Classical and Quantum Solitons in the Symmetric Space Sine-Gordon Theories

TL;DR

This paper constructs and quantizes soliton solutions in symmetric space sine-Gordon theories, revealing their internal moduli space, spectrum, and quantum states as fuzzy geometric co-adjoint orbits.

Contribution

It introduces a method to quantize solitons in symmetric space sine-Gordon theories, linking classical solutions to quantum states via co-adjoint orbits.

Findings

Solitons are kinks with internal moduli space as co-adjoint orbits.

Classical spectrum includes perturbative fluctuations as kink charge diminishes.

Quantization yields quantum states as symmetric tensor representations, forming a fuzzy geometric structure.

Abstract

We construct the soliton solutions in the symmetric space sine-Gordon theories. The latter are a series of integrable field theories in 1+1-dimensions which are associated to a symmetric space F/G, and are related via the Pohlmeyer reduction to theories of strings moving on symmetric spaces. We show that the solitons are kinks that carry an internal moduli space that can be identified with a particular co-adjoint orbit of the unbroken subgroup H of G. Classically the solitons come in a continuous spectrum which encompasses the perturbative fluctuations of the theory as the kink charge becomes small. We show that the solitons can be quantized by allowing the collective coordinates to be time-dependent to yield a form of quantum mechanics on the co-adjoint orbit. The quantum states correspond to symmetric tensor representations of the symmetry group H and have the interpretation of a fuzzy geometric version of the co-adjoint orbit. The quantized finite tower of soliton states includes the perturbative modes at the base.