Abstract
We prove that, for M theory or type II, generic Minkowski flux backgrounds preserving \( \mathcal{N} \) supersymmetries in dimensions D ≥ 4 correspond precisely to integrable generalised \( {G}_{\mathcal{N}} \) structures, where \( {G}_{\mathcal{N}} \) is the generalised structure group defined by the Killing spinors. In other words, they are the analogues of special holonomy manifolds in \( {E}_{d(d)}\times {\mathbb{R}}^{+} \) generalised geometry. In establishing this result, we introduce the Kosmann-Dorfman bracket, a generalisation of Kosmann’s Lie derivative of spinors. This allows us to write down the internal sector of the Killing superalgebra, which takes a rather simple form and whose closure is the key step in proving the main result. In addition, we find that the eleven-dimensional Killing superalgebra of these backgrounds is necessarily the supertranslational part of the \( \mathcal{N} \) -extended super-Poincaré algebra.
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References
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
M. Gualtieri, Generalized complex geometry, DPhil thesis, Oxford University, Oxford U.K. (2004) [math/0401221] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].
C.M. Hull, Generalised geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].
P. Pires Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, \( {E_d}_{(d)}\times {\mathbb{R}}^{+} \) generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry II: \( {E_d}_{(d)}\times {\mathbb{R}}^{+} \) and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].
D. Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry, J. Geom. Phys. 62 (2012) 903 [arXiv:1101.0856] [INSPIRE].
P. Bouwknegt, Courant algebroids and generalizations of geometry, talk given at String-Math, http://hans.math.upenn.edu/StringMath2011/notes/Bouwknegt StringMath2011 talk.pdf, UPenn, U.S.A. (2011).
N. Hitchin, Generalised geometry of type B n , talk given at String-Math, http://www.hcm.uni-bonn.de/fileadmin/stringmath2012/plenary talks/03-Hitchin.pdf, Bonn Germany (2012).
R. Rubio, B n -generalized geometry and G 22 -structures, J. Geom. Phys. 73 (2013) 150 [arXiv:1301.3330].
C. Strickland-Constable, Subsectors, Dynkin diagrams and new generalised geometries, arXiv:1310.4196 [INSPIRE].
K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, arXiv:1401.3360 [INSPIRE].
M. Garcia-Fernandez, Torsion-free generalized connections and heterotic supergravity, Commun. Math. Phys. 332 (2014) 89 [arXiv:1304.4294] [INSPIRE].
A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, JHEP 11 (2014) 160 [arXiv:1407.7542] [INSPIRE].
M. Garcia-Fernandez and C.S. Shahbazi, Self-dual generalized metrics for pure N = 1 six-dimensional supergravity, arXiv:1505.03088 [INSPIRE].
F. Ciceri, A. Guarino and G. Inverso, The exceptional story of massive IIA supergravity, JHEP 08 (2016) 154 [arXiv:1604.08602] [INSPIRE].
D. Cassani, O. de Felice, M. Petrini, C. Strickland-Constable and D. Waldram, Exceptional generalised geometry for massive IIA and consistent reductions, JHEP 08 (2016) 074 [arXiv:1605.00563] [INSPIRE].
J.P. Gauntlett, D. Martelli, S. Pakis and D. Waldram, G structures and wrapped NS5-branes, Commun. Math. Phys. 247 (2004) 421 [hep-th/0205050] [INSPIRE].
J.P. Gauntlett and S. Pakis, The geometry of D = 11 Killing spinors, JHEP 04 (2003) 039 [hep-th/0212008] [INSPIRE].
D. Martelli and J. Sparks, G structures, fluxes and calibrations in M-theory, Phys. Rev. D 68 (2003) 085014 [hep-th/0306225] [INSPIRE].
M. Graña, R. Minasian, M. Petrini and A. Tomasiello, Supersymmetric backgrounds from generalized Calabi-Yau manifolds, JHEP 08 (2004) 046 [hep-th/0406137] [INSPIRE].
M. Graña, R. Minasian, M. Petrini and A. Tomasiello, Generalized structures of N = 1 vacua, JHEP 11 (2005) 020 [hep-th/0505212] [INSPIRE].
M. Graña and F. Orsi, N = 1 vacua in exceptional generalized geometry, JHEP 08 (2011) 109 [arXiv:1105.4855] [INSPIRE].
M. Graña and F. Orsi, N = 2 vacua in generalized geometry, JHEP 11 (2012) 052 [arXiv:1207.3004] [INSPIRE].
M. Graña, R. Minasian, M. Petrini and A. Tomasiello, A scan for new N = 1 vacua on twisted tori, JHEP 05 (2007) 031 [hep-th/0609124] [INSPIRE].
D. Andriot, New supersymmetric flux vacua with intermediate SU(2) structure, JHEP 08 (2008) 096 [arXiv:0804.1769] [INSPIRE].
F. Apruzzi, M. Fazzi, D. Rosa and A. Tomasiello, All AdS 7 solutions of type-II supergravity, JHEP 04 (2014) 064 [arXiv:1309.2949] [INSPIRE].
F. Apruzzi, M. Fazzi, A. Passias, D. Rosa and A. Tomasiello, AdS 6 solutions of type-II supergravity, JHEP 11 (2014) 099 [Erratum ibid. 05 (2015) 012] [arXiv:1406.0852] [INSPIRE].
F. Apruzzi, M. Fazzi, A. Passias and A. Tomasiello, Supersymmetric AdS 5 solutions of massive IIA supergravity, JHEP 06 (2015) 195 [arXiv:1502.06620] [INSPIRE].
A. Rota and A. Tomasiello, AdS 4 compactifications of AdS 7 solutions in type-II supergravity, JHEP 07 (2015) 076 [arXiv:1502.06622] [INSPIRE].
R. Minasian, M. Petrini and A. Zaffaroni, Gravity duals to deformed SYM theories and generalized complex geometry, JHEP 12 (2006) 055 [hep-th/0606257] [INSPIRE].
A. Butti, D. Forcella, L. Martucci, R. Minasian, M. Petrini and A. Zaffaroni, On the geometry and the moduli space of β-deformed quiver gauge theories, JHEP 07 (2008) 053 [arXiv:0712.1215] [INSPIRE].
M. Gabella, J.P. Gauntlett, E. Palti, J. Sparks and D. Waldram, AdS 5 solutions of type IIB supergravity and generalized complex geometry, Commun. Math. Phys. 299 (2010) 365 [arXiv:0906.4109] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supersymmetric backgrounds and generalised special holonomy, Class. Quant. Grav. 33 (2016) 125026 [arXiv:1411.5721] [INSPIRE].
M.J. Duff and J.T. Liu, Hidden space-time symmetries and generalized holonomy in M-theory, Nucl. Phys. B 674 (2003) 217 [hep-th/0303140] [INSPIRE].
C. Hull, Holonomy and symmetry in M-theory, hep-th/0305039 [INSPIRE].
A. Coimbra and C. Strickland-Constable, Generalised structures for N = 1 AdS backgrounds, arXiv:1504.02465 [INSPIRE].
A. Ashmore and D. Waldram, Exceptional Calabi-Yau spaces: the geometry of N = 2 backgrounds with flux, arXiv:1510.00022 [INSPIRE].
M. Graña, J. Louis, A. Sim and D. Waldram, E 7(7) formulation of N = 2 backgrounds, JHEP 07 (2009) 104 [arXiv:0904.2333] [INSPIRE].
A. Ashmore, M. Petrini and D. Waldram, The exceptional generalised geometry of supersymmetric AdS flux backgrounds, arXiv:1602.02158 [INSPIRE].
M. Graña and P. Ntokos, Generalized geometric vacua with eight supercharges, JHEP 08 (2016) 107 [arXiv:1605.06383] [INSPIRE].
A. Ashmore, M. Gabella, M. Graña, M. Petrini and D. Waldram, Exactly marginal deformations from exceptional generalised geometry, arXiv:1605.05730 [INSPIRE].
D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, Exactly marginal deformations and global symmetries, JHEP 06 (2010) 106 [arXiv:1005.3546] [INSPIRE].
Y. Kosmann, Dérivées de Lie des spineurs (in French), Ann. Matemat. Pura Appl. 91 (1971) 317.
J.M. Figueroa-O’Farrill, P. Meessen and S. Philip, Supersymmetry and homogeneity of M-theory backgrounds, Class. Quant. Grav. 22 (2005) 207 [hep-th/0409170] [INSPIRE].
J.M. Figueroa-O’Farrill, E. Hackett-Jones and G. Moutsopoulos, The Killing superalgebra of ten-dimensional supergravity backgrounds, Class. Quant. Grav. 24 (2007) 3291 [hep-th/0703192] [INSPIRE].
J. Figueroa-O’Farrill and N. Hustler, The homogeneity theorem for supergravity backgrounds, JHEP 10 (2012) 014 [arXiv:1208.0553] [INSPIRE].
J. Figueroa-O’Farrill and N. Hustler, The homogeneity theorem for supergravity backgrounds II: the six-dimensional theories, JHEP 04 (2014) 131 [arXiv:1312.7509] [INSPIRE].
J. Figueroa-O’Farrill, Symmetric M-theory backgrounds, Central Eur. J. Phys. 11 (2013) 1 [arXiv:1112.4967] [INSPIRE].
J. Figueroa-O’Farrill and N. Hustler, Symmetric backgrounds of type IIB supergravity, Class. Quant. Grav. 30 (2013) 045008 [arXiv:1209.4884] [INSPIRE].
M. Graña, R. Minasian, M. Petrini and D. Waldram, T-duality, generalized geometry and non-geometric backgrounds, JHEP 04 (2009) 075 [arXiv:0807.4527] [INSPIRE].
J. Figueroa-O’Farrill, E. Hackett-Jones, G. Moutsopoulos and J. Simon, On the maximal superalgebras of supersymmetric backgrounds, Class. Quant. Grav. 26 (2009) 035016 [arXiv:0809.5034] [INSPIRE].
Ü. Ertem and Ö. Açik, Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors, Class. Quant. Grav. 33 (2016) 165002 [arXiv:1601.03356] [INSPIRE].
P. Candelas and D.J. Raine, Spontaneous compactification and supersymmetry in d = 11 supergravity, Nucl. Phys. B 248 (1984) 415 [INSPIRE].
P. Candelas, Compactification and supersymmetry of chiral N = 2 D = 10 supergravity, Nucl. Phys. B 256 (1985) 385 [INSPIRE].
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
S. Ivanov and G. Papadopoulos, A no go theorem for string warped compactifications, Phys. Lett. B 497 (2001) 309 [hep-th/0008232] [INSPIRE].
J.B. Gutowski and G. Papadopoulos, Supersymmetry of AdS and flat backgrounds in M-theory, JHEP 02 (2015) 145 [arXiv:1407.5652] [INSPIRE].
S.W. Beck, J.B. Gutowski and G. Papadopoulos, Supersymmetry of AdS and flat IIB backgrounds, JHEP 02 (2015) 020 [arXiv:1410.3431] [INSPIRE].
S. Beck, J.B. Gutowski and G. Papadopoulos, Supersymmetry of IIA warped flux AdS and flat backgrounds, JHEP 09 (2015) 135 [arXiv:1501.07620] [INSPIRE].
U. Gran, J.B. Gutowski and G. Papadopoulos, On supersymmetric anti-de-Sitter, de-Sitter and Minkowski flux backgrounds, arXiv:1607.00191 [INSPIRE].
M. Gabella, D. Martelli, A. Passias and J. Sparks, N = 2 supersymmetric AdS 4 solutions of M-theory, Commun. Math. Phys. 325 (2014) 487 [arXiv:1207.3082] [INSPIRE].
K. Lee, C. Strickland-Constable and D. Waldram, New gaugings and non-geometry, arXiv:1506.03457 [INSPIRE].
A.R. Frey and J. Polchinski, N = 3 warped compactifications, Phys. Rev. D 65 (2002) 126009 [hep-th/0201029] [INSPIRE].
R. Haag, J.T. Lopuszanski and M. Sohnius, All possible generators of supersymmetries of the S matrix, Nucl. Phys. B 88 (1975) 257 [INSPIRE].
M.F. Sohnius, Introducing supersymmetry, Phys. Rept. 128 (1985) 39 [INSPIRE].
B. de Wit, H. Samtleben and M. Trigiante, On Lagrangians and gaugings of maximal supergravities, Nucl. Phys. B 655 (2003) 93 [hep-th/0212239] [INSPIRE].
B. de Wit, H. Samtleben and M. Trigiante, Gauging maximal supergravities, Fortsch. Phys. 52 (2004) 489 [hep-th/0311225] [INSPIRE].
M. Garcia-Fernandez, R. Rubio and C. Tipler, Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry, arXiv:1503.07562 [INSPIRE].
L.B. Anderson, J. Gray and E. Sharpe, Algebroids, heterotic moduli spaces and the Strominger system, JHEP 07 (2014) 037 [arXiv:1402.1532] [INSPIRE].
X. de la Ossa and E.E. Svanes, Holomorphic bundles and the moduli space of N = 1 supersymmetric heterotic compactifications, JHEP 10 (2014) 123 [arXiv:1402.1725] [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
H. Triendl and J. Louis, Type II compactifications on manifolds with SU(2) × SU(2) structure, JHEP 07 (2009) 080 [arXiv:0904.2993] [INSPIRE].
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Coimbra, A., Strickland-Constable, C. Supersymmetric backgrounds, the Killing superalgebra, and generalised special holonomy. J. High Energ. Phys. 2016, 63 (2016). https://doi.org/10.1007/JHEP11(2016)063
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DOI: https://doi.org/10.1007/JHEP11(2016)063