Abstract
In this paper, we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated with a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; and (iii) the lattice VOA associated with a positive definite even lattice. We prove that in all these cases, the first cohomology \(H^1(V, W)\) consists of zero-mode derivations for every \({\mathbb {N}}\)-graded V-module W (where the grading is not necessarily given by the L(0) operator). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. The relationship between the first cohomology of the VOA and that of the associated Zhu’s algebra is also discussed.
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Acknowledgements
The author would like to thank Yi-Zhi Huang for his long-term constant support, especially for the valuable suggestions and comments that greatly improved this paper. The author would also like to thank the Pacific Institute of Mathematical Science for offering the opportunity of teaching a network-wide graduate course on vertex algebras and Ethan Armitage, Ben Garbuz, Zach Goldthorpe, Nicholas Lai, and Mihai Marian for their participation. Many ideas for this paper arose during lectures and discussions.
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Qi, F. First cohomologies of affine, Virasoro, and lattice vertex operator algebras. Lett Math Phys 112, 56 (2022). https://doi.org/10.1007/s11005-022-01548-9
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DOI: https://doi.org/10.1007/s11005-022-01548-9