A Supermanifold structure on Spaces of Morphisms between Supermanifolds

Abstract

The aim of this work is the construction of a "supermanifold of morphisms $X\to Y$", given two finite-dimensional supermanifolds $X$ and $Y$. More precisely, we will define an object ${\underline{SC}}^{\infty }(X,Y)$ in the category of supermanifolds proposed by Molotkov and Sachse. Initially, it is given by the set-valued functor characterised by the adjunction formula $\mathrm{Hom}(P\times X,Y)\cong \mathrm{Hom}(P,{\underline{SC}}^{\infty }(X,Y))$ where $P$ ranges over all superpoints. We determine the structure of this functor in purely geometric terms: We show that it takes values in the set of certain differential operators and establish a bijective correspondence to the set of sections in certain vector bundles associated to $X$ and $Y$. Equipping these spaces of sections with infinite-dimensional manifold structures using the convenient setting by Kriegl and Michor, we obtain at a supersmooth structure on ${\underline{SC}}^{\infty }(X,Y)$, i.e. a supermanifold of all morphisms $X\to Y$.

51 pages

References in corpus (3)

• A Categorical Formulation of Superalgebra and Supergeometry
• Global Analytic Approach to Super Teichmueller Spaces
• Lectures on Differential Geometry of Modules and Rings