Spectral triple with real structure on fuzzy sphere

TL;DR

This paper constructs a real structure on the fuzzy sphere with KO dimension-4, exploring its spectral properties and implications for modeling physics beyond the standard model.

Contribution

It introduces a real structure on the fuzzy sphere that satisfies KO dimension-4 conditions and discusses its potential for toy models of gauge theories beyond the standard model.

Findings

Real structure consistent with spectral data on fuzzy sphere

Enlarged symmetry group from SO(3) to O(3)

First order condition violation suggests new physics models

Abstract

Here we have illustrated the construction of a real structure on fuzzy sphere $S^2_*$ in its spin-1/2 representation. Considering the SU(2) covariant Dirac and chirality operator on $S^2_*$ given by Watamura et. al. in [6], we have shown that the real structure is consistent with other spectral data for KO dimension-4 fulfilling the zero order condition, where we find it necessary to enlarge the symmetry group from SO(3) to the full orthogonal group O(3). However the first order condition is violated thus paving the way to construct a toy model for an SU(2) gauge theory to capture some features of physics beyond standard model following Connes et.al. [9].