Target Space Duality between Simple Compact Lie Groups and Lie Algebras under the Hamiltonian Formalism: I. Remnants of Duality at the Classical Level

TL;DR

This paper explores classical target-space duality between simple compact Lie groups and their Lie algebras within the Hamiltonian framework, analyzing the canonical transformations and geometric structures involved.

Contribution

It constructs and examines a canonical transformation that demonstrates classical target-space duality between Lie groups and Lie algebras, highlighting features of the duality at the Hamiltonian level.

Findings

Canonical transformation $\

Analysis of the domain and image of the transformation

Insights into the geometry of the T-dual structure on the Lie algebra

Abstract

It has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie group $G$ with a bi-invariant metric and a generating function $\Gamma$ suggested in the physics literature, we follow the above line of thought and work out the canonical transformation $\Phi$ generated by $\Gamma$ together with an $\Ad$-invariant metric and a B-field on the associated Lie algebra $\frak g$ of $G$ so that $G$ and $\frak g$ form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation $\Phi$ including a careful analysis of its domain and image. The geometry of the T-dual structure on $\frak g$ is lightly touched.