Spatial Geometry of Non-Abelian Gauge Theory in \(2 + 1\) Dimensions

TL;DR

This paper reformulates 2+1 dimensional SU(2) Yang-Mills theory using gauge-invariant geometric variables, revealing a connection to topological field theory and addressing issues with degenerate configurations.

Contribution

It introduces a gauge-invariant geometric framework for 2+1D Yang-Mills theory, linking physical states to topological field theory and analyzing degenerate configurations.

Findings

Wave functionals factorize into phase and geometric parts.

Degenerate configurations lead to infinite energy density.

Topological field theory provides insights into gauge theory wave functionals.

Abstract

The Hamiltonian dynamics of \(2 + 1\) dimensional Yang-Mills theory with gauge group SU(2) is reformulated in gauge invariant, geometric variables, as in earlier work on the \(3 + 1\) dimensional case. Physical states in electric field representation have the product form \(\Psi_{\mathrm{phys}} [E^{a i}] = \exp ( i \Omega [ E ] / g ) F [G_{ij}]\), where the phase factor is a simple local functional required to satisfy the Gauss law constraint, and \(G_{ij}\) is a dynamical metric tensor which is bilinear in \(E^{a k}\). The Hamiltonian acting on \(F [ G_{ij} ]\) is local, but the energy density is infinite for degenerate configurations where \(\det G (x)\) vanishes at points in space, so wave functionals must be specially constrained to avoid infinite total energy. Study of this situation leads to the further factorization \(F [G_{ij} ] = F_c [ G_{ij} ] \mathcal R [ G_{ij} ]\), and the product \(\Psi_c [E] \equiv \exp (i \Omega [ E ] / g ) F_c [G_{ij}]\) is shown to be the wave functional of a topological field theory. Further information from topological field theory may illuminate the question of the behavior of physical gauge theory wave functionals for degenerate fields.