where ∂ ∂ μ μ = ∂ ∂ / ∂ ∂ x μ μ is taken to be an orthonormal vector space basis on T M and ∂ ∂ i = ∂ ∂ / ∂ ∂ q i the vector space basis on T Φ Φ . The d ϕ ϕ is a differential form. The sigma model action is then just the conventional inner product on vector-valued k -forms