Cartan Homotopy Operator

Consider a group G with generators T ^I, the gauge potential 1-form A = A _m ^I T ^I dx ^m and the curvature 2-form F = dA + A ². The Cartan homotopy operator [1,2,3] acting on polynomials (F _t , A _t ), with A _t interpolating between two gauge potentials A ₀ and A ₁ as

is defined to be

with the operator l _t defined to act on arbitrary polynomials by

and the convention that l _t is defined to act as an antiderivation. That is l _t (Λ _p Σ _q ) = (l _t Λ _p )Σ _q + (−1)^pΛ _p (l _t Σ _q ) where Λ _p is a p-form and Σ _q is a q-form.

An important relationship is the Cartan homotopy formula

which follows by integrating

over t from 0 to 1.