2d Conformal Field Theories on Magic Triangle

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 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 Cvitanović–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.

💡 Research Summary

The paper establishes a comprehensive bridge between the Cvitanović‑Deligne‑Gross “magic triangle” of Lie groups and two‑dimensional rational conformal field theories (RCFTs). Starting from the original Freudenthal‑Tits magic square, the authors extend the construction to a full two‑dimensional array parameterised by a pair of rational numbers (µ, ν) with µν ≥ 1. They provide universal formulas for the dimension d(µ, ν) and dual Coxeter number h∨(µ, ν) that hold for all entries, including the previously missing cases ν = 1/4 and ν = 4, which introduce the intermediate algebras A₁/₂ and E₇+½.

At level k = 1 the central result is a two‑parameter family of fourth‑order holomorphic modular linear differential equations (MLDEs) (eq. 3.9). The parameters (α, β) control the coefficients built from Eisenstein series, and the four independent solutions are precisely the affine characters of the Wess–Zumino–Witten (WZW) models, Virasoro minimal models, compact bosons and their non‑diagonal (VOA) extensions that populate the extended magic triangle (Table 2, Appendix A).

A universal “magic coset” relation (eq. 3.16) generalises the well‑known dual‑pair structure with respect to (E₈)₁ in the Cvitanović–Deligne exceptional series. For any entry g(µ, ν) the corresponding RCFT can be written as a coset \