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

As D=11 supergravity dimensionally reduced on a torus (most authors immediately consider this as 10D type IIB supergravity via T-duality):

• Paul S. Aspinwall; section 3 in: Some Relationships Between Dualities in String Theory, Nucl. Phys. Proc. Suppl. 46 (1996) 30–38 [doi:10.1016/0920-5632(96)00004-7, arXiv:hep-th/9508154]

• John H. Schwarz: The Power of M Theory, Phys. Lett. B 367 (1996) 97–103 [doi:10.1016/0370-2693(95)01429-2, arXiv:hep-th/9510086]

and with focus on the M2-brane wrapped on that torus:

• Maria P. Garcia del Moral, Joselen M. Pena, Alvaro Restuccia, Supermembrane origin of type II gauged supergravities in 9D, JHEP 1209 (2012) 063 [arXiv:1203.2767, doi:10.1007/JHEP09(2012)063]

• Maria P. Garcia del Moral, Joselen M. Pena, Alvaro Restuccia: Classification of M2-brane 2-torus bundles, U-duality invariance and type II gauged supergravities, Phys. Rev. D 100 (2019) 026005 [doi:10.1103/PhysRevD.100.026005, arXiv:1604.02579]

Direct construction of 9D SuGra via Noether method (not by dim-reduction):

• Hitoshi Nishino, Subhash Rajpoot: Gauged $N=2$ Supergravity in Nine-Dimensions and Domain Wall Solutions, Phys.Lett. B 546 (2002) 261–272 [doi:10.1016/S0370-2693(02)02704-1, arXiv:hep-th/0207246]

More on $SL_2$ S-duality in 9D SuGra:

• Eric Bergshoeff, Chris M. Hull, Tomás Ortín: Duality in the Type–II Superstring Effective Action, Nucl. Phys. B 451 (1995) 547–578 [doi:10.1016/0550-3213(95)00367-2, arXiv:hep-th/9504081]

• Ashok Das, Shibaji Roy: On M-Theory and the Symmetries of Type II String Effective Actions, Nucl. Phys. B 482 (1996) 119–141 [doi:10.1016/S0550-3213(96)00530-5, arXiv:hep-th/9605073]

• 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]

More on gauged 9D supergravity:

• J. J. Fernandez-Melgarejo, Tomás Ortín, E. Torrente-Lujan: The general gaugings of maximal d=9 supergravity, J. High Energ. Phys. 2011 68 (2011) [doi:10.1007/JHEP10(2011)068, arXiv:1106.1760]

The duality-symmetric flux Bianchi identities via cyclification:

• Hisham Sati, Alexander Voronov; Ex. 2.6 in: Mysterious Triality and Rational Homotopy Theory, Comm. Math. Phys. 400 (2023) 1915-1960 [arXiv:2111.14810 hep-th, doi:10.1007/s00220-023-04643-7]