Skip to main content
SpringerLink
Log in

Computation of cohomology of vertex algebras

  • Published:
Japanese Journal of Mathematics Aims and scope

Abstract

We review cohomology theories corresponding to the chiral and classical operads. The first one is the cohomology theory of vertex algebras, while the second one is the classical cohomology of Poisson vertex algebras (PVA), and we construct a spectral sequence relating them. Since in “good” cases the classical PVA cohomology coincides with the variational PVA cohomology and there are well-developed methods to compute the latter, this enables us to compute the cohomology of vertex algebras in many interesting cases. Finally, we describe a unified approach to integrability through vanishing of the first cohomology, which is applicable to both classical and quantum systems of Hamiltonian PDEs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Bakalov, A. D’Andrea and V.G. Kac, Theory of finite pseudoalgebras, Adv. Math., 162 (2001), 1–140.

    Article  MathSciNet  Google Scholar 

  2. B. Bakalov, A. De Sole, R. Heluani and V.G. Kac, An operadic approach to vertex algebra and Poisson vertex algebra cohomology, Jpn. J. Math., 14 (2019), 249–342.

    Article  MathSciNet  Google Scholar 

  3. B. Bakalov, A. De Sole, R. Heluani and V.G. Kac, Chiral versus classical operad, Int. Math. Res. Not. IMRN, 2020 (2020), no. 19, 6463–6488.

    Article  MathSciNet  Google Scholar 

  4. B. Bakalov, A. De Sole, V.G. Kac and V. Vignoli, Poisson vertex algebra cohomology and differential Harrison cohomology, Progr. Math. (to appear); preprint, arXiv:1907.06934.

  5. B. Bakalov, A. De Sole, R. Heluani, V.G. Kac and V. Vignoli, Classical and variational Poisson cohomology, in preparation.

  6. B. Bakalov, A. De Sole and V.G. Kac, Computation of cohomology of Lie conformal and Poisson vertex algebras, Selecta Math. (N.S.), 26 (2020), 50.

    Article  MathSciNet  Google Scholar 

  7. B. Bakalov and V.G. Kac, Field algebras, Int. Math. Res. Not., 2003 (2003), 123–159.

    Article  MathSciNet  Google Scholar 

  8. B. Bakalov, V.G. Kac and A.A. Voronov, Cohomology of conformal algebras, Comm. Math. Phys., 200 (1999), 561–598.

    Article  MathSciNet  Google Scholar 

  9. A. Barakat, A. De Sole and V.G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, Jpn. J. Math., 4 (2009), 141–252.

    Article  MathSciNet  Google Scholar 

  10. A. Beilinson and V. Drinfeld, Chiral Algebras, Amer. Math. Soc. Colloq. Publ., 51, Amer. Math. Soc., Providence, RI, 2004.

    Google Scholar 

  11. R.E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A., 83 (1986), 3068–3071.

    Article  MathSciNet  Google Scholar 

  12. A. D’Andrea and V.G. Kac, Structure theory of finite conformal algebras, Selecta Math. (N.S.), 4 (1998), 377–418.

    Article  MathSciNet  Google Scholar 

  13. A. De Sole and V.G. Kac, Freely generated vertex algebras and nonlinear Lie conformal algebras, Comm. Math. Phys., 254 (2005), 659–694.

    Article  MathSciNet  Google Scholar 

  14. A. De Sole and V.G. Kac, Finite vs. affine W-algebras, Jpn. J. Math., 1 (2006), 137–261.

    Article  MathSciNet  Google Scholar 

  15. A. De Sole and V.G. Kac, Lie conformal algebra cohomology and the variational complex, Comm. Math. Phys., 292 (2009), 667–719.

    Article  MathSciNet  Google Scholar 

  16. A. De Sole and V.G. Kac, Essential variational Poisson cohomology, Comm. Math. Phys., 313 (2012), 837–864.

    Article  MathSciNet  Google Scholar 

  17. A. De Sole and V.G. Kac, The variational Poisson cohomology, Jpn. J. Math., 8 (2013), 1–145.

    Article  MathSciNet  Google Scholar 

  18. D.K. Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc., 104 (1962), 191–204.

    Article  MathSciNet  Google Scholar 

  19. G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2), 46 (1945), 58–67.

    Article  MathSciNet  Google Scholar 

  20. V.G. Kac, Vertex Algebras for Beginners. 2nd ed., Univ. Lecture Ser., 10, Amer. Math. Soc., Providence, RI, 1998.

    MATH  Google Scholar 

  21. V.G. Kac and M. Wakimoto, Quantum reduction and representation theory of superconformal algebras, Adv. Math., 185 (2004), 400–458.

    Article  MathSciNet  Google Scholar 

  22. I.S. Krasil’shchik, Schouten bracket and canonical algebras, In: Global Analysis—Studies and Applications. III, Lecture Notes in Math., 1334, Springer-Verlag, 1988, pp. 79–110.

  23. H. Li, Vertex algebras and vertex Poisson algebras, Commun. Contemp. Math., 6 (2004), 61–110.

    Article  MathSciNet  Google Scholar 

  24. J. McCleary, A User’s Guide to Spectral Sequences. 2nd ed., Cambridge Stud. Adv. Math., 58, Cambridge Univ. Press, 2001.

  25. A. Nijenhuis and R.W. Richardson. Jr., Deformations of Lie algebra structures, J. Math. Mech., 17 (1967), 89–105.

    MathSciNet  MATH  Google Scholar 

  26. P.J. Olver, Bi-Hamiltonian systems, In: Ordinary and Partial Differential Equations, Pitman Res. Notes Math. Ser., 157, Longman Sci. Tech., New York, 1987, pp. 176–193.

    Google Scholar 

  27. D. Tamarkin, Deformations of chiral algebras, In: Proceedings of the International Congress of Mathematicians. Vol. II, Beijing, 2002, pp. 105–116.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially conducted during the authors’ visits to MIT and to the University of Rome La Sapienza. The first author was supported in part by a Simons Foundation grant 584741. The second author was partially supported by the national PRIN fund n. 2015ZWST2C_001 and the University funds n. RM116154CB35DFD3 and RM11715C7FB74D63. All three authors were supported in part by the Bert and Ann Kostant fund.

Author information

Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Bojko Bakalov

  2. Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185, Rome, Italy

    Alberto De Sole

  3. Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, MA, 02139, USA

    Victor G. Kac

Authors
  1. Bojko Bakalov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alberto De Sole
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Victor G. Kac
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Bojko Bakalov, Alberto De Sole or Victor G. Kac.

Additional information

Communicated by: Yasuyuki Kawahigashi

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bakalov, B., De Sole, A. & Kac, V.G. Computation of cohomology of vertex algebras. Jpn. J. Math. 16, 81–154 (2021). https://doi.org/10.1007/s11537-020-2034-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11537-020-2034-9

Keywords and phrases

Mathematics Subject Classification (2020)

Search

Navigation