  We introduce a novel class of two-dimensional non-unitary rational conformal
field theories (RCFTs) whose modular data are identical to the generalized
Haagerup-Izumi modular data. Via the bulk-boundary correspondence, they are
related to the three-dimensional non-unitary Haagerup topological field
theories, recently constructed by a topological twisting of three-dimensional
${\cal N}=4$ rank-zero superconformal field theories (SCFTs), called S-fold
SCFTs. We propose that, up to the overall factors, the half-indices of the
rank-zero SCFTs give the explicit Nahm representation of four conformal
characters of the RCFTs including the vacuum character. Using the theory of
Bantay-Gannon, we can successfully complete them into the full admissible
conformal characters of the RCFTs.
