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Localizing gauge theories from noncommutative geometry

We recall the emergence of a generalized gauge theory from a noncommutative Riemannian spin manifold, viz. a real spectral triple $(A,H,D;J)$. This includes a gauge group determined by the unitaries in the $*$-algebra $A$ and gauge fields arising from a so-called perturbation semigroup which is associated to $A$. Our main new result is the interpretation of this generalized gauge theory in terms of an upper semi-continuous $C^*$-bundle on a (Hausdorff) base space $X$. The gauge group acts by vertical automorphisms on this $C^*$-bundle and can (under some mild conditions) be identified with the space of continuous sections of a group bundle on $X$. This then allows for a geometrical description of the group of inner automorphisms of $A$. We exemplify our construction by Yang-Mills theory and toric noncommutative manifolds and show that they actually give rise to continuous $C^*$-bundles. Moreover, in these examples the corresponding inner automorphism groups can be realized as spaces of sections of group bundles that we explicitly determine.