Abstract
We characterize compact eight-manifolds M which arise as internal spaces in \( \mathcal{N}=1 \) flux compactifications of M-theory down to AdS3 using the theory of foliations, for the case when the internal part ξ of the supersymmetry generator is everywhere non-chiral. We prove that specifying such a supersymmetric background is equivalent with giving a codimension one foliation \( \mathrm{\mathcal{F}} \) of M which carries a leafwise G 2 structure, such that the O’Neill-Gray tensors, non-adapted part of the normal connection and the torsion classes of the G 2 structure are given in terms of the supergravity four-form field strength by explicit formulas which we derive. We discuss the topology of such foliations, showing that the C * algebra \( C\left(M/\mathrm{\mathcal{F}}\right) \) is a noncommutative torus of dimension given by the irrationality rank of a certain cohomology class constructed from G, which must satisfy the Latour obstruction. We also give a criterion in terms of this class for when such foliations are fibrations over the circle. When the criterion is not satisfied, each leaf of \( \mathrm{\mathcal{F}} \) is dense in M .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Becker, A Note on compactifications on spin(7)-holonomy manifolds, JHEP 05 (2001) 003 [hep-th/0011114] [INSPIRE].
M. Becker, D. Constantin, S.J. Gates Jr., I. Linch, William Divine, W. Merrell et al., M theory on spin(7) manifolds, fluxes and 3-D, N = 1 supergravity, Nucl. Phys. B 683 (2004) 67 [hep-th/0312040] [INSPIRE].
D. Constantin, Flux compactification of M-theory on compact manifolds with spin(7) holonomy, Fortsch. Phys. 53 (2005) 1272 [hep-th/0507104] [INSPIRE].
D. Martelli and J. Sparks, G structures, fluxes and calibrations in M-theory, Phys. Rev. D 68 (2003) 085014 [hep-th/0306225] [INSPIRE].
D. Tsimpis, M-theory on eight-manifolds revisited: N = 1 supersymmetry and generalized spin(7) structures, JHEP 04 (2006) 027 [hep-th/0511047] [INSPIRE].
F. Bonetti, T.W. Grimm and T.G. Pugh, Non-Supersymmetric F-theory Compactifications on spin(7) Manifolds, JHEP 01 (2014) 112 [arXiv:1307.5858] [INSPIRE].
M. Graña, C.S. Shahbazi and M. Zambon, Spin(7)-manifolds in compactifications to four dimensions, JHEP 11 (2014) 046 [arXiv:1405.3698] [INSPIRE].
M. Fernandez and A. Gray, Riemannian manifolds with structure group G 2, Ann. Mat. Pura Appl. 132 (1982) 19.
T. Friedrich and S. Ivanov, Parallel spinors and connections with skew symmetric torsion in string theory, Asian J. Math 6 (2002) 303 [math/0102142] [INSPIRE].
T. Friedrich and S. Ivanov, Killing spinor equations in dimension 7 and geometry of integrable G 2 manifolds, J. Geom. Phys. 48 (2003) 1 [math/0112201] [INSPIRE].
S. Grigorian, Short-time behaviour of a modified Laplacian coflow of G2-structures, Adv. Math. 248 (2013) 378.
C.-I. Lazaroiu, E.-M. Babalic and I.-A. Coman, Geometric algebra techniques in flux compactifications (I), arXiv:1212.6766 [INSPIRE].
C.-I. Lazaroiu and E.-M. Babalic, Geometric algebra techniques in flux compactifications (II), JHEP 06 (2013) 054 [arXiv:1212.6918] [INSPIRE].
C.I. Lazaroiu, E.M. Babalic and I.A. Coman, The geometric algebra of Fierz identities in arbitrary dimensions and signatures, JHEP 09 (2013) 156 [arXiv:1304.4403] [INSPIRE].
A. Candel and L. Conlon, Graduate Studies in Mathematics. Vol. 23: Foliations I, AMS Press, Providence U.S.A. (2000).
A. Candel and L. Conlon, Graduate Studies in Mathematics. Vol. 60: Foliations II, AMS Press, Providence U.S.A. (2003).
P. Tondeur, Monographs in Mathematics. Vol. 90: Geometry of foliations, Birkhauser, Boston U.S.A. (1997).
I. Moerdijk and J. Mrcun, Cambdridge Studies in Advanced Mathematics. Vol. 91: Introduction to Foliations and Lie Groupoids, Cambridge University Press, Cambridge U.K. (2003).
A. Bejancu and H.R. Farran, Mathematics and Its Applications. Vol. 580: Foliations and Geometric Structures, Springer, Heidelberg Germany (2005).
E. Cremmer, B. Julia and J. Scherk, Supergravity Theory in Eleven-Dimensions, Phys. Lett. B 76 (1978) 409 [INSPIRE].
D.V. Alekseevsky, V. Cortes, Classification of N -(super)-extended Poincare algebras and bilinear invariants of the spinor representation of Spin(p, q), Comm. Math. Phys. 183 (1997) 477.
D.V. Alekseevsky, V. Cortes, C. Devchand and A. Van Proeyen, Polyvector superPoincaré algebras, Commun. Math. Phys. 253 (2004) 385 [hep-th/0311107] [INSPIRE].
J.P. Gauntlett and S. Pakis, The Geometry of D = 11 Killing spinors, JHEP 04 (2003) 039 [hep-th/0212008] [INSPIRE].
I.A. Bandos, J.A. de Azcarraga, J.M. Izquierdo, M. Picón and O. Varela, On BPS preons, generalized holonomies and D = 11 supergravities, Phys. Rev. D 69 (2004) 105010 [hep-th/0312266] [INSPIRE].
I.A. Bandos, J.A. de Azcarraga, M. Picón and O. Varela, Generalized curvature and the equations of D = 11 supergravity, Phys. Lett. B 615 (2005) 127 [hep-th/0501007] [INSPIRE].
J. Bellorın and T. Ortín, A note on simple applications of the Killing Spinor Identities, Phys. Lett. B 616 (2005) 118 [hep-th/0501246] [INSPIRE].
M.J. Duff, J.T. Liu and R. Minasian, Eleven-dimensional origin of string-string duality: A One loop test, Nucl. Phys. B 452 (1995) 261 [hep-th/9506126] [INSPIRE].
C. Puhle, Spin (7)-Manifolds with Parallel Torsion Form, Commun. Math. Phys. 291 (2009) 303.
I. Agricola, The Srni lectures on non-integrable geometries with torsion, Arch. Math. 42 (2006) 5.
I. Agricola and T. Friedrich, Killing spinors in supergravity with 4 fluxes, Class. Quant. Grav. 20 (2003) 4707 [math/0307360] [INSPIRE].
C. Puhle, The Killing spinor equation with higher order potentials, J. Geom. Phys. 58 (2008) 1355 [arXiv:0707.2217] [INSPIRE].
C.J. Isham, C.N. Pope and N.P. Warner, Nowhere Vanishing Spinors and Triality Rotations in Eight Manifolds, Class. Quant. Grav. 5 (1988) 1297 [INSPIRE].
C.J. Isham and C.N. Pope, Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the Nsr and Gs Superstrings, Class. Quant. Grav. 5 (1988) 257 [INSPIRE].
S. Karigiannis, Deformations of G 2 and Spin(7) Structures on Manifolds, Canadian J. Math. 57 (2005) 1012.
T. Friedrich, I. Kath, A. Moroianu and U. Semmelmann, On nearly parallel G2-structures, J. Geom. Phys. 23 (1997) 256.
D.D. Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford U.K. (2000).
R.L. Bryant, Some remarks on G 2 structures, in Proceeding of Gokova Geometry-Topology Conference, Gökova Bay Turkey (2005), S. Akbulut, T. Onder and R.J. Stern eds., International Press, Boston U.S.A. (2006).
B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966) 459.
A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967) 715.
A.M. Naveira, A classification of Riemannian almost product manifolds, Rend. Mat. Appl. 3 (1983) 577.
F. Latour, Existence de l-formes fermées non singulières dans une classe de cohomologie de de Rham, Publ. Math. IHES 80 (1994) 135.
M. Farber, Mathematical Surveys and Monographs. Vol. 108: Topology of Closed One-Forms, AMS Press, Providence U.S.A. (2004).
K. Tod, All Metrics Admitting Supercovariantly Constant Spinors, Phys. Lett. B 121 (1983) 241 [INSPIRE].
F.T. Farrell, The obstruction to fibering a manifold over a circle, in Actes du Congres International des Mathematiciens. Tome 2, Nice France (1970), Gauthier-Villars, Paris France (1971), pg. 69.
L. Siebenmann, A total Whitehead torsion obstruction to fibering over the circle, Comment. Math. Helv. 45 (1970) 1.
D. Schütz, Finite domination, Novikov homology and nonsingular closed 1-forms, Math. Z. 252 (2006) 623.
D. Schütz, On the Whitehead group of Novikov rings associated to irrational homomorphisms, J. Pure Appl. Algebra 208 (2007) 449.
J.-Cl. Sikorav, Points fixes de diffeomorphismes symplectiques, intersections de sous-varietes lagrangiennes et singularites de un-formes fermees, Ph.D. Thesis, Université Paris-Sud, Paris France (1987).
A. Ranicki, Finite domination and Novikov rings, Topology 34 (1995) 619.
S. Karigiannis, Flows of G 2 -Structures, I, Quart. J. Math. 60 (2009) 487.
J.M. Lee, D. Lear, J. Roth, J. Coskey and L. Nave, Ricci — A Mathematica package for doing tensor calculations in differential geometry, available from http://www.math.washington.edu/ lee/Ricci/.
E.M. Babalic and C.I. Lazaroiu, Singular Foliations for M-theory compactifications, preprint.
G. Reeb, Sur la courbure moyenne des varietes integrales d’une equation de Pfaff w = 0, C.R. Acad. Sci. Paris 231 (1950) 101.
A. Haefliger, Varietes feuilletees, Ann. E.N.S. Pisa 19 (1962) 367.
R. Moussu, Feuilletage sans holonomie d’une veriete fermee, C.R. Acad. Sci. Paris 270 (1970) 1308.
D. Tischler, On fibering certain foliated manifolds over S 1, Topology 9 (1970) 153.
R. Sacksteder, Foliations and pseudo-groups, Am. J. Math. 87 (1965) 79.
S.P. Novikov, Topology of foliations, Trudy Mosk. Math. Obshch. 14 (1965) 248.
H. Imanishi, On the theorem of Denjoy-Sacksteder for codimension one foliations without holonomy, J. Math. Kyoto Univ. 14 (1974) 607.
V. Guillemin, E. Miranda and A.R. Pires, Codimension one symplectic foliations and regular Poisson structures, Bull. Braz. Math. Soc. (N.S.) 42 (2011) 607.
A. Connes, A survey of foliations and operator algebras, Operator algebras and applications. Part 1, Proc. Sympos. Pure Math. 38 (1982) 521.
A. Connes, Noncommutative Geometry, Academic Press, San Diego U.S.A. 1994.
M. Macho-Stadler, La conjecture de Baum-Connes pour un feuilletage sans holonomie de codimension un sur une variete fermee, Publ. Math. 33 (1989) 445.
T. Natsume, The C*-algebras of codimension one foliations without holonomy, Math. Scand. 56 (1985) 96.
L.G. Brown, P. Green and M.A. Rieffel, Stable isomorphism and strong Morita equivalence of C * -algebras, Pacific J. Math. 71 (1977) 349.
T. Natsume, Topological K-theory for codimension one foliations without holonomy, in Proceedings of a Symposium Held at the University of Tokyo, Tokyo Japan (1983), Adv. Stud. Pure Math. 5 (1985) 15.
E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].
K. Peeters, J. Plefka and S. Stern, Higher-derivative gauge field terms in the M-theory action, JHEP 08 (2005) 095 [hep-th/0507178] [INSPIRE].
Y. Hyakutake and S. Ogushi, Higher derivative corrections to eleven dimensional supergravity via local supersymmetry, JHEP 02 (2006) 068 [hep-th/0601092] [INSPIRE].
Y. Hyakutake, Toward the Determination of R 3 F 2 Terms in M-theory, Prog. Theor. Phys. 118 (2007) 109 [hep-th/0703154] [INSPIRE].
H. Looyestijn, E. Plauschinn and S. Vandoren, New potentials from Scherk-Schwarz reductions, JHEP 12 (2010) 016 [arXiv:1008.4286] [INSPIRE].
V. Rowenski and P. Walczak, Topics in Extrinsic Geometry of Codimension-One Foliations, Springer briefs in Mathematics, Springer, Heidelberg Germany (2011).
P. Bouwknegt, K. Hannabuss and V. Mathai, Nonassociative tori and applications to T-duality, Commun. Math. Phys. 264 (2006) 41 [hep-th/0412092] [INSPIRE].
E. Plauschinn, T-duality revisited, JHEP 01 (2014) 131 [arXiv:1310.4194] [INSPIRE].
D. Andriot, M. Larfors, D. Lüst and P. Patalong, (Non-)commutative closed string on T-dual toroidal backgrounds, JHEP 06 (2013) 021 [arXiv:1211.6437] [INSPIRE].
I. Bakas and D. Lüst, 3-Cocycles, Non-Associative Star-Products and the Magnetic Paradigm of R-Flux String Vacua, JHEP 01 (2014) 171 [arXiv:1309.3172] [INSPIRE].
R.L. Bryant, Metrics with Exceptional Holonomy, Ann. Math. 126 (1987) 525.
S. Grigorian and S.-T. Yau, Local geometry of the G 2 moduli space, Commun. Math. Phys. 287 (2009) 459 [arXiv:0802.0723] [INSPIRE].
S. Grigorian, Deformations of G2-structures with torsion, arXiv:1108.2465 [INSPIRE].
O. Gil-Medrano, Geometric Properties of some Classes of Riemannian Almost Product Manifolds, Rend. Circ. Mat. Palermo (2) XXXII (1983) 315.
V. Miquel, Some examples of Riemannian Almost Product Manifolds, Pacific J. Math. 111 (1984) 163.
A. Montesinos, On Certain Classes of Almost Product Structures, Michigan Math. J. 30 (1983) 31.
M. Spivak, A comprehensive introduction to differential geometry, third edition, Publish or Perish, Houston U.S.A. (1999).
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: 1411.3148
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Babalic, E.M., Lazaroiu, C.I. Foliated eight-manifolds for M-theory compactification. J. High Energ. Phys. 2015, 140 (2015). https://doi.org/10.1007/JHEP01(2015)140
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2015)140