Graphs and Reflection Groups

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 $\mathcal{N}=2$ supersymmetric theories and topological field theories are discussed.

42 pages TEX file, harvmac and epsf macros, AMS fonts optional, uuencoded, 8 figures included

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• From CFT's to Graphs
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• Boundary Rings and N=2 Coset Models
• On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective
• Determination of quantum symmetries for higher ADE systems from the modular T matrix
• Fusion rules for admissible representations of affine algebras: the case of $A_2^{(1)}$
• The Hilbert Space of large $N$ Chern-Simons matter theories
• Exactly solvable potentials of Calogero type for q-deformed Coxeter groups
• An Observation on Finite Groups and WZW Modular Invariants
• Pasquier Models at c=1: Cylinder Partition Functions and the role of the Affine Coxeter Element
• Schubert Calculus and Threshold Polynomials of Affine Fusion
• On the signature of certain intersection forms
• Primitive forms for Gepner singularities