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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.

Thank you!

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Related: – Qmechanic Nov 8 '12 at 6:37
@Qmechanic: Why is that question related? – user1504 Nov 8 '12 at 20:00

I wondered about this one myself a while back. I'm not absolutely positive about this but it is definitely in the ballpark.

Here's what I know for the background:

  • I believe the first paper on exotic spheres in physics was by Witten [Commun. Math. Phys., 100, 197–229 (1985)] and centered around the idea that exotic spheres can be interpreted as gravitational instantons. You can download the paper here.

  • Randy Baadhio spent some quality time in the early 90's on this idea and the physical interpretation of exotic spheres as instantons. I believe this paper was the first one he wrote on the subject.

  • Baez also did a pretty good walk-through of exotic spheres in week141 and eventually provides a bit of physical motivation in a string theory context about how they pop up.

More or less here's the interpretation: (and keep in mind Witten gives the basic physical interpretation you are asking about in the 2nd paragraph of the abstract to his paper.)

If you are stuck with gravitational anomalies in your theory (probably caused by a one loop diagram somewhere in there), these global gravitational anomalies lead to restrictions on the fermion zero modes of an instanton field and those instantons can be put into correspondence with exotic spheres.

Hopefully someone else can jump in and sharpen that explanation up for me.

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Thanks for the reference, I will read the reference you pointed out and see if they can help me understand more. – Yingfei Gu Mar 29 '15 at 18:31
I guess in general, i am curious about the difference between topology and differential topology. I guess in most case(for some deep topological reason), we are able to assign a smooth structure to continuous stuff. However, there seems to appear some exotic examples that we are not able to, for example this exotic sphere. Another example that appears to me recently is the subject of cobordism theory as a generalized (co)homology theory. I am curious about whether there are some simple physical model can clearly show the difference. – Yingfei Gu Mar 29 '15 at 18:37

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