Algebraic connections on parallel universes

TL;DR

This paper introduces a novel $bZ$-graded differential algebra on manifolds, connecting noncommutative geometry with gauge theories like Yang-Mills and Higgs fields without relying on Dirac operators.

Contribution

It presents a new algebraic framework for noncommutative geometry that generalizes existing approaches by not depending on Dirac-Yukawa operators.

Findings

Defines a $bZ$-graded differential algebra $oldsymbol{ ext{ extXi}}$ for any manifold.

Establishes a connection between this algebra and gauge fields such as Yang-Mills and Higgs.

Provides a new perspective on noncommutative geometry independent of Dirac operators.

Abstract

For any manifold $M$, we introduce a $\ZZ $-graded differential algebra $\Xi$, 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 $\Xi$ does not rely on a given Dirac-Yukawa operator acting on a space of spinors.