Harmonic Twistor Formalism and Transgression on Hyperkähler Manifolds

In this paper we continue our study of the transgression of characteristic classes of hyperholo-morphic bundles on hyperkähler manifolds [2]. In the previous paper the global construction for the fourth order transgression of the Chern character form on a compact hyperkähler manifold was proposed. In addition, the explicit expression for the transgression of the Chern character arising in the application of the local families index theorem was found. This construction was local over the base of the fibration. It is natural to look for a local derivation of the transgression of the Chern character forms for an arbitrary hyperholomorphic bundle. In this paper we give the general local construction for an arbitrary hyperholomorphic bundle on a four-dimensional dimensional hyperkähler manifold. Note that in d = 4 the condition on a hermitian bundle to be hyperholomorphic is equivalent to the (anti)self-duality condition on the corresponding connection. We propose an explicit expression for the fourth order transgression T (E) of the top degree part of the Chern character form for an arbitrary vector bundle E supplied with a self-dual connection. The construction is local and thus is applicable to an arbitrary four dimensional hyperkähler manifold M. Locally the Chern character form is exact and we have: ch [2] (E) = vol M ∆ 2 T for a volume form vol M. Remarkably, the explicit expression for T (E) is non-trivial even for a linear bundle E. In our derivation we essentially use the harmonic twistor approach, the variant of the twistor formalism developed in [8, 3, 4]. In twistor approach [9] one codes the information about self-dual connections on a vector bundle in terms of holomorphic structures on a bundle over the twistor fibration Z M → M with a fiber being S 2. Remarkably, the proposed expression for T (E) is given in terms of the determinant of the ∂ A-operator in the sense of Quillen [7] acting on sections of the holomorphic bundle restricted to the fibers. This implies that the results of this paper may be connected with the local families index for the twistor fibration. Rather straightforwardly, the construction described in this paper may be generalized to hy-perkähler manifolds of an arbitrary dimension. We are going to discuss the general construction connecting the approaches of this paper and of [2] in the future publication.