Formal Naive Dirac Operators and Graph Topology

TL;DR

This paper investigates the relationship between zero modes of formal Dirac operators on graphs and the Betti numbers of the underlying space, providing proofs and bounds for graphs with commuting translations.

Contribution

It proves a conjecture relating zero modes to Betti numbers for certain graphs and extends it to bounds on individual Betti numbers, linking graph topology and algebraic structures.

Findings

Confirmed the conjecture for graphs with commuting translations.

Established bounds on individual Betti numbers.

Connected zero modes to graph homology and representation theory.

Abstract

Motivated by a recent conjecture of Misumi and Yumoto relating the number of zero modes of lattice Dirac operators to the sum of the Betti numbers of the underlying spacetime manifold, we study formal Dirac operators on a class of graphs admitting such in terms of their zero modes. Our main result is that for graphs on which translations commute, the conjecture of Misumi and Yumoto can be shown and indeed can be strengthened to obtain bounds on the individual Betti numbers rather than merely on their sum. Interpretations of the zero modes in terms of graph quotients and of the representation theory of abelian groups are given, and connections with a homology theory for such graphs are highlighted.