Graphs and Reflection Groups

TL;DR

This paper explores how certain graphs, extending ADE Dynkin diagrams, encode root system geometries and relate to reflection groups, with implications for conformal field theories and supersymmetric models.

Contribution

It introduces a bilinear form on root systems derived from these graphs and studies the associated reflection groups, linking graph structures to physical theories.

Findings

Graphs generalizing ADE diagrams encode root system geometry.

Non integrally laced graphs relate to subgroups of reflection groups.

Connections to conformal field theories and supersymmetric theories are discussed.

Abstract

It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the conditions satisfied by these graphs, we define a bilinear form on a root system in terms of the adjacency matrices of these graphs and undertake the study of the group generated by the reflections in the hyperplanes orthogonal to these roots. Some ``non integrally laced " graphs are shown to be associated with subgroups of these reflection groups. The empirical relevance of these graphs in the classification of conformal field theories or in the construction of integrable lattice models is recalled, and the connections with recent developments in the context of ${\cal N}=2$ supersymmetric theories and topological field theories are discussed.