Here, the g − − 1 ∂ ∂ μ μ g is the pullback of the Maurer–Cartan form, for g ∈ ∈ G , onto the spacetime manifold. The π π m is a projection onto the odd-parity piece of the Cartan involution. That is, given the Lie algebra g of G , the involution decomposes the space into odd and even parity components g = m ⊕ ⊕ h corresponding to the two eigenstates of the involution. The sigma model Lagrangian can then be written as