Basically an exotic sphere is topologically a sphere, but doesn't look like a one. Or more accurately:
homeomorphic but not diffeomorphic to the standard Euclidean n-sphere
The first exotic spheres were constructed by John Milnor (1956) in dimension n = 7 as S3-bundles over S4.
It looks like an SU(2) gauge field on S4? But I don't know more about this? I guess there is lots of expertise here. Could you give a physical interpretation of this 7 sphere? Just to make it more intuitive.