QUANTUM INTEGRABLE MODELS AND DISCRETE CLASSICAL HIROTA EQUATIONS

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's biline ar difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic s olutions of Hirota's equation give a complete set of eigenvalues of th e quantum transfer matrices. Eigenvalues of Baxter's Q-operator are so lutions to the auxiliary linear problems for classical Hirota's equati on. The elliptic solutions relevant to the Bethe ansatz are studied. T he nested Bethe ansatz equations for A(k-1)-type models appear as disc rete time equations of motions for zeros of classical tau-functions an d Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a ne w determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T-Q-relation are derived.