$Z_2\times Z_2$ Lattice as a Connes-Lott-Quantum Group Model

TL;DR

This paper combines quantum group methods with noncommutative geometry on a $Z_2\times Z_2$ lattice, interpreting Higgs fields as discrete spacetime components and analyzing the resulting geometric and algebraic structures.

Contribution

It introduces a novel approach merging quantum groups with Connes' noncommutative geometry on a discrete lattice, providing new insights into the structure of elementary particle models.

Findings

Derived a natural Dirac operator on the $Z_2\times Z_2$ lattice.

Interpreted Higgs fields as discrete spacetime elements.

Computed noncommutative cohomology and moduli of flat connections.

Abstract

We apply quantum group methods for noncommutative geometry to the $Z_2\times Z_2$ lattice to obtain a natural Dirac operator on this discrete space. This then leads to an interpretation of the Higgs fields as the discrete part of spacetime in the Connes-Lott formalism for elementary particle Lagrangians. The model provides a setting where both the quantum groups and the Connes approach to noncommutative geometry can be usefully combined, with some of Connes' axioms, notably the first-order condition, replaced by algebraic methods based on the group structure. The noncommutative geometry has nontrivial cohomology and moduli of flat connections, both of which we compute.