Quantum integrable models in discrete 2 + 1-dimensional space-time: Auxiliary linear problem on a lattice, zero-curvature representation, isospectral deformation of the Zamolodchikov-Bazhanov-Baxter model

Physics of Particles and Nuclei

Volumn 35, Issue 5, 2004, Pages 567-597

  Sergeev, S M ^a,b  

^a JOINT INSTITUTE FOR NUCLEAR RESEARCH   (Russian Federation)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 8744313731     PISSN: 10637796     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (42)

1. S. M. Sergeev, "Quantum 2 + 1 Evolution Model," J. Phys. A: Math. Gen. 32, 5639-5714 (1999).
2. S. M. Sergeev, "A Three-Dimensional Quantum Integrable Mapping," Teor. Mat. Fiz. 118 (3), 479-487 (1999).
3. S. Sergeev, "On Exact Solution of a Classical 3d Integrable Model," J. Nonlinear Math. Phys. 7 (1), 57-72 (2000).
4. S. Sergeev, "Auxiliary Transfer Matrices for Three Dimensional Integrable Models," Teor. Mat. Fiz. 124, 1187-1201 (2000).
5. S. Sergeev, "Some Properties of Local Linear Problem's Matrices," Zap. Nauchn. Semin. POMI (Notes of PDMI) 269, 292-307 (2000).
6. S. Sergeev, "Integrable Three Dimensional Models in Wholly Discrete Space-Time," in Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory, Ed. by S. Pakuliak and G. von Gehlen (Kluwer Academic, Dordrecht, 2001), NATO Sci. Ser. 2: Math. Phys. Chem., Vol. 35.
7. S. Sergeev, "Complex of Three Dimensional Integrable Models," J. Phys. A: Math. Gen. 34, 10 493-10 503 (2001).
8. S. Sergeev, Preprint MPIM 2002-46, Teor. Mat. Fiz. (in press).
9. S. Sergeev, Preprint MPIM 2002-139, Teor. Mat. Fiz. (in press).
10. A. P. Isaev and S. M. Sergeev, "Quantum Lax Operators and Discrete 2 + 1-Dimensional Integrable Models," Lett. Math. Phys. (2003) (in press).
11. S. Sergeev and S. Pakuliak, Preprint MPIM 2002-85, Teor. Mat. Fiz. (in press).
12. S. Pakuliak and S. Sergeev, "Quantum Relativistic Toda Chain at Root of Unity: Isospectrality, Modified Q-Operator and Functional Bethe Ansatz," Int. J. Math. Math. Sci. 31 (9), 513-553 (2002).
13. G. von Gehlen, S. Pakuliak, and S. Sergeev, Preprint MPIM 2003-57 (submitted to J. Phys. A).
14. R. J. Baxter, "On Zamolodchikov's Solution of the Tetrahedron Equation," Commun. Math. Phys. 88, 185-205 (1983); "Partition Function of the Three-Demensional Zamolodchikov Model," Phys. Rev. Lett. 53, 1795 (1984); "The Yang-Baxter Equations and the Zamolodchikov Model," Physica D (Amsterdam) 18, 321-347 (1986).
15. R. J. Baxter, "On Zamolodchikov's Solution of the Tetrahedron Equation," Commun. Math. Phys. 88, 185-205 (1983); "Partition Function of the Three-Demensional Zamolodchikov Model," Phys. Rev. Lett. 53, 1795 (1984); "The Yang-Baxter Equations and the Zamolodchikov Model," Physica D (Amsterdam) 18, 321-347 (1986).
16. R. J. Baxter, "On Zamolodchikov's Solution of the Tetrahedron Equation," Commun. Math. Phys. 88, 185-205 (1983); "Partition Function of the Three-Demensional Zamolodchikov Model," Phys. Rev. Lett. 53, 1795 (1984); "The Yang-Baxter Equations and the Zamolodchikov Model," Physica D (Amsterdam) 18, 321-347 (1986).
17. A. B. Zamolodchikov, "Tetrahedron Equations and Integrable Systems in Three Dimensional Space," Zh. Eksp. Teor. Fiz. 79, 641-664 (1980) [Sov. Phys. JETP 52, 325 (1980)]; "Tetrahedron Equations and the Relativistic S Matrix of Straight Strings in 2 + 1 Dimensions," Commun. Math. Phys. 79, 489-505 (1981).
18. A. B. Zamolodchikov, "Tetrahedron Equations and Integrable Systems in Three Dimensional Space," Zh. Eksp. Teor. Fiz. 79, 641-664 (1980) [Sov. Phys. JETP 52, 325 (1980)]; "Tetrahedron Equations and the Relativistic S Matrix of Straight Strings in 2 + 1 Dimensions," Commun. Math. Phys. 79, 489-505 (1981).
19. A. B. Zamolodchikov, "Tetrahedron Equations and Integrable Systems in Three Dimensional Space," Zh. Eksp. Teor. Fiz. 79, 641-664 (1980) [Sov. Phys. JETP 52, 325 (1980)]; "Tetrahedron Equations and the Relativistic S Matrix of Straight Strings in 2 + 1 Dimensions," Commun. Math. Phys. 79, 489-505 (1981).
20. V. V. Bazhanov and R. J. Baxter, "New Solvable Lattice Models in Three Dimensions," J. Stat. Phys. 69,453-485 (1992).
21. R. J. Baxter and P. J. Forrester, "Is the Zamolodchikov Model Critical?" J. Phys. A 18, 1483-1497 (1985).
22. J.-M. Maillet and F. Nijhoff, "Integrability for Multidimensional Lattice Models," Phys. Lett. B 224, 389-396 (1989).
23. R. M. Kashaev, "On Discrete Three-Dimensional Equations Associated with the Local Yang-Baxter Relation," Lett. Math. Phys. 35, 389-397 (1996).
24. S. M. Sergeev, "Solutions of the Functional Tetrahedron Equation Connected with the Local Yang-Baxter Equation for the Ferro-electric Condition," Lett. Math. Phys. 45 (2), 113-119 (1997); I. G. Korepanov, R. M. Kashaev, and S. M. Sergeev, "Functional Tetrahedron Equation," Teor. Mat. Fiz. 117 (3), 370-384 (1998).
25. S. M. Sergeev, "Solutions of the Functional Tetrahedron Equation Connected with the Local Yang-Baxter Equation for the Ferro-electric Condition," Lett. Math. Phys. 45 (2), 113-119 (1997); I. G. Korepanov, R. M. Kashaev, and S. M. Sergeev, "Functional Tetrahedron Equation," Teor. Mat. Fiz. 117 (3), 370-384 (1998).
26. S. M. Sergeev, V. V. Mangazeev, and Yu. G. Stroganov, "Vertex Reformulation of the Bazhanov-Baxter Model," J. Stat. Phys. 82, 31-50 (1996).
27. V. Bazhanov, A. Bobenko, and N. Reshetikhin, "Quantum Discrete Sine-Gordon Model at Roots of 1," Commun. Math. Phys. 175, 377-400 (1996).
28. R. Hirota, "Discrete Analogue of a Generalized Toda Equation," Proc. Jpn. Acad. 50 (11), 3785-3791 (1981).
29. E. K. Sklyanin, in Functional Bethe Ansatz in Integrable and Superintegrable Systems, Ed. by B. A. Kupershmidt (World Sci., Singapore, 1990), pp. 8-33; "Quantum Inverse Scattering Method. Selected Topics," in Quantum Group and Quantum Integrable Systems, Nankai Lectures in Mathematical Physics, Ed. by Mo-Lin Ge (World Sci., Singapore, 1992), pp. 63-97; "Separation of Variables. New Trends," Progr. Theor. Phys. 118 (Suppl), 35-60 (1995).
30. E. K. Sklyanin, in Functional Bethe Ansatz in Integrable and Superintegrable Systems, Ed. by B. A. Kupershmidt (World Sci., Singapore, 1990), pp. 8-33; "Quantum Inverse Scattering Method. Selected Topics," in Quantum Group and Quantum Integrable Systems, Nankai Lectures in Mathematical Physics, Ed. by Mo-Lin Ge (World Sci., Singapore, 1992), pp. 63-97; "Separation of Variables. New Trends," Progr. Theor. Phys. 118 (Suppl), 35-60 (1995).
31. E. K. Sklyanin, in Functional Bethe Ansatz in Integrable and Superintegrable Systems, Ed. by B. A. Kupershmidt (World Sci., Singapore, 1990), pp. 8-33; "Quantum Inverse Scattering Method. Selected Topics," in Quantum Group and Quantum Integrable Systems, Nankai Lectures in Mathematical Physics, Ed. by Mo-Lin Ge (World Sci., Singapore, 1992), pp. 63-97; "Separation of Variables. New Trends," Progr. Theor. Phys. 118 (Suppl), 35-60 (1995).
32. F. A. Smirnov, Preprint (2001), math-ph/0109013.
33. T. Miwa, "On Hirota's Difference Equations," Proc. Jpn. Acad., Ser. A 58, 9-12 (1982).
34. L. D. Faddeev, "Discrete Heisenberg-Weyl Group and Modular Group," Lett. Math. Phys. 34, 249-254 (1995); "Modular Double of Quantum Group," in Proceedings of Conférence Moshé Flato, Dijon, France, 1999 (Kluwer Academic, Dordrecht, 2000), Vol. 1, pp. 149-156, Math. Phys. Stud., Vol. 21.
35. L. D. Faddeev, "Discrete Heisenberg-Weyl Group and Modular Group," Lett. Math. Phys. 34, 249-254 (1995); "Modular Double of Quantum Group," in Proceedings of Conférence Moshé Flato, Dijon, France, 1999 (Kluwer Academic, Dordrecht, 2000), Vol. 1, pp. 149-156, Math. Phys. Stud., Vol. 21.
36. q(s-fraktur sign and l-fraktur sign(2, ℝ), the Modular Double, and the Multiparticle q-Deformed Toda Chains," Commun. Math. Phys. 225, 573-609 (2002).
37. L. D. Faddeev, R. M. Kashaev, and A. Yu. Volkov, "Strongly Coupled Quantum Discrete Liouville Theory. I: Algebraic Approach and Duality," Commun. Math. Phys. 219, 199-219 (2001); L. D. Faddeev and R. M. Kashaev, "Strongly Coupled Quantum Discrete Liouville Theory. II: Geometric Interpretation of the Evolution Operator," J. Phys. A 35, 4043-4048 (2002).
38. L. D. Faddeev, R. M. Kashaev, and A. Yu. Volkov, "Strongly Coupled Quantum Discrete Liouville Theory. I: Algebraic Approach and Duality," Commun. Math. Phys. 219, 199-219 (2001); L. D. Faddeev and R. M. Kashaev, "Strongly Coupled Quantum Discrete Liouville Theory. II: Geometric Interpretation of the Evolution Operator," J. Phys. A 35, 4043-4048 (2002).
39. V. V. Mangazeev and Yu. G. Stroganov, "Elliptic Solution for Modified Tetrahedron Equation," Mod. Phys. Lett. A 8, 3475-3482 (1993); V. V. Mangazeev, S. M. Sergeev, and Yu. G. Stroganov, "New Series of 3d Lattice Integrable Models," Int. J. Mod. Phys. A 9, 5517-5530 (1994).
40. V. V. Mangazeev and Yu. G. Stroganov, "Elliptic Solution for Modified Tetrahedron Equation," Mod. Phys. Lett. A 8, 3475-3482 (1993); V. V. Mangazeev, S. M. Sergeev, and Yu. G. Stroganov, "New Series of 3d Lattice Integrable Models," Int. J. Mod. Phys. A 9, 5517-5530 (1994).
41. J.-M. Maillard and S. Sergeev, "Three Dimensional Integrable Models Based on Modified Tetrahedron Equations and Quantum Dilogarithm," Phys. Lett. B 405, 55-63 (1997).
42. G. von Gehlen, S. Pakuliak, and S. Sergeev, Preprint MPIM 2002-102, J. Phys. A (in press).