Аннотация
Рассматривается комплексное рациональное вырождение гиперболической модели Руйсенарса, возникающее в пределе ω1 + ω2 → 0 (или b → i в 2d конформной теории поля), и детально изучается случай двухчастичной системы. Соответствующие волновые функции описываются комплексными гипергеометрическими функциями в представлении Меллина-Барнса. Найдены их дуальное интегральное представление и симметрия отражения по константе взаимодействия. Кроме того, рассмотрен комплексный предел гиперболического Q-оператора Бакстера. Другое комплексное вырождение гиперболической модели Руйсенарса получено переходом к специальному пределу ω1 − ω2 → 0 (или b → 1). Дополнительно представлены два новых вырождения к комплексным моделям типа Калоджеро-Сазерленда.Поддерживающие организации
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).
Библиографические ссылки
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[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.
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[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.