Yang--Mills Configurations from 3D Riemann--Cartan Geometry

TL;DR

This paper establishes a geometric interpretation of 3D Yang--Mills configurations using Riemann--Cartan geometry, deriving an Ashtekar-type mapping through Clifford algebra decomposition, revealing topological Lagrangian structures.

Contribution

It introduces a novel geometric mapping between Yang--Mills fields and 3D Riemann--Cartan geometry using Clifford algebra techniques, highlighting topological Lagrangian components.

Findings

Mapping relates Yang--Mills fields to Riemann--Cartan geometry

Derived Lagrangian includes Chern--Simons and cosmological terms

Topological nature of the Lagrangian is established

Abstract

Recently, the {\it spacelike} part of the $SU(2)$ Yang--Mills equations has been identified with geometrical objects of a three--dimensional space of constant Riemann--Cartan curvature. We give a concise derivation of this Ashtekar type (``inverse Kaluza--Klein") {\it mapping} by employing a $(3+1)$--decomposition of {\it Clifford algebra}--valued torsion and curvature two--forms. In the subcase of a mapping to purely axial 3D torsion, the corresponding Lagrangian consists of the translational and Lorentz {\it Chern--Simons term} plus cosmological term and is therefore of purely topological origin.