Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 086, 24 pages      arXiv:2303.11461      https://doi.org/10.3842/SIGMA.2023.086

Unitarity of the SoV Transform for $\mathrm{SL}(2,\mathbb C)$ Spin Chains

Alexander N. Manashov
Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany

Received March 30, 2023, in final form October 20, 2023; Published online November 04, 2023

Abstract
We prove the unitarity of the separation of variables transform for $\mathrm{SL}(2,\mathbb C)$ spin chains by a method based on the use of Gustafson integrals.

Key words: spin chains; separation of variables; Gustafson's integrals.

pdf (619 kb)   tex (66 kb)  

References

  1. Bartels J., Lipatov L.N., Prygarin A., Integrable spin chains and scattering amplitudes, J. Phys. A 44 (2011), 454013, 29 pages, arXiv:1104.0816.
  2. Bytsko A.G., Teschner J., Quantization of models with non-compact quantum group symmetry: modular $XXZ$ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006), 12927-12981, arXiv:hep-th/0602093.
  3. Cavaglià A., Gromov N., Levkovich-Maslyuk F., Separation of variables and scalar products at any rank, J. High Energy Phys. 2019 (2019), no. 9, 052, 28 pages, arXiv:1907.03788.
  4. Derkachov S., Kazakov V., Olivucci E., Basso-Dixon correlators in two-dimensional fishnet CFT, J. High Energy Phys. 2019 (2019), no. 4, 032, 31 pages, arXiv:1811.10623.
  5. Derkachov S., Olivucci E., Exactly solvable magnet of conformal spins in four dimensions, Phys. Rev. Lett. 125 (2020), 031603, 7 pages, arXiv:1912.07588.
  6. Derkachov S., Olivucci E., Conformal quantum mechanics & the integrable spinning Fishnet, J. High Energy Phys. 2021 (2021), no. 11, 060, 29 pages, arXiv:2103.01940.
  7. Derkachov S., Olivucci E., Exactly solvable single-trace four point correlators in $\chi \mathrm{CFT}_4$, J. High Energy Phys. 2021 (2021), no. 2, 146, 86 pages, arXiv:2007.15049.
  8. Derkachov S.E., Baxter's $Q$-operator for the homogeneous $XXX$ spin chain, J. Phys. A 32 (1999), 5299-5316, arXiv:solv-int/9902015.
  9. Derkachov S.E., Korchemsky G.P., Manashov A.N., Noncompact Heisenberg spin magnets from high-energy QCD. I. Baxter $Q$-operator and separation of variables, Nuclear Phys. B 617 (2001), 375-440, arXiv:hep-th/0107193.
  10. Derkachov S.E., Korchemsky G.P., Manashov A.N., Baxter $\mathbb Q$-operator and separation of variables for the open $\mathrm{SL}(2,{\mathbb R})$ spin chain, J. High Energy Phys. 2003 (2003), no. 10, 053, 31 pages, arXiv:hep-th/0309144.
  11. Derkachov S.E., Korchemsky G.P., Manashov A.N., Separation of variables for the quantum $\mathrm{SL}(2,\mathbb R)$ spin chain, J. High Energy Phys. 2003 (2003), no. 7, 047, 30 pages, arXiv:hep-th/0210216.
  12. Derkachov S.E., Kozlowski K.K., Manashov A.N., On the separation of variables for the modular $XXZ$ magnet and the lattice sinh-Gordon models, Ann. Henri Poincaré 20 (2019), 2623-2670, arXiv:1806.04487.
  13. Derkachov S.E., Kozlowski K.K., Manashov A.N., Completeness of SoV representation for $\mathrm{SL}(2,\mathbb R)$ spin chains, SIGMA 17 (2021), 063, 26 pages, arXiv:2102.13570.
  14. Derkachov S.E., Manashov A.N., Iterative construction of eigenfunctions of the monodromy matrix for $\mathrm{SL}(2,\mathbb C)$ magnet, J. Phys. A 47 (2014), 305204, 25 pages, arXiv:1401.7477.
  15. Derkachov S.E., Manashov A.N., Spin chains and Gustafson's integrals, J. Phys. A 50 (2017), 294006, 20 pages, arXiv:1611.09593.
  16. Derkachov S.E., Manashov A.N., On complex gamma-function integrals, SIGMA 16 (2020), 003, 20 pages, arXiv:1908.01530.
  17. Derkachov S.E., Valinevich P.A., Separation of variables for the quantum $\mathrm{SL}(3,\mathbb C)$ spin magnet: eigenfunctions of Sklyanin $B$-operator, J. Math. Sci. 242 (2019), 658-682, arXiv:1807.00302.
  18. Faddeev L.D., How algebraic Bethe ansatz works for integrable model, in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 149-219, arXiv:hep-th/9605187.
  19. Faddeev L.D., Korchemsky G.P., High energy QCD as a completely integrable model, Phys. Lett. B 342 (1195), 311-322, arXiv:hep-th/9404173.
  20. Gel'fand I.M., Graev M.I., Retakh V.S., Hypergeometric functions over an arbitrary field, Russian Math. Surveys 59 (2004), 831-905.
  21. Gel'fand I.M., Graev M.I., Vilenkin N.Y., Generalized functions. Vol. 5. Integral geometry and representation theory, Academic Press, New York, 1966.
  22. Gromov N., Levkovich-Maslyuk F., Ryan P., Determinant form of correlators in high rank integrable spin chains via separation of variables, J. High Energy Phys. 2021 (2021), no. 5, 169, 79 pages, arXiv:2011.08229.
  23. Gromov N., Levkovich-Maslyuk F., Sizov G., New construction of eigenstates and separation of variables for $\mathrm {SU}(N)$ quantum spin chains, J. High Energy Phys. 2017 (2017), no. 9, 111, 39 pages, arXiv:1610.08032.
  24. Gromov N., Primi N., Ryan P., Form-factors and complete basis of observables via separation of variables for higher rank spin chains, J. High Energy Phys. 2022 (2022), no. 11, 039, 53 pages, arXiv:2202.01591.
  25. Gustafson R.A., Some $q$-beta and Mellin-Barnes integrals with many parameters associated to the classical groups, SIAM J. Math. Anal. 23 (1992), 525-551.
  26. Gustafson R.A., Some $q$-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras, Trans. Amer. Math. Soc. 341 (1994), 69-119.
  27. Gutzwiller M.C., The quantum mechanical Toda lattice. II, Ann. Physics 133 (1981), 304-331.
  28. Izergin A.G., Korepin V.E., The quantum inverse scattering method approach to correlation functions, Comm. Math. Phys. 94 (1984), 67-92, arXiv:cond-mat/9301031.
  29. Kharchev S., Lebedev D., Integral representation for the eigenfunctions of a quantum periodic Toda chain, Lett. Math. Phys. 50 (1999), 53-77, arXiv:hep-th/9910265.
  30. Kharchev S., Lebedev D., Integral representations for the eigenfunctions of quantum open and periodic Toda chains from the QISM formalism, J. Phys. A 34 (2001), 2247-2258, arXiv:hep-th/0007040.
  31. Kitanine N., Maillet J.M., Terras V., Correlation functions of the $XXZ$ Heisenberg spin-$\frac{1}{2}$ chain in a magnetic field, Nuclear Phys. B 567 (2000), 554-582, arXiv:math-ph/9907019.
  32. Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
  33. Kozlowski K.K., Unitarity of the SoV transform for the Toda chain, Comm. Math. Phys. 334 (2015), 223-273, arXiv:1306.4967.
  34. Kulish P.P., Sklyanin E.K., Quantum spectral transform method. Recent developments, in Integrable Quantum Field Theories (Tvärminne, 1981), Lecture Notes in Phys., Vol. 151, Springer, Berlin, 1982, 61-119.
  35. Lipatov L.N., High-energy scattering in QCD and in quantum gravity and two-dimensional field theories, Nuclear Phys. B 365 (1991), 614-632.
  36. Lipatov L.N., Asymptotic behavior of multicolor QCD at high energies in connection with exactly solvable spin models, JETP Lett. 59 (1994), 596-599, arXiv:hep-th/9311037.
  37. Lipatov L.N., Integrability of scattering amplitudes in $\mathcal{N}=4$ SUSY, J. Phys. A 42 (2009), 304020, 25 pages, arXiv:0902.1444.
  38. Maillet J.M., Niccoli G., On quantum separation of variables, J. Math. Phys. 59 (2018), 091417, 47 pages, arXiv:1807.11572.
  39. Maillet J.M., Niccoli G., Complete spectrum of quantum integrable lattice models associated to $\mathcal U_q(\widehat{{\mathfrak{gl}}_n)}$ by separation of variables, J. Phys. A 52 (2019), 315203, 39 pages, arXiv:1811.08405.
  40. Maillet J.M., Niccoli G., On quantum separation of variables beyond fundamental representations, SciPost Phys. 10 (2021), 026, 38 pages, arXiv:1903.06618.
  41. Maillet J.M., Niccoli G., Vignoli L., On scalar products in higher rank quantum separation of variables, SciPost Phys. 9 (2020), 086, 64 pages, arXiv:2003.04281.
  42. Ryan P., Integrable systems, separation of variables and the Yang-Baxter equation, Ph.D. Thesis, Trinity College Dublin, 2021, arXiv:2201.12057.
  43. Ryan P., Volin D., Separated variables and wave functions for rational $\mathfrak{gl}(N)$ spin chains in the companion twist frame, J. Math. Phys. 60 (2019), 032701, 23 pages, arXiv:1810.10996.
  44. Ryan P., Volin D., Separation of variables for rational $\mathfrak{gl}(n)$ spin chains in any compact representation, via fusion, embedding morphism and Bäcklund flow, Comm. Math. Phys. 383 (2021), 311-343, arXiv:2002.12341.
  45. Sarkissian G.A., Elliptic and complex hypergeometric integrals in quantum field theory, Phys. Part. Nuclei Lett. 20 (2023), 281-286.
  46. Semenov-Tian-Shansky M.A., Quantization of open Toda lattices, in Dynamical Systems VII, Encyclopaedia Math. Sci., Springer, Berlin, 1994, 226-259.
  47. Silantyev A.V., Transition function for the Toda chain, Teoret. and Math. Phys. 150 (2007), 315-331, arXiv:nlin.SI/0603017.
  48. Sklyanin E.K., The quantum Toda chain, in Nonlinear Equations in Classical and Quantum Field Theory (Meudon/Paris, 1983/1984), Lecture Notes in Phys., Vol. 226, Springer, Berlin, 1985, 196-233.
  49. Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Quantum Group and Quantum Integrable Systems, Nankai Lectures Math. Phys., World Scientific, River Edge, NJ, 1992, 63-97.
  50. Sklyanin E.K., Tahtadzhan L.A., Faddeev L.D., Quantum inverse problem method. I, Teoret. and Math. Phys. 40 (1980), 688-706.
  51. Slavnov N.A., Calculation of scalar products of wave functions and form-factors in the framework of the algebraic Bethe ansatz, Teoret. and Math. Phys. 79 (1989), 232-240.
  52. Tahtadzhan L.A., Faddeev L.D., The quantum method for the inverse problem and the $XYZ$ Heisenberg model, Russian Math. Surveys 34 (1979), no. 5, 11-68.
  53. Valinevich P.A., Derkachev S.E., Kulish P.P., Uvarov E.M., Construction of eigenfunctions for a system of quantum minors of the monodromy matrix for an $\mathrm{SL}(n,\mathbb{C})$-invariant spin chain, Teoret. and Math. Phys. 189 (2016), 1529-1553.
  54. Wallach N.R., Real reductive groups. II, Pure Appl. Math., Vol. 132, Academic Press, Inc., Boston, MA, 1992.

Previous article  Next article  Contents of Volume 19 (2023)