Soliton Solutions on Noncommutative Orbifold $T^{2N}/G$

TL;DR

This paper constructs soliton solutions on noncommutative orbifolds by developing a generalized representation and projection operators, providing a comprehensive set of solutions in the noncommutative geometric setting.

Contribution

It introduces a generalized $Kq$ representation and derives all soliton solutions on the noncommutative orbifold $T^{2N}/G$, extending the understanding of projections in noncommutative geometry.

Findings

Constructed common eigenstates of translation operators.

Established the generalized $Kq$ representation on $T^{2N}.

Derived the complete set of projection operators for $T^{2N}/G$.

Abstract

In this paper, we construct the common eigenstates of "translation" operators $\{U_{s}\}$ and establish the generalized $Kq$ representation on integral noncommutative torus $T^{2N}$. We then study the finite rotation group $G$ in noncommutative space as a mapping in the $Kq$ representation and prove a Blocking Theorem. We finally obtain the complete set of projection operators on the integral noncommutative orbifold $T^{2N}/G$ in terms of the generalized $Kq$ representation. Since projectors are soliton solutions on noncommutative space in the limit $\alpha ^{\prime}B_{ij}\to \infty (\Theta_{ij}/\alpha ^{\prime}\to 0)$, we thus obtain all soliton solutions on that orbifold $T^{2N}/G$.