Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1995-07-10
Commun.Math.Phys. 179 (1996) 265-294
Physics
High Energy Physics
High Energy Physics - Theory
42 pages TEX file, harvmac and epsf macros, AMS fonts optional, uuencoded, 8 figures included
Scientific paper
10.1007/BF02102590
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.
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