The equivariant Toda lattice (Q1885500)

The article provides a succinct but rather comprehensive insight into some topical problems concerning Toda lattices. The fundamental structures are adaptations from \textit{K. Ueno} and \textit{K. Takashi} [Adv. Stud. Pure Math. 4, 1--95 (1984; Zbl 0577.58020)] or \textit{B. A. Kupershmidt} [Astérisque, 123, Paris: Société Mathématique de France (1985; Zbl 0565.58024)]. The author thoroughly deals with a new kind of reduction of Toda lattice, called the equivariant Toda lattices, defined by the constraint \((\delta_1-\overline \delta_1)L= \nu L\) on the Lax operator \(L\). The equivariant Toda lattice provides a description of the equivariant Gromov-Witten invariants in \(\mathbb{C}\mathbb{P}^1\) in terms of a Lax operator whose coefficients are obtained by explicit recursion (the Toda conjecture). In the paper, the dispersionless limit of the equivariant Toda lattice is related to the equivariant genus 0 Gromov-Witten invariants by explicit formulae. Finally, the equivariant Toda lattices are related to the dressing operator formalisms and a Hamiltonian structure is determined by using a perturbation theory.