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

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 $A_{\mu i}$ and 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 $A_{\mu i}$ 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 $M_{2j+1}(C)$ for $j$ any non-negative half-integer, and in this case only integer spins $0\le l\le 2j$ 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 $0,1,\sqrt{3}$ respectively, which we conjecture could arise in connection with an approximate $SU(2)$ flavour symmetry.