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Cyclic orbifolds of lattice vertex operator algebras having group-like fusions

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Abstract

Let L be an even (positive definite) lattice and \(g\in O(L)\). In this article, we prove that the orbifold vertex operator algebra \(V_{L}^{{\hat{g}}}\) has group-like fusion if and only if g acts trivially on the discriminant group \({\mathcal {D}}(L)=L^*/L\) (or equivalently \((1-g)L^*<L\)). We also determine their fusion rings and the corresponding quadratic space structures when g is fixed point free on L. By applying our method to some coinvariant sublattices of the Leech lattice \(\Lambda \), we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic vertex operator algebras of central charge 24 using the orbifold vertex operator algebra \(V_{\Lambda _g}^{{\hat{g}}}\).

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Acknowledgements

The author thanks Hiroki Shimakura for simulating discussion and comments. He also thanks the referee for very helpful comments. A preliminary version of this article has been reported in a publication of RIMS, Kyoto University.

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  1. Institute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan

    Ching Hung Lam

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Correspondence to Ching Hung Lam.

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C. H. Lam was partially supported by a research Grant AS-IA-107-M02 of Academia Sinica and MoST Grant 104-2115-M-001-004-MY3 of Taiwan.

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Lam, C.H. Cyclic orbifolds of lattice vertex operator algebras having group-like fusions. Lett Math Phys 110, 1081–1112 (2020). https://doi.org/10.1007/s11005-019-01251-2

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