  In this paper we continue our study of the fourth order transgression on
hyper\"ahler manifolds introduced in the previous paper. We give a local
construction for the fourth-order transgression of the Chern character form of
an arbitrary vector bundle supplied with a self-dual connection on a four
dimensional hyperk\"ahler manifold. The construction is based on the harmonic
twistor formalism. Remarkably, the resulted expression for the fourth order
transgression is given in terms of the determinant of the
$\bar{\partial}$-operator defined on fibers of the twistor fibration.
