Transgression on Hyperkähler Manifolds and Generalized Higher Torsion Forms

Transgression of the characteristic classes taking values in the differential forms is a reach source of the interesting algebraic objects. The examples include Chern-Simons and Bott-Chern forms which are given by the transgression of the Chern character form. Chern-Simons forms are defined for a vector bundle over an arbitrary real manifold and are connected with the representation of combinations of Chern classes by the exact form ω = dφ. Bott-Chern forms are defined for holomorphic hermitian vector bundles over Kähler manifolds. These additional structures allow to use the double transgression ω = ∂∂φ to define this invariant. Basically the existence of this representation is a consequence of the action of the multiplicative group of complex numbers C * on the cohomology of an arbitrary Kähler manifold. It is natural to guess that in the case when there is a bigger group acting on the cohomology one should look for more involved objects associated with vector bundles. In this paper we consider the case of the action of the multiplicative group of quaternions H * on the cotangent bundle which induces the action of H * on the cohomology of the manifold. Supplying the manifold with a metric compatible with the action of H * we get a hyperkähler manifold. We propose a new invariant of a hyperholomorphic bundle over a hyperkähler manifold connected with the Chern character form by the fourth order " transgression " ω = dd I d J d K φ. It takes values in differential forms and its zero degree part is hyprholomorphic analog of the logarithm of the holomorphic torsion (holomorphic torsion is trivial for hyperkähler manifolds). This new hypertorsion seems first have appeared in the physical literature [5]. The expression for the hypertorsion in terms of the integration over quaternionic projective plane proposed in this paper is a direct generalization of the formula for the double transgression [11]. The double transgression of the Chern character form in terms of the integration over complex projective plane provides the first example of the series of the regulator maps in algebraic K-theory. We believe that the results of this paper imply (among other interesting applications) that there is a generalization of the regulator maps in algebraic K-theory with the basic simplex being the configuration of linear subspaces in the quaternionic linear spaces. The paper is organized as follows. In the first part we propose the generalization of …