Abstract
In supergravity compactifications, there is in general no clear prescription on how to select a finite-dimensional family of metrics on the internal space, and a family of forms on which to expand the various potentials, such that the lower-dimensional effective theory is supersymmetric. We propose a finite-dimensional family of deformations for regular Sasaki-Einstein seven-manifolds M7, relevant for M-theory compactifications down to four dimensions. It consists of integrable Cauchy-Riemann structures, corresponding to complex deformations of the Calabi-Yau cone M8 over M7. The non-harmonic forms we propose are the ones contained in one of the Kohn-Rossi cohomology groups, which is finite-dimensional and naturally controls the deformations of Cauchy-Riemann structures. The same family of deformations can be also described in terms of twisted cohomology of the base M6, or in terms of Milnor cycles arising in deformations of M8. Using existing results on SU(3) structure compactifications, we briefly discuss the reduction of M-theory on our class of deformed Sasaki-Einstein manifolds to four-dimensional gauged supergravity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Gurrieri, J. Louis, A. Micu and D. Waldram, Mirror symmetry in generalized Calabi-Yau compactifications, Nucl. Phys. B 654 (2003) 61 [hep-th/0211102] [INSPIRE].
M. Graña, J. Louis and D. Waldram, Hitchin functionals in N = 2 supergravity, JHEP 01 (2006) 008 [hep-th/0505264] [INSPIRE].
A.-K. Kashani-Poor and R. Minasian, Towards reduction of type-II theories on SU(3) structure manifolds, JHEP 03 (2007) 109 [hep-th/0611106] [INSPIRE].
A.-K. Kashani-Poor, Nearly Kähler reduction, JHEP 11 (2007) 026 [arXiv:0709.4482] [INSPIRE].
J.P. Gauntlett, S. Kim, O. Varela and D. Waldram, Consistent supersymmetric Kaluza-Klein truncations with massive modes, JHEP 04 (2009) 102 [arXiv:0901.0676] [INSPIRE].
D. Cassani and A.-K. Kashani-Poor, Exploiting N = 2 in consistent coset reductions of type IIA, Nucl. Phys. B 817 (2009) 25 [arXiv:0901.4251] [INSPIRE].
D. Cassani, P. Koerber and O. Varela, All homogeneous N = 2 M-theory truncations with supersymmetric AdS 4 vacua, JHEP 11 (2012) 173 [arXiv:1208.1262] [INSPIRE].
I. Bena, G. Giecold, M. Graña, N. Halmagyi and F. Orsi, Supersymmetric consistent truncations of IIB on T 1,1, JHEP 04 (2011) 021 [arXiv:1008.0983] [INSPIRE].
D. Cassani and A.F. Faedo, A supersymmetric consistent truncation for conifold solutions, Nucl. Phys. B 843 (2011) 455 [arXiv:1008.0883] [INSPIRE].
R. D’Auria, S. Ferrara, M. Trigiante and S. Vaula, Gauging the Heisenberg algebra of special quaternionic manifolds, Phys. Lett. B 610 (2005) 147 [hep-th/0410290] [INSPIRE].
T. House and E. Palti, Effective action of (massive) IIA on manifolds with SU(3) structure, Phys. Rev. D 72 (2005) 026004 [hep-th/0505177] [INSPIRE].
A. Micu, E. Palti and P.M. Saffin, M-theory on seven-dimensional manifolds with SU(3) structure, JHEP 05 (2006) 048 [hep-th/0602163] [INSPIRE].
R. Eager, J. Schmude and Y. Tachikawa, Superconformal indices, Sasaki-Einstein manifolds and cyclic homologies, Adv. Theor. Math. Phys. 18 (2014) 129 [arXiv:1207.0573] [INSPIRE].
R. Eager and J. Schmude, Superconformal indices and M 2-branes, JHEP 12 (2015) 062 [arXiv:1305.3547] [INSPIRE].
G. Székelyhidi, The Kähler-Ricci flow and K-stability, Amer. J. Math. 132 (2010) 1077 [arXiv:0803.1613].
P. Candelas and X. de la Ossa, Moduli space of Calabi-Yau manifolds, Nucl. Phys. B 355 (1991) 455 [INSPIRE].
B. Dubrovin, Integrable systems in topological field theory, Nucl. Phys. B 379 (1992) 62.
G. Dall’Agata and N. Prezas, N = 1 geometries for M-theory and type IIA strings with fluxes, Phys. Rev. D 69 (2004) 066004 [hep-th/0311146] [INSPIRE].
K. Behrndt, M. Cvetič and T. Liu, Classification of supersymmetric flux vacua in M-theory, Nucl. Phys. B 749 (2006) 25 [hep-th/0512032] [INSPIRE].
X.-X. Chen, S. Donaldson and S. Sun, Kähler-einstein metrics and stability, arXiv:1210.7494.
P. Candelas and X.C. de la Ossa, Comments on conifolds, Nucl. Phys. B 342 (1990) 246 [INSPIRE].
M.B. Stenzel, Ricci-flat metrics on the complexification of a compact rank one symmetric space., Manuscr. Math. 80 (1993) 151.
M. Cvetič, G.W. Gibbons, H. Lü and C.N. Pope, Ricci flat metrics, harmonic forms and brane resolutions, Commun. Math. Phys. 232 (2003) 457 [hep-th/0012011] [INSPIRE].
K. Saito, Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1982) 775.
A.M. Nadel, Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Proc. Natl. Acad. Sci. 86 (1989) 7299.
P. Candelas, Yukawa couplings between (2, 1) forms, Nucl. Phys. B 298 (1988) 458 [INSPIRE].
V.A. Iskovskikh et al., Algebraic geometry: Fano varieties. V, Springer, Germany (1999).
S. Blesneag, E.I. Buchbinder, P. Candelas and A. Lukas, Holomorphic Yukawa couplings in heterotic string theory, JHEP 01 (2016) 152 [arXiv:1512.05322] [INSPIRE].
T. Akahori and P.M. Garfield, Hamiltonian flow over deformations of ordinary double points, J. Math. Anal. Appl. 333 (2007) 24.
J. Schmude, Laplace operators on Sasaki-Einstein manifolds, JHEP 04 (2014) 008 [arXiv:1308.1027] [INSPIRE].
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Gravitational multi-instantons, Phys. Lett. 78B (1978) 430 [INSPIRE].
A. Dimca, Topics on real and complex singularities: an introduction, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Germany (1987).
R.J. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J. 162 (2013) 2855 [arXiv:1205.6347] [INSPIRE].
A.S. Gusein-Zade and A. Varchenko, Singularities of differentiable maps, volume II, Birkhäuser, Geramny (2012).
C. Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge University Press, Cambridge U.K. (2002).
K. Saito, The higher residue pairings K ( k) F for a family of hypersurface singular points, Singularities. Part 2 40 (1981) 441.
K. Aleshkin and A. Belavin, Special geometry on the 101 dimesional moduli space of the quintic threefold, arXiv:1710.11609 [INSPIRE].
F. Forstnerič, Stein manifolds and holomorphic mappings: the homotopy principle in complex analysis, Springer, Germany (2011).
W. Ebeling, Functions of several complex variables and their singularities, American Mathematical Society, U.S.A. (2007).
P. Albin, Analysis on non-compact manifolds, https://old.math.illinois.edu/~palbin/18158/18158May26.pdf (2008).
S. Cappell et al., Cohomology of harmonic forms on Riemannian manifolds with boundary, Forum Math. 18 (2006) 923.
C. Shonkwiler, Poincaré duality angles for Riemannian manifolds with boundary, arXiv:0909.1967.
N.J. Hitchin, The geometry of three-forms in six dimensions, J. Diff. Geom. 55 (2000) 547 [math/0010054] [INSPIRE].
S. Dragomir and G. Tomassini, Differential geometry and analysis on CR manifolds, Springer, Germany (2007).
S.S.T. Yau, Kohn-Rossi cohomology and its application to the complex Plateau problem. I, Ann. Math. 113 (1981) 67.
H.R.J.J. Kohn, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math. 81 (1965) 451.
H.S. Luk et al., Holomorphic de Rham cohomology of strongly pseudoconvex cr manifolds with s1-actions, J. Diff. Geom. 63 (2003) 155.
I. Naruki, On Hodge structure of isolated singularity of complex hypersurface, Proc. Jpn. Acad. 50 (1974) 334.
N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya, Japan (1975).
X. Huang, H.S. Luk and S.S.T. Yau, Punctured local holomorphic de Rham cohomology, J. Math. Soc. Jpn. 55 (2003) 633.
M. Kuranishi, Application of \( {\overline{\partial}}_b \) to deformation of isolated singularities, Several complex Variables, Proc. Symp. Pure Math. 30 (1977) 97.
T. Akahori, Intrinsic Formula for Kuranishi’s \( \overline{\partial}\phi \), Publ. Res. Inst. Math. Sci. 14 (1978) 615.
T. Akahori and K. Miyajima, Complex analytic construction of the Kuranishi family on a normal strongly pseudo-convex manifold, II, Publ. Res. Inst. Math. Sci. 16 (1980) 811.
T. Akahori, The canonical Kaehler potential on the parameter space of the versal family of CR structures, J. Math. Anal. Appl. 300 (2004) 43.
T. Akahori, P.M. Garfield and J.M. Lee, Deformation theory of five-dimensional CR structures and the Rumin complex, math/0104056.
T. Akahori, Homogeneous polynomial hypersurface isolated singularities, J. Korean Math. Soc. 40 (1980) 667.
J. Cao and S.-C. Chang, Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces, math/0609312.
J. Cao and S.-C. Chang, The modified Calabi-Yau problems for CR-manifolds and applications, arXiv:0801.3431.
D.C. Chang, S.C. Chang and J. Tie, Calabi-Yau theorem and Hodge-Laplacian heat equation in a closed strictly pseudoconvex CR manifold, J. Diff. Geom. 97 (2014) 395.
R.J. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, II, arXiv:1301.5312 [INSPIRE].
R. J. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, III, arXiv:1405.7140.
G. Tian and S. T. Yau, Complete Kähler manifolds with zero Ricci curvature II, Inv. Math. 106 (1991) 27.
J. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988) 157.
E. Cremmer, B. Julia and J. Scherk, Supergravity theory in eleven-dimensions, Phys. Lett. B 76 (1978) 409 [INSPIRE].
J. Louis and A. Micu, Type 2 theories compactified on Calabi-Yau threefolds in the presence of background fluxes, Nucl. Phys. B 635 (2002) 395 [hep-th/0202168] [INSPIRE].
B. de Wit, H. Samtleben and M. Trigiante, Magnetic charges in local field theory, JHEP 09 (2005) 016 [hep-th/0507289] [INSPIRE].
B. de Wit and M. van Zalk, Electric and magnetic charges in N = 2 conformal supergravity theories, JHEP 10 (2011) 050 [arXiv:1107.3305] [INSPIRE].
S. Ferrara and S. Sabharwal, Quaternionic manifolds for Type II superstring vacua of Calabi-Yau spaces, Nucl. Phys. B 332 (1990) 317 [INSPIRE].
K. Hristov, H. Looyestijn and S. Vandoren, Maximally supersymmetric solutions of D = 4 N = 2 gauged supergravity, JHEP 11 (2009) 115 [arXiv:0909.1743] [INSPIRE].
J. Louis, P. Smyth and H. Triendl, Supersymmetric vacua in N = 2 supergravity, JHEP 08 (2012) 039 [arXiv:1204.3893] [INSPIRE].
S. de Alwis, J. Louis, L. McAllister, H. Triendl and A. Westphal, Moduli spaces in AdS 4 supergravity, JHEP 05 (2014) 102 [arXiv:1312.5659] [INSPIRE].
H. Erbin and N. Halmagyi, Abelian hypermultiplet gaugings and BPS vacua in \( \mathcal{N}=2 \) supergravity, JHEP 05 (2015) 122 [arXiv:1409.6310] [INSPIRE].
M. Shmakova, Calabi-Yau black holes, Phys. Rev. D 56 (1997) 540 [hep-th/9612076] [INSPIRE].
O. Aharony, M. Berkooz, J. Louis and A. Micu, Non-Abelian structures in compactifications of M-theory on seven-manifolds with SU(3) structure, JHEP 09 (2008) 108 [arXiv:0806.1051] [INSPIRE].
H. Looyestijn, E. Plauschinn and S. Vandoren, New potentials from Scherk-Schwarz reductions, JHEP 12 (2010) 016 [arXiv:1008.4286] [INSPIRE].
W. Ebeling, Monodromy, math/0507171.
G.G.M. Hamm and A. Helmut., Invarianten quasihomogener vollständiger durchschnitte., Inv. Math. 49 (1978) 67.
R. Randell, The Milnor number of some isolated complete intersection singularities with C ∗ -action, Proc. Amer. Math. Soc. 72 (1978) 375.
C. Arezzo, A. Ghigi and G. P. Pirola, Symmetries, quotients and Kähler-Einstein metrics, J. Reine Agew. Math. 2006 (2006) 177 [math/0402316] .
R. Dervan, On K-stability of finite covers, Bull. London Math. Soc. 48 (2016) 717 [arXiv:1505.07754].
H. Süß, Fano threefolds with 2-torus action — A picture book, arXiv:1308.2379.
G. Franchetti, Harmonic forms on ALF gravitational instantons, JHEP 12 (2014) 075 [arXiv:1410.2864] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1712.06608
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Katmadas, S., Tomasiello, A. Gauged supergravities from M-theory reductions. J. High Energ. Phys. 2018, 48 (2018). https://doi.org/10.1007/JHEP04(2018)048
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2018)048