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

Sergio Cecotti, Cumrun Vafa|arXiv (Cornell University)|Nov 23, 1992

Nonlinear Waves and Solitons7 references153 citations

TL;DR

This paper establishes a direct correspondence between the soliton spectrum in massive N=2 supersymmetric quantum field theories in two dimensions and the scaling dimensions of chiral primary fields at the associated conformal field theory fixed point. By relating soliton numbers to monodromy phases via the tt* equations, the authors derive a classification of N=2 theories through generalized Dynkin diagrams, recovering the A–D–E classification for minimal models and extending it to higher central charge theories.

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.

Motivation & Objective

• To establish a general relation between soliton degeneracies in massive N=2 theories and scaling dimensions of chiral primary fields at the conformal point.
• To show that the reality condition on chiral field charges imposes strong constraints on soliton numbers, leading to a classification program.
• To generalize the A–D–E classification of minimal N=2 models by associating soliton structures to bilinear forms and Dynkin-like diagrams.
• To extend the classification to non-Landau-Ginzburg theories using tt* geometry and monodromy data.
• To provide a framework for classifying N=2 conformal theories admitting massive deformations, including those related to Calabi-Yau sigma models.

Proposed method

• Use the topological-anti-topological (tt*) equations to relate the monodromy of the flat connection to the U(1) charges of chiral primary fields.
• Map the phase of monodromy eigenvalues to the U(1) charges of chiral operators, establishing a link between soliton numbers and scaling dimensions.
• Define a bilinear form from soliton numbers between vacua, with 2 on the diagonal, to encode the soliton structure.
• Apply Picard-Lefschetz theory to Landau-Ginzburg models, relating soliton numbers to intersection numbers of vanishing cycles.
• Use the reality of chiral field charges to constrain allowed soliton configurations, leading to classification via generalized Dynkin diagrams.
• Leverage complex multiplication and Galois actions on period maps to predict UV OPE coefficients in terms of gamma functions and algebraic numbers.

Experimental results

Research questions

• RQ1 How are the scaling dimensions of chiral primary fields in N=2 superconformal field theories related to the soliton spectrum in their massive deformations?
• RQ2 What constraints do the reality of chiral field charges impose on the number of solitons between vacua in massive N=2 theories?
• RQ3 Why does the A–D–E classification naturally arise in the context of N=2 minimal models, and can this be generalized beyond minimal models?
• RQ4 Can the soliton structure of massive N=2 theories be encoded in a generalized Dynkin diagram formalism, and what are the properties of such diagrams?
• RQ5 How can the UV OPE coefficients of chiral fields be predicted from soliton data and monodromy in the tt* framework?

Key findings

• The phase of monodromy eigenvalues of the tt* connection corresponds exactly to the U(1) charges of chiral primary fields at the conformal point.
• The soliton numbers between vacua determine a bilinear form with 2 on the diagonal, and its signature is directly related to the chiral ring structure.
• For minimal N=2 models, the reality of chiral field charges restricts soliton numbers to at most one between any pair of vacua, leading to the A–D–E classification.
• The generalized Dynkin diagrams derived from soliton data classify N=2 theories beyond minimal models, with the A–D–E case arising as a special instance.
• In Landau-Ginzburg models, soliton numbers match the intersection numbers of vanishing cycles, confirming the Picard-Lefschetz correspondence.
• The UV OPE coefficients of chiral operators are predicted by a conjectured formula involving gamma functions and algebraic numbers, with explicit agreement in A_n minimal models.