Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model

Abstract

Using the framework of quantum Riemannian geometry, we show that gravity on the product of spacetime and a fuzzy sphere is equivalent under minimal assumptions to gravity on spacetime, an su2-valued Yang-Mills field Aμi and a real-symmetric-matrix valued Liouville-sigma model field hij for gravity on the fuzzy sphere. Moreover, a massless real scalar field on the product appears as a tower of scalar fields on spacetime, with one for each internal integer spin l representation of SU(2), minimally coupled to Aμi and with mass depending on l and the fuzzy sphere size. For discrete values of the deformation parameter, the fuzzy spheres can be reduced to matrix algebras M2j+1(ℂ) for j any non-negative half-integer, and in this case only integer spins 0 ≤ l ≤ 2j appear in the multiplet. Thus, for j = 1 a massless field on the product appears as a massless SU(2) internal spin 0 field, a massive internal spin 1 field and a massive internal spin 2 field, in mass ratio 0, 1,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{3} $$\\end{document} respectively, which we conjecture could arise in connection with an approximate SU(2) flavour symmetry.