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]

Dual graviton
CompositionElementary particle
Electric chargee

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 gravityEdit

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][8]


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][8]


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 theoryEdit

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 gravitoelectromagnetismEdit

In similar manner while we define gravitomagnetic and gravitoelectic for the graviton, we can define electric and magnetic fields for the dual graviton.[9] 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  :[10][8]


and scalar curvature   with dual scalar curvature  :[10]


where   denotes the Hodge dual.

Dual graviton in conformal gravityEdit

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.[11]

See alsoEdit


  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. ^ a b c Danehkar, A. (2019). "Electric-magnetic duality in gravity and higher-spin fields". Frontiers in Physics. 6: 146. Bibcode:2019FrP.....6..146D. doi:10.3389/fphy.2018.00146.
  9. ^ 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.
  10. ^ a b Henneaux, M., "E10 and gravitational duality"
  11. ^ 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.