Superconnection character forms and the Cayley transform
Let \(E=E^+\oplus E^-\) be a super vector bundle over a smooth manifold M equipped with an inner product, and let D be a connection on E compatible with the grading and the inner product. Let T: \(E^+\to E^- \) denote a vector bundle map and \(X=\left( \begin{matrix} 0\\ T\end{matrix} \begin{matrix} -T^*\\ 0\end{matrix} \right)\). In this situation the author introduced the superconnection character form \(\Phi\) (E,D,X) which is a closed form in \(\Omega\) (M) [Topology 24, 89-95 (1985; Zbl 0569.58030)]. In the present paper he describes the construction of this form in new notations, which allow to treat simultaneously the graded and ungraded cases. He discusses the Cayley transform sending a skew adjoint operator to a unitary operator. There is a correspondence between subbundles W of E and unitary automorphisms g of E which are inverted under conjugation by the involution \(\epsilon\) defined by the grading of E. For any automorphism g of E a form \(\Phi\) (E,D,g) is constructed which agrees with \(\Phi\) (E,D,X) in the case that g is the Cayley transform of X. It is shown that the form \(\Phi\) (E,D,g) is induced from the universal superconnection character form over a sufficiently large Grassmannian Gr(V) or unitary group U(V). Character forms \(\gamma_{2k}\) on Gr(V) and \(\gamma_{2k-1}\) on U(V) are introduced, which are closed forms and invariant under the action of U(V). The class of \(\gamma_{2k}\) in \(H^{2k}(Gr(V))\) is up to a factor equal to the Chern character class \(ch_ k\) of the canonical subbundle over Gr(V). These invariant classes are studied in detail and related to the universal superconnection character form. The last part deals with the problem of extending the results to infinite dimensions. For this reason the unitary group \(U^ p\) and Grassmannian \(Gr^ p\) of a Hilbert space H relative to the Schatten class \({\mathcal L}^ p(H)\) are defined. They are Banach manifolds for which the de Rham theorem holds, and they are classifying spaces for K-theory. The author proves that the super connection forms are defined on these manifolds and he proves a theorem relating the super connection and invariant character forms in this case.
universal superconnection character form
scientific article; zbMATH DE number 4098308