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We classify two-dimensional local conformal nets with parity symmetry andcentral charge less than 1, up to isomorphism. The maximal ones are in abijective correspondence with the pairs of A-D-E Dynkin diagrams with thedifference of their Coxeter numbers equal to 1. In our previous classificationof one-dimensional local conformal nets, Dynkin diagrams D_{2n+1} and E_7 donot appear, but now they do appear in this classification of two-dimensionallocal conformal nets. Such nets are also characterized as two-dimensional localconformal nets with mu-index equal to 1 and central charge less than 1. Ourmain tool, in addition to our previous classification results forone-dimensional nets, is 2-cohomology vanishing for certain tensor categoriesrelated to the Virasoro tensor categories with central charge less than 1.Comment: 40 pages, LaTeX 2

Classification of Two-Dimensional Local Conformal Nets with c < 1 and 2-Cohomology Vanishing for Tensor Categories