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Aref’eva, I. : The gauge field as chiral field on the path and its integrability, Lett. Math. Phys. 3 (1979) 241–247 2. Arms, J. , Marsden, J. : Symmetry and bifurcations of momentum mappings, Commun. Math. Phys. 78 (1981) 455–478 3. : Lectures on non-perturbative gravity, World Scientific, Singapore, 1991 4. : Mathematical problems of non-perturbative quantum general relativity, Les Houches lecture notes, preprint Syracuse SUGP-92/11-2 5. Ashtekar, A. and Isham, C. : Representations of the holonomy algebras of gravity and non-Abelian gauge theories, Classical & Quantum Gravity 9 (1992) 1433–1467 6.

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].

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