Yang-Mills connections on surfaces and representations of the path group

Morrison, Kent

doi:10.1090/S0002-9939-1991-1069292-7

Yang-Mills connections on surfaces and representations of the path group
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Proc. Amer. Math. Soc. 112 (1991), 1101-1106 Request permission

Abstract:

We prove that Yang-Mills connections on a surface are characterized as those with the property that the holonomy around homotopic closed paths only depends on the oriented area between the paths. Using this we have an alternative proof for a theorem of Atiyah and Bott that the Yang-Mills connections on a compact orientable surface can be characterized by homomorphisms to the structure group from an extension of the fundamental group of the surface. In addition, for $M = {S^2}$, we obtain the results that the Yang-Mills connections on ${S^2}$ are isolated and correspond with the conjugacy classes of closed geodesies through the identity in the structure group.

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