Communications in Mathematical Physics

, Volume 349, Issue 1, pp 361–392 | Cite as

Abelian Duality on Globally Hyperbolic Spacetimes

  • Christian Becker
  • Marco BeniniEmail author
  • Alexander Schenkel
  • Richard J. Szabo


We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger–Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism in a manifestly covariant way and without the assumption of compact Cauchy surfaces. We construct semi-classical configuration spaces and corresponding presymplectic Abelian groups of observables, which are quantized by the CCR-functor to the category of C*-algebras. We demonstrate explicitly how duality is implemented as a natural isomorphism between quantum field theories. We apply this formalism to develop a fully covariant quantum theory of self-dual fields.


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  1. Bar15.
    Bär, C.: Green-hyperbolic operators on globally hyperbolic spacetimes. Commun. Math. Phys. 333, 1585 (2015). arXiv:1310.0738 [math-ph]
  2. BB14.
    Bär C., Becker C.: Differential Characters. Lect. Notes Math., vol. 2112. Springer, Berlin (2014)Google Scholar
  3. BF09.
    Bär, C., Fredenhagen, K.: (eds.): Quantum Field Theory on Curved Spacetimes. Lect. Notes Phys. vol. 786 (2009)Google Scholar
  4. BGP07.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, Zürich (2007). arXiv:0806.1036 [math.DG]
  5. BBSS15.
    Becker, C., Benini, M., Schenkel, A., Szabo, R.J.: Cheeger–Simons differential characters with compact support and Pontryagin duality. arXiv:1511.00324 [math.DG]
  6. BSS14.
    Becker, C., Schenkel, A., Szabo, R.J.: Differential cohomology and locally covariant quantum field theory. arXiv:1406.1514 [hep-th]
  7. BEE96.
    Beem J.K., Ehrlich P., Easley K.: Global Lorentzian Geometry. CRC Press, Boca Raton (1996)zbMATHGoogle Scholar
  8. BM06.
    Belov, D., Moore, G.W.: Holographic action for the self-dual field. arXiv:hep-th/0605038
  9. 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]
  10. BDS14a.
    Benini, M., Dappiaggi, C., Schenkel, A.: Quantized Abelian principal connections on Lorentzian manifolds. Commun. Math. Phys. 330, 123 (2014). arXiv:1303.2515 [math-ph]
  11. BDS14b.
    Benini, M., Dappiaggi C., Schenkel, A.: Quantum field theory on affine bundles. Ann. Henri Poincaré 15, 171 (2014). arXiv:1210.3457 [math-ph]
  12. BSS15.
    Benini, M., Schenkel, A., Szabo, R.J.: Homotopy colimits and global observables in Abelian gauge theory. Lett. Math. Phys. 105, 1193 (2015). arXiv:1503.08839 [math-ph]
  13. BS05.
    Bernal A.N., Sanchez M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257, 43 (2005) arXiv:gr-qc/0401112 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. BS06.
    Bernal A.N., Sanchez M.: Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys. 77, 183 (2006) arXiv:gr-qc/0512095 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. BFV03.
    Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31 (2003) arXiv:math-ph/0112041 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. CS85.
    Cheeger J., Simons J.: Differential characters and geometric invariants. Lect. Notes Math. 1167, 50 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 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]
  18. DS13.
    Dappiaggi, C., Siemssen, D.: Hadamard states for the vector potential on asymptotically flat spacetimes. Rev. Math. Phys. 25, 1350002 (2013). arXiv:1106.5575 [gr-qc]
  19. Ell14.
    Elliott, C.: Abelian duality for generalised Maxwell theories. arXiv:1402.0890 [math.QA]
  20. Few13.
    Fewster, C.J.: Endomorphisms and automorphisms of locally covariant quantum field theories. Rev. Math. Phys. 25, 1350008 (2013). arXiv:1201.3295 [math-ph]
  21. FL14.
    Fewster, C.J., Lang, B.: Dynamical locality of the free Maxwell field. Ann. Henri Poincaré 17, 401 (2016). arXiv:1403.7083 [math-ph]
  22. Fre00.
    Freed D.S.: Dirac charge quantization and generalized differential cohomology. Surv. Diff. Geom. VII, 129 (2000) arXiv:hep-th/0011220 zbMATHMathSciNetGoogle Scholar
  23. FMS07a.
    Freed D.S., Moore G.W., Segal G.: The uncertainty of fluxes. Commun. Math. Phys. 271, 247 (2007) arXiv:hep-th/0605198 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  24. FMS07b.
    Freed D.S., Moore G.W., Segal G.: Heisenberg groups and noncommutative fluxes. Ann. Phys. 322, 236 (2007) arXiv:hep-th/0605200 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. Hat02.
    Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  26. HLZ03.
    Harvey, F.R., Lawson, H.B.Jr., Zweck, J.: The de Rham–Federer theory of differential characters and character duality. Am. J. Math. 125, 791 (2003). arXiv:math.DG/0512251
  27. M+73.
    Manuceau J., Sirugue M., Testard D., Verbeure A.: The smallest C*-algebra for canonical commutation relations. Commun. Math. Phys. 32, 231 (1973)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. O’N83.
    O’Neill B.: Semi–Riemannian Geometry with Applications to Relativity. Academic Press, Cambridge (1983)zbMATHGoogle Scholar
  29. 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]
  30. SS08.
    Simons J., Sullivan D.: Axiomatic characterization of ordinary differential cohomology. J. Topol. 1, 45 (2008) arXiv:math.AT/0701077 CrossRefzbMATHMathSciNetGoogle Scholar
  31. Sza12.
    Szabo, R.J.: Quantization of higher Abelian gauge theory in generalized differential cohomology. PoS ICMP 2012, 009 (2012). arXiv:1209.2530 [hep-th]

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany
  2. 2.Department of MathematicsHeriot-Watt UniversityEdinburghUK
  3. 3.Maxwell Institute for Mathematical SciencesEdinburghUK
  4. 4.Higgs Centre for Theoretical PhysicsEdinburghUK

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