Cyclotomicity of squared unitary F-matrix entries

Prove that, for the unitary F-matrix associated with a rational conformal field theory, every squared entry satisfies \(\lvert\bar F_{ij}\rvert^2\in\mathbb{Q}(\zeta_{2\tilde N})\), where \(\tilde N\) is the least common multiple of the denominators of the exponents of the associated Fuchsian ordinary differential equation at \(z=0\).

Background

The paper computes braiding F-matrices as connection matrices between Frobenius bases of Fuchsian differential equations for Virasoro minimal-model conformal blocks. After rescaling by products of three-point structure constants, the resulting matrices are made unitary or orthogonal. This normalization is useful because the numerical matrices can then be converted into exact algebraic expressions.

The authors observe in all of their examples that the squares of the entries of the unitary F-matrix lie in a cyclotomic extension of the rational numbers. The conjecture proposes a specific containing field, determined by the denominators of the local exponents of the Fuchsian equation, and is intended to support the conversion of numerical F-matrix estimates into exact values.