Minimal models, modular linear differential equations and modular forms of fractional weights

Abstract: We show that modular forms of fractional weights on principal congruence subgroups of odd levels, which are found by T. Ibukiyama, naturally appear as characters being multiplied $\eta^{c_{\text{eff}}}$ of the so-called minimal models of type $(2, p)$, where $c_{\text{eff}}$ is the effective central charge of a minimal model. Using this fact and modular invariance property of the space of characters, we give a different proof of the result showned by Ibukiyama that is the explicit formula representing $\mathrm{SL}{2}(\mathbb{Z})$ on the space of the ibukiyama modular forms. We also find several pairs of spaces of the Ibukiyama modular forms on $\Gamma(m)$ and $\Gamma(n)$ with $m|n$ having the property that the former are included in the latter. Finally, we construct vector-valued modular forms of weight $k\in\frac{1}{5}\mathbb{Z}{>0}$ starting from the Ibukiyama modular forms of weight $k\in\frac{1}{5}\mathbb{Z}_{>0}$ with some multiplier system by a symmetric tensor product of this representation and show that the components functions coincides with the solution space of some monic modular linear differential equation of weight $k$ and the order $1+5k$.