Abstract
We consider a complex rational degeneration of the hyperbolic Ruijsenaars model emerging in the limit ω1 + ω2 → 0 (or b → i in 2d CFT) and investigate the two-particle case in detail. Corresponding wave functions are described by complex hypergeometric functions in the Mellin–Barnes representation. Their dual integral representation and reflection symmetry in the coupling constant are established. Besides, a complex limit of the hyperbolic Baxter Q-operators is considered. Another complex degeneration of the hyperbolic Ruijsenaars model is obtained by taking a special ω1 − ω2 → 0 (or b → 1) limit. Additionally, two new degenerations to the complex Calogero–Sutherland type models are described.
Acknowledgements
The authors thank S. Derkachov and S. Khoroshkin for interesting discussions. This study has been partially supported by the Russian Science Foundation (grant 24-21-00466).
References
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[32] R. Kashaev, The quantum dilogarithm and Dehn twists in quantum Teichmuller theory, in: Inte-grable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory, KluwerAcad. Publ., Dordrecht, 2001, pp. 211–221.
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[36] N. Belousov, S. Derkachov, S. Kharchev, and S. Khoroshkin, BaxterQ-operators in Ruijsenaars–Sutherland hyperbolic systems: One- and two-particle cases, Zap. Nauchn. Sem. POMI 520 (2023)50–123.arXiv:2309.06108.
[37] Yu. A. Neretin, Index hypergeometric integral transform (Russian), addendum to the Russiantranslation of G. E. Andrews, R. Askey, and R. Roy “Special functions”, MCCME Moscow (2013),607–624; English version.arXiv:1208.3342.
[38] Ya. A. Smorodinsky and M. Huszar, Representations of the Lorentz group and generalization ofhelicity states, Theor. Math. Phys. 4 (3) (1970) 867–876.
[39] N. M. Belousov, G. A. Sarkissian, and V. P. Spiridonov, From hyperbolic to complex Eulerintegrals, in preparation.[40] P. V. Antonenko, N. M. Belousov, S. E. Derkachov, and P. A. Valinevich, Reflection opera-tor and hypergeometry II:SL(2,C)spin chain, Zap. Sem. Nauchn. POMI 532 (2024) 47–79.arXiv:2406.19864.[41] Yu. A. Neretin, Barnes–Ismagilov integrals and hypergeometric functions of the complex field,SIGMA 16 (2020) 072.
[42] L. D. Faddeev, R. M. Kashaev, and A. Y. Volkov, Strongly coupled quantum discrete Liouvilletheory. 1. Algebraic approach and duality, Commun. Math. Phys. 219 (2001) 199–219.
[43] M. Halln ̈as and S. Ruijsenaars, A recursive construction of joint eigenfunctions for the hyperbolicnonrelativistic Calogero–Moser Hamiltonians, IMRN 2015 (20) (2015) 10278–10313.
[44] M. Halln ̈as and S. Ruijsenaars, Product formulas for the relativistic and nonrelativistic conicalfunctions, Adv. Stud. Pure Math. 76 (2018) 195–245.
[45] L. D. Faddeev, Current-like variables in massive and massless integrable models, in: Quantumgroups and their applications in physics (Varenna, 1994), Proc. Intern. School Phys. Enrico Fermi127, IOS Press, Amsterdam, 1994, pp. 117–135.arXiv:hep-th/9408041
[46] V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, Exact solution of the Faddeev–Volkovmodel, Phys. Lett. A 372 (2008) 1547–1550.
[47] I. M. Gelfand, M. I. Graev, and V. S. Retakh, Hypergeometric functions over an arbitrary field,Russ. Math. Surv. 59 (5) (2004) 831–905.
[2] M. Halln ̈as, Calogero–Moser–Sutherland systems (2023).arXiv:2312.12932.
[3] S. N. M. Ruijsenaars, Complete integrability of relativistic Calogero–Moser systems and ellipticfunction identities, Commun. Math. Phys. 110 (1987) 191–213.
[4] J. F. van Diejen, Integrability of difference Calogero–Moser systems, J. Math. Phys. 35 (1994)2983–3004.
[5] S. Ruijsenaars, A relativistic conical function and its Whittaker limits, SIGMA 7 (2011) 101.
[6] M. Halln ̈as and S. Ruijsenaars, Joint eigenfunctions for the relativistic Calogero–Moser Hamilto-nians of hyperbolic type. I. First steps, IMRN 2014 (16) (2014) 4400–4456.
[7] M. Halln ̈as and S. Ruijsenaars, Joint eigenfunctions for the relativistic Calogero–Moser Hamilto-nians of hyperbolic type. II. The two- and three-variable cases, IMRN 2018 (14) (2018) 4404–4449.
[8] M. Halln ̈as and S. Ruijsenaars, Joint eigenfunctions for the relativistic Calogero–Moser Hamilto-nians of hyperbolic type. III. Factorized asymptotics, IMRN 2021 (6) (2021) 4679–4708.
[9] N. Belousov, S. Derkachov, S. Kharchev, and S. Khoroshkin, Baxter operators in Ruijsenaarshyperbolic system I: Commutativity ofQ-operators, Ann. Henri Poincar ́e 25 (2024) 3207–3258;arXiv:2303.06383.
[10] N. Belousov, S. Derkachov, S. Kharchev and S. Khoroshkin, Baxter operators in Ruijsenaarshyperbolic system II: Bispectral wave functions, Ann. Henri Poincar ́e 25 (2024) 3259–3296.arXiv:2303.06382.
[11] N. Belousov, S. Derkachov, S. Kharchev, and S. Khoroshkin, Baxter operators in Ruijsenaarshyperbolic system III: Orthogonality and completeness of wave functions, Ann. Henri Poincar ́e 25(2024) 3297–3332.arXiv:2307.16817.
[12] N. Belousov, S. Derkachov, S. Kharchev and S. Khoroshkin, Baxter operators in Ruijsenaarshyperbolic system IV: Coupling constant reflection symmetry, Commun. Math. Phys. 405 (4)(2024) 106.arXiv:2308.07619.
[13] V. P. Spiridonov, Elliptic hypergeometric functions and Calogero–Sutherland type models, Theor.Math. Phys. 150 (2) (2007) 266–277.
[14] F. Atai and M. Noumi, Eigenfunctions of the van Diejen model generated by gauge and integraltransformations, Adv. Math. 412 (2023) 108816.
[15] O. Chalykh, Bethe ansatz for the Ruijsenaars model ofBC1-type, SIGMA 3 (2007) 028.
[16] V. P. Spiridonov, Essays on the theory of elliptic hypergeometric functions, Russ. Math. Surv.63 (3) (2008), 405–472.
[17] F. J. van de Bult, E. M. Rains, and J. V. Stokman, Properties of generalized univariate hyperge-ometric functions, Commun. Math. Phys. 275 (2007) 37–95.
[18] E. Apresyan, G. Sarkissian, and V. P. Spiridonov, A parafermionic hypergeometric function andsupersymmetric6j-symbols, Nucl. Phys. B 990 (2023) 116170.
[19] S. E. Derkachov, G. A. Sarkissian, and V. P. Spiridonov, Elliptic hypergeometric function and6j-symbols for theSL(2,CCC)group, Theor. Math. Phys. 213 (1) (2022) 1406–1422.
[20] E. M. Rains, Limits of elliptic hypergeometric integrals, Ramanujan J. 18 (3) (2009) 257–306.
[21] G. A. Sarkissian and V. P. Spiridonov, The endless beta integrals, SIGMA 16 (2020) 074.
[22] G. A. Sarkissian and V. P. Spiridonov, Complex hypergeometric functions and integrable many-body problems, J. Phys. A: Math. Theor. 55 (2022) 385203.
[23] M. Nishizawa and K. Ueno, Integral solutions ofq-difference equations of the hypergeometrictype with|q|= 1, in: Proc. Workshop “Infinite Analysis — Integral Systems and RepresentationTheory”, pp. 247–255.arXiv:q-alg/9612014.
[24] S. N. M. Ruijsenaars, Systems of Calogero–Moser Type, in: Proc. of the Summer School “Particlesand Fields”, Banff, Canada, 1994, Springer, New York, 1999, pp. 251–352.
[25] J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces (2003).arXiv:hep-th/0308031.
[26] E. Apresyan and G. Sarkissian,S-move matrix in the NS sector ofN= 1super Liouville fieldtheory, JHEP 07 (2024) 127.
[27] Ph. Di Francesco, R. Kedem, S. Khoroshkin, G. Schrader, and A. Shapiro, Ruijsenaars wavefunc-tions as modular group matrix coefficients, Lett. Math. Phys. 114 (2024) 136.
[28] K. Hosomichi, S. Lee, and J. Park, AGT on theS-duality wall, JHEP 12 (2010) 079.
[29] M. Bullimore, H.-C. Kim, and P. Koroteev, Defects and quantum Seiberg–Witten geometry, JHEP05 (2015) 095.
[30] F. A. H. Dolan, V. P. Spiridonov, and G. S. Vartanov, From4dsuperconformal indices to3dpartition functions, Phys. Lett. B 704 (3) (2011) 234–241.
[31] N. M. Belousov, G. A. Sarkissian, and V. P. Spiridonov, Complex binomial theorem and pentagonidentities, Theor. Math. Phys. (2025), to appear.arXiv:2412.07562.
[32] R. Kashaev, The quantum dilogarithm and Dehn twists in quantum Teichmuller theory, in: Inte-grable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory, KluwerAcad. Publ., Dordrecht, 2001, pp. 211–221.
[33] V. F. Molchanov and Yu. A. Neretin, A pair of commuting hypergeometric operators on thecomplex plane and bispectrality, J. Spectr. Theory 11 (2021) 509–586.
[34] B. Ponsot and J. Teschner, Clebsch–Gordan and Racah–Wigner coefficients for a continuous seriesof representations ofUq(sl(2,R)), Commun. Math. Phys. 224 (2001) 613–655.
[35] N. Belousov, S. Khoroshkin, Ruijsenaars spectral transform, Lett. Math. Phys. 115 (2025) 66.
[36] N. Belousov, S. Derkachov, S. Kharchev, and S. Khoroshkin, BaxterQ-operators in Ruijsenaars–Sutherland hyperbolic systems: One- and two-particle cases, Zap. Nauchn. Sem. POMI 520 (2023)50–123.arXiv:2309.06108.
[37] Yu. A. Neretin, Index hypergeometric integral transform (Russian), addendum to the Russiantranslation of G. E. Andrews, R. Askey, and R. Roy “Special functions”, MCCME Moscow (2013),607–624; English version.arXiv:1208.3342.
[38] Ya. A. Smorodinsky and M. Huszar, Representations of the Lorentz group and generalization ofhelicity states, Theor. Math. Phys. 4 (3) (1970) 867–876.
[39] N. M. Belousov, G. A. Sarkissian, and V. P. Spiridonov, From hyperbolic to complex Eulerintegrals, in preparation.[40] P. V. Antonenko, N. M. Belousov, S. E. Derkachov, and P. A. Valinevich, Reflection opera-tor and hypergeometry II:SL(2,C)spin chain, Zap. Sem. Nauchn. POMI 532 (2024) 47–79.arXiv:2406.19864.[41] Yu. A. Neretin, Barnes–Ismagilov integrals and hypergeometric functions of the complex field,SIGMA 16 (2020) 072.
[42] L. D. Faddeev, R. M. Kashaev, and A. Y. Volkov, Strongly coupled quantum discrete Liouvilletheory. 1. Algebraic approach and duality, Commun. Math. Phys. 219 (2001) 199–219.
[43] M. Halln ̈as and S. Ruijsenaars, A recursive construction of joint eigenfunctions for the hyperbolicnonrelativistic Calogero–Moser Hamiltonians, IMRN 2015 (20) (2015) 10278–10313.
[44] M. Halln ̈as and S. Ruijsenaars, Product formulas for the relativistic and nonrelativistic conicalfunctions, Adv. Stud. Pure Math. 76 (2018) 195–245.
[45] L. D. Faddeev, Current-like variables in massive and massless integrable models, in: Quantumgroups and their applications in physics (Varenna, 1994), Proc. Intern. School Phys. Enrico Fermi127, IOS Press, Amsterdam, 1994, pp. 117–135.arXiv:hep-th/9408041
[46] V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, Exact solution of the Faddeev–Volkovmodel, Phys. Lett. A 372 (2008) 1547–1550.
[47] I. M. Gelfand, M. I. Graev, and V. S. Retakh, Hypergeometric functions over an arbitrary field,Russ. Math. Surv. 59 (5) (2004) 831–905.