Algebraic connections on parallel universes

Abstract

For any manifold $M$, we introduce a $\ZZ $-graded differential algebra $Ξ$, which, in particular, is a bi-module over the associative algebra $C(M\cup M)$. We then introduce the corresponding covariant differentials and show how this construction can be interpreted in terms of Yang-Mills and Higgs fields. This is a particular example of noncommutative geometry. It differs from the prescription of Connes in the following way: The definition of $Ξ$ does not rely on a given Dirac-Yukawa operator acting on a space of spinors.