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I heard a saying that $\mathbb{R}^4$ having infinitely many differential structures which are not diffeomorphic to each other has a relationship with Yang-Mills field. Does anyone can explain it, and give me some references.

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Relevant : 1. Donaldson theory 2. Seiberg-Witten invariants – user10001 May 4 '14 at 8:02
@user10001 Can you speak more explicitly? Thanks! – user34669 May 4 '14 at 11:48
up vote 2 down vote accepted

A concise account of the Donaldson's theorem relating smooth structures on 4-manifolds to the Yang-Mills theory can be found in the following review by C. Nash (section 5). This theorem is based on previous theorems proven by mathematicians.

Donaldson's theorem sets a much more strict characterization on the allowed forms of the intersection matrix of smoothable 4-manifolds. (Please see the intersection matrix definition in equation 5.1.).

In the proof Donaldson used the fact that the number of singularities in the one instanton moduli space is equal to the number of unit eigenvalues of the intersection matrix.

Later, Donaldson investigated higher instanton moduli spaces and discovered invariants which are sensitive to the smooth structure. Witten found a topological field theory (based on the Seiberg-Witten moduli space) whose correlation functions give these invariants. This was a great achievement, since, it was hard to compute these invariants using to the existing methods. The references to these works are given in Nash’s article.

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