nLab D=8 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	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

• 10-dimensional type II supergravity, heterotic supergravity

References

• Abdus Salam, Ergin Sezgin, d=8 supergravity, Nuclear Physics B 258 (1985) 284-304 [doi:10.1016/0550-3213(85)90613-3]

• 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 U-duality in 8d:

• James Liu, Ruben Minasian, U-branes and $T^3$ 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]

• Guillaume Bossard, Bingxin Lao, Ruben Minasian: Uniqueness of $D=8$ minimal supergravity with two vector multiplets [arXiv:2605.19002]