Free Field Realizations of Superelliptic Affine Lie Algebras

We study Wakimoto-type free field constructions for superelliptic affine Lie algebras associated with coordinate rings $A=\mathbb{C}[t^{\pm1},u \mid u^m = p(t)]$, focusing on $\mathfrak{sl}_2$. We construct explicit operators on a tensor product of $m$ ghost Fock spaces, recovering the standard Wakimoto operator product expansions in the even sector and the correct $h^{(0)}$-charge relations in the odd sector. We then prove that the remaining mixed-sector brackets are obstructed within this class by two independent mechanisms: a charge-residue obstruction, arising from the K"{a}hler differential recurrence, and a Heisenberg branch-cut obstruction, caused by non-integer exponents in vertex operator products. These results yield a unified obstruction theorem for Wakimoto-type constructions in the superelliptic setting, explaining the failure of na"{i}ve free field realizations beyond the classical affine case.