanti, H. Poghosyan, and R. Poghossian, T, Q and periods in SU(3) N=2 SYM, J. High Energy Phys. 03 (2020) 049.
  44. F. Yan, Exact WKB and the quantum Seiberg-Witten curve for 4d N=2 pure SU(3) Yang-Mills. Abelianization, J. High Energy Phys. 03 (2022) 164.
  45. A. Klemm, W. Lerche, S. Yankielowicz, and S. Theisen, Simple singularities and N=2 supersymmetric Yang-Mills theory, Phys. Lett. B 344, 169 (1995).
  46. P. C. Argyres and A. E. Faraggi, The vacuum structure and spectrum of N=2 supersymmetric SU(N) gauge theory, Phys. Rev. Lett. 74, 3931 (1995).
  47. P. Dorey, C. Dunning, and R. Tateo, The ODE/IM correspondence, J. Phys. A 40, R205 (2007).
  48. A. Grassi and M. Mariño, A solvable deformation of quantum mechanics, SIGMA 15, 025 (2019).
  49. F. Del Monte and P. Longhi, The threefold way to quantum periods: WKB, TBA equations and q-Painlevé, SciPost Phys. 15, 112 (2023).
  50. S. Baldino, WKB methods for finite difference Schrodinger equations, Ph.D. thesis, Lisbon, IST Maths, 2023, arXiv:2410.14628.
  51. F. Del Monte and P. Longhi, Monodromies of second order q-difference equations from the WKB approximation, arXiv:2406.00175.
  52. J. Gu and M. Marino, Thou shalt not tunnel: Complex instantons and tunneling suppression in deformed quantum mechanics, arXiv:2602.20576.
  53. J. Baerman, G. Ravazzini, and J. Teschner, Higher-rank Mathieu opers, Toda chain, and analytic Langlands correspondence, arXiv:2512.18744.

Outline

Information

Phys. Rev. D 114, 066020– Published 28 September, 2026
  • Received 21 June 2026
  • Accepted 9 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation