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

A bstract 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 su 2 -valued Yang-Mills field A μi and a real-symmetric-matrix valued Liouville-sigma model field h ij 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 A μi 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 M 2 j +1 (ℂ) for j any non-negative half-integer, and in this case only integer spins 0 ≤ l ≤ 2 j 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 3 respectively, which we conjecture could arise in connection with an approximate SU(2) flavour symmetry.