Abstract
De Rham cohomology with spacelike compact and timelike compact supports has recently been noticed to be of importance for understanding the structure of classical and quantum Maxwell theory on curved spacetimes. Similarly, causally restricted cohomologies of different differential complexes play a similar role in other gauge theories. We introduce a method for computing these causally restricted cohomologies in terms of cohomologies with either compact or unrestricted supports. The calculation exploits the fact that the de Rham–d’Alembert wave operator can be extended to a chain map that is homotopic to zero and that its causal Green function fits into a convenient exact sequence. As a first application, we use the method on the de Rham complex, then also on the Calabi (or Killing–Riemann–Bianchi) complex, which appears in linearized gravity on constant curvature backgrounds. We also discuss applications to other complexes, as well as generalized causal structures and functoriality.
Article PDF
Similar content being viewed by others
References
Ashtekar A., Sen A.: On the role of spacetime topology in quantum phenomena: Superselection of charge and emergence of nontrivial vacua. J. Math. Phys. 21, 526–533 (2008)
Baer, C.: Green-hyperbolic operators on globally hyperbolic spacetimes (2013). arXiv:1310.0738
Baer, C., Ginoux, N., Pfaeffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. Vol. 2 of ESI Lectures in Mathematics and Physics. European Mathematical Society (2007). arXiv:0806.1036
Beem, J.K., Ehrlich, P., Easley, K.: Global Lorentzian Geometry. Vol. 202 of Pure and Applied Mathematics. Marcel Dekker, New York (1996)
Benini, M.: Optimal space of linear classical observables for Maxwell k-forms via spacelike and timelike compact de Rham cohomologies (2014). arXiv:1401.7563
Benini, M., Dappiaggi, C., Schenkel, A.: Quantized Abelian principal connections on Lorentzian manifolds. Commun. Math. Phys. 330, 123–152 (2013). arXiv:1303.2515
Bergery, L.B., Bourguignon, J.-P., Lafontaine, J.: Déformations localement triviales des variétés riemanniennes. In: Differential Geometry, Part 1, Vol. 27 of Proceedings of Symposia in Pure Mathematics, pp. 3–32. AMS, Providence, RI (1975)
Bernal, A., Sánchez, M.: Further results on the smoothability of cauchy hypersurfaces and cauchy time functions. Lett. Math. Phys. 77, 183–197 (2006). arXiv:gr-qc/0512095
Bernal, A.N., Sánchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257, 43–50 (2005). arXiv:gr-qc/0401112
Bini D., Cherubini C., Jantzen R.T., Ruffini R.: de Rham wave equation for tensor valued p-forms. Int. J. Modern Phys. D 12, 1363–1384 (2003)
Bott R., Tu L.W.: Differential Forms in Algebraic Topology. Vol. 82 of Graduate Texts in Mathematics. Springer, New York (1982)
Bredon G.E.: Sheaf Theory. Graduate Texts in Mathematics. Springer, New York (1997)
Calabi, E.: On compact, Riemannian manifolds with constant curvature. I. In: Allendoerfer, C.B. (ed.) Differential Geometry, Vol. 3 of Proceedings of Symposia in Pure Mathematics, pp. 155–180. AMS, Providence (1961)
Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space. Vol. 41 of Princeton Mathematical Series. Princeton University Press, Princeton (1993)
Dappiaggi, C., Lang, B.: Quantization of Maxwell’s equations on curved backgrounds and general local covariance. Lett. Math. Phys. 101, 265–287 (2012). arXiv:1104.1374
Eastwood, M.: Variations on the de Rham complex. Notices Am. Math. Soc. 46, 1368–1376 (1999). http://www.ams.org/notices/199911/fea-eastwood
Fathi A., Siconolfi A.: On smooth time functions. Math. Proc. Camb. Philos. Soc. 152, 303–339 (2011)
Fewster, C.J., Hunt, D.S.: Quantization of linearized gravity in cosmological vacuum spacetimes. Rev. Math. Phys. 25, 1330003 (2013). arXiv:1203.0261
Fewster, C.J., Lang, B.: Dynamical locality of the free Maxwell field (2014). arXiv:1403.7083
Fulling S.A., King R.C., Wybourne B.G., Cummins C.J.: Normal forms for tensor polynomials. I. The Riemann tensor. Class. Quantum Gravity 9, 1151 (1992)
Fulton, W.: Young Tableaux: With Applications to Representation Theory and Geometry. Vol. 35 of London Mathematical Society Student Texts. Cambridge University Press, New York (1997)
García-Parrado, A., Senovilla, J.M.M.: Causal structures and causal boundaries. Class. Quantum Gravity 22, R1–R84 (2005). arXiv:gr-qc/0501069
Gasqui J., Goldschmidt H.: Déformations infinitésimales des espaces riemanniens localement symétriques. I. Adv. Math. 48, 205–285 (1983)
Gasqui, J., Goldschmidt, H.: Complexes of differential operators and symmetric spaces. In: Hazewinkel, M., Gerstenhaber, M. (eds.) Deformation Theory of Algebras and Structures and Applications, Vol. 247 of NATO ASI Series, pp. 797–827. Kluwer, Dordrecht (1988)
Geroch R.: Domain of dependence. J. Math. Phys. 11, 437–449 (1970)
Ginoux, N.: Linear wave equations. In: Baer, C., Fredenhagen, K. (eds.) Quantum Field Theory on Curved Spacetimes: Concepts and Methods, Vol. 786 of Lecture Notes in Physics. Springer, Berlin (2009)
Goldberg S.I.: Curvature and Homology. Dover, Mineola (1998)
Goldschmidt H.: Existence theorems for analytic linear partial differential equations. Ann. Math. 86, 246–270 (1967)
Goldschmidt H.: Duality theorems in deformation theory. Trans. Am. Math. Soc. 292, 1 (1985)
Hack, T.-P.: Quantization of the linearised Einstein–Klein–Gordon system on arbitrary backgrounds and the special case of perturbations in inflation (2014). arXiv:1403.3957.
Hack, T.-P., Schenkel, A.: Linear bosonic and fermionic quantum gauge theories on curved spacetimes. Gen. Relativ. Gravit. 45, 877–910 (2013). arXiv:1205.3484
Hawking S.W., Ellis G.F.R.: The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1973)
Jost J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2011)
Khavkine, I.: Characteristics, conal geometry and causality in locally covariant field theory (2012). arXiv:1211.1914
Khavkine, I.: The Calabi complex and Killing sheaf cohomology (2014). arXiv:1409.7212
Khavkine, I.: Covariant phase space, constraints, gauge and the Peierls formula. Int. J. Mod. Phys. A 29, 1430009 (2014). arXiv:1402.1282
Kronheimer E.H., Penrose R.: On the structure of causal spaces. Math. Proc. Camb. Philos. Soc. 63, 481–501 (1967)
Lawson J.D.: Ordered manifolds, invariant cone fields, and semigroups. Forum Math. 1, 273–308 (1989)
Leyland P., Roberts J.E.: The cohomology of nets over Minkowski space. Commun. Math. Phys. 62, 173–189 (1978)
Lichnerowicz, A.: Propagateurs, commutateurs et anticommutateurs en relativité générale. In: DeWitt, C., DeWitt, B.S. (eds.) Relativity, Groups and Topology, pp. 821–861. Gordon and Breach, New York (1964)
Misner C.W., Thorne K.S., Wheeler J.A.: Gravitation. Physics Series. W. H. Freeman, San Francisco (1973)
Neeb K.H.: Conal orders on homogeneous spaces. Invent. Math. 104, 467–496 (1991)
O’Neill B.: Semi-Riemannian Geometry With Applications to Relativity. Vol. 103 of Pure and Applied Mathematics. Academic Press, San Diego (1983)
Pommaret J.-F.: Systems of Partial Differential Equations and Lie Pseudogroups. Vol. 14 of Mathematics and its Applications. Gordon and Breach, New York (1978)
Quillen, D.G.: Formal properties of over-determined systems of partial differential equations. PhD thesis, Harvard University (1964)
Sanders, K.: A note on spacelike and timelike compactness. Class. Quantum Gravity 30, 115014 (2012). arXiv:1211.2469
Sanders, K., Dappiaggi, C., Hack, T.-P.: Electromagnetism, local covariance, the Aharonov–Bohm effect and Gauss’ law. Commun. Math. Phys. 328, 625–667 (2014), arXiv:1211.6420
Serre J.-P.: Un théorème de dualité. Commentarii Mathematici Helvetici 29, 9–26 (1955)
Stepanov S.E.: The Killing–Yano tensor. Theor. Math. Phys. 134, 333–338 (2003)
Sullivan D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Inventiones Mathematicae 36, 225–255 (1976)
Tarkhanov N.N.: Complexes of Differential Operators. Vol. 340 of Mathematics and its Applications. Kluwer, Dordrecht (1995)
Wald R.M.: General Relativity. University of Chicago Press, Chicago (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Karl-Henning Rehren.
Rights and permissions
About this article
Cite this article
Khavkine, I. Cohomology with Causally Restricted Supports. Ann. Henri Poincaré 17, 3577–3603 (2016). https://doi.org/10.1007/s00023-016-0481-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-016-0481-x