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MODULI SPACES OF G₂ MANIFOLDS

SERGEY GRIGORIAN

Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, D-14476 Golm, Germany

Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794, USA

https://doi.org/10.1142/S0129055X10004132Cited by:15 (Source: Crossref)

Abstract

This paper is a review of current developments in the study of moduli spaces of G₂ manifolds. G₂ manifolds are seven-dimensional manifolds with the exceptional holonomy group G₂. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi–Yau manifolds in seven dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M-theory compactifications. Despite the physical motivation, many of the results are of purely mathematical interest. Here we cover the basics of G₂ manifolds, local deformation theory of G₂ structures and the local geometry of the moduli spaces of G₂ structures.

Keywords:

AMSC: 53C25, 53C29, 53Z05

References

B. Acharya and E. Witten, Chiral fermions from manifolds of G(2) holonomy , arXiv:hep-th/0109152 . Google Scholar
B. S. Acharya and S. Gukov, Phys. Rept. 392, 121 (2004), arXiv:hep-th/0409191 DOI: 10.1016/j.physrep.2003.10.017. Crossref, Web of Science, ADS, Google Scholar
M. Aganagic, A. Klemm and C. Vafa, Z. Naturforsch. A 57, 1 (2002), arXiv:hep-th/0105045. Crossref, Web of Science, ADS, Google Scholar
D. V. Alekseevsky and V. Cortes, Geometric construction of the r-map: From affine special real to special Kähler manifolds , arXiv:0811.1658 . Google Scholar
W. Ambrose and I. M. Singer, Trans. Amer. Math. Soc. 75, 428 (1953). Crossref, Web of Science, Google Scholar
J. Baez, Bull. Amer. Math. Soc. (N.S.) 39, 145 (2002), DOI: 10.1090/S0273-0979-01-00934-X. Crossref, Web of Science, Google Scholar
C. Beasley and E. Witten, JHEP 07, (2002), arXiv:hep-th/0203061. Google Scholar
K. Becker , M. Becker and J. H. Schwarz , String Theory and M-Theory: A Modern Introduction ( Cambridge University Press , 2007 ) . Google Scholar
M. Berger, Bull. Soc. Math. France 83, 279 (1955). Google Scholar
E. Bonan, C. R. Acad. Sci. Paris 262, 127 (1966). Google Scholar
R. Bryant and S. Salamon, Duke Math. J. 58, 829 (1989), DOI: 10.1215/S0012-7094-89-05839-0. Crossref, Web of Science, Google Scholar
R. L. Bryant, Ann. of Math. (2) 126(3), 525 (1987), DOI: 10.2307/1971360. Crossref, Web of Science, Google Scholar
R. L. Bryant, Some remarks on G_2-structures , arXiv:math.DG/0305124 . Google Scholar
P. Candelas and X. de la Ossa, Nucl. Phys. B 355, 455 (1991), DOI: 10.1016/0550-3213(91)90122-E. Crossref, Web of Science, ADS, Google Scholar
M. Cveticet al., Phys. Rev. Lett. 88, 4 (2002), arXiv:hep-th/0112098 DOI: 10.1103/PhysRevLett.88.121602. Google Scholar
J. de Boer, A. Naqvi and A. Shomer, The topological , arXiv:hep-th/0506211 . Google Scholar
M. J. Duff, M-theory on manifolds of G(2) holonomy: The first twenty years , arXiv:hep-th/0201062 . Google Scholar
M. Fernández and A. Gray, Ann. Mat. Pura Appl. (4) 132, 19 (1982). Crossref, Web of Science, Google Scholar
D. S. Freed, Comm. Math. Phys. 203, 31 (1999), arXiv:hep-th/9712042 DOI: 10.1007/s002200050604. Crossref, Web of Science, ADS, Google Scholar
S. Grigorian, Betti numbers of a class of barely G2 manifolds , arXiv:0909.4681 . Google Scholar
S. Grigorian and S.-T. Yau, Comm. Math. Phys. 287, 459 (2009), arXiv:0802.0723 DOI: 10.1007/s00220-008-0595-1. Crossref, Web of Science, ADS, Google Scholar
S. Gukov, S.-T. Yau and E. Zaslow, Duality and fibrations on G(2) manifolds , arXiv:hep-th/0203217 . Google Scholar
J. Gutowski and G. Papadopoulos, Nucl. Phys. B 615, 237 (2001), arXiv:hep-th/0104105 DOI: 10.1016/S0550-3213(01)00419-9. Crossref, Web of Science, ADS, Google Scholar
F. R. Harvey , Spinors and Calibrations ( Academic Press , 1990 ) . Google Scholar
J. A. Harvey and G. W. Moore, Superpotentials and membrane instantons , arXiv:hep-th/9907026 . Google Scholar
N. J. Hitchin, The geometry of three-forms in six and seven dimensions , arXiv:math.DG/0010054 . Google Scholar
K. Hori et al. , Mirror Symmetry ( Amer. Math. Soc. , 2003 ) . Google Scholar
D. D. Joyce, J. Differential Geom. 43, 291 (1996). Crossref, Web of Science, Google Scholar
D. D. Joyce, J. Differential Geom. 43, 329 (1996). Crossref, Web of Science, Google Scholar
D. D. Joyce , Compact Manifolds with Special Holonomy , Oxford Mathematical Monographs ( Oxford University Press , 2000 ) . Crossref, Google Scholar
S. Karigiannis, Canad. J. Math. 57, 1012 (2005), arXiv:math.DG/0301218. Crossref, Web of Science, Google Scholar
S. Karigiannis, Geometric flows on manifolds with G_2 structure, I , arXiv:math.DG/0702077 . Google Scholar
S. Karigiannis, Geom. Topol. 13(3), 1583 (2009), arXiv:0807.3346 DOI: 10.2140/gt.2009.13.1583. Crossref, Web of Science, Google Scholar
S. Karigiannis and N. C. Leung, Hodge theory for G2-manifolds: Intermediate Jacobians and Abel–Jacobi maps , arXiv:0709.2987 . Google Scholar
A. Kovalev, Twisted connected sums and special Riemannian holonomy , arXiv:math.DG/0012189 . Google Scholar
A. Kovalev and J. Nordström, Asymptotically cylindrical 7-manifolds of holonomy G2 with applications to compact irreducible G2-manifolds , arXiv:0907.0497 . Google Scholar
J.-H. Lee and N. C. Leung, Geometric structures on G(2) and Spin(7)-manifolds , arXiv:math.DG/0202045 . Google Scholar
J. Nordström, Math. Proc. Cambridge Philos. Soc. 145(2), 311 (2008), arXiv:0705.4444. Crossref, Web of Science, Google Scholar
H. Shima , The Geometry of Hessian Structures ( World Scientific Publishing , 2007 ) . Link, Google Scholar
A. Strominger, Comm. Math. Phys. 133, 163 (1990), DOI: 10.1007/BF02096559. Crossref, Web of Science, ADS, Google Scholar
A. Strominger, S.-T. Yau and E. Zaslow, Nucl. Phys. B 479, 243 (1996), arXiv:hep-th/9606040 DOI: 10.1016/0550-3213(96)00434-8. Crossref, Web of Science, ADS, Google Scholar
G. Tian, Mathematical Aspects of String Theory, Adv. Ser. Math. Phys. 1 (World Sci. Publishing, 1987) pp. 629–646. Link, Google Scholar
A. Todorov, Comm. Math. Phys. 126, 325 (1989), DOI: 10.1007/BF02125128. Crossref, Web of Science, ADS, Google Scholar
P. K. Townsend, Phys. Lett. B 350, 184 (1995), arXiv:hep-th/9501068. Crossref, Web of Science, ADS, Google Scholar
P. C. West , Introduction to Supersymmetry and Supergravity ( World Scientific Publishing , Singapore , 1990 ) . Link, Google Scholar
E. Witten, Nucl. Phys. B 443, 85 (1995), arXiv:hep-th/9503124 DOI: 10.1016/0550-3213(95)00158-O. Crossref, Web of Science, ADS, Google Scholar