Canonical equivalence of non-isometric sigma models and Poisson-Lie T-duality

TL;DR

This paper proves that Poisson-Lie T-duality between certain sigma models is a canonical transformation, demonstrating invariance of Poisson brackets and extending understanding of dualities without isometries.

Contribution

It establishes the canonical nature of Poisson-Lie T-duality transformations and derives the conditions for generating functionals in non-isometric sigma models.

Findings

Poisson-Lie T-duality is a canonical transformation.

Explicit invariance of classical Poisson brackets is demonstrated.

Conditions for generating functionals in non-isometric models are derived.

Abstract

We prove that a transformation, conjectured in our previous work, between phase-space variables in $\s$-models related by Poisson-Lie T-duality is indeed a canonical one. We do so by explicitly demonstrating the invariance of the classical Poisson brackets. This is the first example of a class of $\s$-models with no isometries related by canonical transformations. In addition we discuss generating functionals of canonical transformations in generally non-isometric, bosonic and supersymmetric $\s$-models and derive the complete set of conditions that determine them. We apply this general formalism to find the generating functional for Poisson-Lie T-duality. We also comment on the relevance of this work to D-brane physics and to quantum aspects of T-duality.