Verification of the folding-embedding conjecture for the pair {A_{2r−1},B_r}

Establish, for general integers r≥2, the conformal embedding predicted by the folding conjecture, namely CFT(A1,B_r)⊂CFT(A1,A_{2r−1}), by first identifying the CFT associated with (A1,B_r) and then proving the embedding for all r.

Background

The authors verify the folding-embedding conjecture CFT(A1,G′)⊂CFT(A1,G) for several foldings, including {E6,F4}, {D4,G2}, and {D_r,C_{r−1}}. However, the family {A_{2r−1},B_r} remains unverified in general.

The obstacle is the lack of a general CFT identification for (A1,B_r), which prevents a direct check of the proposed embedding beyond small values r=2,3 where it can be confirmed.

References

Currently we could not verify the cases {A_{2r-1},B_r} for general r, due to the lack of CFT understanding for the type (A_1,B_r). Nevertheless, for r=2,3, we can determine \mathrm{\bf CFT}(A_1,B_r) and confirm the conjecture.

— Dynkin diagrams, generalized Nahm sums and 2d CFTs  (2604.00847 - Sun et al., 1 Apr 2026) in Section 3 (Generalized Nahm sums and CFTs), paragraph after Conjecture \ref{conj:A1}