# nLab 5-dimensional supergravity

### Context

#### Gravity

gravity, supergravity

# Contents

## Idea

supergravity in dimension 5. For $N = 1$ 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)

$\array{ 11d \; SuGra\; on\; S^1 \times Y_6 \times X_4 &\longrightarrow& 5d \; SuGra\; on\; S^1 \times X_4 && strong \; coupling \\ \downarrow && \downarrow \\ 10d\; tpye \;IIA\; Sugra\; on \; Y_6 \times X_4 &\longrightarrow& 10d\; Sugra\; on \; X_4 && weak \; coupling }$

## Properties

### 5d Chern-Simons term

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

$d F_3 = F_2 \wedge F_2$

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

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

$d G_7 = G_4 \wedge G_4$

(up to prefactors) with $G_7 = \star G_4$.

### U-duality

supergravity gauge group (split real form)T-duality group (via toroidal KK-compactification)U-dualitymaximal gauged supergravity
$SL(2,\mathbb{R})$1$SL(2,\mathbb{Z})$ S-duality10d 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})$$E_{9(9)}(\mathbb{Z})$2d supergravityE8-equivariant elliptic cohomology
E10$E_{10(10)}$$O(9,9;\mathbb{Z})$$E_{10(10)}(\mathbb{Z})$
E11$E_{11(11)}$$O(10,10;\mathbb{Z})$$E_{11(11)}(\mathbb{Z})$

## References

### General

Basic discussion includes

### 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).

### Gauged sugra

For 5d gauged supergravity:

• M. Pernici, K. Pilch, Peter van Nieuwenhuizen, Gauged $N=8$ $D=5$ Supergravity, Nucl.Phys. B259 (1985) 460 (spire)

• M. Gunaydin, L.J. Romans, N.P. Warner, Compact and Noncompact Gauged Supergravity Theories in Five-Dimensions, Nucl.Phys. B272 (1986) 598-646 (spire)

• Murat Gunaydin, Marco Zagermann, The Gauging of Five-dimensional, $N=2$ Maxwell-Einstein Supergravity Theories coupled to Tensor Multiplets, Nucl.Phys.B572:131-150,2000 (arXiv:hep-th/9912027)

• Murat Gunaydin, Marco Zagermann, The Vacua of 5d, $N=2$ Gauged Yang-Mills/Einstein/Tensor Supergravity: Abelian Case, Phys.Rev.D62:044028,2000 (arXiv:hep-th/0002228)

• A. Ceresole, Gianguido Dall'Agata, General matter coupled $N=2$, $D=5$ gauged supergravity, Nucl.Phys. B585 (2000) 143-170 (arXiv:hep-th/0004111)

### Horava-Witten compactification

Discussion of KK-compactification on $S^1/(\mathbb{Z}/2)$-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 balck holes to 5d black holes and embedding as black holes in string theory includes

• Henriette Elvang, Roberto Emparan, David Mateos, Harvey S. Reall, A supersymmetric black ring, Phys.Rev.Lett.93:211302,2004 (arXiv:hep-th/0407065)

• I. Bena and P. Kraus, Microscopic description of black rings in AdS/CFT JHEP 12 (2004) 070, hep-th/0408186.

• I. Bena and P. Kraus, Microstates of the D1-D5-KK system Phys. Rev. D72 (2005) 025007, hep-th/0503053

• Davide Gaiotto, Andrew Strominger, and X. Yin, 5D black rings and 4D black holes JHEP 02 (2006) 023 (hep-th/0504126)

• Davide Gaiotto, Andrew Strominger, and X. Yin, New connections between 4D and 5D black holes, JHEP 02 (2006) 024 (hep-th/0503217)

Revised on August 26, 2016 07:36:39 by Urs Schreiber (89.204.153.62)