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Geometric structure on the orbit space of gauge connections. (English) Zbl 0624.53055

The orbit space C/G - for the action of the group G of vertical automorphisms of a principal bundle P on the space C of connections on P - is understood as the configuration space for Yang-Mills field theory. The topology of C/G was first approached by I. M. Singer [Commun. Math. Phys. 60, 7-12 (1978; Zbl 0379.53009)], who emphasized its non- triviality, even if P is trivial. He considered the covenient free actions of the group G/Center, on the subset of irreducible connections, or the action on C of the group \(G/G_ 0\), where \(G_ 0\) consists of vertical automorphisms fixing a given point of P, and obtained two principal fibrations. Differential geometrical aspects of these fibrations are investigated by other authors, who assumed the technical attitude to consider the extended group \(G^{k+1}\) of vertical automorphisms of P in the Sobolev class \(H^{k+1}\), acting on the space \(C^ k\) of \(H^ k\)-connections, for large k. The corresponding quotient spaces are supplied with a Hilbert manifold structure. However, the action of G on C is no longer free and the topology of the configuration space C/G is more complicated. The more realistic study of \(C^ k/G^{k+1}\) (for a compact P) was initiated by the first author and J. S. Rogulski [On the stratification of the orbit space for the action of automorphisms on connections, Diss. Math. 250, 67 p. (1986; Zbl 0614.57025)], who proved that this space admits just a stratification onto Hilbert manifolds.
In the present paper the authors present a slightly different approach. They prove that the orbit space \(C^ k/G^{k+1}\) is homeomorphic to the quotient space of \(C^ k/G_ 0^{k+1}\) with respect to the action of the group \(G^{k+1}/G_ 0^{k+1}\), which is isomorphic to the structure group (hence it is a compact group). This provides some simplifications; a slice theorem is proved, which leads to a stratification structure on \(C^ k/G^{k+1}\), which is simpler than the one obtained previously.
Reviewer: A.Manià

MSC:

53C80 Applications of global differential geometry to the sciences
81T08 Constructive quantum field theory
55R05 Fiber spaces in algebraic topology
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