nLab D=6 supergravity

Idea

supergravity in dimension 6

Properties

U-duality

	supergravity gauge group (split real form)	T-duality group (via toroidal KK-compactification)	U-duality	maximal gauged supergravity
	$SL(2,\mathbb{R})$	1	$SL(2,\mathbb{Z})$ S-duality	D=10 type IIB supergravity
	SL$(2,\mathbb{R}) \times$ O(1,1)	$\mathbb{Z}_2$	$SL(2,\mathbb{Z})$ $\times \mathbb{Z}_2$	D=9 supergravity
SU(3)$\times$ SU(2)	SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$	$O(2,2;\mathbb{Z})$	$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$	D=8 supergravity
SU(5)	$SL(5,\mathbb{R})$	$O(3,3;\mathbb{Z})$	$SL(5,\mathbb{Z})$	D=7 supergravity
Spin(10)	$Spin(5,5)$	$O(4,4;\mathbb{Z})$	$O(5,5,\mathbb{Z})$	D=6 supergravity
E₆	$E_{6(6)}$	$O(5,5;\mathbb{Z})$	$E_{6(6)}(\mathbb{Z})$	D=5 supergravity
E₇	$E_{7(7)}$	$O(6,6;\mathbb{Z})$	$E_{7(7)}(\mathbb{Z})$	D=4 supergravity
E₈	$E_{8(8)}$	$O(7,7;\mathbb{Z})$	$E_{8(8)}(\mathbb{Z})$	D=3 supergravity
E₉	$E_{9(9)}$	$O(8,8;\mathbb{Z})$	$E_{9(9)}(\mathbb{Z})$	D=2 supergravity	E₈-equivariant elliptic cohomology
E₁₀	$E_{10(10)}$	$O(9,9;\mathbb{Z})$	$E_{10(10)}(\mathbb{Z})$
E₁₁	$E_{11(11)}$	$O(10,10;\mathbb{Z})$	$E_{11(11)}(\mathbb{Z})$

(Hull-Townsend 94, table 1, table 2)

Related concepts

• supergravity

  • D=2 supergravity

  • D=3 supergravity

  • D=4 supergravity

  • D=5 supergravity

  • D=6 supergravity

  • D=7 supergravity

  • D=8 supergravity

  • D=9 supergravity

  • D=10 supergravity

  • D=11 supergravity

    • D=11 N=1 supergravity

• D=6 N=(1,0) SCFT

• D=6 N=(2,0) SCFT

• self-dual string in 6d

• 3-brane in 6d

References

General

Background on $D=6$ spin representations and supersymmetry:

• Paul Howe, Germán Sierra, Paul Townsend, Supersymmetry in six dimensions, Nuclear Physics B

  221 2 (1983) 331-348 [doi:10.1016/0550-3213(83)90582-5]

Formulation of $D=6$ supergravity on superspace:

• Moustafa A. Awada, Paul Townsend, Germán Sierra: Six-dimensional simple and extended chiral supergravity in superspace, Class. Quantum Grav. 2 (1985) L85 [doi:10.1088/0264-9381/2/4/005]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §III.7 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [§III.7: pdf]

  (in D'Auria-Fré-Regge formulation of supergravity)

• Gianguido Dall'Agata, Kurt Lechner, $N = 1$, $D = 6$ Supergravity: Duality and non Minimal Couplings, Nucl. Phys. B 511 (1998) 326-352 [arXiv:hep-th/9707236, doi:10.1016/S0550-3213(97)00719-0]

• William D. Linch III, Gabriele Tartaglino-Mazzucchelli, Six-dimensional Supergravity and Projective Superfields, JHEP 1208 (2012) 075 [arXiv:1204.4195, doi:10.1007/JHEP08(2012)075]

• Cesar Arias, William D. Linch III, Alexander K. Ridgway: Appendix A in: Superforms in Six-dimensional Superspace, J. High Energ. Phys. 2016 16 (2016). [arXiv:1402.4823, doi:10.1007/JHEP05(2016)016]

• Cesar Arias, The Geometry of 6D, $\mathcal{N} = (1,0)$ Superspace and its Matter Couplings [arXiv:1511.00335, spire:1402402]

Further discussion:

• Sergio Ferrara, Fabio Riccioni, Augusto Sagnotti, Tensor and Vector Multiplets in Six-Dimensional Supergravity, Nucl. Phys. B 519 (1998) 115-140 [arXiv:hep-th/9711059, doi:10.1016/S0550-3213(97)00837-7]

• Fabio Riccioni, Augusto Sagnotti, Some Properties of Tensor Multiplets in Six-Dimensional Supergravity, Nucl. Phys. Proc. Suppl. 67 (1998) 68-73 [arXiv:hep-th/9711077, doi:10.1016/S0920-5632(98)00121-2]

• Fabio Riccioni, All Couplings of Minimal Six-dimensional Supergravity_, Nucl. Phys. B 605 (2001) 245-265 [arXiv:hep-th/0101074, doi:10.1016/S0550-3213(01)00199-7]

• Vijay Kumar, Daniel S. Park, Washington Taylor, 6D supergravity without tensor multiplets, JHEP 1104:080,2011 (arXiv:1011.0726)

• Ángel Murcia, Carlos S. Shahbazi, Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity, Journal of Geometry and Physics 158 (2020) [arXiv:1912.08723, doi:10.1016/j.geomphys.2020.103868]

  (relation to contact geometry)

• Edoardo Lauria, Antoine Van Proeyen, $\mathcal{N}=2$ Supergravity in $D=4,5,6$ Dimensions (arXiv:2004.11433)

• Yuta Hamada, Gregory J. Loges, Towards a complete classification of 6D supergravities [arXiv:2311.00868]

• Gregory J. Loges, A new infinite class of 6d, $\mathcal{N}=(1,0)$ supergravities [arXiv:2402.04371]

For “exotic” $\mathcal{N} = (4,0)$-supersymmetry:

• Chris Hull, Strongly Coupled Gravity and Duality, Nucl.Phys. B 583 (2000) 237-259 [arXiv:hep-th/0004195, doi:10.1016/S0550-3213(00)00323-0]

• Chris Hull, Symmetries and Compactifications of $(4,0)$ Conformal Gravity, JHEP 0012: 007 (2000) [arXiv:hep-th/0011215, doi:10.1088/1126-6708/2000/12/007]

• Yannick Bertrand, Stefan Hohenegger, Olaf Hohm, Henning Samtleben, Toward Exotic 6D Supergravities, Phys. Rev. D 103 (2021) 046002 [arXiv:2007.11644, doi:10.1103/PhysRevD.103.046002]

• Yannick Bertrand, Stefan Hohenegger, Olaf Hohm, Henning Samtleben, Supersymmetric action for 6D $(4,0)$ supergravity, JHEP (2022) [arXiv:2206.04100]

• Chris Hull, Gravity, Duality and Conformal Symmetry, in Quantum gravity, branes and M-theory [arXi:2209.11716]

• Paul S. Howe, U. Lindström: Exotic gravity theory in loop space [arXiv:2412.16292]

Killing spinors

Discussion of Killing spinors/enhancement of number of supersymmetries in D=6 supergravity:

• Mehmet Akyol, George Papadopoulos, Spinorial geometry and Killing spinor equations of 6-D supergravity, Class. Quant. Grav. 28:105001, 2011 (arXiv:1010.2632)

• Mehmet Akyol, George Papadopoulos, $(1,0)$ superconformal theories in six dimensions and Killing spinor equations, JHEP 07 (2012) 070 (arXiv:1204.2167)

KK-reduction to JT gravity

Realization of JT-gravity as Kaluza-Klein reduction of D=6 supergravity on the worldvolume of D1-D5 brane bound states or M2-M5 brane bound states:

• Yue-Zhou Li, Shou-Long Li, H. Lu, Exact Embeddings of JT Gravity in Strings and M-theory, Eur. Phys. J. C (2018) 78: 791 (arXiv:1804.09742)

• Iosif Bena, Pierre Heidmann, David Turton, $AdS_2$ Holography: Mind the Cap, JHEP 1812 (2018) 028 (arXiv:1806.02834)