nLab 8-dimensional supergravity

Idea

supergravity in dimension 8

Properties

U-duality

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

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

• 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, E. Sezgin, d=8 supergravity, Nuclear Physics B, Volume 258, p. 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=8 gauged supergravities, JHEP 1201:029, 2012 (arXiv:1110.2886)

Discussion of the U-duality in 8d includes

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