Induced QCD from the Noncommutative Geometry of a Supermanifold

TL;DR

This paper explores the noncommutative geometric structure of a supermanifold, deriving a super Yang--Mills action and connecting it to the continuum limit of induced QCD in the Kazakov--Migdal model.

Contribution

It introduces a novel noncommutative geometric framework for supermanifolds that leads to a new understanding of induced QCD via super Yang--Mills theory.

Findings

Super Yang--Mills action coupled to a scalar field derived from supermanifold curvature.

Modified Dirac operator yields an action matching the continuum limit of induced QCD.

Establishes a link between noncommutative geometry of supermanifolds and lattice gauge theories.

Abstract

We study 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. 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 Kazakov--Migdal model.