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The Virasoro vertex algebra and factorization algebras on Riemann surfaces

Abstract

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello–Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta–gamma system using the method of effective BV quantization.

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Notes

  1. 1.

    Some care is needed to define this category correctly. We refer the interested reader to [9]

References

  1. 1.

    Atiyah, M.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc. 85, 181–207 (1957)

  2. 2.

    Beilinson, A., Drinfeld, V.: Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51. American Mathematical Society, Providence, RI (2004). doi:10.1090/coll/051

  3. 3.

    Beĭlinson, A.A., Schechtman, V.V.: Determinant bundles and Virasoro algebras. Commun. Math. Phys. 118(4), 651–701 (1988). http://projecteuclid.org/euclid.cmp/1104162170

  4. 4.

    Butson, D., Yoo, P.: Degenerate classical field theories and boundary theories. Available at https://arxiv.org/abs/1611.00311

  5. 5.

    Cliff, E.: Universal d-modules and stacks of étale germs of n-dimensional varieties. Available at https://arxiv.org/abs/1410.8457

  6. 6.

    Cliff, E.: Universal factorization spaces and algebras. Available at https://arxiv.org/abs/1608.08122

  7. 7.

    Costello, K.: A geometric construction of the Witten genus, II. Available at http://arxiv.org/abs/1112.0816

  8. 8.

    Costello, K.: Renormalization and Effective Field Theory, Mathematical Surveys and Monographs, vol. 170. American Mathematical Society, New York (2011)

  9. 9.

    Costello, K., Gwilliam, O.: Factorization Algebras in Quantum Field Theory, vol. 1. Cambridge University Press, Cambridge (2016). doi:10.1017/9781316678626. https://www.cambridge.org/core/books/factorization-algebras-in-quantum-field-theory/9597AFE8E8767F8F38A73C74B8F2501B

  10. 10.

    Costello, K., Gwilliam, O.: Factorization Algebras in Quantum Field Theory, vol. II. Cambridge University Press, Cambridge (2016)

  11. 11.

    Feigin, B., Frenkel, E.: Semi-infinite weil complex and the Virasoro algebra. Commun. Math. Phys. 137(3), 617–639 (1991). http://projecteuclid.org/euclid.cmp/1104202744

  12. 12.

    Francis, J., Gaitsgory, D.: Chiral Koszul duality. Sel. Math. (N.S.) 18(1), 27–87 (2012). doi:10.1007/s00029-011-0065-z

  13. 13.

    Frenkel, E., Ben-Zvi, D.: Vertex Algebras, Mathematical Surveys and Monographs, vol. 88, 2nd edn. AMS, New York (2001)

  14. 14.

    Fuks, D. B. Cohomology of infinite-dimensional \({\rm L}\)ie algebras. Contemporary Soviet Mathematics, pp xii+339. Consultants Bureau, New York (1986). [Translated from the Russian by A. B. Sosinskiĭ]

  15. 15.

    Iohara, K., Koga, Y.: Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics. Springer, London (2011). doi:10.1007/978-0-85729-160-8

  16. 16.

    Kac, V.G., Wakimoto, M.: Unitarizable highest weight representations of the Virasoro, Neveu-Schwarz and Ramond algebras. In: Conformal Groups and Related Symmetries: Physical Results and Mathematical Background (Clausthal-Zellerfeld, 1985), Lecture Notes in Physics, vol. 261, pp. 345–371. Springer, Berlin (1986). doi:10.1007/3540171630_93

  17. 17.

    Lawrence, A., Nekrasov, N., Vafa, C.: On conformal field theories in four dimensions. Nucl. Phys. B 533(1–3), 199–209 (1998). doi:10.1016/S0550-3213(98)00495-7

  18. 18.

    Losev, A., Moore, G., Nekrasov, N., Shatashvili, S.: Chiral Lagrangians, anomalies, supersymmetry, and holomorphy. Nucl. Phys. B 484(1–2), 196–222 (1997). doi:10.1016/S0550-3213(96)00612-8

  19. 19.

    Manetti, M.: On some formality criteria for DG-Lie algebras. arXiv:1310.3048 (2013)

  20. 20.

    Segal, G.B.: Differential Geometrical Methods in Theoretical Physics. Springer, Netherlands (1988)

  21. 21.

    Serre, J.P.: Quelques problémes globaux relatifs aux variétés de stein. Colloque sur les fonctions de plusieurs variables, pp. 57–68 (1953)

  22. 22.

    Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9(1), 237–302 (1996). doi:10.1090/S0894-0347-96-00182-8

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Acknowledgements

I’d like to thank Owen Gwilliam and Ryan Grady for their shared interest in this project and generosity in providing detailed comments and discussion pertaining to this work. I have also benefited from useful comments from and discussions with Chris Elliott, Ben Knudsen, and Philsang Yoo. Finally, I’d like to thank Kevin Costello for sparking my interest in this project.

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Correspondence to Brian Williams.

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Williams, B. The Virasoro vertex algebra and factorization algebras on Riemann surfaces. Lett Math Phys 107, 2189–2237 (2017). https://doi.org/10.1007/s11005-017-0982-7

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Keywords

  • Factorization algebras
  • Virasoro vertex algebra
  • BV quantization
  • Conformal field theory

Mathematics Subject Classification

  • 81R10
  • 17B65
  • 18G55