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

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.