By Rau J., Mueller B.
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To illustrate this point, in Appendix A we restrict ourselves to U (1) connections and, by exploiting the Abelian character of this group, show how one can obtain the main results of this paper using piecewise C 1 loops. Whether similar constructions are possible in the non-Abelian case is, however, an open question. Finally, in Appendix 26 Abhay Ashtekar and Jerzy Lewandowski B we consider another extension. In the main body of the paper, Σ is a 3-manifold and the gauge group G is taken to be SU (2).
Note that the affine space structure of A is lost in this projection; A/G is a genuinely non-linear space with a rather complicated topological structure. The next notion we need is that of closed loops in Σ. e. maps p : [0, s1 ] ∪ . . 2) which are continuous on the whole domain and C ω on the closed intervals [sk , sk+1 ]. Given two paths p1 : [0, 1] → Σ and p2 : [0, 1] → Σ such that p1 (1) = p2 (0), we denote by p2 ◦ p1 the natural composition: p2 ◦ p1 (s) = p1 (2s), for s ∈ [0, 12 ] p2 (2s − 1), for s ∈ [ 21 , 1].