Selecta Mathematica

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

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

Open Access
Article
  • 62 Downloads

Abstract

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 

Notes

Acknowledgements

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.

References

  1. 1.
    Barthel, T., May, J.P., Riehl, E.: Six model structures for DG-modules over DGAs: model category theory in homological action. N. Y. J. Math. 20, 1077–1159 (2014). arXiv: 1310.1159 MathSciNetMATHGoogle Scholar
  2. 2.
    Barwick, C., Kan, D.M.: Relative categories: another model for the homotopy theory of homotopy theories. Indag. Math. (N.S.) 23(1–2), 42–68 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bashkirov, D., Voronov, A.A.: The BV formalism for L1-algebras. J. Homotopy Relat. Struct. arXiv:1410.6432
  4. 4.
    Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and quantization. Phys. Lett. B 102(1), 27–31 (1981)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Batalin, I.A., Vilkovisky, G.A.: Quantization of gauge theories with linearly dependent generators. Phys. Rev. D (3) 28(10), 2567–2582 (1983)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Batalin, I.A., Vilkovisky, G.A.: Existence theorem for gauge algebra. J. Math. Phys. 26(1), 172–184 (1985)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Behrend, K., Fantechi, B.: Gerstenhaber and Batalin–Vilkovisky structures on Lagrangian intersections, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Inc., Boston, MA, pp. 1–47 (2009)Google Scholar
  8. 8.
    Beilinson, A., Drinfeld, V.: Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51. American Mathematical Society, Providence, RI (2004)MATHGoogle Scholar
  9. 9.
    Ben-Bassat, O., Brav, C., Bussi, V., Joyce, D.: A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications. Geom. Topol. 19(3), 1287–1359 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bousfield, A.K., Gugenheim, V.K.A.M.: On PL de Rham theory and rational homotopy type. Mem. Am. Math. Soc. 8, 179 (1976)MathSciNetMATHGoogle Scholar
  11. 11.
    Braun, C., Lazarev, A.: Homotopy BV algebras in Poisson geometry. Trans. Mosc. Math. Soc. 74, 217–227 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Calaque, D., Pantev, T., Toën, B., Vaquié, M., Vezzosi, G.: Shifted Poisson structures and deformation quantization. J. Topol. 10(2), 483–584 (2017). arXiv:1506.03699 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cattaneo, A.S.: From topological field theory to deformation quantization and reduction, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, pp. 339–365. (2006) MR2275685Google Scholar
  14. 14.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Classical and quantum Lagrangian field theories with boundary (2012). arXiv:1207.0239
  15. 15.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Semiclassical quantization of classical field theories. Mathematical aspects of quantum field theories. Math. Phys. Stud., Springer, Cham, pp. 275–324 (2015)Google Scholar
  16. 16.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Perturbative BV theories with Segal-like gluing (2016). arXiv:1602.00741
  17. 17.
    Chu, H., Haugseng, R.: Enriched 1-operads (2017). arXiv:1707.08049
  18. 18.
    Costello, K.: Renormalization and Effective Field Theory. Mathematical Surveys and Monographs, vol. 170. American Mathematical Society, Providence, RI (2011)MATHGoogle Scholar
  19. 19.
    Costello, K., Gwilliam, O.: Factorization algebras in quantum field theory. Vol. 1, New Mathematical Monographs, vol. 31. Cambridge University Press, Cambridge (2017). http://people.mpim-bonn.mpg.de/gwilliam
  20. 20.
    Duskin, J.W.: Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories. Theory Appl. Categ. 9(10), 198–308 (2002). (electronic)MathSciNetMATHGoogle Scholar
  21. 21.
    Dwyer, W.G., Kan, D.M.: Simplicial localizations of categories. J. Pure Appl. Algebra 17(3), 267–284 (1980)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Dwyer, W.G., Kan, D.M.: Function complexes in homotopical algebra. Topology 19(4), 427–440 (1980)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fiorenza, D.: An introduction to the Batalin–Vilkovisky formalism. Comptes Rendus des Rencontres Mathematiques de Glanon (2003). arXiv:math/0402057
  24. 24.
    Fresse, B.: Koszul duality complexes for the cohomology of iterated loop spaces of spheres. An alpine expedition through algebraic topology. Contemp. Math., vol. 617, Amer. Math. Soc., Providence, RI, pp. 165–188 (2014)  https://doi.org/10.1090/conm/617/12281
  25. 25.
    Gaitsgory, D., Rozenblyum, N.: A study in derived algebraic geometry. Vol. I. Correspondences and duality, Mathematical Surveys and Monographs, vol. 221, American Mathematical Society, Providence, RI (2017). http://www.math.harvard.edu/~gaitsgde/GL
  26. 26.
    Gaitsgory, D., Rozenblyum, N.: A study in derived algebraic geometry. Vol. II. Deformations, Lie theory and formal geometry, Mathematical Surveys and Monographs, vol. 221, American Mathematical Society, Providence, RI (2017). http://www.math.harvard.edu/~gaitsgde/GL
  27. 27.
    Gepner, D., Haugseng, R.: Enriched 1-categories via non-symmetric 1-operads. Adv. Math. 279, 575–716 (2015). arXiv:1312.3178 MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Gepner, D., Haugseng, R., Nikolaus, T.: Lax colimits and free fibrations in 1-categories. Doc. Math. 22, 1225–1266 (2017). arXiv:1501.02161 MathSciNetMATHGoogle Scholar
  29. 29.
    Gwilliam, O.: Factorization algebras and free field theories. Thesis (Ph.D.)—Northwestern University (2012). http://people.mpim-bonn.mpg.de/gwilliam
  30. 30.
    Gwilliam, O., Johnson-Freyd, T.: How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism (2012). arXiv:1202.1554
  31. 31.
    Haugseng, R.: The higher Morita category of En-algebras. Geom. Topol. 21, 1631–1730 (2017). arXiv:1412.8459 MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Haugseng, R.: \(\infty \)-Operads via Day convolution (2017). arXiv:1708.09632
  33. 33.
    Hennion, B.: Tangent Lie algebra of derived Artin stacks (2013). arXiv:1312.3167
  34. 34.
    Hinich, V.: Homological algebra of homotopy algebras. Commun. Algebra 25(10), 3291–3323 (1997)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Hinich, V.: DG coalgebras as formal stacks. J. Pure Appl. Algebra 162(2–3), 209–250 (2001). arXiv:math/9812034 MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence, RI (1999)Google Scholar
  37. 37.
    Knudsen, B.: Higher enveloping algebras (2016). arXiv:1605.01391
  38. 38.
    Loday, J.-L., Vallette, B.: Algebraic Operads. Grundlehren der Mathematischen Wissenschaften, vol. 346. Springer, Heidelberg (2012)MATHGoogle Scholar
  39. 39.
    Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009). http://math.harvard.edu/~lurie
  40. 40.
    Lurie, J.: Derived algebraic geometry V: structured spaces (2009). http://math.harvard.edu/~lurie
  41. 41.
    Lurie, J.: Derived algebraic geometry VII: spectral schemes (2011). http://math.harvard.edu/~lurie
  42. 42.
    Lurie, J.: Higher Algebra (2014). http://math.harvard.edu/~lurie
  43. 43.
    Lurie, J.: Spectral algebraic geometry (2017). http://math.harvard.edu/~lurie
  44. 44.
    Markarian, N.: Weyl n-algebras. Commun. Math. Phys 350(2), 421–442 (2017). arXiv:1504.01931 MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Mazel-Gee, A.: Quillen adjunctions induce adjunctions of quasicategories. N. Y. J. Math. 22, 57–93 (2016). available at arXiv:1501.03146 MathSciNetMATHGoogle Scholar
  46. 46.
    Pantev, T., Toën, B., Vaquié, M., Vezzosi, G.: Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci. 117, 271–328 (2013)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Pavlov, D., Scholbach, J.: Admissibility and rectification of colored symmetric operads (2014). arXiv:1410.5675
  48. 48.
    Pavlov, D., Scholbach, J.: Homotopy theory of symmetric powers (2015). arXiv:1510.04969
  49. 49.
    Pridham, J.P.: Deformation quantisation for (\(-1\))-shifted symplectic structures and vanishing cycles (2015). arXiv:1508.07936
  50. 50.
    Safronov, P.: Poisson reduction as a coisotropic intersection (2015). arXiv:1509.08081
  51. 51.
    Schwede, S., Shipley, B.E.: Algebras and modules in monoidal model categories. Proc. Lond. Math. Soc. (3) 80(2), 491–511 (2000)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Toën, B., Vezzosi, G.: Homotopical algebraic geometry II: geometric stacks and applications. Mem. Am. Math. Soc. 193, 902 (2008). arXiv:math/0404373 MathSciNetMATHGoogle Scholar
  53. 53.
    Toën, B.: Operations on derived moduli spaces of branes (2013). arXiv:1307.0405
  54. 54.
    Vezzosi, G.: Quadratic forms and Clifford algebras on derived stacks. Adv. Math. 301, 161–203 (2016). arXiv:1309.1879 MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Wallbridge, J.: Homotopy theory in a quasi-abelian category (2015). arXiv:1510.04055
  56. 56.
    Witten, E.: A note on the antibracket formalism. Mod. Phys. Lett. A 5(7), 487–494 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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

Personalised recommendations