This expression can be unpacked a bit. The field space Φ Φ can be chosen to be any Riemannian manifold. Historically, this is the "sigma" of the sigma model; the historically-appropriate symbol σ σ is avoided here to prevent clashes with many other common usages of σ σ in geometry. Riemannian manifolds always come with a metric tensor g . Given an atlas of charts on Φ Φ , the field space can always be locally trivialized, in that given U ⊂ ⊂ Φ Φ in the atlas, one may write a map U → → R n giving explicit local coordinates ϕ ϕ = ( ϕ ϕ 1 , ⋯ ⋯ , ϕ ϕ n ) on that patch. The metric tensor on that patch is a matrix having components g i j ( ϕ ϕ ) .