The global symmetries act as internal symmetries on the group-valued field g ( x ) as ρ ρ L ( g ′ ) g ( x ) = g ′ g ( x ) and ρ ρ R ( g ) g ( x ) = g ( x ) g ′ . The corresponding conserved currents from Noether's theorem are L α α = g − − 1 ∂ ∂ α α g , R α α = ∂ ∂ α α g g − − 1 . The equations of motion turn out to be equivalent to conservation of the currents, ∂ ∂ α α L α α = ∂ ∂ α α R α α = 0 or in coordinate-free form d ∗ ∗ L = d ∗ ∗ R = 0. The currents additionally satisfy the flatness condition, d L + 1 2 [ L , L ] = 0 or in coordinates ∂ ∂ α α L β β − − ∂ ∂ β β L α α + [ L α α , L β β ] = 0 , and therefore the equations of motion can be formulated entirely in terms of the currents.