A group of loops [Formula: see text] is associated to every smooth pointed manifold M using a strong homotopy relation. It is shown that the holonomy of a connection on a principal G-bundle may be presented as a group morphism [Formula: see text] and that every such morphism satisfying a natural smoothness condition is the holonomy of some unique connection up to isomorphism.
We define the thin fundamental Gray 3-groupoid S3(M ) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M ) → C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.
Abstract. We define the thin fundamental categorical group P 2 (M, * ) of a based smooth manifold (M, * ) as the categorical group whose objects are rank-1 homotopy classes of based loops on M and whose morphisms are rank-2 homotopy classes of homotopies between based loops on M . Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂ : E → G, ). We construct categorical holonomies, defined to be smooth morphisms P 2 (M, * ) → C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1-form on P , a principal G bundle over M , and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.
In this paper, we establish a one-to-one correspondence between Uð1Þ-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group Uð1Þ on a simply connected manifold M is a group morphism from the thin second homotopy group to Uð1Þ; satisfying a smoothness condition, where a homotopy between maps from ½0; 1 2 to M is thin when its derivative is of rank 42: For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. # 2002 Elsevier Science (USA)
We describe the dynamics of euclideanized SO(4)-symmetric Einstein-Yang-Mills (EYM) systems with arbitrary compact gauge groups [Formula: see text]. For the case of SO(n) and SU(n) gauge groups and simple embeddings of the isotropy group in [Formula: see text], we show that in the resulting dynamical system, the Friedmann equation decouples from the Yang-Mills equations. Furthermore, the latter can be reduced to a system of two second-order differential equations. This allows us to find a broad class of instanton (wormhole) solutions of the EYM equations. These solutions are not afflicted by the giant-wormhole catastrophe.
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