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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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]]
Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, Zürich (2007). [arXiv:0806.1036 [math.DG]]
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]]
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]]
Benini, M.: Locality in Abelian gauge theories over globally hyperbolic spacetimes. Dissertation for Ph.D., University of Pavia (2015). [arXiv:1503.00131 [math-ph]]
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]]
Benini M., Dappiaggi C., Schenkel A.: Quantized Abelian principal connections on Lorentzian manifolds. Commun. Math. Phys. 330, 123 (2014) [arXiv:1303.2515 [math-ph]]
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]]
Berger C., Moerdijk I.: Resolution of coloured operads and rectification of homotopy algebras. Contemp. Math. 431, 31–58 (2007) [arXiv:math/0512576 [math.AT]]
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]]
Brunetti R., Ruzzi G.: Superselection sectors and general covariance. I. Commun. Math. Phys. 237(1-2), 31 (2007) [arXiv:gr-qc/0511118 [gr-qc]]
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]]
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]]
Choquet-Bruhat Y.: Global existence theorems for hyperbolic harmonic maps. Ann. Inst. H. Poincaré Phys. Théor. 46(1), 97–111 (1987)
Choquet-Bruhat Y.: Yang–Mills–Higgs fields in three space time dimensions. Mém. Soc. Math. Fr. 46, 73–97 (1991)
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)
Cisinski D.-C.: Images directes cohomologiques dans les catégories de modèles. Ann. Math. Blaise Pascal 10(2), 195–244 (2003)
Cisinski D.-C.: Locally constant functors. Math. Proc. Camb. Philos. Soc. 147(3), 593–614 (2009) [arXiv:0803.4342 [math.AT]]
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
Crainic, M.: Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes. Comment. Math. Helv. 78(4), 681–721 (2003)
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]]
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]]
Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations I. Commun. Math. Phys. 13, 1 (1969)
Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations. 2. Commun. Math. Phys. 15, 173 (1969)
Dugger, D.: A primer on homotopy colimits. http://pages.uoregon.edu/ddugger/hocolim.pdf
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)
Fewster C.J.: Endomorphisms and automorphisms of locally covariant quantum field theories. Rev. Math. Phys. 25, 1350008 (2013) [arXiv:1201.3295 [math-ph]]
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]]
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]]
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]]
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]]
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)
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)
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)
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)
Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hirschhorn, P.S.: Model Categories and Their Localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI (2003)
Hollander S.: A homotopy theory for stacks. Israel J. Math. 163, 93–124 (2008) [arXiv:math.AT/0110247]]
Jardine J.F.: A closed model structure for differential graded algebras. Fields Inst. Commun. 17, 55 (1997)
Mac Lane S.: Categories for the Working Mathematician, Graduate Texts in Mathematics. Springer, New York (1998)
Rodríguez-González B.: Realizable homotopy colimits. Theory Appl. Categ. 29(22), 609–634 (2014) [arXiv:1104.0646 [math.AG]]
Ruzzi G.: Homotopy, net-cohomology and superselection sectors in globally hyperbolic spacetimes. Rev. Math. Phys. 17, 1021 (2014) [arXiv:math-ph/0412014]]
Sanders K.: The locally covariant Dirac field. Rev. Math. Phys. 22, 381 (2010) [arXiv:0911.1304 [math-ph]]
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]]
Schenkel, A., Zahn, J.: Global anomalies on Lorentzian space-times. Ann. Henri Poincaré 18(8), 2693 (2017). [arXiv:1609.06562 [hep-th]]
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]]
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)
Walter, B.: Rational homotopy calculus of functors. Dissertation for Ph.D., Brown University (2005). [arXiv:math/0603336 [math.AT]]
Zahn J.: The renormalized locally covariant Dirac field. Rev. Math. Phys. 26(1), 1330012 (2014) [arXiv:1210.4031 [math-ph]]
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
Open Access This 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.
About this article
Cite this article
Benini, M., Schenkel, A. Quantum Field Theories on Categories Fibered in Groupoids. Commun. Math. Phys. 356, 19–64 (2017). https://doi.org/10.1007/s00220-017-2986-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2986-7