Relativistic Toda systems

Abstract

Abstract. We present and study Poincare-invariant generalizations of the Galilei-invariant Toda systems. The classical nonperiodic systems are solved by means of an explicit action-angle transformation.

Exact quantization conditions for the relativistic Toda lattice

“…[29,67]). 5 Another interesting problem is the relation between the g quantization conditions obtained for the relativistic Toda lattice (and also, presumably, for the Goncharov-Kenyon integrable system) and the approach based on quantizing the mirror curve presented in [29], which involves a single quantization condition encoded in the vanishing of a (quantum) theta function.…”

“…The corresponding monodromy matrix is 5) and it can be shown that 2t(z; R) = Tr T(z; R) (2.6) satisfies the commutation relation…”

“…In the case of the Toda lattice, these conditions were obtained in [1][2][3][4], by finding appropriate solutions of the one-dimensional quantum Baxter equation associated to the integrable system. The Toda lattice has a relativistic generalization [5], and the techniques of separation of variables lead to a Baxter equation involving difference operators [6,7]. In the relativistic case, the solution to the quantum Baxter equation is not known, even for the two-particle lattice.…”

Toda Systems, Cluster Characters, and Spectral Networks

“…Following [GK13], by a cluster integrable system we mean an integrable system whose phase space is a cluster variety equipped with its canonical Poisson structure. Examples include those studied in [GK13,FM14,HKKR00,Wil13a], and generally encompass those referred to as relativistic integrable systems in the literature [Rui90]. Roughly, cluster varieties are Poisson varieties whose coordinate rings (cluster algebras) are equipped with a canonical partial basis of functions called cluster variables [FZ02].…”

Off-diagonal Bethe Ansatz solution of the τ 2-model

“…Among quantum integrable models, the τ 2 (BBS)-model [1] plays a special role for its unique properties, e.g., it is one of the simplest quantum integrable models associated with cyclic representation of the Weyl algebra; it allows to include multiple inhomogeneity parameters on each single site without breaking the integrability of the model; and more interestingly, the τ 2 -model under certain parameter constraint is highly related to some other integrable models such as the chiral Potts model [2][3][4][5][6][7] and the relativistic quantum Toda chain model [8]. Many papers have appeared in literature for such connections and many efforts have been made to obtain the solutions of chiral Potts model by solving the τ 2 -model with a recursive functional relation [9][10][11][12].…”