Localizing gauge theories from noncommutative geometry

2014, January 1
W.D. van Suijlekom

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.