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Yang-Mills theory A nonabelian gauge theory. For a fixed compact Lie group G with Lie algebra g, the field in this theory is vector potentialA, i.e., a connection 1-form on some principal G bundle. Let FA be the curvature 2-form of the connection A. The fundamental Lagrangian in pure gauge theory is

L(A) = 1 2 |FA||dnx|,

where |FA| denotes the norm in the Lie algebra. From the variational principle we get the equations of motion, the Yang-Mills equations

dA ∗ FA = 0, where dA is the covariant derivative with respect to A and ∗ is the Hodge star operator. For any A we also have the Bianchi identity dAFA = 0. In local coordinates we have:

Fµν = ∂µAν − ∂νAν + i[Aµ,Aν], and L = T r(FµνFµν).