Free Field Realizations of Superelliptic Affine Lie Algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915930312802304 |
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| author | Santos, Felipe Albino dos |
| author_facet | Santos, Felipe Albino dos |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Free Field Realizations of Superelliptic Affine Lie Algebras Santos, Felipe Albino dos Representation Theory 17B67, 17B69 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. |
| title | Free Field Realizations of Superelliptic Affine Lie Algebras |
| topic | Representation Theory 17B67, 17B69 |
| url | https://arxiv.org/abs/2604.09461 |