Dual graviton

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Dual graviton
CompositionElementary particle
Electric chargee

In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality predicted by some formulations of supergravity in eleven dimensions.[3]

The dual graviton was first hypothesized in 1980.[4] It was theoretically modeled in 2000s,[1][2] which was then predicted in eleven-dimensional mathematics of SO(8) supergravity in the framework of electric-magnetic duality.[3] It again emerged in the E11 generalized geometry in eleven dimensions,[5] and the E7 generalized vielbeine-geometry in eleven dimensions.[6] While there is no local coupling between graviton and dual graviton, the field introduced by dual graviton may be coupled to a BF model as non-local gravitational fields in extra dimensions.[7]

Dual linearized gravity[edit]

The dual formulations of linearized gravity are described by a mixed Young symmetry tensor , the so-called dual graviton, in any spacetime dimension D > 4 with the following characters:[2]

where square brackets show antisymmetrization.

For 5-D spacetime, the spin-2 dual graviton is described by the Curtright field . The symmetry properties imply that

The Lagrangian action for the spin-2 dual graviton in 5-D spacetime, the Curtright field, becomes[2]

where is defined as

and the gauge symmetry of the Curtright field is

The dual Riemann curvature tensor of the dual graviton is defined as follows:[2]

and the dual Ricci curvature tensor and scalar curvature of the dual graviton become, respectively

They fulfill the following Bianchi identities

where is the 5-D spacetime metric.

Dual graviton coupling with BF theory[edit]

Dual gravitons have interaction with topological BF model in D = 5 through the following Lagrangian action[7]


Here, is the curvature form, and is the background field.

In principle, it should similarly be coupled to a BF model of gravity as the linearized Einstein–Hilbert action in D > 4:

where is the determinant of the metric tensor matrix, and is the Ricci scalar.

Dual gravitoelectromagnetism[edit]

In similar manner while we define gravitomagnetic and gravitoelectic for the graviton, we can define electric and magnetic fields for the dual graviton.[8] There are the following relation between the gravitoelectic field and gravitomagnetic field of the graviton and the gravitoelectic field and gravitomagnetic field of the dual graviton :[9]

and scalar curvature with dual scalar curvature :[9]

where denotes the Hodge dual.

Dual graviton in conformal gravity[edit]

The free (4,0) conformal gravity in D = 6 is defined as

where is the Weyl tensor in D = 6. The free (4,0) conformal gravity can be reduced to the graviton in the ordinary space, and the dual graviton in the dual space in D = 4.[10]

See also[edit]


  1. ^ a b Hull, C. M. (2001). "Duality in Gravity and Higher Spin Gauge Fields". Journal of High Energy Physics. 2001 (9): 27. arXiv:hep-th/0107149. Bibcode:2001JHEP...09..027H. doi:10.1088/1126-6708/2001/09/027.
  2. ^ a b c d e Bekaert, X.; Boulanger, N.; Henneaux, M. (2003). "Consistent deformations of dual formulations of linearized gravity: A no-go result". Physical Review D. 67 (4): 044010. arXiv:hep-th/0210278. Bibcode:2003PhRvD..67d4010B. doi:10.1103/PhysRevD.67.044010.
  3. ^ a b de Wit, B.; Nicolai, H. (2013). "Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions". Journal of High Energy Physics. 2013 (5): 77. arXiv:1302.6219. Bibcode:2013JHEP...05..077D. doi:10.1007/JHEP05(2013)077.
  4. ^ Curtright, T. (1985). "Generalised Gauge Fields". Physics Letters B. 165 (4–6): 304. Bibcode:1985PhLB..165..304C. doi:10.1016/0370-2693(85)91235-3.
  5. ^ West, P. (2012). "Generalised geometry, eleven dimensions and E11". Journal of High Energy Physics. 2012 (2): 18. arXiv:1111.1642. Bibcode:2012JHEP...02..018W. doi:10.1007/JHEP02(2012)018.
  6. ^ Godazgar, H.; Godazgar, M.; Nicolai, H. (2014). "Generalised geometry from the ground up". Journal of High Energy Physics. 2014 (2): 75. arXiv:1307.8295. Bibcode:2014JHEP...02..075G. doi:10.1007/JHEP02(2014)075.
  7. ^ a b Bizdadea, C.; Cioroianu, E. M.; Danehkar, A.; Iordache, M.; Saliu, S. O.; Sararu, S. C. (2009). "Consistent interactions of dual linearized gravity in D = 5: couplings with a topological BF model". European Physical Journal C. 63 (3): 491–519. arXiv:0908.2169. Bibcode:2009EPJC...63..491B. doi:10.1140/epjc/s10052-009-1105-0.
  8. ^ Henneaux, M.; Teitelboim, C. (2005). "Duality in linearized gravity". Physics Letters B. 71 (2): 024018. arXiv:gr-qc/0408101. Bibcode:2005PhRvD..71b4018H. doi:10.1103/PhysRevD.71.024018.
  9. ^ a b Henneaux, M., "E10 and gravitational duality" http://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_workshops_conferences/jointerc_2014/henneaux.pdf
  10. ^ Hull, C. M. (2000). "Symmetries and Compactifications of (4,0) Conformal Gravity". Journal of High Energy Physics. 2000 (0012): 007. arXiv:hep-th/0011215. Bibcode:2000JHEP...12..007H. doi:10.1088/1126-6708/2000/12/007.