Bosonic and Fermionic Singularities in Diffeology

TL;DR

This paper investigates the differential geometry of the quadrant with subset diffeology, revealing a dichotomy between differential forms and symmetric tensors, and introduces the concept of singular capacity to measure tensor singularities.

Contribution

It uncovers a fundamental dichotomy in the behavior of forms and tensors on a diffeological space and defines singular capacity as a new measure of tensor singularities.

Findings

Differential forms are restrictions of smooth forms on R^2, showing Fermionic behavior.

Symmetric tensors exhibit Bosonic behavior with accumulating singularities.

A decomposition theorem identifies the purely axial singular parts.

Abstract

We explore the differential geometry of the quadrant $C_2 = [0,\infty[^2$, equipped with the subset diffeology of $\mathbf{R}^2$. We show a striking dichotomy between differential forms and symmetric tensors. While differential forms on $C_2$ are simply restrictions of smooth forms on $\mathbf{R}^2$ (a "Fermionic" behavior where singularities are hidden), symmetric tensors exhibit a "Bosonic" behavior where singularities accumulate. We prove a decomposition theorem identifying exactly the singular parts: they are purely axial. Surprisingly, the mixed interaction term is forced to be regular by the symmetries of the corner. Finally, we introduce the notion of \emph{singular capacity} to quantify the order of singularity a tensor can support.