nLab D=5 supergravity




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supergravity in dimension 5. For N=1N = 1 this arises from 11-dimensional supergravity by KK-compactification on a Calabi-Yau manifold of complex dimension 3 (see at M-theory on Calabi-Yau manifolds), hence serves as the low-energy effective field theory of the strong-coupling version of Calabi-Yau compactifications of type IIA string theory (see supersymmetry and Calabi-Yau manifolds)

11dSuGraonS 1×Y 6×X 4 5dSuGraonS 1×X 4 strongcoupling 10dtpyeIIASugraonY 6×X 4 10dSugraonX 4 weakcoupling \array{ 11d \; SuGra\; on\; S^1 \times Y_6 \times X_4 &\longrightarrow& 5d \; SuGra\; on\; S^1 \times X_4 && strong \; coupling \\ \downarrow && \downarrow \\ 10d\; tpye \;IIA\; Sugra\; on \; Y_6 \times X_4 &\longrightarrow& 10d\; Sugra\; on \; X_4 && weak \; coupling }


5d Chern-Simons term

This theory has a 2-form field strength F 2F_2, locally F 2=dAF_2 = d A, with a 5d Chern-Simons theory action functional locally of the form XF 2F 2A\propto \int_X F_2 \wedge F_2 \wedge A (e.g. Castellani-D’Auria-Fre (III.5.70), Gauntlet-Myers-Townsend 98, p. 3, GGHPR 02 (2.1), Bonetti-Grimm-Hohenegger 13). Hence its equation of motion is of the non-linear form

dF 3=F 2F 2 d F_3 = F_2 \wedge F_2

with F 3F 2F_3 \coloneqq \star F_2 the Hodge dual of F 2F_2 (GGHPR 02 (2.2)).

This is reflected in the corresponding cochains on super Minkowski spacetime

μ D0,5d=ψ¯ Aψ AAAAμ string,5d=ψ¯ AΓ aψ Ae a \mu_{D0,5d} = \overline{\psi}_A \psi_A \phantom{AAA} \mu_{string,5d} = \overline{\psi}_A\Gamma_a \psi_A \wedge e^a


dμ string,5d=μ D0,5dμ D0,5d. d \mu_{string,5d} = \mu_{D0,5d} \wedge \mu_{D0,5d} \,.

due to the Fierz identity in Castellani-D’Auria-Fré 91 (III.5.50a), this example:

(the other Fierz identity (III.5.50a) gives the cocycle for the membrane (super 2-brane in 5d) μ membrane,5di2ψ¯ AΓ abψe ae b\mu_{membrane,5d} \coloneqq \frac{i}{2}\overline{\psi}_A \Gamma_{a b} \psi \wedge e^a \wedge e^b, dμ membrane,5d=0d \mu_{membrane,5d} = 0, that appears already in the old brane scan. )

This is a lower dimensional analogue to the situation for the C-field G 4G_4 (locally G 4=dCG_4 = d C) in 11-dimensional supergravity, which has a Chern-Simons term locally of the form G 4G 4C\propto \int G_4 \wedge G_4 \wedge C and hence the equation of motion

dG 7=12G 4G 4 d G_7 \;=\; -\tfrac{1}{2}G_4 \wedge G_4

with G 7=G 4G_7 = \star G_4.

Black holes and black rings

The first black ring solution in 5d sugra was found in (Elvang-Emparan-Mateos-Reall 04, Elvang-Emparan-Mateos-Reall 05).

Supersymmetric black holes exist precisely only in dimensions 4 and 5 (Gauntlett-Myers-Townsend 98). These play a key role in the discussion of black holes in string theory.

(There are supersymmetric particle-like solutions of d>5d \gt 5 supergravity theories that are sometimes called black holes, but these are always singular. There are also supersymmetric black holes in d=3d = 3, but the spacetime in that case is asymptotically anti-de Sitter spacetime rather than asymptotically flat. Of course, there are non-singular supersymmetric black brane solutions in various d4d \geq 4 supergravity theories but these are neither ‘particle-like’ nor, strictly speaking, asymptotically flat.)

Via Calabi-Yau compactification of 11d supergravity

Discussion of 5d supergravity as a KK-compactification of 11-dimensional supergravity on a Calabi-Yau manifold of complex dimension 3 (M-theory on Calabi-Yau manifolds) is discussed in

(Hull-Townsend 95, p.30-31, Cadavid-Ceresole-D’Auria-Ferrara 95 Ferrara-Khuria-Minasian 96, Ferrara-Minasian-Sagnotti 96). See also (Mizoguchi-Ohta 98).


supergravity gauge group (split real form)T-duality group (via toroidal KK-compactification)U-dualitymaximal gauged supergravity
SL(2,)SL(2,\mathbb{R})1 SL ( 2 , ) SL(2,\mathbb{Z}) S-duality10d type IIB supergravity
SL(2,)×(2,\mathbb{R}) \times O(1,1) 2\mathbb{Z}_2 SL ( 2 , ) SL(2,\mathbb{Z}) × 2\times \mathbb{Z}_29d supergravity
SU(3)×\times SU(2)SL(3,)×SL(2,)(3,\mathbb{R}) \times SL(2,\mathbb{R})O(2,2;)O(2,2;\mathbb{Z})SL(3,)×SL(2,)SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})8d supergravity
SU(5)SL(5,)SL(5,\mathbb{R})O(3,3;)O(3,3;\mathbb{Z})SL(5,)SL(5,\mathbb{Z})7d supergravity
Spin(10)Spin(5,5)Spin(5,5)O(4,4;)O(4,4;\mathbb{Z})O(5,5,)O(5,5,\mathbb{Z})6d supergravity
E6E 6(6)E_{6(6)}O(5,5;)O(5,5;\mathbb{Z})E 6(6)()E_{6(6)}(\mathbb{Z})5d supergravity
E7E 7(7)E_{7(7)}O(6,6;)O(6,6;\mathbb{Z})E 7(7)()E_{7(7)}(\mathbb{Z})4d supergravity
E8E 8(8)E_{8(8)}O(7,7;)O(7,7;\mathbb{Z})E 8(8)()E_{8(8)}(\mathbb{Z})3d supergravity
E9E 9(9)E_{9(9)}O(8,8;)O(8,8;\mathbb{Z})E 9(9)()E_{9(9)}(\mathbb{Z})2d supergravityE8-equivariant elliptic cohomology
E10E 10(10)E_{10(10)}O(9,9;)O(9,9;\mathbb{Z})E 10(10)()E_{10(10)}(\mathbb{Z})
E11E 11(11)E_{11(11)}O(10,10;)O(10,10;\mathbb{Z})E 11(11)()E_{11(11)}(\mathbb{Z})

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

N=2N = 2 supersymmetry

Consider super Lie algebra cocoycles on N=2N =2 5d super-Minkowski spacetime (as in the brane scan).

With the notation as used at super Minkowski spacetime – Canonical coordinates, there are now two copies of spinor-valued 1-forms, denoted ψ 1\psi_1 and ψ 2\psi_2. We use indices of the form A,B,A,B, \cdots for these. Then the non-trivial bit of the Chevalley-Eilenberg algebra differential for N=2N = 2, d=5d = 5 super Minkowski spacetime is

d CEe a=i2ψ¯ AΓ aψ A d_{CE} e^a = - \tfrac{i}{2} \overline{\psi}_A \wedge \Gamma^a \psi_A

where summation over repeated indices is understood.

There is a Fierz identity

ψ¯ Aψ Aψ¯ Bψ B=ψ¯ AΓ aψ Aψ¯ BΓ aψ B. \overline{\psi}_A \wedge \psi_A \wedge \overline{\psi}_B \wedge \psi_B \;=\; \overline{\psi}_A \wedge \Gamma_a \psi_A \wedge \overline{\psi}_B \wedge \Gamma^a \psi_B \,.

(Castellani-D’Auria-Fré (III.5.50a))

This implies that

d CE(ψ¯ AΓ aψ Ae a)(ψ¯ AΓ aψ A)(ψ¯ BΓ aψ B). d_{CE} (\overline{\psi}_A \Gamma^a \psi_A \wedge e_a) \;\propto\; (\overline{\psi}_A \wedge \Gamma^a \psi_A) \wedge (\overline{\psi}_B \wedge \Gamma^a \psi_B) \,.

There is a 4-cocycle of the form

μ 2=ϵ ABψ¯ AΓ abψ Be ae b. \mu_2 = \epsilon^{A B} \overline{\psi}_A \wedge \Gamma^{a b} \psi_B \wedge e_a \wedge e_b \,.

(Castellani-D’Auria-Fré (III.5.50b), (III.5.53c))


Construction of 5d gauged supergravity via D'Auria-Fré formulation of supergravity is in

surveyed in


General discussion:

See also:

Via M-theory on Calabi-Yau 3-folds

Discussion via KK-compactification as M-theory on Calabi-Yau manifolds includes

Further discussion of the 5d Chern-Simons term includes

(one-loop corrections).

Via type IIB theory

Gauged sugra

The maximal 5d gauged supergravity was first constructed in

See (ACFG 01).

Horava-Witten compactification

Discussion of KK-compactification on S 1/(/2)S^1/(\mathbb{Z}/2)-orbifolds (the version of Horava-Witten theory after dimensional reduction) is discussed in

  • Filipe Paccetti Correia, Michael G. Schmidt, Zurab Tavartkiladze, 4D Superfield Reduction of 5D Orbifold SUGRA and Heterotic M-theory (arXiv:hep-th/0602173)

Black hole solutions

Discussion of lifts of 4d black holes to 5d black holes and black rings and embedding as black holes in string theory includes


Further defect branes:

Last revised on January 5, 2024 at 20:53:19. See the history of this page for a list of all contributions to it.