I am thinking about Kaluza Klein theory in the 3 dimensional lens spaces. These have an isometry group SU(2)xU(1), generically, and in some way interpolate between the extreme cases of manifolds $S^2 \times S^1$ and $S^3$ (or, arguably depending of your parametrisation, across $S^2 \times S^1 ... S^3 ... S^1 \times S^2$).
My question is, simply, if the application of Kaluza Klein construction forces different coupling constants to the two groups $SU(2)$ and $U(1)$, and then a kind of discrete Weinberg angle, or if they have always the same coupling.
Note that when you build a lens space bundle over CP2 (which is equivalent to a $S^1$ bundle over $CP^2$ x $CP^1$, if I read correctly a remark of Atiyah reported by Kreck and Stolz), you get an isometry group of SU(3) times SU(2) times (1), so the same question could be asked here, in this more widely known setup. Of course that odd dimensional spaces do not carry chirality, but my question is just about having different couplings, that is all. It could be interesting, as a plus, to compare with coupling relationships coming from Non Commutative Geometry...
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