Bethe Ansatz and Classical Hirota Equations 1

A brief non-technical review of the recent study [1] of classical integrable structures in quantum integrable systems is given. It is explained how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's Q-operators, etc) with elements of classical non-linear integrable difference equations (τ-functions, Baker-Akhiezer functions, etc). The nested Bethe ansatz equations for A k−1-type models emerge as discrete time equations of motion for zeros of classical τ-functions and Baker-Akhiezer functions. The connection with discrete time Ruijsenaars-Schneider system of particles is discussed. At present it is argued that classical and quantum integrable models have much deeper interrelations than any kind of a naive " classical limit ". In [1], a particular aspect of this phenomenon has been analysed in detail: Bethe equations, which are usually considered as a tool inherent to quantum integrability, arise naturally as a result of solving entirely classical non-linear discrete time integrable equations. This suggests an intriguing connection between integrable quantum field theories and classical soliton equations in discrete time. Here we outline main ideas of the paper [1] omitting all technical details. To save the space, the material is organized as a " quantum-classical dictionary " supplied with brief comments. 1. Eigenvalues of quantum transfer matrices = (classical) τ-functions. Due to the Yang-Baxter equation the transfer matrices commute for all values of the spectral parameters in the auxiliary space (AS) 2 : [T A (u), T A ′ (u ′)] = 0. This property allows one to diagonalize them simultaneously. From now on we use this diagonal representation. The identification with τ-function is justified in the next item. 2. Fusion rules = Hirota's difference equation. The fusion procedure in the AS gives rise to a family of commuting transfer matrices T A (u) with the same quantum space. They obey a number of fusion relations [4] which can be recast into the model-independent bilinear form [5]. Let T a s (u) be the transfer matrix for the rectangular Young diagram of length s and height a, then it holds T a s (u + 1)T a s (u − 1) − T a s+1 (u)T a s−1 (u) = T a+1 s (u)T a−1 s (u). (1) Remarkably, this equation coincides with Hirota's bilinear difference equation (HBDE) [6] which is known to unify the majority of soliton equations, both discrete and continuous. Fusion of more complicated representations in the AS is described by …