Quaternionic Transgression in Hyperkähler Geometry

Abstract: It is known, that hyperkähler varieties being a special case of Riemann and Kähler manifolds, admit more symmetries. For this reason, all important structures possibly admit hyperkähler generalization. Some of these generalizations make a theme of this thesis.The main concept we used is the transgression of differential forms and its application to characteristic classes. If the manifold is endowed with a Kähler structure then the multiplicative group CX acts on differential forms and cohomology. This allows us to construct the Bott-Chern secondary classes which are the descendants of the double transgression ω=∂∂φ for exact forms of pure Hodge type.In this thesis, 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 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=ddIdJdKφ. This fourth-order transgression is given in terms of the chiral determinant of the ∂-operator acting on sections of the holomorphic bundle restricted to the fibers. The double transgression of the Chern character form in terms of the integration over complex projective plane proposed in this thesis provides the first example of the series of the regulator maps in the algebraic K-theory.We hope that the results of this thesis imply (among other applications) that there is a generalization of regulator maps in algebraic K-theory where the basic simplex is a configuration of linear subspaces in the quaternionic linear spaces. We give the explicit local construction of this new invariant for the important example of the infinite dimensional bundle that arises in the construction of the local index theorem for families. We define a higher analytic hypertorsion for families of hyperholomorphic bundles on compact hyperkähler manifolds. An explicit formula for the zeroth-degree part of it is given in terms of the Laplace operators acting on sections of the vector bundle twisted by the bundle of differential forms. This torsion-like invariant coincides with the quaternionic torsion introduced by Leung and Yi.