The Multivalued Free-Field Maps of Liouville and Toda Gravities

TL;DR

This paper explores the relationship between interacting and free fields in Liouville and Toda gravity theories, providing explicit constructions of wavefunctions and analyzing their invariance under Weyl groups.

Contribution

It introduces explicit canonical transformations using intertwining operators, revealing invariance properties and detailed operator-state mappings in these gravity models.

Findings

Wavefunctions constructed explicitly and shown to be Weyl group invariant

Analysis of operator-state maps and Seiberg bounds

Spectra derived from free-field spectra divided by Weyl group actions

Abstract

Liouville and Toda gravity theories with non-vanishing interaction potentials have spectra obtained by dividing the free-field spectra for these cases by the Weyl group of the corresponding $A_1$ or $A_2$ Lie algebra. We study the canonical transformations between interacting and free fields using the technique of intertwining operators, giving explicit constructions for the wavefunctions and showing that they are invariant under the corresponding Weyl groups. These explicit constructions also permit a detailed analysis of the operator-state maps and of the nature of the Seiberg bounds.