Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs

Abstract

Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.

References

  1. 1.

    Kreimer, D.: Adv. Theor. Math. Phys. 2, 303 (1998). https://doi.org/10.4310/ATMP.1998.v2.n2.a4

    MathSciNet  Article  Google Scholar 

  2. 2.

    Connes, A., Kreimer, D.: Comm. Math. Phys. 210(1), 249 (2000). https://doi.org/10.1007/s002200050779

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Connes, A., Kreimer, D.: Comm. Math. Phys. 216(1), 215 (2001). https://doi.org/10.1007/PL00005547

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Kreimer, D.: Phys. Rept. 363, 387 (2002). https://doi.org/10.1016/S0370-1573(01)00099-0

    ADS  Article  Google Scholar 

  5. 5.

    Kreimer, D.: Annals Phys. 321, 2757 (2006). https://doi.org/10.1016/j.aop.2006.01.004

    ADS  Article  Google Scholar 

  6. 6.

    van Suijlekom, W.D.: Comm. Math. Phys. 276(3), 773 (2007). https://doi.org/10.1007/s00220-007-0353-9

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Kreimer, D.: Annals Phys. 323, 49 (2008). https://doi.org/10.1016/j.aop.2007.06.005

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Borinsky, M.: Graphs in perturbation theory: algebraic structure and asymptotics. Springer Theses Recognizing Outstanding Ph.D Research. Springer International Publishing, Berlin (2018). https://doi.org/10.1007/978-3-030-03541-9

    Book  Google Scholar 

  9. 9.

    Yeats, K.: Growth estimates for Dyson-Schwinger equations. Ph.D. thesis Boston University (2008)

  10. 10.

    Yeats, K.: A combinatorial perspective on quantum field theory, Springer Briefs in Mathematical Physics, vol. 15. Springer International Publishing, Cham (2017). https://doi.org/10.1007/978-3-319-47551-6

    Book  Google Scholar 

  11. 11.

    Tanasa, A., Vignes-Tourneret, F.: J. Noncommut. Geomet. 2, 125 (2008). https://doi.org/10.4171/JNCG/17

    Article  Google Scholar 

  12. 12.

    Tanasa, A., Kreimer, D.: J. Noncommut. Geomet. 7, 255 (2013). https://doi.org/10.4171/JNCG/116

    Article  Google Scholar 

  13. 13.

    Raasakka, M., Tanasa, A.: , vol. 70. https://doi.org/10.1007/s00220-012-1549-1 (2014)

  14. 14.

    Avohou, R.C., Rivasseau, V., Tanasa, A.: J. Phys. A 48, 485204 (2015). https://doi.org/10.1088/1751-8113/48/48/485204

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kontsevich, M.: Comm. Math. Phys. 147, 1 (1992). https://doi.org/10.1007/BF02096792

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Grosse, H., Wulkenhaar, R.: Lett. Math. Phys. 71, 13 (2005). https://doi.org/10.1007/s11005-004-5116-3

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Grosse, H., Wulkenhaar, R.: Comm. Math. Phys. 256, 305 (2005). https://doi.org/10.1007/s00220-004-1285-2

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Wulkenhaar, R. In: Chamseddine, A.H., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds.) Advances in Noncommutative Geometry, pp 607–690. Springer International Publishing (2019). https://doi.org/10.1007/978-3-030-29597-4_11

  19. 19.

    Hock, A.: Matrix field theory. Ph.D. thesis, WWU Münster (2020)

    MATH  Google Scholar 

  20. 20.

    Ben Geloun, J., Rivasseau, V.: Comm. Math. Phys. 318, 69 (2013). https://doi.org/10.1007/s00220-012-1549-1

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    Ben Geloun, J., Rivasseau, V.: Ann. Henri Poincaré, 19, 3357 (2018). https://doi.org/10.1007/s00023-018-0712-4

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    Ben Geloun, J.: Comm. Math. Phys. 332, 117 (2014). https://doi.org/10.1007/s00220-014-2142-6

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Carrozza, S.: Tensorial methods and renormalization in Group Field Theories. Ph.D. thesis, Université Paris-Sud 11 Paris Orsay (2013)

  24. 24.

    De Pietri, R., Freidel, L., Krasnov, K., Rovelli, C.: Nucl. Phys. B B574, 785 (2000). https://doi.org/10.1016/S0550-3213(00)00005-5

    ADS  Article  Google Scholar 

  25. 25.

    Oriti, D.: Spin foam models of quantum spacetime. Ph.D. thesis, Cambridge University (2003)

  26. 26.

    Freidel, L.: Int. J. Theor. Phys. 44, 1769 (2005). https://doi.org/10.1007/s10773-005-8894-1

    Article  Google Scholar 

  27. 27.

    Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: Phys. Rept. 254, 1 (1995). https://doi.org/10.1016/0370-1573(94)00084-G

    ADS  Article  Google Scholar 

  28. 28.

    Gurau, R.: Comm. Math. Phys. 304, 69 (2011). https://doi.org/10.1007/s00220-011-1226-9

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Gurau, R.: Class. Quant. Grav. 27, 235023 (2010). https://doi.org/10.1088/0264-9381/27/23/235023

    ADS  Article  Google Scholar 

  30. 30.

    Gurau, R.: Random tensors. Oxford University Press, Oxford (2016). https://doi.org/10.1093/acprof:oso/9780198787938.001.0001

    Book  Google Scholar 

  31. 31.

    Oriti, D., Ryan, J.P., Thürigen, J.: New J. Phys. 17, 023042 (2015). https://doi.org/10.1088/1367-2630/17/2/023042

    ADS  MathSciNet  Article  Google Scholar 

  32. 32.

    Grosse, H., Sako, A., Wulkenhaar, R.: Nucl. Phys. B 926, 20 (2018). https://doi.org/10.1016/j.nuclphysb.2017.10.022

    ADS  Article  Google Scholar 

  33. 33.

    Thürigen, J.: arXiv:2103.01136 (2021)

  34. 34.

    Bonzom, V., Gurau, R., Rivasseau, V.: Phys. Rev. D 85, 084037 (2012). https://doi.org/10.1103/PhysRevD.85.084037

    ADS  Article  Google Scholar 

  35. 35.

    Kaminski, W., Kisielowski, M., Lewandowski, J.: Class. Quant. Grav. 27, 095006 (2010). https://doi.org/10.1088/0264-9381/29/4/049502

    ADS  Article  Google Scholar 

  36. 36.

    Eynard, B.: Counting surfaces, progress in mathematical physics, vol. 70. Springer, Berlin (2016). https://doi.org/10.1007/978-3-7643-8797-6

    Google Scholar 

  37. 37.

    Reidemeister, K.: Topologie der Polyeder. Mathematik und ihre Anwendungen in Monographien und Lehrbüchern. Akademische Verlagesellschaft M. B. H. (1938)

  38. 38.

    Thürigen, J.: Discrete quantum geometries and their effective dimension. Ph.D. thesis, Humboldt-Universität zu Berlin. https://doi.org/10.18452/17309 (2015)

  39. 39.

    Seifert, H., Threlfall, W.: A textbook of topology pure and applied mathematics. Academic Press (1980)

  40. 40.

    Das, S.R., Dhar, A., Sengupta, A.M., Wadia, S.R.: Mod. Phy. Lett. A 5(1), 1041 (1990). https://doi.org/10.1142/S0217732390001165

    ADS  Article  Google Scholar 

  41. 41.

    Korchemsky, G.P.: Mod. Phys. Lett. A 7, 3081 (1992). https://doi.org/10.1142/S0217732392002470

    ADS  MathSciNet  Article  Google Scholar 

  42. 42.

    Aldous, D.: In: Stochastic analysis (Durham 1990), pp 23–70. Cambridge Univ. Press, Cambridge (1991). https://doi.org/10.1017/CBO9780511662980.003

  43. 43.

    Borinsky, M.: Comput. Phys. Commun. 185(12), 3317 (2014). https://doi.org/10.1016/j.cpc.2014.07.023

    ADS  Article  Google Scholar 

  44. 44.

    Manchon, D.: In: Comptes Rendus Des Rencontres Mathematiques De Glanon 2001 (2003)

  45. 45.

    Cvitanovic, P.: Field theory (nordita lecture notes). http://chaosbook.org/FieldTheory/ (1983)

  46. 46.

    Grosse, H., Wulkenhaar, R.: Comm. Math. Phys. 254, 91 (2004). https://doi.org/10.1007/s00220-004-1238-9

    ADS  Article  Google Scholar 

  47. 47.

    Gurau, R.G.: Ann. Math. 12(5), 829 (2011). https://doi.org/10.1007/s00220-005-1440-4

    Google Scholar 

  48. 48.

    Gurau, R.: Ann. Henri Poincaré, 13, 399 (2012). https://doi.org/10.1007/s00023-011-0118-z

    ADS  MathSciNet  Article  Google Scholar 

  49. 49.

    Ryan, J.P.: Phys. Rev. D 85, 024010 (2011). https://doi.org/10.1103/PhysRevD.85.024010

    ADS  Article  Google Scholar 

  50. 50.

    Ousmane Samary, D., Vignes-Tourneret, F.: Comm. Math. Phys. 329, 545 (2014). https://doi.org/10.1007/s00220-014-1930-3

    ADS  MathSciNet  Article  Google Scholar 

  51. 51.

    Ambjorn, J., Jurkiewicz, J., Makeenko, Y.M.: Phys. Lett. B 251(4), 517 (1990). https://doi.org/10.1016/0370-2693(90)90790-D

    ADS  MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

I thank M. Borinsky, J. Ben Geloun, A. Hock, D. Kreimer, A. Pithis and R. Wulkenhaar for discussions and comments on the paper, in particular M. Borinsky for very helpful discussions on the various subalgebra structures occuring in Hopf algebras of diagrams. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in two ways, primarily under the project number 418838388 and furthermore under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.

Funding

Open Access funding enabled and organized by Projekt DEAL.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Johannes Thürigen.

Additional information

Publisher’s Note

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

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Thürigen, J. Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs. Math Phys Anal Geom 24, 19 (2021). https://doi.org/10.1007/s11040-021-09390-6

Download citation

Keywords

  • Hopf algebra
  • Non-local field theory
  • Renormalization
  • Random geometry

Mathematics Subject Classification (2010)

  • MSC2020: 81T18
  • 81T15
  • 16T05
  • 16T30
  • 81T32
  • 05C10
  • 81V17