On Time-Reversal Equivariant Hermitian Operator Valued Maps from a 2D-Torus Phase Space

TL;DR

This paper classifies time-reversal symmetric maps from a 2D torus phase space to hermitian operators, analyzing their homotopy classes and how to transition between them with minimal degenerations.

Contribution

It provides a classification of equivariant hermitian operator maps on a torus phase space and explores controlled degenerations for homotopy transitions.

Findings

Classification of equivariant hermitian operator maps

Description of homotopy classes and degenerations

Methods for implementing homotopy jumps with small degenerations

Abstract

In this paper we classify maps from a torus phase space $X$ to $\mathcal{H}_n^*$, the space of $n \times n$, non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on $X$ and an involution on $\mathcal{H}_n^*$ defining a certain symmetry class. Furthermore we study certain controlled degenerations of such maps and describe how one can implement jumps from one homotopy class to another while allowing only "small" degenerations.