Complete classification of rational conformal field theories (RCFTs)

Develop a complete classification of two‑dimensional rational conformal field theories, including all possible chiral algebras, modular invariants, and consistent spectra consistent with unitarity and modular invariance.

Background

RCFTs play a central role in exactly solvable models and in the study of 2D critical phenomena. While many families (e.g., WZW models, cosets, orbifolds, rational points of marginal deformations) are known, a comprehensive classification is lacking.

A full classification would systematize the landscape of solvable 2D CFTs and clarify relations among known constructions (cosets, orbifolds, tensor products) and modular invariants (ADE‑type and beyond).

References

A complete classification of RCFTs is not known.

— Modern Approach to 2D Conformal Field Theory  (2412.18307 - Kusuki, 2024) in Section 6.6 (RCFT and ICFT)

A central question is which of the constructed spectra admit OPE coefficients defining a unitary compact CFT.

— Large gaps and BTZ entropy in modular spectra with positive integer degeneracies  (2609.39930 - Chang et al., 30 Sep 2026) in Section 6, Discussion, paragraph “Towards a full CFT realization”