D=8 supergravity (changes) in nLab

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Contents

Idea

supergravity in dimension 8

Properties

U-duality

The U-duality group of maximax 8d SuGra is SL(3,ℤ)×SL(2,ℤ)SL(3,\mathbb{Z})\times SL(2,\mathbb{Z}) (e.g. Liu-Minasian 97)

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

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

• 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

• 11-dimensional supergravity

• 10-dimensional type II supergravity, heterotic supergravity

• 8-dimensional supergravity

• 7-dimensional supergravity

• 5-dimensional supergravity

• 4-dimensional supergravity

• 3-dimensional supergravity

References

• Abdus Salam, Ergin Sezgin, d=8 supergravity, Nuclear Physics B 258 (1985) 284-304 []

• M. Awada, Paul Townsend, d = 8 Maxwell/Einstein supergravity, Physics Letters B Volume 156, Issues 1–2, 13 June 1985

• Mees de Roo, Giuseppe Dibitetto, Yihao Yin, Critical points of maximal D=8D=8 gauged supergravities, JHEP 1201:029 (2012) [[arXiv:1110.2886](http://arxiv.org/abs/1110.2886)]

Discussion of U-duality in 8d:

• James Liu, Ruben Minasian, U-branes and T 3T^3 fibrations, Nucl.Phys. B510 (1998) 538-554 (arXiv:hep-th/9707125)

Further discussion:

• Bing-Xin Lao, Ruben Minasian, Consistency of eight-dimensional supergravities: Anomalies, Lattices and Counterterms [[arXiv:2306.07766](https://arxiv.org/abs/2306.07766)]