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Local Geometry of the G 2 Moduli Space

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Abstract

We consider deformations of torsion-free G 2 structures, defined by the G 2-invariant 3-form φ and compute the expansion of \({\ast \varphi }\) to fourth order in the deformations of φ. By considering M-theory compactified on a G 2 manifold, the G 2 moduli space is naturally complexified, and we get a Kähler metric on it. Using the expansion of \({\ast \varphi }\), we work out the full curvature of this metric and relate it to the Yukawa coupling.

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References

  1. Candelas P., de la Ossa X.: Moduli space of Calabi-Yau manifolds. Nucl. Phys. B355, 455–481 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  2. Strominger A.: Special geometry. Commun. Math. Phys. 133, 163–180 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Yau S.-T.: On the Ricci curvature of a compact Kaehler manifold and the complex monge-ampère equation. I. Comm. Pure Appl. Math. 31, 339–411 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Joyce, D.D.: Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford: Oxford University Press, 2000

  5. Kovalev, A.: Twisted connected sums and special Riemannian holonomy. math/0012189

  6. Gibbons G.W., Page D.N., Pope C.N.: Einstein metrics on S 3, R 3 and R 3 bundles. Commun. Math. Phys. 127, 529 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Bryant, R.L.: Some remarks on G_2-structures. In: Proc. of GoKovaGeometry - Topology Conf. (2005), S. Akbulut, T. Onder, R.J. Slem, Somerville, MA: Intle Press, 2006, pp 95–109

  8. de Boer, J., Naqvi, A., Shomer, A.: The topological G(2) string. http://arXiv.org/abs/hep-th/0506211, 2005

  9. Karigiannis, S., Leung, N.C.: Hodge theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps. http://arXiv.org/abs/0709.2987, 2007

  10. Karigiannis S.: Deformations of G_2 and Spin(7) Structures on Manifolds. Canadian J. Math. 57, 1012 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Fernández M., Gray A.: Riemannian manifolds with structure group G 2. Ann. Mat. Pura Appl. (4) 132, 19–45 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Berger, M.: Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83 (1955)

  13. Karigiannis, S.: Geometric Flows on Manifolds with G_2 Structure, I. http://arXiv.org/list/math/0702077, 2007

  14. House T., Micu A.: M-theory compactifications on manifolds with G(2) structure. Class. Quant. Grav. 22, 1709–1738 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Acharya B.S., Gukov S.: M theory and Singularities of Exceptional Holonomy Manifolds. Phys. Rept. 392, 121–189 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  16. Cremmer E., Julia B., Scherk J.: Supergravity theory in 11 dimensions. Phys. Lett. B76, 409 (1978)

    ADS  Google Scholar 

  17. Witten E.: On flux quantization in M-theory and the effective action. J. Geom. Phys. 22, 1–13 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Harvey, J.A., Moore, G.W.: Superpotentials and membrane instantons. http://arXiv.org/list/hep-th/9907026, 1999

  19. Beasley C., Witten E.: A note on fluxes and superpotentials in M-theory compactifications on manifolds of G(2) holonomy. JHEP 07, 046 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  20. Gutowski J., Papadopoulos G.: Moduli spaces and brane solitons for M theory compactifications on holonomy G(2) manifolds. Nucl. Phys. B615, 237–265 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  21. Papadopoulos G., Townsend P.K.: Compactification of d = 11 supergravity on spaces of exceptional holonomy. Phys. Lett. B357, 300–306 (1995)

    ADS  MathSciNet  Google Scholar 

  22. Portugal R.: The Riegeom package: abstract tensor calculation. Comput. Phys. Commun. 126, 261–268 (2002)

    Article  Google Scholar 

  23. Hitchin, N.J.: The geometry of three-forms in six and seven dimensions. http://arXiv.org/list/math/0010054, 2000

  24. Karigiannis, S., Lin, C.: Curvature of the moduli space of G_2-manifolds. In preparation.

  25. Lee J.-H., Leung, N.C.: Geometric structures on G(2) and Spin(7)-manifolds. http://arXiv.org/list/math/0202045, 2002

  26. Gukov, S., Yau, S.-T., Zaslow, E.: Duality and fibrations on G(2) manifolds. http://arXiv.org/list/hep-th/0203217, 2002

  27. Aganagic M., Klemm A., Vafa C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A57, 1–28 (2002)

    MathSciNet  Google Scholar 

  28. Strominger A., Yau S.-T., Zaslow E.: Mirror symmetry is T-duality. Nucl. Phys. B479, 243–259 (1996)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom

    Sergey Grigorian

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Shing-Tung Yau

Authors
  1. Sergey Grigorian
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  2. Shing-Tung Yau
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Correspondence to Sergey Grigorian.

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Communicated by G.W. Gibbons

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Grigorian, S., Yau, ST. Local Geometry of the G 2 Moduli Space. Commun. Math. Phys. 287, 459–488 (2009). https://doi.org/10.1007/s00220-008-0595-1

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  • DOI: https://doi.org/10.1007/s00220-008-0595-1

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