Hirota-Miwa difference equations

introduction

how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's \(Q\)-operators, etc.) with elements of classical nonlinear integrable difference equations (\(\tau\)-functions, Baker-Akhiezer functions, etc.).

The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with the classical Hirota bilinear difference equation.
This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries.
Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices.
The elliptic solutions relevant to the Bethe ansatz are studied.
The nested Bethe ansatz equations for \(A_{k−1}\)-type models appear as discrete time equations of motions for zeros of classical \(\tau\)-functions and Baker-Akhiezer functions.
Determinant representations of the general solution to the bilinear discrete Hirota equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained.

Eigenvalues of Baxter's \(Q\)-operator are solutions to the auxiliary linear problems for the classical Hirota equation.
Difference equations for eigenvalues of the \(Q\)-operators which generalize Baxter's three-term \(T-Q\)-relation are derived

Zabrodin, A. 2012. “Bethe Ansatz and Hirota Equation in Integrable Models.” arXiv:1211.4428 [hep-Th, Physics:math-Ph] (November 19). http://arxiv.org/abs/1211.4428.
Zabrodin, A. V. 1998. “Hirota Equation and Bethe Ansatz.” Theoretical and Mathematical Physics 116 (1) (July 1): 782–819. doi:10.1007/BF02557123.
Wiegmann, P. 1997. “Bethe Ansatz and Classical Hirota Equation.” International Journal of Modern Physics B 11 (01n02) (January 20): 75–89. doi:10.1142/S0217979297000101.
Zabrodin, A. V. 1997. “Hirota’s Difference Equations.” Theoretical and Mathematical Physics 113 (2) (November 1): 1347–1392. doi:10.1007/BF02634165. http://arxiv.org/abs/solv-int/9704001
Zabrodin, A. V. 1997. “Bethe Ansatz and Classical Hirota Equations.” In Proceedings of the Second International A. D. Sakharov Conference on Physics (Moscow, 1996), 627–632. World Sci. Publ., River Edge, NJ. http://arxiv.org/abs/hep-th/9607162.

Krichever, I., O. Lipan, P. Wiegmann, and A. Zabrodin. 1997. “Quantum Integrable Models and Discrete Classical Hirota Equations.” Communications in Mathematical Physics 188 (2): 267–304. doi:10.1007/s002200050165. http://arxiv.org/abs/hep-th/9604080.
T. Miwa, Proc. Japan. Acad. 58, 9 (1982).
R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Jpn. 50, 3785 (1981).