Geodesic
Reflection operator and hypergeometry II: $SL(2,\mathbb{C})$ spin chain
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 30, Tome 532 (2024), pp. 47-79 
P. V. Antonenko; N. M. Belousov; S. È. Derkachov; P. A. Valinevich. Reflection operator and hypergeometry II: $SL(2,\mathbb{C})$ spin chain. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 30, Tome 532 (2024), pp. 47-79. http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a1/
@article{ZNSL_2024_532_a1,
     author = {P. V. Antonenko and N. M. Belousov and S. \`E. Derkachov and P. A. Valinevich},
     title = {Reflection operator and hypergeometry {II:} $SL(2,\mathbb{C})$ spin chain},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {47--79},
     year = {2024},
     volume = {532},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a1/}
}
TY  - JOUR
AU  - P. V. Antonenko
AU  - N. M. Belousov
AU  - S. È. Derkachov
AU  - P. A. Valinevich
TI  - Reflection operator and hypergeometry II: $SL(2,\mathbb{C})$ spin chain
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2024
SP  - 47
EP  - 79
VL  - 532
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a1/
LA  - en
ID  - ZNSL_2024_532_a1
ER  - 
%0 Journal Article
%A P. V. Antonenko
%A N. M. Belousov
%A S. È. Derkachov
%A P. A. Valinevich
%T Reflection operator and hypergeometry II: $SL(2,\mathbb{C})$ spin chain
%J Zapiski Nauchnykh Seminarov POMI
%D 2024
%P 47-79
%V 532
%U http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a1/
%G en
%F ZNSL_2024_532_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We consider noncompact open $SL(2, \mathbb{C})$ spin chain and construct eigenfunctions of $B$-element of monodromy matrix for the simplest case of the chain with one site. The reflection operator appearing in this construction can be used to express eigenfunction for $n$ sites in terms of the eigenfunction for $n-1$ sites, this general result is briefly announced. We prove orthogonality and completeness of constructed eigenfunctions in the case of one site, express them in terms of the hypergeometric function of the complex field and derive the equation for the reflection operator with the general $SL(2,\mathbb{C})$-invariant $\mathbb{R}$-operator.

[1] P. Antonenko, N. Belousov, S. Derkachov, S. Khoroshkin, Reflection operator and hypergeometry I: $SL(2, \mathbb{R})$ spin chain, to appear, 2024 | MR

[2] I. V. Cherednik, “Factorizing Particles on a Half Line and Root Systems”, Theor. Math. Phys., 61 (1984), 977–983 | DOI | MR | Zbl

[3] D. Chicherin, S. Derkachov, D. Karakhanyan, R. Kirschner, “Baxter operators for arbitrary spin”, Nuclear Physics B, 854:2 (2012), 393–432 | DOI | MR | Zbl

[4] S. E. Derkachov, G. P. Korchemsky, A. N. Manashov, “Noncompact Heisenberg spin magnets from high-energy QCD: I.V Baxter Q-operator and separation of variables”, Nuclear Physics B, 617 (2001), 375–440 | DOI | MR | Zbl

[5] S. Derkachov, V. Kazakov, E. Olivucci, “Basso-Dixon correlators in two-dimensional fishnet CFT”, J. High Energy Physics, 2019 (2019), 32 | DOI | MR

[6] St. Petersburg Math. J., 21:4 (2010), 513–577 | DOI | MR | Zbl

[7] S. E. Derkachov, A. N. Manashov, “Iterative construction of eigenfunctions of the monodromy matrix for $SL(2,\mathbb{C})$ magnet”, J. Phys. A: Math. Theor., 47:30 (2014) | DOI | MR

[8] S. E. Derkachov, A. N. Manashov, “On Complex Gamma-Function Integrals”, SIGMA, 16 (2020), 003 | MR | Zbl

[9] S. Derkachov, A. Manashov, P. Valinevich, “$\mathrm{SL}(2,\mathbb{C})$ Gustafson Integrals”, SIGMA, 14 (2018), 030 | MR | Zbl

[10] L. D. Faddeev, “How algebraic Bethe ansatz works for integrable model”, Quantum Symmetries/Symetries Quantiques, Proceedings of the Les Houches Summer School, 64, 1998, 149–211 | MR

[11] L. D. Faddeev, G. P. Korchemsky, “High energy QCD as a completely integrable model”, Physics Letters B, 342 (1995), 311–322 | DOI

[12] I. M. Gelfand, M. I. Graev, V. S. Retakh, “Hypergeometric functions over an arbitrary field”, Russian Mathematical Surveys, 59:5 (2004), 831–905 | DOI | MR | Zbl

[13] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Generalized functions, v. 5, Integral geometry and representation theory, AMS Chelsea Publishing, 1966 | MR

[14] P. P. Kulish, E. K. Sklyanin, “Algebraic structures related to the reflection equations”, J. Phys. A, 25 (1992), 5963–5976 | DOI | MR

[15] L. N. Lipatov, “High energy asymptotics of multi-colour QCD and two-dimensional conformal field theories”, Physics Letters B, 309:3–4 (1993), 394–396 | DOI | MR

[16] L. N. Lipatov, “High-energy asymptotics of multicolor QCD and exactly solvable lattice models”, JETP Lett., 59 (1994), 596–599

[17] A. N. Manashov, “Unitarity of the SoV Transform for $\mathrm{SL}(2,\mathbb{C})$ Spin Chains”, SIGMA, 19 (2023), 086 | MR | Zbl

[18] V. F. Molchanov, Yu. A. Neretin, “A pair of commuting hypergeometric operators on the complex plane and bispectrality”, J. Spectral Theory, 11:2 (2021), 509–586 | DOI | MR | Zbl

[19] Y.A. Neretin, “Barnes–Ismagilov Integrals and Hypergeometric Functions of the Complex Field”, SIGMA, 16 (2020), 072 | MR | Zbl

[20] E. K. Sklyanin, “The Quantum Toda Chain”, Lect. Notes Phys., 226 (1985), 196–233 | DOI | MR | Zbl

[21] E. K. Sklyanin, “Separation of Variables: New Trends”, Prog. Theor. Phys. Suppl., 118 (1995), 35–60 | DOI | MR | Zbl

[22] E. K. Sklyanin, “Boundary Conditions for Integrable Quantum Systems”, J. Phys. A, 21 (1988), 2375–2389 | DOI | MR | Zbl

[23] G. A. Sarkissian, V. P. Spiridonov, “The endless beta integrals”, SIGMA, 16 (2020), 074 | MR | Zbl