A noncommutative sigma-model

TL;DR

This paper explores a novel noncommutative sigma-model where both the space-time and string world-sheet are noncommutative, focusing on noncommutative tori to analyze maps, equations, critical points, and partition functions.

Contribution

It introduces a noncommutative sigma-model framework and provides detailed analysis for noncommutative tori, including existence of maps, equations, and critical point classification.

Findings

Determined conditions for maps between noncommutative tori.

Derived Euler-Lagrange equations for the model.

Classified critical points and studied the partition function.

Abstract

We begin to study a sigma-model in which both the space-time manifold and the two-dimensional string world-sheet are made noncommutative. The most precise results apply to the case where both the space-time manifold and the two-dimensional string world-sheet are replaced by noncommutative tori. In this situation, we are able to determine when maps between such noncommutative tori exist, to derive the Euler-Lagrange equations, to classify many of the critical points of the Lagrangian, and to study the associated partition function.