Non-easy On The Rail-shaped Projection Operator

With the development of string theory, noncommutative field theory has played a more and more important role in modern physics, especially in understanding the structureof space time, quantum Hall effect and high temperature superconductivity. In this thesis, we concentrate to the construction of project operators on noncommutative orbifoldT²/Z_N, which correspond to the solitonic solution of field theory on such orbifold.First, we introduce the Groenewold-Moyal *-product operation, which is a basic construction in the description of fields on n dimensional noncommutative flat space R_θⁿ. Using such *-product instead of ordinary one, we can obtain an noncommutative version of field theory, i. e. an field theory on noncommutative flat space. Next, we will discuss the structure of noncommutative orbifold T² / Z_N thoroughly. Because of the flatness of noncommutative orbifold T²/Z_N, the field theory on such space have many similarities to that of defined on noncommutative flat space. As an example, we analyze the field theory on one dimensional orbifold T¹/Z₂, which can be generalized to two dimensional orbifold T²/Z_N easily. After endowed with noncommutative structure, many properties of this model has been discussed in detail. For the noncommutative torus with noncommutative parameters inverse each other, we construct explicitly their structure algebras A_θand A_1/θ respectively. Due to the fact that they are equivalence in the sense of Morita, we construct the Morita equivalence bimodule, which play an essential role in construction of project operator.In the process of construction of solitonic solutions, we introduce the |k,q> representationwith Z₆ symmetry. We give exclusive attention to the behavior of such |k,q> representation under the action of cycle group Z₆, i. e. the action of rotation of 2π/6. As a consequence, with the double rotation ofπ/3 and rotation of 2π/3 respectively, the resulted representations should coincide with each other, and this leads to a nontrivial identity- Gauss summation formula, for which the proof is very difficult to comprehendin the framework of number theory. The following chapters is attributed to the construction of projector. By means of Morita equivalence of A_θand A_1/θ, we construct a projector on noncommutative orbifold T²/Z_N. For T²/Z₄, we give the matrix elements of projector explicitly in |k, q> representation. In the explicit form of projector, the inverse of element of A_1/θ algebra b^-1 has been used. In the case that noncommutative parameters is smaller than one, we verify that b^-1 exists and belongs to A_1/θ.