2d Conformal Field Theories on Magic Triangle

Abstract

The magic triangle due to Cvitanovi\'c and Deligne--Gross is an extension of the Freudenthal--Tits magic square of semisimple Lie algebras. In this paper, we identify all two-dimensional 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 uncover a two-parameter family of fourth-order modular linear differential equation whose solutions yield the affine characters of all elements in the magic triangle. We further establish a universal coset relation for the whole triangle, generalizing the dual-pair structure with respect to $(E_8)_1$ in the Cvitanovi\'c--Deligne exceptional series. This coset structure determines the dimensions and degeneracies of all primary fields and leads to five atomic models from which all theories in the triangle can be constructed. At level two, we find that a distinghuished row of the triangle -- the subexceptional series -- exhibits emergent $N=1$ supersymmetry. The corresponding Neveu--Schwarz/Ramond characters satisfy a one-parameter family of fermionic modular linear differential equations. In addition, we find several new uniform coset constructions involving WZW models at higher levels.