We made the following study on algebraic quantum foeld theory.
We gave a classification of 2-dimensional local conformal nets of operator algebras with Longo in [40]. The basic invariant is the central charge, and we have given a complete classification for the case where this value is less than 1 and the net has parity symmetry and maximality with respect to inclusions. The conditions on parity symmetry and maximality can be dropped, but we then simply have more combinatorial complexity. This is based on our previous classification of 1-dimensional nets in [36], also with Longo. The new key tool is 2-cohomology vanishing for certain tensor categories, which also gives a new and operator algebraic proof of uniqueness of a certain algebraic system for which we had simply cited an argument on vertex operator algebras in [36].