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

SIGMA 10 (2014), 006, 18 pages      arXiv:1310.6988
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy

Anton Zabrodin a, b, c, d
a) Institute of Biochemical Physics, 4 Kosygina, 119334, Moscow, Russia
b) ITEP, 25 B. Cheremushkinskaya, 117218, Moscow, Russia
c) National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia
d) MIPT, Institutskii per. 9, 141700, Dolgoprudny, Moscow region, Russia

Received October 18, 2013, in final form January 08, 2014; Published online January 11, 2014

Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310], we show how to construct the master $T$-operator for the quantum inhomogeneous ${\rm GL}(N)$ $XXX$ spin chain with twisted boundary conditions. It satisfies the bilinear identity and Hirota equations for the classical mKP hierarchy. We also characterize the class of solutions to the mKP hierarchy that correspond to eigenvalues of the master $T$-operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum spin chain and the classical Ruijsenaars-Schneider system of particles.

Key words: quantum integrable spin chains; classical many-body systems; quantum-classical correspondence; master $T$-operator; tau-function.

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  1. Airault H., McKean H.P., Moser J., Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), 95-148.
  2. Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., Classical tau-function for quantum spin chains, J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310.
  3. Alexandrov A., Leurent S., Tsuboi Z., Zabrodin A., The master $T$-operator for the Gaudin model and the KP hierarchy, arXiv:1306.1111.
  4. Bazhanov V.V., Reshetikhin N., Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A: Math. Gen. 23 (1990), 1477-1492.
  5. Cherednik I.V., An analogue of the character formula for Hecke algebras, Funct. Anal. Appl. 21 (1987), 172-174.
  6. Date E., Kashiwara M., Jimbo M., Miwa T., Transformation groups for soliton equations, in Nonlinear Integrable Systems - Classical Theory and Quantum Theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, 39-119.
  7. Gaiotto D., Koroteev P., On three dimensional quiver gauge theories and integrability, J. High Energy Phys. 2013 (2013), no. 5, 126, 59 pages, arXiv:1304.0779.
  8. Gorsky A., Zabrodin A., Zotov A., Spectrum of quantum transfer matrices via classical many-body systems, J. High Energy Phys., to appear, arXiv:1310.6958.
  9. Harnad J., Ènol'skii V.Z., Schur function expansion of KP $\tau$-functions associated with algebraic curves, Russ. Math. Surv. 66 (2011), 767-807, arXiv:1012.3152.
  10. Hikami K., Kulish P.P., Wadati M., Construction of integrable spin systems with long-range interactions, J. Phys. Soc. Japan 61 (1992), 3071-3076.
  11. Hirota R., Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan 50 (1981), 3785-3791.
  12. Iliev P., Rational Ruijsenaars-Schneider hierarchy and bispectral difference operators, Phys. D 229 (2007), 184-190, math-ph/0609011.
  13. Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943-1001.
  14. Kazakov V., Leurent S., Tsuboi Z., Baxter's $Q$-operators and operatorial Bäcklund flow for quantum (super)-spin chains, Comm. Math. Phys. 311 (2012), 787-814, arXiv:1010.4022.
  15. Kazakov V., Sorin A., Zabrodin A., Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics, Nuclear Phys. B 790 (2008), 345-413, hep-th/0703147.
  16. Kazakov V., Vieira P., From characters to quantum (super)spin chains via fusion, J. High Energy Phys. 2008 (2008), no. 10, 050, 31 pages, arXiv:0711.2470.
  17. Krichever I., Rational solutions of the Zakharov-Shabat equations and completely integrable systems of $N$ particles on a line, J. Sov. Math. 21 (1983), 335-345.
  18. Krichever I., General rational reductions of the Kadomtsev-Petviashvili hierarchy and their symmetries, Funct. Anal. Appl. 29 (1995), 75-80.
  19. Krichever I., Lipan O., Wiegmann P., Zabrodin A., Quantum integrable models and discrete classical Hirota equations, Comm. Math. Phys. 188 (1997), 267-304, hep-th/9604080.
  20. Krichever I., Zabrodin A., Spin generalization of the Ruijsenaars-Schneider model, the nonabelian two-dimensionalized Toda lattice, and representations of the Sklyanin algebra, Russ. Math. Surv. 50 (1995), 1101-1150, hep-th/9505039.
  21. Kuniba A., Nakanishi T., Suzuki J., Functional relations in solvable lattice models. I. Functional relations and representation theory, Internat. J. Modern Phys. A 9 (1994), 5215-5266, hep-th/9309137.
  22. Kuniba A., Ohta Y., Suzuki J., Quantum Jacobi-Trudi and Giambelli formulae for $U_q(B^{(1)}_r)$ from the analytic Bethe ansatz, J. Phys. A: Math. Gen. 28 (1995), 6211-6226, hep-th/9506167.
  23. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  24. Miwa T., On Hirota's difference equations, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 9-12.
  25. Mukhin E., Tarasov V., Varchenko A., Bethe subalgebras of the group algebra of the symmetric group, arXiv:1004.4248.
  26. Mukhin E., Tarasov V., Varchenko A., Gaudin Hamiltonians generate the Bethe algebra of a tensor power of the vector representation of $\mathfrak{gl}_N$, St. Petersburg Math. J. 22 (2011), 463-472, arXiv:0904.2131.
  27. Mukhin E., Tarasov V., Varchenko A., KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians, SIGMA 8 (2012), 072, 11 pages, arXiv:1201.3990.
  28. Mukhin E., Tarasov V., Varchenko A., Spaces of quasi-exponentials and representations of the Yangian $Y(gl_N)$, arXiv:1303.1578.
  29. Nekrasov N., Rosly A., Shatashvili S., Darboux coordinates, Yang-Yang functional, and gauge theory, Nuclear Phys. B Proc. Suppl. 216 (2011), 69-93, arXiv:1103.3919.
  30. Nijhoff F.W., Ragnisco O., Kuznetsov V.B., Integrable time-discretisation of the Ruijsenaars-Schneider model, Comm. Math. Phys. 176 (1996), 681-700, hep-th/9412170.
  31. Orlov A.Y., Shiota T., Schur function expansion for normal matrix model and associated discrete matrix models, Phys. Lett. A 343 (2005), 384-396, math-ph/0501017.
  32. Ruijsenaars S.N.M., Schneider H., A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), 370-405.
  33. Sato M., Sato Y., Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, in Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), North-Holland Math. Stud., Vol. 81, North-Holland, Amsterdam, 1983, 259-271.
  34. Shiota T., Calogero-Moser hierarchy and KP hierarchy, J. Math. Phys. 35 (1994), 5844-5849, hep-th/9402021.
  35. Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Quantum Group and Quantum Integrable Systems, Nankai Lectures Math. Phys., World Sci. Publ., River Edge, NJ, 1992, 63-97, hep-th/9211111.
  36. Sklyanin E.K., Separation of variables. New trends, Progr. Theoret. Phys. Suppl. 118 (1995), 35-60, solv-int/9504001.
  37. Wilson G., Collisions of Calogero-Moser particles and an adelic Grassmannian, Invent. Math. 133 (1998), 1-41.
  38. Zabrodin A., Discrete Hirota's equation in quantum integrable models, Internat. J. Modern Phys. B 11 (1997), 3125-3158, hep-th/9610039.
  39. Zabrodin A., The Hirota equation and the Bethe ansatz, Theoret. and Math. Phys. 116 (1998), 782-819.
  40. Zabrodin A., Bäcklund transformation for the Hirota difference equation, and the supersymmetric Bethe ansatz, Theoret. and Math. Phys. 155 (2008), 567-584, arXiv:0705.4006.
  41. Zabrodin A., Hirota equation and Bethe ansatz in integrable models, Suuri-kagaku J. (2013), no. 596, 7-12, arXiv:1211.4428.
  42. Zabrodin A., The master $T$-operator for vertex models with trigonometric $R$-matrices as classical $\tau$-function, Theoret. and Math. Phys. 171 (2013), 52-67, arXiv:1205.4152.

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