Skip to main content
SpringerLink
Log in

A Remark on the q-Hypergeometric Integral Bailey Pair and the Solution to the Star-Triangle Equation

  • PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
  • Published:
Physics of Particles and Nuclei Letters Aims and scope Submit manuscript

Abstract

We rewrite the recently constructed q-hypergeometric integral Bailey pair in a general form. Then with the help of the Bailey pair and \(q\)-beta hypergeometric sum-integral, we construct the star-triangle relation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge University Press, 1999; Izd. Mosk. Tsentra Nepreryvnogo Matem.Obrazov., Moscow, 2013).

  2. M. J. Schlosser, “Hypergeometric and Basic Hypergeometric Series and Integrals Associated with Root Systems,” in Encyclopedia of Special Functions (The Askey–Bateman Project, 2020)), pp. 122–158.

  3. E. Catak, I. Gahramanov, and M. Mullahasanoglu, “Hyperbolic and trigonometric hypergeometric solutions to the star-star equation,” Eur. Phys. J. C 82, 789 (2022). arXiv:2107.06880 [hep-th].

  4. I. Gahramanov and H. Rosengren, “Basic hypergeometry of supersymmetric dualities,” Nucl. Phys. B 913, 747–768 (2016). arXiv:1606.08185 [hep-th].

  5. D. N. Bozkurt and I. Gahramanov, “Pentagon identities arising in supersymmetric gauge theory computations,” Theor. Math. Phys. 198, 189 (2019). arXiv: 1803.00855 [math-ph].

  6. M. Mullahasanoglu and N. Tas, “Lens partition functions and integrability properties,” arXiv:2112.15161 [hep-th].

  7. H. Rosengren, “Rahman’s biorthogonal functions and superconformal indices,” Constructive Approximation 47, 529–52 (2018). arXiv:1612.05051 [math.CA].

  8. S. O. Warnaar, “50 Years of Bailey’s Lemma,” in Algebraic Combinatorics and Applications (Springer, 2001), pp. 333–347.

    Google Scholar 

  9. V. Spiridonov, “An elliptic incarnation of the bailey chain,” Intl. Math. Res. Notices 2002, 1945–1977 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  10. V. P. Spiridonov, “A bailey tree for integrals,” Theor. Math. Phys. 139, 536–541 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Gahramanov and V. P. Spiridonov, “The star-triangle relation and 3d superconformal indices,” J. High Energy Phys. 08, 040 (2015). arXiv:1505.00765 [hep-th].

  12. F. Brünner and V. P. Spiridonov, “A duality web of linear quivers,” Phys. Lett. B 761, 261–264 (2016)., arXiv: 1605.06991 [hep-th].

  13. F. Brünner and V. P. Spiridonov, “4d \(\mathcal{N}\) = 1 quiver gauge theories and the An Bailey lemma,” J. High Energy Phys. 03, 105 (2018). arXiv:1712.07018 [hep-th].

  14. V. P. Spiridonov, “The rarefied elliptic Bailey lemma and the Yang–Baxter equation,” J. Phys. A: Gen. Phys. 52, 355201 (2019). arXiv:1904.12046 [math-ph].

  15. R. Kashaev, F. Luo, and G. Vartanov, “A TQFT of Turaev–Viro type on shaped triangulations,” Ann. Henri Poincare 17, 1109–1143 (2016). arXiv:1210.8393 [math.QA].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. I. Gahramanov, B. Keskin, D. Kosva, and M. Mullahasanoglu, “On Bailey pairs for \(\mathcal{N}\) = 2 supersymmetric gauge theories on \({{S_{b}^{3}} \mathord{\left/ {\vphantom {{S_{b}^{3}} {{{\mathbb{Z}}_{r}}}}} \right. \kern-0em} {{{\mathbb{Z}}_{r}}}}\),” arXiv:2210.11455 [hep-th].

  17. I. Gahramanov and O. E. Kaluc, “Bailey pairs for the q-hypergeometric integral pentagon identity,” arXiv: 2111.14793 [math-ph].

  18. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Elsevier, 2016).

    MATH  Google Scholar 

  19. R. J. Baxter, “Star-triangle and star-star relations in statistical mechanics,” Intl. J. Mod. Phys. B 11, 27–37 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. A. P. Kels, “New solutions of the star–triangle relation with discrete and continuous spin variables,” J. Phys. A: Ge. Phys. 48, 435201 (2015). arXiv:1504.07074 [math-ph].

  21. K. Y. Magadov and V. P. Spiridonov, “Matrix Bailey lemma and the star-triangle relation,” SIGMA 14, 121 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  22. T. Dimofte, D. Gaiotto, and S. Gukov, “3-Manifolds and 3d Indices,” Adv. Theor. Math. Phys. 17, 975–1076 (2013). arXiv:1112.5179 [hep-th].

Download references

ACKNOWLEDGMENTS

We would like to thank Ilmar Gahramanov for providing us with the problem and for his many useful comments. We are also grateful to Mustafa Mullahasanoğlu for the valuable discussions and to Boğazişi University for hospitality where part of this work was done.

Funding

The work is supported by the 3501-TUBITAK Career Development Program under grant number 122F451.

Author information

Authors and Affiliations

  1. Department of Physics, Istanbul University, 34134, Istanbul, Turkey

    E. Catak

Authors
  1. E. Catak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Catak.

Ethics declarations

The author declares that he has no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Catak, E. A Remark on the q-Hypergeometric Integral Bailey Pair and the Solution to the Star-Triangle Equation. Phys. Part. Nuclei Lett. 20, 1357–1360 (2023). https://doi.org/10.1134/S1547477123060080

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1547477123060080

Keywords:

Search

Navigation