Induced QCD from the noncommutative geometry of a supermanifold

Abstract

The noncommutative geometry of a two-leaf Parisi–Sourlas supermanifold in Connes’ formalism using different K cycles over the Grassmann algebra-valued functions on the supermanifold is studied. We find that the curvature of the trivial noncommutative vector bundle defines in the simplest case the super Yang–Mills action coupled to a scalar field. By considering a modified Dirac operator and a suitable limit of its parameters we then obtain an action that turns out to be the continuum limit of the induced QCD in the Kazakov–Migdal model.