nLab D=9 supergravity

Idea

supergravity in dimension 9, hence on a 9-manifold

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

References

• Maria P. Garcia del Moral, J. M. Pena, Alvaro Restuccia, Supermembrane origin of type II gauged supergravities in 9D, JHEP 1209 (2012) 063 [arXiv:1203.2767]

On $SL_2$ S-duality in 9D SuGra:

• Patrick Meessen, Tomás Ortín: An $Sl(2,\mathbb{Z})$ Multiplet of Nine-Dimensional Type II Supergravity Theories, Nucl. Phys. B 541 (1999) 195–245 [doi:10.1016/S0550-3213(98)00780-9, arXiv:hep-th/9806120]

• Mohab Abou-Zeid, Bernard de Wit, Dieter Lüst, Hermann Nicolai: Space-Time Supersymmetry, IIA/B Duality and M-Theory, Phys. Lett. B 466 (1999) 144–152 [doi:10.1016/S0370-2693(99)01114-4, arXiv:hep-th/9908169]