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Communications in Mathematical Physics

, Volume 356, Issue 1, pp 19–64 | Cite as

Quantum Field Theories on Categories Fibered in Groupoids

  • Marco Benini
  • Alexander Schenkel
Open Access
Article

Abstract

We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories.

References

  1. BG11.
    Bär, C., Ginoux, N.: Classical and quantum fields on Lorentzian manifolds. In: Bär, C., Lohkamp, J., Schwarz, M. (eds.) Global Differential Geometry. Springer Proceedings in Mathematics, vol. 17 (2011). [arXiv:1104.1158 [math-ph]]
  2. BGP07.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, Zürich (2007). [arXiv:0806.1036 [math.DG]]
  3. BBSS17.
    Becker C., Benini M., Schenkel A., Szabo R.J.: Abelian duality on globally hyperbolic spacetimes. Commun. Math. Phys. 349(1), 361 (2017) [arXiv:1511.00316 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. BSS16.
    Becker C., Schenkel A., Szabo R.J.: Differential cohomology and locally covariant quantum field theory. Rev. Math. Phys. 29(01), 1750003 (2016) [arXiv:1406.1514 [hep-th]]MathSciNetCrossRefzbMATHGoogle Scholar
  5. Ben15.
    Benini, M.: Locality in Abelian gauge theories over globally hyperbolic spacetimes. Dissertation for Ph.D., University of Pavia (2015). [arXiv:1503.00131 [math-ph]]
  6. BDHS14.
    Benini M., Dappiaggi C., Hack T.P., Schenkel A.: A C *-algebra for quantized principal U(1)-connections on globally hyperbolic Lorentzian manifolds. Commun. Math. Phys. 332, 477 (2014) [arXiv:1307.3052 [math-ph]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. BDS14.
    Benini M., Dappiaggi C., Schenkel A.: Quantized Abelian principal connections on Lorentzian manifolds. Commun. Math. Phys. 330, 123 (2014) [arXiv:1303.2515 [math-ph]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. BDS14.
    Benini M., Schenkel A., Szabo R.J.: Homotopy colimits and global observables in Abelian gauge theory. Lett. Math. Phys. 105(9), 1193 (2015) [arXiv:1503.08839 [math-ph]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. BM07.
    Berger C., Moerdijk I.: Resolution of coloured operads and rectification of homotopy algebras. Contemp. Math. 431, 31–58 (2007) [arXiv:math/0512576 [math.AT]]MathSciNetCrossRefzbMATHGoogle Scholar
  10. BFV03.
    Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: a new paradigm for local quantum field theory. Commun. Math. Phys. 237(1-2), 31 (2003) [arXiv:math-ph/0112041]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. BR07.
    Brunetti R., Ruzzi G.: Superselection sectors and general covariance. I. Commun. Math. Phys. 237(1-2), 31 (2007) [arXiv:gr-qc/0511118 [gr-qc]]ADSMathSciNetzbMATHGoogle Scholar
  12. BCRV16.
    Buchholz, D., Ciolli, F., Ruzzi, G., Vasselli, E.: The universal C *-algebra of the electromagnetic field. Lett. Math. Phys. 106(2), 269 (2016); Erratum: [Lett. Math. Phys. 106(2), 287 (2016)] [arXiv:1506.06603 [math-ph]]
  13. BCRV17.
    Buchholz D., Ciolli F., Ruzzi G., Vasselli E.: The universal C *-algebra of the electromagnetic field II. Topological charges and spacelike linear fields. Lett. Math. Phys. 107(2), 201 (2017) [arXiv:1610.03302 [math-ph]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. C-B87.
    Choquet-Bruhat Y.: Global existence theorems for hyperbolic harmonic maps. Ann. Inst. H. Poincaré Phys. Théor. 46(1), 97–111 (1987)MathSciNetzbMATHGoogle Scholar
  15. C-B91.
    Choquet-Bruhat Y.: Yang–Mills–Higgs fields in three space time dimensions. Mém. Soc. Math. Fr. 46, 73–97 (1991)MathSciNetzbMATHGoogle Scholar
  16. CS97.
    Chrusciel P.T., Shatah J.: Global existence of solutions of the Yang–Mills equations on globally hyperbolic four dimensional Lorentzian manifolds. Asian J. Math. 1, 530 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Cis03.
    Cisinski D.-C.: Images directes cohomologiques dans les catégories de modèles. Ann. Math. Blaise Pascal 10(2), 195–244 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Cis09.
    Cisinski D.-C.: Locally constant functors. Math. Proc. Camb. Philos. Soc. 147(3), 593–614 (2009) [arXiv:0803.4342 [math.AT]]MathSciNetCrossRefzbMATHGoogle Scholar
  19. CG16.
    Costello, K., Gwilliam, O.: Factorization Algebras in Quantum Field Theory, New Mathematical Monographs, vol. 31, Cambridge University Press (2016). Book draft available at http://people.mpim-bonn.mpg.de/gwilliam/vol1may8.pdf
  20. Cra03.
    Crainic, M.: Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes. Comment. Math. Helv. 78(4), 681–721 (2003)Google Scholar
  21. DHP09.
    Dappiaggi C., Hack T.-P., Pinamonti N.: The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor. Rev. Math. Phys. 21(10), 1241 (2009) [arXiv:0904.0612 [math-ph]]MathSciNetCrossRefzbMATHGoogle Scholar
  22. DL12.
    Dappiaggi C., Lang B.: Quantization of Maxwell’s equations on curved backgrounds and general local covariance. Lett. Math. Phys. 101, 265 (2012) [arXiv:1104.1374 [gr-qc]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. DHR69a.
    Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations I. Commun. Math. Phys. 13, 1 (1969)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. DHR69b.
    Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations. 2. Commun. Math. Phys. 15, 173 (1969)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. Dug.
    Dugger, D.: A primer on homotopy colimits. http://pages.uoregon.edu/ddugger/hocolim.pdf
  26. DS95.
    Dwyer, W.G., Spalinski, J.: Homotopy theories and model categories. In: James, I.M. (ed.) Handbook of algebraic topology, pp. 73–126. North-Holland, Amsterdam (1995)Google Scholar
  27. Few13.
    Fewster C.J.: Endomorphisms and automorphisms of locally covariant quantum field theories. Rev. Math. Phys. 25, 1350008 (2013) [arXiv:1201.3295 [math-ph]]MathSciNetCrossRefzbMATHGoogle Scholar
  28. Few16a.
    Fewster, C.J.: On the spin-statistics connection in curved spacetimes. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds.) Quantum Mathematical Physics: A Bridge Between Mathematics and Physics. Birkhäuser, Basel (2016). [arXiv:1503.05797 [math-ph]]
  29. Few16a.
    Fewster C.J.: Locally covariant quantum field theory and the spin-statistics connection. Int. J. Mod. Phys. D 25(06), 1630015 (2016) [arXiv:1603.01036 [gr-qc]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. FS15.
    Fewster C.J., Schenkel A.: Locally covariant quantum field theory with external sources. Ann. Henri Poincaré 16(10), 2303 (2015) [arXiv:1402.2436 [math-ph]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. FS15.
    Fewster C.J., Verch R.: Dynamical locality and covariance: what makes a physical theory the same in all spacetimes?. Ann. Henri Poincaré 13, 1613 (2012) [arXiv:1106.4785 [math-ph]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. Fre90.
    Fredenhagen, K.: Generalizations of the theory of superselection sectors. In: Kastler, D. (ed.) The Algebraic Theory of Superselection Sectors: Introduction and Recent Results, vol. 379. World Scientific Publishing, Singapore (1990)Google Scholar
  33. Fre93.
    Fredenhagen, K.: Global observables in local quantum physics. In: Araki, H., Ito, K.R., Kishimoto, A., Ojima, I. (eds.) Quantum and Non-commutative Analysis: Past, Present and Future Perspectives, vol. 41. Kluwer Academic Publishers, Dordrecht (1993)Google Scholar
  34. FRS92.
    Fredenhagen K., Rehren K.-H., Schroer B.: Superselection sectors with braid group statistics and exchange algebras II: geometric aspects and conformal covariance. Rev. Math. Phys. 4, 113 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. GB78.
    Greub, W., Petry, H.R.: On the lifting of structure groups. In: Bleuler, K., Petry, H.R., Reetz, A. (eds.) Differential Geometric Methods in Mathematical Physics II. Lecture Notes on Mathematics, vol. 676, Springer, Berlin (1978)Google Scholar
  36. Hat02.
    Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  37. Hir03.
    Hirschhorn, P.S.: Model Categories and Their Localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI (2003)Google Scholar
  38. Hol08.
    Hollander S.: A homotopy theory for stacks. Israel J. Math. 163, 93–124 (2008) [arXiv:math.AT/0110247]]MathSciNetCrossRefzbMATHGoogle Scholar
  39. Jar97.
    Jardine J.F.: A closed model structure for differential graded algebras. Fields Inst. Commun. 17, 55 (1997)MathSciNetzbMATHGoogle Scholar
  40. MacL98.
    Mac Lane S.: Categories for the Working Mathematician, Graduate Texts in Mathematics. Springer, New York (1998)zbMATHGoogle Scholar
  41. Rod14.
    Rodríguez-González B.: Realizable homotopy colimits. Theory Appl. Categ. 29(22), 609–634 (2014) [arXiv:1104.0646 [math.AG]]MathSciNetzbMATHGoogle Scholar
  42. Ruz05.
    Ruzzi G.: Homotopy, net-cohomology and superselection sectors in globally hyperbolic spacetimes. Rev. Math. Phys. 17, 1021 (2014) [arXiv:math-ph/0412014]]CrossRefzbMATHGoogle Scholar
  43. San10.
    Sanders K.: The locally covariant Dirac field. Rev. Math. Phys. 22, 381 (2010) [arXiv:0911.1304 [math-ph]]MathSciNetCrossRefzbMATHGoogle Scholar
  44. SDH14.
    Sanders K., Dappiaggi C., Hack T.P.: Electromagnetism, local covariance, the Aharonov–Bohm effect and Gauss’ law. Commun. Math. Phys. 328, 625 (2014) [arXiv:1211.6420 [math-ph]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. SZ16.
    Schenkel, A., Zahn, J.: Global anomalies on Lorentzian space-times. Ann. Henri Poincaré 18(8), 2693 (2017). [arXiv:1609.06562 [hep-th]]
  46. Ver01.
    Verch R.: A spin statistics theorem for quantum fields on curved space-time manifolds in a generally covariant framework. Commun. Math. Phys. 223, 261 (2001) [arXiv:math/0102035]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. Vis05.
    Vistoli, A.: Grothendieck topologies, fibered categories and descent theory. In: Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S.L., Nitsure, N., Vistoli, A. (eds.) Fundamental Algebraic Geometry, Mathematical Surveys and Monographs, vol. 123, pp. 1–104, American Mathematical Society, Providence, RI (2005)Google Scholar
  48. Wal05.
    Walter, B.: Rational homotopy calculus of functors. Dissertation for Ph.D., Brown University (2005). [arXiv:math/0603336 [math.AT]]
  49. Zah14.
    Zahn J.: The renormalized locally covariant Dirac field. Rev. Math. Phys. 26(1), 1330012 (2014) [arXiv:1210.4031 [math-ph]]MathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2017

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.Institut für MathematikUniversität PotsdamPotsdamGermany
  2. 2.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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