On a manifold (considered as the spacetime) M and a choice of compact Lie group G, the field content is a function U : M → → G . This defines a related field j μ μ = U − − 1 ∂ ∂ μ μ U , a g -valued vector field (really, covector field) which is the Maurer–Cartan form. The principal chiral model is defined by the Lagrangian density L = κ κ 2 t r ( ∂ ∂ μ μ U − − 1 ∂ ∂ μ μ U ) = − − κ κ 2 t r ( j μ μ j μ μ ) , where κ κ is a dimensionless coupling. In differential-geometric language, the field U is a section of a principal bundle π π : P → → M with fibres isomorphic to the principal homogeneous space for M (hence why this defines the principal chiral model).