Free Field Realizations of Superelliptic Affine Lie Algebras

TL;DR

This paper develops Wakimoto-type free field constructions for superelliptic affine Lie algebras, revealing fundamental obstructions to their realization beyond classical cases.

Contribution

It introduces explicit operators for superelliptic affine Lie algebras and identifies two key obstructions preventing naive free field realizations.

Findings

Constructed explicit Wakimoto operators for superelliptic affine Lie algebras.

Identified charge-residue and branch-cut obstructions in mixed-sector brackets.

Proved a unified obstruction theorem explaining limitations of free field realizations.

Abstract

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.