Singular automorphic products and BKM algebras

Abstract

After proving the moonshine conjecture, Borcherds asked in 1995 whether there exist only finitely many holomorphic automorphic products of singular weight. We show that, up to conjugation, there are exactly 27 such products on lattices of the form 2U⊕ L. The classification proceeds by identifying each product at a standard $1$-dimensional cusp with the superdenominator of a BKM superalgebra that carries a graded module structure over an affine Lie superalgebra ĝ. The auxiliary algebras ĝ are shown to satisfy strong conditions that reduce the possibilities to an initial list of 1372 candidates. An elimination argument leaves exactly 85 algebras ĝ that correspond to the 27 products. This classification is closely connected with certain distinguished families of vertex operator superalgebras. Our results provide a systematic framework for extending affine Lie superalgebras to hyperbolic ones with modularity, generalizing the work of Feingold and Frenkel (1983).

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BibTeX RIS

BibTeX

@misc{sun2026singular,
  title = {{Singular automorphic products and BKM algebras}},
  author = {Kaiwen Sun and Haowu Wang and Brandon Williams},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15658},
  eprint = {2609.15658},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15658},
}

RIS

TY  - UNPB
TI  - Singular automorphic products and BKM algebras
AU  - Kaiwen Sun
AU  - Haowu Wang
AU  - Brandon Williams
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15658
UR  - https://arxiv.org/abs/2609.15658
AB  - After proving the moonshine conjecture, Borcherds asked in 1995 whether there exist only finitely many holomorphic automorphic products of singular weight. We show that, up to conjugation, there are exactly 27 such products on lattices of the form 2U⊕ L. The classification proceeds by identifying each product at a standard $1$-dimensional cusp with the superdenominator of a BKM superalgebra that carries a graded module structure over an affine Lie superalgebra ĝ. The auxiliary algebras ĝ are shown to satisfy strong conditions that reduce the possibilities to an initial list of 1372 candidates. An elimination argument leaves exactly 85 algebras ĝ that correspond to the 27 products. This classification is closely connected with certain distinguished families of vertex operator superalgebras. Our results provide a systematic framework for extending affine Lie superalgebras to hyperbolic ones with modularity, generalizing the work of Feingold and Frenkel (1983).
ER  -

Abstract

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