Noncommutative Supertori in Two Dimensions

TL;DR

This paper explores the construction of noncommutative supertori in two-dimensional superspaces, focusing on how different supersymmetry and spin structures influence their realizability.

Contribution

It demonstrates the conditions under which noncommutative supertori with various spin structures can be constructed in N=(1,1) and N=(2,2) supersymmetric settings.

Findings

Noncommutative supertori with odd spin structure only in N=(2,2) broken to N=(1,1)

Noncommutative supertori with even spin structures in both N=(1,1) and N=(2,2)

Implementation of spin structures via translational properties in operator formalism

Abstract

First we consider the deformations of superspaces with N=(1,1) and N=(2,2) supersymmetries in two dimensions. Among these the construction of noncommutative supertorus with odd spin structure is possible only in the case of N=(2,2) supersymmetry broken down to N=(1,1). However, for the even spin structures the construction of noncommutative supertorus is possible for both N=(1,1) and N=(2,2) cases. The spin structures are realized by implementing the translational properties along the cycles of commutative supertorus in the operator version: Odd spin structure is realized by the translation in the fermionic direction in the same manner as in the construction of noncommutative torus, and even spin structures are realized with appropriate versions of the spin angular momentum operator.