Discrete Morse functions for graph configuration spaces

TL;DR

This paper introduces a novel approach using discrete Morse functions to analyze two-particle configuration spaces on graphs, providing insights relevant to quantum statistics and anyon-like behaviors.

Contribution

It presents an alternative to existing methods by applying Morse functions with physical interpretations, advancing the understanding of quantum particles on networks.

Findings

Morse functions offer a new perspective on two-particle graph configuration spaces.

The approach connects discrete Morse theory with quantum statistics on networks.

Potential implications for generalized anyon statistics in quantum systems.

Abstract

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. Our motivation comes from the problem of quantum statistics for particles on networks, for which generalized versions of anyon statistics can appear.