On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version)

TL;DR

This paper establishes a connection between soliton spectra in N=2 supersymmetric quantum field theories in two dimensions and the scaling dimensions of chiral fields at conformal points, leading to a classification scheme using generalized Dynkin diagrams.

Contribution

It introduces a classification framework for symmetric N=2 conformal theories and their deformations based on soliton spectra and Dynkin diagram generalizations, including Landau-Ginzburg models.

Findings

Relation between soliton spectrum and scaling dimensions at conformal points

Restrictions on soliton numbers from reality conditions of scaling dimensions

Classification of theories via generalized Dynkin diagrams

Abstract

We find a relation between the spectrum of solitons of massive $N=2$ quantum field theories in $d=2$ and the scaling dimensions of chiral fields at the conformal point. The condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetric $N=2$ conformal theories and their massive deformations in terms of a suitable generalization of Dynkin diagrams (which coincides with the A--D--E Dynkin diagrams for minimal models). The Landau-Ginzburg theories are a proper subset of this classification. In the particular case of LG theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity; the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of Picard-Lefschetz theory.