Induced C*-Complexes in Metaplectic Geometry

Abstract

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of Kostant's symplectic spinor bundle. Defining a Hilbert C * -structure on this bundle for a suitable C * -algebra, we obtain an elliptic C * -complex in the sense of Mishchenko-Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert C * -modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles. * E-mail address: Svatopluk.Krysl@mff.cuni.cz † The author thanks for financial supports from the founding No. 17-01171S granted by the Czech Science Foundation. We thank to the anonymous reviewer for his comments and suggestions.[14]. In [66] and [61], C * -compact perturbations of the complexes' differentials are allowed. For the infinite rank bundles, one cannot expect that, in general, the space of harmonic elements represents the cohomology groups homeomorphically (quotient topology) and linearly (quotient projections). Therefore, it makes sense to find conditions when this happens. This was investigated by the author in the past years (see [35], [36] or an overview in [37]).In the connection to sheaf cohomology, Banach and Fréchet bundles are studied in the papers of Illusie [25] and Röhrl [57]. To present a sample of the broad context (foremost connected to C * -algebras, K-theory and homological algebra) in which such bundles are considered, let us mention the papers of Maeda, Rosenberg [43], Freed, Lott [16], Larraín-Hubach [39], the author [36] and Fathizadeh, Gabriel [12]. There are also works in which holomorphic Banach bundles and their sheaf cohomology groups are treated. See Lempert [40] and Kim [27]. A further reason for an investigation of these complexes might originate in ), which we do not touch here explicitly, although we give a modest interpretation of the introduced structures in the realm of quantum theory.The aim of our paper is to study elliptic complexes for the case of infinite rank Hilbert C * -bundles which are induced by Lie group representations, and to show as well how the theory of connections and appropriately generalized elliptic complexes apply in this situation. In order to make the considered situation not too general, we choose a specific Lie group representation (the Segal-Shale-Weil representation of the metaplectic group). We hope that the reader may follow the text easier. Our further aim is to find examples for the theory of Mishchenko and Fomenko ([14]) that would not be trivial or covered by the theory developed around the classical Atiyah-Singer index theorem, and to show how the Hodge theory for certain C * -bundles (derived in [35] and [36]) make us able to describe the cohomology of elliptic complexes on Hilbert bundles easily.Let (V, ω 0 ) be a symplectic vector space and L be a Lagrangian subspace of it. The Segal-Shale-Weil representation is a non-trivial representation of the connected double cover G...