Generalized Chern-Simons Form and Descent Equation

TL;DR

This paper introduces a generalized Chern-Simons form and descent equation incorporating scalar fields, derived via noncommutative geometry techniques but applicable within standard differential geometry.

Contribution

It develops a method to extend Chern-Simons forms and descent equations to include scalar fields using noncommutative geometry concepts.

Findings

Derived generalized Chern-Simons forms with scalar fields.

Established that resulting equations are independent of noncommutative geometry.

Provided algebraic justification within ordinary differential geometry.

Abstract

We present the general method to introduce the generalized Chern-Simons form and the descent equation which contain the scalar field in addition to the gauge fields. It is based on the technique in a noncommutative differential geometry (NCG) which extends the $N$-dimensional Minkowski space $M_N$ to the discrete space such as $M_N\times Z_2$ with two point space $Z_2$. However, the resultant equations do not depend on NCG but are justified by the algebraic rules in the ordinary differential geometry.