Quasi-classical limit of Toda hierarchy and W-infinity symmetries

TL;DR

This paper extends the quasi-classical limit and W-infinity symmetries from the KP hierarchy to the Toda hierarchy, redefining key notions and analyzing the contraction of symmetries in the dispersionless limit.

Contribution

It introduces the quasi-classical limit for the Toda hierarchy, redefining fundamental concepts and exploring the realization and contraction of W-infinity symmetries.

Findings

W-infinity symmetries are realized via rescaled vertex operators.

Symmetries contract to w-infinity in the dispersionless limit.

Redefinitions of dressing operators, Baker-Akhiezer functions, and tau functions are provided.

Abstract

Previous results on quasi-classical limit of the KP hierarchy and its W-infinity symmetries are extended to the Toda hierarchy. The Planck constant $\hbar$ now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions and tau function, are redefined. $W_{1+\infty}$ symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted to $w_{1+\infty}$ symmetries of the dispersionless hierarchy through their action on the tau function. (A few errors in the earlier version is corrected.)