Elliptic Algebra and Integrable Models for Solitons on Noncummutative Torus ${\cal T}$

TL;DR

This paper explores the algebraic structure of solitons on a noncommutative torus, linking elliptic algebra, integrable models, and theta functions to advance understanding of soliton dynamics in noncommutative geometry.

Contribution

It introduces a novel algebraic framework for solitons on noncommutative tori, embedding the Heisenberg group into integrable model transfer matrices and generalizing to generic theta functions.

Findings

Established a basis of the Hilbert space using theta functions.

Embedded the Heisenberg group into transfer matrix representations.

Generalized results to arbitrary theta functions.

Abstract

We study the algebra ${\cal A}_n$ and the basis of the Hilbert space ${\cal H}_n$ in terms of the $\theta$ functions of the positions of $n$ solitons. Then we embed the Heisenberg group as the quantum operator factors in the representation of the transfer matrice of various integrable models. Finally we generalize our result to the generic $\theta$ case.