A Noncommutative Sigma Model

TL;DR

This paper develops a noncommutative generalization of string theory concepts by replacing classical parameter spaces with noncommutative tori, deriving a new action, and analyzing mappings and quantum properties of these spaces.

Contribution

It introduces a noncommutative sigma model, extending classical string theory notions to noncommutative tori, including a generalized Polyakov action and existence theorems for mappings.

Findings

Derived a noncommutative Polyakov action.

Analyzed *-homomorphisms between quantum tori.

Calculated partition functions and path integrals.

Abstract

We replaced the classical string theory notions of parameter space and world-time with noncommutative tori and consider maps between these spaces. The dynamics of mappings between different noncommutative tori were studied and a noncommutative generalization of the Polyakov action was derived. The quantum torus was studied in detail as well as *-homomorphisms between different quantum tori. A finite dimensional representation of the quantum torus was studied and the partition function and other path integrals were calculated. At the end we proved existence theorems for mappings between different noncommutative tori.