Infinite-dimensional supermanifolds over arbitrary base fields

Abstract

In his recent investigation of a super Teichmüller space, Sachse (2007), based on work of Molotkov (1984), has proposed a theory of Banach supermanifolds using the `functor of points' approach of Bernstein and Schwarz. We prove that the the category of Berezin-Kostant-Leites supermanifolds is equivalent to the category of finite-dimensional Molotkov-Sachse supermanifolds. Simultaneously, using the differential calculus of Bertram-Glöckner-Neeb (2004), we extend Molotkov-Sachse's approach to supermanifolds modeled on Hausdorff topological super-vector spaces over an arbitrary non-discrete Hausdorff topological base field of characteristic zero. We also extend to locally k-omega base fields the `DeWitt' supermanifolds considered by Tuynman in his monograph (2004), and prove that this leads to a category which is isomorphic to the full subcategory of Molokov-Sachse supermanifolds modeled on locally k-omega spaces.