Variational Dual Solutions of Chern-Simons Theory

TL;DR

This paper introduces a variational approach to Chern-Simons theory, establishing existence of solutions via minimization of action functionals and analyzing geometric aspects for the SU(2) gauge group.

Contribution

It develops a scheme for generating lower semi-continuous action functionals and proves the existence of solutions through coercivity and minimization techniques.

Findings

Existence of minimizers for the action functional in appropriate spaces

Establishment of coercivity for the variational functional

Geometric analysis relating connection forms to dual schemes for SU(2)

Abstract

A scheme for generating weakly lower semi-continuous action functionals corresponding to the Euler-Lagrange equations of Chern-Simons theory is described. Coercivity is deduced for such a functional in appropriate function spaces to prove the existence of a minimizer, which constitutes a solution to the Euler-Lagrange equations of Chern-Simons theory in a relaxed sense. A geometric analysis is also made, especially for the gauge group SU(2), relating connection forms on the bundle to corresponding forms in the dual scheme.