Edge states of $(2+1)D$ $BF$ theory as the effective field theory of translational invariant ${\bf Z}_{2}$ spin liquids*

We classify (2+1)-dimensional $BF$ field theory as the effective field theory for the achiral and translation invariant ${\bf Z}_{2}$ spin liquids on the square lattice. We use the pattern of the crystal momenta of $BF$ field theory to find the corresponding ${\bf Z}_{2}$ spin liquid. Then, we show that some classes of ${\bf Z}_{2}$ spin liquids can support gapless helical edge states depending on the projective symmetry group of the effective $BF$ theory. We supplement this effective theory by studying possible projective symmetry group of ${\bf Z}_{2}$ spin liquids.