Pseudo-bundles of exterior algebras as diffeological Clifford modules

TL;DR

This paper explores the structure and behavior of diffeological pseudo-bundles of exterior algebras and Clifford modules, focusing on their compatibility under gluing and duality conditions in a diffeological setting.

Contribution

It introduces conditions under which the dual of a glued pseudo-bundle can be obtained by gluing duals, ensuring compatibility and isometry in the diffeological context.

Findings

The dual of the gluing map being a diffeomorphism ensures the commutativity condition.

The natural map used in the construction is an isometry for the pseudo-metrics.

Compatibility of pseudo-metrics and duality conditions are established under specific assumptions.

Abstract

We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals (this is not automatic in the diffeological context). We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a very natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved.