Annales Henri Poincaré

, Volume 16, Issue 1, pp 255–288 | Cite as

Topology, Rigid Cosymmetries and Linearization Instability in Higher Gauge Theories

  • Igor KhavkineEmail author


We consider a class of non-linear PDE systems, whose equations possess Noether identities (the equations are redundant), including non-variational systems (not coming from Lagrangian field theories), where Noether identities and infinitesimal gauge transformations need not be in bijection. We also include theories with higher stage Noether identities, known as higher gauge theories (if they are variational). Some of these systems are known to exhibit linearization instabilities: there exist exact background solutions about which a linearized solution is extendable to a family of exact solutions only if some non-linear obstruction functionals vanish. We give a general, geometric classification of a class of these linearization obstructions, which includes as special cases all known ones for relativistic field theories (vacuum Einstein, Yang–Mills, classical N = 1 supergravity, etc.). Our classification shows that obstructions arise due to the simultaneous presence of rigid cosymmetries (generalized Killing condition) and non-trivial de Rham cohomology classes (spacetime topology). The classification relies on a careful analysis of the cohomologies of the on-shell Noether complex (consistent deformations), adjoint Noether complex (rigid cosymmetries) and variational bicomplex (conserved currents). An intermediate result also gives a criterion for identifying non-linearities that do not lead to linearization instabilities.


Gauge Theory Linearization Instability Asymptotic Boundary Condition High Gauge Theory Deformation Current 
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  1. 1.
    Fischer, A.E., Marsden, J.E., Moncrief, V.: The structure of the space of solutions of Einstein’s equations. I. One Killing field. Annales de l’institut Henri Poincaré A 33(2), 147–194 (1980).
  2. 2.
    Arms J.M., Marsden J.E., Moncrief V.: The structure of the space of solutions of Einstein’s equations II: several Killing fields and the Einstein–Yang–Mills equations. Ann. Phys. 144(1), 81–106 (1982)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Choquet-Bruhat, Y., Deser, S.: Stabilité initiale de l’espace temps de Minkowski. Comptes Rendus de l’Académie des Sciences de Paris Série A-B 275(20), A1019–A1021 (1972).
  4. 4.
    Choquet-Bruhat, Y., Fischer, A.E., Marsden, J.E.: Maximal hypersurfaces and positivity of mass. In: Ehlers, J. (ed.) Isolated Gravitating Systems in General Relativity: Proceedings of the International School of Physics “Enrico Fermi”, Course LXVII, pp. 396–456. Italian Physical Society/North-Holland, Amsterdam (1979)Google Scholar
  5. 5.
    Moncrief V.: Spacetime symmetries and linearization stability of the Einstein equations. I. J. Math. Phys. 16(3), 493–498 (1975)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Moncrief V.: Space-time symmetries and linearization stability of the Einstein equations. II. J. Math. Phys. 17(10), 1893–1902 (1976)ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    Moncrief V.: Gauge symmetries of Yang–Mills fields. Ann. Phys. 108(2), 387–400 (1977)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    Arms J.M.: Linearization stability of gravitational and gauge fields. J. Math. Phys. 20(3), 443–453 (1979)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bao D.: A sufficient condition for the linearization stability of N =  1 supergravity: a preliminary report. Ann. Phys. 158(1), 211–278 (1984)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Sardanashvily G.: Noether identities of a differential operator: the Koszul–Tate complex. Int. J. Geom. Methods Mod. Phys. 02(05), 873–886 (2005)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Kazinski P.O., Lyakhovich S.L., Sharapov A.A.: Lagrange structure and quantization. J. High Energy Phys. 2005(07), 076 (2005) arXiv:hep-th/0506093 CrossRefMathSciNetGoogle Scholar
  12. 12.
    Barnich G., Brandt F.: Covariant theory of asymptotic symmetries, conservation laws and central charges. Nucl. Phys. B 633(1–2), 3–82 (2002) arXiv:hep-th/0111246 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Verbovetsky, A.: Notes on the horizontal cohomology. In: Henneaux, M., Joseph, K., Vinogradov, A. (eds.) Secondary Calculus and Cohomological Physics, vol. 219, pp. 211–231. American Mathematical Society, Providence (1998). arXiv:math/9803115
  14. 14.
    Krasil’shchik, J., Verbovetsky, A.: Homological Methods in Equations of Mathematical Physics. Advanced Texts in Mathematics. Open Education and Sciences, Opava, (1998). arXiv:math/9808130
  15. 15.
    Goldschmidt H.: Existence theorems for analytic linear partial differential equations. Ann. Math. 86(2), 246–270 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Pommaret, J.F.: Partial Differential Equations and Group Theory: New Perspectives for Applications. Mathematics and Its Applications, vol. 293. Springer, Berlin (2010)Google Scholar
  17. 17.
    Sergyeyev, A.: On recursion operators and nonlocal symmetries of evolution equations. In: Krupka, D. (ed.) Proceedings of the Seminar on Differential Geometry. pp. 159–173. Silesian University at Opava, Opava, 2000. arXiv:nlin/0012011
  18. 18.
    Fischer A.E., Marsden J.E.: Linearization stability of the Einstein equations. Bull. Am. Math. Soc. 79(5), 997–1004 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Losic B., Unruh W.G.: On leading order gravitational backreactions in de Sitter spacetime. Phys. Rev. D 74(2), 023511 (2006) arXiv:gr-qc/0604122 ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    Arms J.M., Anderson I.M.: Perturbations of conservation laws in field theories. Ann. Phys. 167(2), 354–389 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Olver, P.J.: Applications of Lie groups to differential equations, 2nd edn. Graduate Texts in Mathematics, vol. 107. Springer, New York (1993)Google Scholar
  22. 22.
    Anderson I.M.: Introduction to the variational bicomplex. In: Gotay, M.J., Marsden, J.E., Moncrief, V. (eds.) Mathematical Aspects of Classical Field Theory. Contemporary Mathematics, vol. 132, pp. 51–73. American Mathematical Society, Providence (1992)CrossRefGoogle Scholar
  23. 23.
    Anderson, I.M.: The variational bicomplex. Unpublished draft (1989)Google Scholar
  24. 24.
    Vinogradov, A.M., Krasilshchik, I.S. (eds.): Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Translations of Mathematical Monographs, vol. 182. American Mathematical Society, Providence (1999)Google Scholar
  25. 25.
    Goldschmidt H.: Integrability criteria for systems of nonlinear partial differential equations. J. Differ. Geom. 1(3-4), 269–307 (1967)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Seiler W.M., Tucker R.W.: Involution and constrained dynamics. I. the Dirac approach. J. Phys. A Math. Gen. 28(15), 4431–4451 (1995)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Barnich G., Brandt F., Henneaux M.: Local BRST cohomology in gauge theories. Phys. Rep. 338(5), 439–569 (2000) arXiv:hep-th/0002245 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Hirsch M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33. Springer, Berlin (1976)Google Scholar
  29. 29.
    Kriegl A., Michor P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)CrossRefGoogle Scholar
  30. 30.
    Christodoulou D., Klainerman S.: The global nonlinear stability of the Minkowski space. Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton (1993)Google Scholar
  31. 31.
    Brill D.R., Deser S.: Instability of closed spaces in general relativity. Commun. Math. Phys. 32(4), 291–304 (1973)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Baez, J.C., Schreiber, U.: Higher gauge theory. In: Categories in Algebra, Geometry and Mathematical Physics. Contemporary Mathematics, vol. 431, pp. 7–30. American Mathematical Society, Providence (2007). arXiv:math/0511710
  33. 33.
    Fiorenza D., Rogers C.L., Schreiber U.: A higher Chern–Weil derivation of AKSZ σ-models. Int. J. Geom. Methods Mod. Phys. 10(01), 1250078 (2013) arXiv:1108.4378 CrossRefMathSciNetGoogle Scholar
  34. 34.
    Wikipedia: Poincaré duality—Wikipedia, The Free Encyclopedia. 2013. [Online; Accessed 10-March-2013]Google Scholar
  35. 35.
    Fischer, A.E., Marsden, J.E.: Linearization stability of nonlinear partial differential equations. In: Differential Geometry (Proceedings of Symposia in Pure Mathematics, vol. XXVII, Part 2, Stanford Univiversity, Stanford, California, 1973), pp. 219–263. American Mathematical Society, Providence (1975)Google Scholar
  36. 36.
    Abbott L.F., Deser S.: Stability of gravity with a cosmological constant. Nucl. Phys. B 195(1), 76–96 (1982)ADSCrossRefzbMATHGoogle Scholar
  37. 37.
    Taub A.H.: Approximate stress energy tensor for gravitational fields. J. Math. Phys. 2(6), 787–793 (1961)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Wikipedia: Yang–Mills theory—Wikipedia, The Free Encyclopedia. 2013. [Online; Accessed 10-March-2013]Google Scholar
  39. 39.
    Wikipedia: Chern–Simons theory—Wikipedia, The Free Encyclopedia. 2013. [Online; Accessed 10-March-2013]Google Scholar
  40. 40.
    Goldman W.M.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2), 200–225 (1984)CrossRefzbMATHGoogle Scholar
  41. 41.
    Freedman D.Z., Townsend P.K.: Antisymmetric tensor gauge theories and non-linear σ-models. Nucl. Phys. B 177(2), 282–296 (1981)ADSCrossRefMathSciNetGoogle Scholar
  42. 42.
    Roytenberg D.: AKSZ-BV formalism and Courant algebroid-induced topological field theories. Lett. Math. Phys. 79(2), 143–159 (2006) arXiv:hep-th/060815 ADSCrossRefMathSciNetGoogle Scholar
  43. 43.
    Bergshoeff E., Roo M., Wit B., Nieuwenhuizen P.: Ten-dimensional Maxwell–Einstein supergravity, its currents, and the issue of its auxiliary fields. Nucl. Phys. B 195(1), 97–136 (1982)ADSCrossRefzbMATHGoogle Scholar
  44. 44.
    Henneaux, M.: Consistent interactions between gauge fields: the cohomological approach. In: Henneaux, M., Krasil’shchik, J., Vinogradov, A. (eds.) Secondary Calculus and Cohomological Physics. Contemporary Mathematics, vol. 219, p. 93. AMS (1999). arXiv:hep-th/9712226
  45. 45.
    Fedosov B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40(2), 213–238 (1994)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Pflaum M.J.: On the deformation quantization of symplectic orbispaces. Diff. Geom. Appl. 19(3), 343–368 (2003) arXiv:math-ph/0208020 CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Pflaum, M.J., Posthuma, H., Tang, X.: Quantization of Whitney functions. arXiv:1202.5575
  48. 48.
    Gotay M.J.: Poisson reduction and quantization for the n + 1 photon. J. Math. Phys. 25(7), 2154–2159 (1984)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    Wikipedia: Hodge dual—Wikipedia, The Free Encyclopedia. 2013. [Online; Accessed 10-March-2013]Google Scholar

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© Springer Basel 2014

Authors and Affiliations

  1. 1.Institute for Theoretical Physics UtrechtUtrechtThe Netherlands

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