Superintegrability, symmetry and point particle T-duality

TL;DR

This paper explores how integrability and symmetry reveal that point particle T-duality is more widespread and physically relevant than previously thought, connecting various dynamical systems through a web of dualities.

Contribution

It demonstrates that point particle T-duality applies to physically relevant systems and uncovers a web of dualities among maximally superintegrable backgrounds.

Findings

T-duality connects Coulomb scattering in flat and curved spaces.

T-duality relates Calogero-Moser dynamics in different geometries.

Knowing the Hamiltonian alone does not reveal the space's curvature.

Abstract

We show that the ideas related to integrability and symmetry play an important role not only in the string T-duality story but also in its point particle counterpart. Applying those ideas, we find that the T-duality seems to be a more widespread phenomenon in the context of the point particle dynamics than it is in the string one; moreover, it concerns physically very relevant point particle dynamical systems and not just somewhat exotic ones fabricated for the purpose. As a source of T-duality examples, we consider maximally superintegrable spherically symmetric electro-gravitational backgrounds in $n$ dimensions. We then describe in detail four such spherically symmetric dynamical systems which are all mutually interconnected by a web of point particle T-dualities. In particular, the dynamics of a charged particle scattered by a repulsive Coulomb potential in a flat space is T-dual to the dynamics of the Coulomb scattering in the space of constant negative curvature, but it is also T-dual to the (conformal) Calogero-Moser inverse square dynamics both in flat and hyperbolic spaces. Thus knowing just the Hamiltonian dynamics of the scattered particle cannot give us an information about the curvature of the space in which the particle moves.