Abstract
We present a detailed study of a new mathematical object in E6(6)ℝ+ generalised geometry called an ‘exceptional complex structure’ (ECS). It is the extension of a conventional complex structure to one that includes all the degrees of freedom of M-theory or type IIB supergravity in six or five dimensions, and as such characterises, in part, the geometry of generic supersymmetric compactifications to five-dimensional Minkowkski space. We define an ECS as an integrable U*(6) × ℝ+ structure and show it is equivalent to a particular form of involutive subbundle of the complexified generalised tangent bundle L1 ⊂ Eℂ. We also define a refinement, an SU*(6) structure, and show that its integrability requires in addition a vanishing moment map on the space of structures. We are able to classify all possible ECSs, showing that they are characterised by two numbers denoted ‘type’ and ‘class’. We then use the deformation theory of ECS to find the moduli of any SU*(6) structure. We relate these structures to the geometry of generic minimally supersymmetric flux backgrounds of M-theory of the form ℝ4,1 × M, where the SU*(6) moduli correspond to the hypermultiplet moduli in the lower-dimensional theory. Such geometries are of class zero or one. The former are equivalent to a choice of (non-metric-compatible) conventional SL(3, ℂ) structure and strikingly have the same space of hypermultiplet moduli as the fluxless Calabi-Yau case.
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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, Ph.D. Thesis, Oxford University, Oxford U.K. (2003) [math/0401221] [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].
P. Pires Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].
M. Graña and F. Orsi, N = 1 vacua in Exceptional Generalized Geometry, JHEP 08 (2011) 109 [arXiv:1105.4855] [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].
A. Ashmore and D. Waldram, Exceptional Calabi-Yau spaces: the geometry of \( \mathcal{N} \) = 2 backgrounds with flux, Fortsch. Phys. 65 (2017) 1600109 [arXiv:1510.00022] [INSPIRE].
A. Coimbra and C. Strickland-Constable, Supersymmetric Backgrounds, the Killing Superalgebra, and Generalised Special Holonomy, JHEP 11 (2016) 063 [arXiv:1606.09304] [INSPIRE].
M. Graña and P. Ntokos, Generalized geometric vacua with eight supercharges, JHEP 08 (2016) 107 [arXiv:1605.06383] [INSPIRE].
A. Ashmore, M. Petrini and D. Waldram, The exceptional generalised geometry of supersymmetric AdS flux backgrounds, JHEP 12 (2016) 146 [arXiv:1602.02158] [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].
J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis and H.S. Reall, All supersymmetric solutions of minimal supergravity in five-dimensions, Class. Quant. Grav. 20 (2003) 4587 [hep-th/0209114] [INSPIRE].
A. Lukas, B.A. Ovrut, K.S. Stelle and D. Waldram, The Universe as a domain wall, Phys. Rev. D 59 (1999) 086001 [hep-th/9803235] [INSPIRE].
A. Lukas, B.A. Ovrut, K.S. Stelle and D. Waldram, Heterotic M-theory in five-dimensions, Nucl. Phys. B 552 (1999) 246 [hep-th/9806051] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, Del Pezzo surfaces and type-I-prime theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 1, hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 2, hep-th/9812127 [INSPIRE].
J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS5 solutions of M-theory, Class. Quant. Grav. 21 (2004) 4335 [hep-th/0402153] [INSPIRE].
C.M. Hull, Generalised Geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Ed(d) × ℝ+ generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].
A. Ashmore, C. Strickland-Constable, D. Tennyson and D. Waldram, Generalising G2 geometry: involutivity, moment maps and moduli, JHEP 01 (2021) 158 [arXiv:1910.04795] [INSPIRE].
A. Ashmore, C. Strickland-Constable, D. Tennyson and D. Waldram, Heterotic backgrounds via generalised geometry: moment maps and moduli, JHEP 11 (2020) 071 [arXiv:1912.09981] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as Generalised Geometry II: Ed(d) × ℝ+ and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].
A. Ashmore, M. Gabella, M. Graña, M. Petrini and D. Waldram, Exactly marginal deformations from exceptional generalised geometry, JHEP 01 (2017) 124 [arXiv:1605.05730] [INSPIRE].
A. Coimbra and C. Strickland-Constable, Supersymmetric AdS backgrounds and weak generalised holonomy, arXiv:1710.04156 [INSPIRE].
M. Graña, J. Louis, A. Sim and D. Waldram, E7(7) formulation of N = 2 backgrounds, JHEP 07 (2009) 104 [arXiv:0904.2333] [INSPIRE].
N.J. Hitchin, The Geometry of Three-Forms in Six Dimensions, J. Diff. Geom. 55 (2000) 547 [math/0010054] [INSPIRE].
N.J. Hitchin, Stable forms and special metrics, math/0107101 [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.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].
J.P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004) 086002 [hep-th/0302158] [INSPIRE].
K. Kodaira, Complex manifolds and deformation of complex structures, Springer, Berlin Germany (2006).
K. Behrndt and S. Gukov, Domain walls and superpotentials from M-theory on Calabi-Yau three folds, Nucl. Phys. B 580 (2000) 225 [hep-th/0001082] [INSPIRE].
M. Günaydin, L. Romans and N. Warner, Iib, or not iib: That is the question, Phys. Lett. B 164 (1985) 309.
R. Gilmore, Lie groups, physics, and geometry: an introduction for physicists, engineers and chemists, Cambridge University Press, Cambridge U.K. (2008).
D.S. Berman, M. Cederwall, A. Kleinschmidt and D.C. Thompson, The gauge structure of generalised diffeomorphisms, JHEP 01 (2013) 064 [arXiv:1208.5884] [INSPIRE].
N.J. Hitchin, The Geometry of Three-Forms in Six Dimensions, J. Diff. Geom. 55 (2000) 547 [math/0010054] [INSPIRE].
A. Swann, Hyperkähler and quaternionic kähler geometry, Math. Ann. 289 (1991) 421.
C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys Diff. Geom. 7 (1999) 123 [hep-th/9810250] [INSPIRE].
N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler Metrics and Supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].
G. Habib and K. Richardson, Modified differentials and basic cohomology for riemannian foliations, J. Geom. Anal. 23 (2013) 1314 [arXiv:1007.2955].
G. Tian, Smoothness of the Universal Deformation Space of Compact Calabi-Yau Manifolds and Its Peterson-Weil Metric, Adv. Ser. Math. Phys. 1 (1987) 629.
A.N. Todorov, The Weil-Petersson geometry of the moduli space of SU(n ≧ 3) (Calabi-Yau) manifolds I, Commun. Math. Phys. 126 (1989) 325.
A. Ashmore, Marginal deformations of 3d \( \mathcal{N} \) = 2 CFTs from AdS4 backgrounds in generalised geometry, JHEP 12 (2018) 060 [arXiv:1809.03503] [INSPIRE].
A. Ashmore, M. Petrini, E. Tasker and D. Waldram, to appear.
V. Pestun and E. Witten, The Hitchin functionals and the topological B-model at one loop, Lett. Math. Phys. 74 (2005) 21 [hep-th/0503083] [INSPIRE].
A. Strominger, Loop corrections to the universal hypermultiplet, Phys. Lett. B 421 (1998) 139 [hep-th/9706195] [INSPIRE].
L. Anguelova, M. Roček and S. Vandoren, Quantum corrections to the universal hypermultiplet and superspace, Phys. Rev. D 70 (2004) 066001 [hep-th/0402132] [INSPIRE].
I. Antoniadis, R. Minasian, S. Theisen and P. Vanhove, String loop corrections to the universal hypermultiplet, Class. Quant. Grav. 20 (2003) 5079 [hep-th/0307268] [INSPIRE].
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Tennyson, D., Waldram, D. Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory. J. High Energ. Phys. 2021, 88 (2021). https://doi.org/10.1007/JHEP08(2021)088
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DOI: https://doi.org/10.1007/JHEP08(2021)088