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 membrane cocycle 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 supegravity 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})1SL(2,)SL(2,\mathbb{Z}) S-duality10d type IIB supergravity
SL(2,)×(2,\mathbb{R}) \times O(1,1) 2\mathbb{Z}_2SL(2,)× 2SL(2,\mathbb{Z}) \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

  • Laura Andrianopoli, Francesco Cordaro, Pietro Fré, Leonardo Gualtieri, Non-Semisimple Gaugings of D=5 N=8 Supergravity and FDA.s, Class.Quant.Grav. 18 (2001) 395-414 (arXiv:hep-th/0009048)

surveyed in


General discussion includes

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

Review includes

Last revised on May 13, 2021 at 01:22:00. See the history of this page for a list of all contributions to it.