Abstract
We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces the vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to the classical operad, which produces a Poisson vertex algebra cohomology complex. The latter is closely related to the variational Poisson cohomology studied by two of the authors.
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Acknowledgments
We are grateful to Pavel Etingof for providing a proof of Lemma 6.4. We would like to acknowledge discussions with him as well as with Corrado De Concini, Andrea Maffei and Alexander Voronov.We also thank the referee for thoughtful comments. The research was partially conducted during the authors’ visits to IHES, MIT, SISSA and the University of Rome La Sapienza. We are grateful to these institutions for their kind hospitality. The first author was supported in part by a Simons Foundation grant 279074. The second author was partially supported by the national PRIN fund n. 2015ZWST2C_001 and the University funds n. RM116154CB35DFD3 and RM11715C7FB74D63. The third author was partially supported by the Bert and Ann Kostant fund.
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Communicated by: Yasuyuki Kawahigashi
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Bakalov, B., De Sole, A., Heluani, R. et al. An operadic approach to vertex algebra and Poisson vertex algebra cohomology. Jpn. J. Math. 14, 249–342 (2019). https://doi.org/10.1007/s11537-019-1825-3
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DOI: https://doi.org/10.1007/s11537-019-1825-3
Keywords and phrases
- superoperads
- chiral and classical operads
- vertex algebra and PVA coho-mologies
- variational Poisson cohomology