  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.
