Classification of spin structures on the noncommutative n-torus

TL;DR

This paper classifies all spin structures on the noncommutative n-torus, revealing 2^n possible structures and their relation to classical cases, with a focus on unitary equivalences and the influence of Connes' theorem.

Contribution

It provides a complete classification of spin structures on the noncommutative n-torus, extending classical results and exploring their unitary equivalences and dependence on Connes' theorem.

Findings

Noncommutative n-torus has 2^n spin structures.

Classification aligns with isospectral deformations of classical tori.

For n>3, classification depends on Connes' spin manifold theorem.

Abstract

We classify spin structures on the noncommutative torus, and find that the noncommutative n-torus has 2^n spin structures, corresponding to isospectral deformations of spin structures on the commutative n-torus. For n>3 the classification depends on Connes' spin manifold theorem. In addition, we study unitary equivalences of these spin structures.