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The Virasoro vertex algebra and factorization algebras on Riemann surfaces


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

Mathematics Subject Classification

  • 81R10
  • 17B65
  • 18G55