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

TL;DR

This paper classifies two-dimensional local conformal nets with central charge less than 1, revealing a correspondence with A-D-E Dynkin diagrams and utilizing 2-cohomology vanishing for related tensor categories.

Contribution

It extends previous classifications to two dimensions, incorporating Dynkin diagrams D_{2n+1} and E_7, and employs 2-cohomology vanishing as a key tool.

Findings

Maximal nets correspond to pairs of A-D-E Dynkin diagrams with Coxeter number difference of 1.

D_{2n+1} and E_7 diagrams appear in two-dimensional classification, unlike in one dimension.

Classification achieved for nets with mu-index 1 and central charge less than 1.

Abstract

We classify two-dimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of one-dimensional local conformal nets, Dynkin diagrams D_{2n+1} and E_7 do not appear, but now they do appear in this classification of two-dimensional local conformal nets. Such nets are also characterized as two-dimensional local conformal nets with mu-index equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for one-dimensional nets, is 2-cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.