Bratteli networks and the Spectral Action on quivers

Abstract

In the context of noncommutative geometry, we consider quiver representations -- not on vector spaces, as traditional, but on finite-dimensional prespectral triples (`discrete topological noncommutative spaces'). A similar idea appeared in the original work of Marcolli-van Suijlekom on quiver representations in spectral triples (`discrete noncommutative geometries'), which paved the way for some of our results in independent directions. We introduce Bratteli networks, a structure that yields a neat combinatorial characterisation of the space $\mathrm{Rep}~Q$ of prespectral-triple-representations of a quiver $\mathrm{Rep}~Q$, as well as of the gauge group and of their quotient. Not only these claims that make it possible to `integrate over $\mathrm{Rep}~Q$' are, as we now argue, in line with the spirit of random noncommutative geometry -- formulating path integrals over Dirac operators -- but they also contain a physically relevant case. Namely, the equivalence between quiver representations and path algebra modules, established here for the new category, inspired the following construction: Only from representation theory data, we build a spectral triple for the quiver and evaluate the spectral action functional from a general formula over closed paths. When we apply this construction to lattice-quivers, we obtain not only Wilsonian Yang-Mills lattice gauge theory, but also the Weisz-Wohlert-cells in the context of Symanzik's improved gauge theory. We show that a hermitian (`Higgs') matrix field emerges from the self-loops of the quiver and derive the Yang-Mills--Higgs theory on flat space as a smooth limit.