Representing the Derivative of Trace of Holonomy

Trace of holonomy around a fixed loop defines a function on the space of unitary connections on a hermitian vector bundle over a Riemannian manifold. Using the derivative of trace of holonomy, the loop, and a flat unitary connection, a functional is defined on the vector space of twisted degree 1 cohomology classes with coefficients in skew-hermitian bundle endomorphisms. It is shown that this functional is obtained by pairing elements of cohomology with a degree 1 homology class built directly from the loop and equipped with a flat section obtained from the variation of holonomy around the loop. When the base manifold is closed Kähler, hard Lefschetz duality implies that the space of twisted degree 1 cohomology classes is a symplectic vector space, and, coupled with Poincaré duality, can be identified with the space of twisted degree 1 homology classes. In this case, the functional is obtained by contracting the symplectic pairing on cohomology with the hard Lefschetz dual of the Poincaré dual of the twisted homology class built using the loop