Classification of 2-dimensional conformal field theories and 2-cohomology vanishing for tensor categories

Abstract: This is a joint work with R. Longo. We classify 2-dimensional local conformal nets with c less than 1. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams. In our previous classification of 1-dimensional local conformal nets, D_odd and E_7 did not appear, but now they do appear in this classification. Our new main tool is 2-cohomology vanishing for certain tensor categories.