Physics on manifolds with exotic differential structures

A given topological manifold can sometimes be endowed with inequivalent differential structures. Physically this means that what is meant by a differentiable function (smooth) is simply different for observers using inequivalent differential structures. {The 7-sphere, $\bS^7$, was the first topological manifold where the possibility of inequivalent differential structures was discovered \cite{Milnor}.} In this paper, we examine the import of inequivalent differential structures on the physics of fields obeying the Dirac equation on $\bS^7$. { $\bS^7$ is a fibre bundle of the 3-sphere as a fibre on the 4-sphere as a base. We consider the Kaluza-Klein limit of such a fibre bundle which reduces to a SO(4) Yang-Mills gauge theory over $\bS^4$. We find, for certain specific symmetric set of gauge potentials, that the spectrum of the Dirac operator can be computed explicitly, for each choice of the differential structure. Hence identical topological manifolds have different physical laws. We find this the most important conclusion of our analysis.