Magnetic monopoles, squashed 3-spheres and gravitational instantons from exotic R⁴

We show that, in some limit, gravitational instantons correspond to exotic smooth geometry on R^4. The geometry of this exotic R^4 represents magnetic monopoles of Polyakov-'t Hooft type and the BPS condition allows for the generation of the charges from the gravitational sources of exotic R^4. Higgs field is, as usual, present in the monopole configurations. We indicate some possible scenarios in cosmology, particle physics and condensed matter where pure SU(2) Yang-Mills theory on the exotic R^4 aquires non-zero mass for its gauge boson when the smoothness is changed to the standard R^4 and, then, it is described by Yang-Mills Higgs theory. The derivation of the results is based on the quasi-modularity of the expressions and the geometry of foliations in the limit.