Functional analytic issues in $\mathbb{Z}_2^n$-geometry

Andrew James Bruce,Norbert Poncin

doi:10.33044/revuma.v60n2a21

Abstract

We show that the function sheaf of a $\mathbb{Z}_2^n$-manifold is a nuclear Fr\'echet sheaf of $\mathbb{Z}_2^n$-graded $\mathbb{Z}_2^n$-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a $\mathbb{Z}_2^n$-morphism are all continuous. These results are essential for the existence of categorical products in the category of $\mathbb{Z}_2^n$-manifolds. All proofs are self-contained and explicit.

Highlights

Zn2 -geometry is an emerging framework in mathematics and in mathematical physics, which has been introduced in the foundational papers [8] and [11]
The Zn2 -commutation rule is necessary, and sufficient: it can be shown [7] that any commutation rule, for any finite number m of coordinates, is of the form (1.1), for some n ≥ 2m
Other applications of Zn2 -geometry rely on Zn2 Lie groups and their actions on Zn2 -manifolds, on Zn2 vector bundles and their sections, on the internal Hom in the category of Zn2 -manifolds, etc

Summary

Introduction

Zn2 -geometry is an emerging framework in mathematics and in mathematical physics, which has been introduced in the foundational papers [8] and [11]. Other applications of Zn2 -geometry rely on Zn2 Lie groups (generalized super Lie groups) and their actions on Zn2 -manifolds (which are expected to be of importance in supergravity), on Zn2 vector bundles (generalized super vector bundles) and their sections (these are basic objects needed for instance in the study of Zn2 Lie algebroids), on the internal Hom in the category of Zn2 -manifolds (which is of importance in field theory—Zn-gradings and Zn2 -parities), etc. All these notions are themselves based on products in the category of Zn2 -manifolds. (2) all Zn2 -morphisms and all Zn2 -differential operators are continuous with respect to the locally convex topology of the structure sheaf, and they are continuous with respect to the J -adic topology of the structure sheaf, where J is the kernel of the projection onto base functions

Zn2 -manifolds and their morphisms

Linear Zn2 -algebra

Sheaves of differential operators on a Zn2 -manifold

Functional analytic properties of the function sheaf of a Zn2 -manifold