This makes it explicit and plainly evident that the sigma model is just the kinetic energy of a point particle. From the point of view of the manifold M , the field ϕ ϕ is a scalar, and so d ϕ ϕ can be recognized just the ordinary gradient of a scalar function. The Hodge star is merely a fancy device for keeping track of the volume form when integrating on curved spacetime. In the case that M is flat, it can be completely ignored, and so the action is