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Letters in Mathematical Physics

, Volume 108, Issue 8, pp 1873–1884 | Cite as

BV equivalence between triadic gravity and BF theory in three dimensions

  • A. S. Cattaneo
  • M. Schiavina
  • I. Selliah
Article
  • 76 Downloads

Abstract

The triadic description of general relativity in three dimensions is known to be a BF theory. Diffeomorphisms, as symmetries, are easily recovered on-shell from the symmetries of BF theory. This note describes an explicit off-shell BV symplectomorphism between the BV versions of the two theories, each endowed with their natural symmetries.

Keywords

Equivalence of field theories BV formalism 3d gravity Topological field theories Palatini theory BF theory 

Mathematics Subject Classification

83C47 83C05 70S99 81S10 

References

  1. 1.
    Anselmi, D.: Some reference formulas for the generating functions of canonical transformations. Eur. Phys. J. C 76, 49 (2016)ADSCrossRefGoogle Scholar
  2. 2.
    Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and quantization. Phys. Lett. B 102(1), 27–31 (1981)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Becchi, C., Rouet, A., Stora, R.: The abelian Higgs Kibble model, unitarity of the S-operator. Phys Lett B 52, 344 (1974)ADSCrossRefGoogle Scholar
  4. 4.
    Becchi, C., Rouet, A., Stora, R.: Renormalization of the abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127 (1975)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Becchi, C., Rouet, A., Stora, R.: Renormalization of gauge theories. Ann. Phys. 98, 2 (1976)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Tyutin, I.V.: Lebedev Physics Institute preprint, vol. 39, (1975). arXiv:0812.0580
  7. 7.
    Carlip, S.: Quantum Gravity in \(2+1\) Dimensions, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1998)CrossRefMATHGoogle Scholar
  8. 8.
    Cartan, E.: Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion. C. R. Acad. Sci. 174, 593–595 (1922)MATHGoogle Scholar
  9. 9.
    Cartan, E.: Comptes rendus hebdomadaires des séances de l’Académie des sciences, 174, 437–439, 593–595, 734–737, 857–860, 1104–1107 (1922)Google Scholar
  10. 10.
    Cattaneo, A.S., Dherin, B., Weinstein, A.: Symplectic microgeometry II: generating functions. Bull. Braz. Math. Soc. N. Ser. 42, 507 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical BV theories on manifolds with boundary. Comm. Math. Phys. 332(2), 535–603 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cattaneo, A.S., Schiavina, M.: BV-BFV approach to general relativity: Einstein–Hilbert action. J. Math. Phys. 57(2), 023515 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cattaneo, A.S., Schiavina, M.: The reduced phase space of Palatini–Cartan–Holst theory. arXiv:1707.05351 (2017)
  14. 14.
    Cattaneo, A.S., Schiavina, M.: BV-BFV approach to general relativity: Palatini–Cartan–Holst action. arXiv:1707.06328 (2017)
  15. 15.
    Palatini, A.: Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rend. Circ. Mat. Palermo 43, 203 (1919). [English translation by Hojman, R., Mukku, C. In: Bergmann, P.G. De Sabbata, V. (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)]Google Scholar
  16. 16.
    Piguet, O.: Ghost equations and diffeomorphism invariant theories. Class. Quantum Gravity 17, 3799–3806 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Stasheff, J.: Deformation theory and the Batalin-Vilkovisky master equation. In: Deformation Theory and Symplectic Geometry, Proceedings, Meeting, Ascona, Switzerland, June 16–22, 1996, arXiv:q-alg/9702012
  18. 18.
    Stasheff, J.: Homological reduction of constrained Poisson algebras. J. Diff. Geom. 45, 221–240 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Witten, E.: \(2+1\) dimensional gravity as an exactly soluble system, Nucl. Phys. B 311, 46–78 (1988/89)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikZürichSwitzerland
  2. 2.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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