SpringerLink
Log in

New model category structures for algebraic quantum field theory

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, we define and compare several new Quillen model structures which present the homotopy theory of algebraic quantum field theories. In this way, we expand foundational work of Benini and Schenkel (Fortschr Phys 67(8–9):1910015, 2019) by providing a richer framework to detect and treat homotopical phenomena in quantum field theory. Our main technical tool is a new extension model structure on operadic algebras which is constructed via (right) Bousfield localization.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Notes

  1. Of course, this classification is far from sharp.

  2. These properties are not essential for the definition of AQFTs but they will be so for later results.

  3. The use of trees to described operads is quite useful and it is justified in, for example, [37] or [44].

  4. The localization of \({{\,\mathrm{\textsf{C}}\,}}\) at \({{\,\mathrm{\textsf{S}}\,}}\) is the “initial" functor \({{\,\mathrm{\textsf{C}}\,}}\rightarrow {{\,\mathrm{\textsf{C}}\,}}_{{{\,\mathrm{\textsf{S}}\,}}}\) among those \({{\,\mathrm{\textsf{C}}\,}}\rightarrow {{\,\mathrm{\textsf{D}}\,}}\) that send \({{\,\mathrm{\textsf{S}}\,}}\) to isomorphisms.

  5. We interpret this condition as \(\mathbb {L}\upiota _{\sharp }\) being homotopically fully faithful. It is ensured if \(\upiota \) is fully faithful in the ordinary operadic sense.

References

  1. Ayala, D., Francis, J.: Factorization homology of topological manifolds. J. Topol. 8(4), 1045–1084 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barwick, C., Kan, D.M.: Relative categories: another model for the homotopy theory of homotopy theories. K. Ned. Akad. Wet. Indag. Math. New Ser. 23(1–2), 42–68 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barwick, C.: On left and right model categories and left and right Bousfield localizations. Homol. Homot. Appl. 12(2), 245–320 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Philos. Soc. 129(3), 447–475 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benini, M., Bruinsma, S., Schenkel, A.: Linear Yang–Mills theory as a homotopy AQFT. Commun. Math. Phys. 378(1), 185–218 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Benini, M., Carmona, V., Schenkel, A.: Strictification theorems for the homotopy time-slice axiom, August 2022. ar**v:2208.04344 [hep-th, physics:math-ph]

  7. Benini, M., Dappiaggi, C., Schenkel, A.: Algebraic quantum field theory on spacetimes with timelike boundary. Ann. Henri Poincare J. Theor. Math. Phys. 19(8), 2401–2433 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Benini, M., Giorgetti, L., Schenkel, A.: A skeletal model for 2d conformal AQFTs. Commun. Math. Phys. 395(1), 269–298 (2022)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Benini, M., Perin, M., Schenkel, A.: Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds. Commun. Math. Phys. 377(2), 971–997 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Benini, M., Schenkel, A.: Quantum field theories on categories fibered in groupoids. Commun. Math. Phys. 356(1), 19–64 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Benini, M., Schenkel, A.: Higher structures in algebraic quantum field theory: LMS/EPSRC Durham symposium on higher structures in M-theory. Fortsch. Phys. 67(8–9), 1910015 (2019)

    Article  ADS  Google Scholar 

  12. Benini, M., Schenkel, A., Szabo, R.J.: Homotopy colimits and global observables in Abelian Gauge theory. Lett. Math. Phys. 105(9), 1193–1222 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Benini, M., Schenkel, A., Woike, L.: Homotopy theory of algebraic quantum field theories. Lett. Math. Phys. 109(7), 1487–1532 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Benini, M., Schenkel, A., Woike, L.: Operads for algebraic quantum field theory. Commun. Contemp. Math. 23(2), 2050007 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boardman, J.M., Vogt, R.M.: Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347. Springer, Berlin (1973)

    MATH  Google Scholar 

  16. Bruinsma, S.: Coloring Operads for Algebraic Field Theory (2019). ar**v: 1903.02863 [hep-th]

  17. Bruinsma, S., Fewster, C.J., Schenkel, A.: Relative Cauchy evolution for linear homotopy AQFTs (2021). ar**v:2108.10592 [hep-th, physics:math-ph]

  18. Bruinsma, S., Schenkel, A.: Algebraic field theory operads and linear quantization. Lett. Math. Phys. 109(11), 2531–2570 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Carmona, V.: Envelo** operads and applications. Work in progress

  20. Carmona, V.: When Bousfield localizations and homotopy idempotent functors meet again (2022). ar**v:2203.15849 [math]

  21. Carmona, V., Flores, R., Muro, F.: Localization of (pr)operads. Work in progress

  22. Carmona, V., Flores, R., Muro, F.: A model structure for locally constant factorization algebras (2021). ar**v:2107.14174 [math]

  23. Costello, K., Gwilliam, O.: Factorization Algebras in Quantum Field Theory, volume 31 of New Mathematical Monographs, vol. 1. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  24. Dwyer, W.G., Kan, D.M.: Function complexes in homotopical algebra. Topol. Int. J. Math. 19(4), 427–440 (1980)

    MathSciNet  MATH  Google Scholar 

  25. Fewster, C.J., Verch, R.: Algebraic quantum field theory in curved spacetimes (2015). ar**v:1504.00586 [math-ph]

  26. Fredenhagen, K., Rehren, K.-H., Seiler, E.: Quantum field theory: where we are. In: Approaches to Fundamental Physics, Volume 721 of Lecture Notes in Physics, pp. 61–87. Springer, Berlin (2007)

  27. Fredenhagen, K.: Generalizations of the theory of superselection sectors. In: The Algebraic Theory of Superselection Sectors (Palermo, 1989), pp. 379–387. World Scientific Publishing, River Edge (1990)

  28. Fredenhagen, K.: Global observables in local quantum physics. In: Quantum and Non-commutative Analysis (Kyoto, 1992) Volume 16 of Mathematical Physics Studies, pp. 41–51. Kluwer Academic Publishers, Dordrecht (1993)

  29. Fresse, B.: Homotopy of Operads and Grothendieck–Teichmuller Groups. Part 1, Volume 217 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2017)

    Book  MATH  Google Scholar 

  30. Fresse, B.: Homotopy of Operads and Grothendieck–Teichmuller Groups. Part 2, Volume 217 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2017)

    Book  MATH  Google Scholar 

  31. Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848–861 (1964)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Hirschhorn, P.S.: Model Categories and Their Localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  33. Horel, G.: Factorization homology and calculus a la Kontsevich Soibelman. J. Noncommut. Geom. 11(2), 703–740 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999)

    MATH  Google Scholar 

  35. Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Lurie, J.: On the classification of topological field theories. In: Current Developments in Mathematics, 2008, pp. 129–280. International Press, Somerville (2009)

  37. Markl, M., Shnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  38. Rejzner, K.: Perturbative Algebraic Quantum Field Theory. Mathematical Physics Studies, Springer, Cham (2016)

    Book  MATH  Google Scholar 

  39. Shulman, M.A.: Parametrized spaces model locally constant homotopy sheaves. Topol. Appl. 155(5), 412–432 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Stolz, S., Teichner, P.: Supersymmetric field theories and generalized cohomology. In: Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Volume 83 of Proceedings of Symposia in Pure Mathematics, pp. 279–340. American Mathematical Society, Providence (2011)

  41. Toen, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Weiss, I.: From operads to dendroidal sets. In: Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Volume 83 of Proceedings of Symposia in Pure Mathematics, pp. 31–70. American Mathematical Society, Providence (2011)

  43. White, D., Yau, D.: Bousfield localization and algebras over colored operads. Appl. Categ. Struct. A J. Devot. Appl. Categ. Methods Algebra Anal. Order Topol. Comput. Sci. 26(1), 153–203 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Yau, D.: Homotopical Quantum Field Theory (2019). ar**v:1802.08101 [math-ph]

Download references

Acknowledgements

The author would like to thank his advisors, R. Flores and F. Muro, for their support. He also wants to thank A. Schenkel his helpful comments and interest. C. Maestro also deserves recognition; his help with LaTeX matters is of unquestionable value.

Author information

Authors and Affiliations

  1. Facultad de Matemáticas, Departamento de Álgebra-Imus, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012, Sevilla, Spain

    Victor Carmona

Authors
  1. Victor Carmona
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Carmona.

Additional information

Publisher's Note

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

The author was partially supported by the Spanish Ministry of Economy under the Grant MTM2016-76453-C2-1-P (AEI/FEDER, UE), by the Andalusian Ministry of Economy and Knowledge and the Operational Program FEDER 2014-2020 under the Grant US-1263032, by Grant PID2020-117971GB-C21 of the Spanish Ministry of Science and Innovation, and Grant FQM-213 of the Junta de Andalucía. He was also partly supported by Spanish Ministry of Science, Innovation and Universities Grant FPU17/01871.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carmona, V. New model category structures for algebraic quantum field theory. Lett Math Phys 113, 33 (2023). https://doi.org/10.1007/s11005-023-01644-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-023-01644-4

Keywords

Mathematics Subject Classification

Search

Navigation