Graph-theoretic determination of massless modes in latticized theory-space models

TL;DR

This paper presents a graph-theoretic approach to analyze and construct latticized theory-space models, determining massless modes based solely on the topology of the associated bipartite graphs.

Contribution

Introduces a novel graph-theoretic method using maximum matchings and Dulmage-Mendelsohn decomposition to analyze fermion mass spectra in latticized models.

Findings

Number of massless modes equals the size of a maximum matching.

Wave-function support is restricted to fields reachable by even-length alternating paths.

Method is independent of model parameters and depends only on topology.

Abstract

A graph-theoretic method is introduced for analyzing fermion mass spectra in latticized theory-space models, including chain models arising from dimensional deconstruction. Fermion mass terms are mapped to bipartite graphs, with fields as vertices and nonvanishing mass terms as edges. The number of massless modes is shown to be fixed by the cardinality of a maximum matching of the associated graph. Moreover, the wave-function support of these modes is restricted to fields reachable from exposed or unmatched vertices by even-length maximum-matching-alternating paths, as characterized by the Dulmage-Mendelsohn decomposition. These results depend only on the topology of latticized theory space and are independent of model parameters. The method enables a systematic construction of latticized models with prescribed numbers and localization properties of massless modes.