2d Conformal Field Theories on Magic Triangle

TL;DR

The paper extends the Cvitanović–Deligne magic triangle to two-dimensional rational conformal field theories by identifying the full set of level-one RCFTs and proving they satisfy a universal four-parameter modular linear differential equation, together with a broad coset framework that generalizes known E8 dual pairs. It shows that all level-one theories decompose into five atomic building blocks and obey a magic bilinear relation among characters, revealing a matrix-valued modular structure. At level two, it uncovers emergent N=1 supersymmetry in the subexceptional series and develops uniform fermionic MLDEs for NS/R sectors, alongside new level-two coset constructions across the exceptional series. The work then extends these constructions to arbitrary levels, providing explicit identifications with WZW models, Virasoro minimal and non-diagonal invariants, and a network of coset relations, thereby offering a cohesive, scalable framework for RCFTs tied to the extended magic triangle.

Abstract

The magic triangle due to Cvitanović and Deligne--Gross is an extension of the Freudenthal--Tits magic square of semisimple Lie algebras. In this paper, we identify all 2d rational conformal field theories associated to the magic triangle. These include various Wess--Zumino--Witten (WZW) models, Virasoro minimal models, compact bosons and their non-diagonal modular invariants. At level one, we find a two-parameter family of modular linear differential equation of fourth order whose solutions produce the affine characters of all elements in the magic triangle. We find a universal coset relation for the whole triangle which generalizes the dual pairs with respect to $(E_8)_1$ in the Cvitanović--Deligne exceptional series. This leads to the dimension and degeneracy of each primary field and also to five atomic models which constitute all theories in the triangle. At level two, we find a special row of the triangle -- the subexceptional series has novel $N=1$ supersymmetry, and the Neveu--Schwarz/Ramond characters satisfy a one-parameter family of fermionic modular linear differential equations. Moreover, we find many new coset constructions involving WZW models at higher levels.