The Determinant Line Bundle for Fredholm Operators: Construction, Properties, and Classification

TL;DR

This paper constructs and analyzes determinant line bundles over Fredholm operators, establishing their properties, explicit isomorphisms, and classifying all compatible systems, thus clarifying their role in functional analysis.

Contribution

It provides a comprehensive construction and classification of determinant line bundles over Fredholm operators, including explicit formulas and comparisons with existing approaches.

Findings

Explicit formulas for isomorphisms between line bundles

Complete classification of compatible determinant line bundle systems

Comparison of conventions with existing literature

Abstract

We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possible systems of compatible determinant line bundles and compare the conventions and approaches used elsewhere in the literature.