Infinite-dimensional supermanifolds over arbitrary base fields

TL;DR

This paper establishes an equivalence between different categories of supermanifolds, extending the theory to infinite-dimensional cases over arbitrary base fields and unifying various approaches.

Contribution

It proves the equivalence of Berezin-Kostant-Leites and Molotkov-Sachse supermanifold categories and extends supermanifold theory to infinite-dimensional and more general base fields.

Findings

Equivalence of supermanifold categories established

Extended supermanifold theory to infinite-dimensional spaces

Unified various supermanifold approaches

Abstract

In his recent investigation of a super Teichm\"uller 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-Gl\"ockner-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.