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5-dimensional supergravity

Context

Gravity

String theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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 }

Properties

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), 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 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=G 4G 4 d G_7 = G_4 \wedge G_4

(up to prefactors) with G 7=G 4G_7 = \star G_4.

Via Calabi-Yau compactification of M-theory

(Hull-Townsend 95, p.30-31, Ferrara-Khuria-Minasian 96)

U-duality

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)

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=8N=8 D=5D=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=2N=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=2N=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=2N=2, D=5D=5 gauged supergravity, Nucl.Phys. B585 (2000) 143-170 (arXiv:hep-th/0004111)

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 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)