Non-Archimedean Fr\'echet Algebras and the Loop Space of a Hypersurface Complement

TL;DR

This paper explores the structure of loop spaces into hypersurface complements, revealing that their associated Laurent series algebras are localizations of simpler algebras, using non-Archimedean functional analysis techniques.

Contribution

It introduces a novel application of non-Archimedean functional analysis to study topological algebras of Laurent series in the context of loop spaces.

Findings

The Laurent series algebra is a topological localization of the polynomial algebra.

Non-Archimedean semi-norms generate the topology of the studied algebras.

The approach connects loop space topology with non-Archimedean algebraic structures.

Abstract

We study the space of loops into a hypersurface complement, and show that the corresponding topological algebra of Laurent series with coefficients in $\mathcal{O}(L\mathbf{A}^{d}_{f})$ is a topological localisation of $\mathcal{O}(L\mathbf{A}^{d})$. This requires introducing a small amount of non-Archimedean functional analysis. In particular we work with topological algebras whose topology is generated by a family of sub-multiplicative, non-Archimedean semi-norms.