Topology of graph configuration spaces and quantum statistics

TL;DR

This thesis characterizes abelian quantum statistics on graphs, linking the number of anyon phases to graph connectivity, and introduces a Morse theory approach to analyze two-particle configuration spaces with physical interpretations.

Contribution

It provides a comprehensive classification of quantum statistics on graphs, proves independence of statistics for 2-connected graphs, and introduces a Morse function method with physical insights.

Findings

Number of anyon phases relates to graph connectivity

For 2-connected graphs, quantum statistics are particle number independent

Morse functions offer a physically interpretable approach to configuration spaces

Abstract

In this thesis we develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs. Moreover we present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions, which have a nice physical interpretation as two-body potentials constructed from one-body potentials. We also give a brief introduction to discrete Morse theory.