Soliton Solutions on Noncommutative Orbifold $ T^2/Z_4

TL;DR

This paper constructs explicit soliton solutions on a noncommutative orbifold $T^2/Z_4$ using projectors with $Z_4$ symmetry, providing a comprehensive set of solutions with continuous eigenvalue functions.

Contribution

It introduces a new method to explicitly construct all projectors with minimal trace on $T^2/Z_4$, including their integral and derivative expressions, representing soliton solutions.

Findings

Constructed projectors with $Z_4$ symmetry.

Eigenvalue functions are continuous with respect to parameters.

Provided integral and derivative expressions for projectors.

Abstract

In this paper, we explicitly construct a series of projectors on integral noncommutative orbifold $T^2/Z_4$ by extended $GHS$ constrution. They include integration of two arbitary functions with $Z_4$ symmetry. Our expressions possess manifest $Z_{4}$ symmetry. It is proved that the expression include all projectors with minimal trace and in their standard expansions, the eigen value functions of coefficient operators are continuous with respect to the arguments $k$ and $q$. Based on the integral expression, we alternately show the derivative expression in terms of the similar kernal to the integral one.Since projectors correspond to soliton solutions of the field theory on the noncommutative orbifold, we thus present a series of corresponding solitons.