5-dimensional 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), 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)


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)



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 (