Classification of spin structures on the noncommutative $n$-torus

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 $n\ge 4$ the classification depends on Connes’ spin manifold theorem. In addition, we study unitary equivalences of these spin structures.