Selecta Mathematica

, Volume 24, Issue 2, pp 1247–1313 | Cite as

Linear Batalin–Vilkovisky quantization as a functor of \(\infty \)-categories

  • Owen Gwilliam
  • Rune Haugseng
Open Access


We study linear Batalin–Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of \(\infty \)-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of \(\mathrm {E}_{n}\)-algebras to produce a “higher BV quantization” functor, and when restricted to formal moduli problems, it behaves like a determinant. Along the way we also use our machinery to give an algebraic construction of \(\mathrm {E}_{n}\)-enveloping algebras for shifted Lie algebras.

Mathematics Subject Classification

18G55 18D50 53D55 81T70 55U99 13D10 58J52 14D15 14D23 17B55 17B81 81Q99 



Open access funding provided by Max Planck Society. OG thanks Kevin Costello for teaching him the BV formalism and pointing out that it behaves like a determinant, an idea he pursued in his thesis and that prompted this collaboration. He also thanks Nick Rozenblyum, Toly Preygel, and Thel Seraphim for many helpful conversations around quantization and higher categories. RH thanks Irakli Patchkoria for help with model-categorical technicalities and Dieter Degrijse for some basic homological algebra. Together we thank Theo Johnson-Freyd, David Li-Bland, and Claudia Scheimbauer for stimulating conversations around these topics, particularly the pursuit of higher Weyl quantization. Finally, this work was begun at the Max Planck Institute for Mathematics when RH and OG were both postdocs there, and we deeply appreciate the open and stimulating atmosphere of MPIM that made it so easy to begin our collaboration. Moreover, it is through the MPIM’s great generosity that we were able to continue work and finish the paper during several visits by RH.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Max-Planck-Institut für MathematikBonnGermany
  2. 2.Københavns UniversitetCopenhagenDenmark

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