Localizing gauge theories from noncommutative geometry

We recall the emergence within noncommutative geometry of a generalized gauge theory from a real spectral triple for a unital C⁎-algebra A. This includes a gauge group determined by the unitaries in A and gauge fields arising from a so-called perturbation semigroup which is associated to A. Our main new result is the construction of an upper semi-continuous C⁎-bundle on a (Hausdorff) base space X whose space of continuous sections is isomorphic to A. The gauge group acts by vertical automorphisms on this C⁎-bundle and can (under some conditions) be identified with the space of continuous sections of a group bundle on X. Moreover, in some cases the gauge group is found to coincide with the group of inner automorphisms of A.We discuss two classes of examples of our construction: Yang-Mills theory and toric noncommutative manifolds and we show that they actually give rise to continuous C⁎-bundles. Moreover, in these examples the gauge groups coincide with the inner automorphism groups and can be realized as spaces of sections of group bundles that we explicitly determine.