String Theory Compactifications

  • Mariana GrañaEmail author
  • Hagen Triendl
Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)


String theory is consistently defined in ten dimensions, six of which should be curled up in some small internal compact manifold. The procedure of linking this manifold to four-dimensional physics is called string compactification, and in these lectures we will review it quite extensively. We will start with a very brief introduction to string theory, in particular we will work out its massless spectrum and show how the condition on the number of dimensions arises. We will then dwell on the different possible internal manifolds, starting from the simplest to the most relevant phenomenologically. We will show that these are most elegantly described by an extension of ordinary Riemannian geometry termed generalized geometry, first introduced by Hitchin. We shall finish by discussing (partially) open problems in string phenomenology, such as the embedding of the Standard Model and obtaining de Sitter solutions.


  1. 1.
    M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, vol. I, II (Cambridge University Press, 1987)Google Scholar
  2. 2.
    J. Polchinski, String Theory, vol. I, II (Cambridge University Press, 1998)Google Scholar
  3. 3.
    C.V. Johnson, D-Branes (Cambridge University Press, 2003). C.V. Johnson, D-Brane Primer. arXiv:hep-th/0007170
  4. 4.
    B. Zwiebach, A First Course in String Theory (Cambridge University Press, 2004)Google Scholar
  5. 5.
    K. Becker, M. Becker, J.H. Schwarz, String Theory and M-Theory Google Scholar
  6. 6.
    N. Hitchin, Generalized Calabi–Yau manifolds. arXiv:math.DG/0209099
  7. 7.
    M. Gualtieri, Generalized Complex Geometry, DPhil thesis (Oxford University, 2004). arXiv:math.DG/0401221
  8. 8.
    F. Witt, Special metric structures and closed forms, DPhil thesis (Oxford University, 2004). arXiv:math.DG/0502443
  9. 9.
    M. Grana, Flux compactifications in string theory: a comprehensive review. Phys. Rep. 423, 91 (2006). arXiv:hep-th/0509003 ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    M.R. Douglas, S. Kachru, Flux compactification. Rev. Mod. Phys. 79, 733 (2007). arXiv:hep-th/0610102. R. Blumenhagen, B. Kors, D. Lust, S. Stieberger, Four-dimensional string compactifications with D-branes, orientifolds and fluxes. Phys. Rept. 445, 1 (2007). arXiv:hep-th/0610327. H. Samtleben, Lectures on gauged supergravity and flux compactifications. Class. Quant. Grav. 25, 214002 (2008). arXiv:0808.4076 [hep-th]. P. Koerber, Lectures on generalized complex geometry for physicists. Fortsch. Phys. 59, 169 (2011). arXiv:1006.1536 [hep-th]
  11. 11.
    C.G. Callan Jr., E.J. Martinec, M.J. Perry, D. Friedan, Strings in background fields. Nucl. Phys. B 262, 593 (1985)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    A.M. Polyakov, Quantum geometry of bosonic strings. Phys. Lett. B 103, 207 (1981)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Friedan, E.J. Martinec, S.H. Shenker, Conformal invariance, supersymmetry and string theory. Nucl. Phys. B 271, 93 (1986)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Chodos, C.B. Thorn, Making the massless string massive. Nucl. Phys. B 72, 509 (1974). R.C. Myers, New dimensions for old strings. Phys. Lett. B 199, 371 (1987)Google Scholar
  15. 15.
    J. Polchinski, Dirichlet branes and ramond-ramond charges. Phys. Rev. Lett. 75, 4724 (1995). arXiv:hep-th/9510017 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    F. Gliozzi, J. Scherk, D.I. Olive, Supersymmetry, supergravity theories and the dual spinor model. Nucl. Phys. B 122, 253 (1977)ADSCrossRefGoogle Scholar
  17. 17.
    A. Giveon, M. Porrati, E. Rabinovici, Target space duality in string theory. Phys. Rept. 244, 77 (1994). arXiv:hep-th/9401139 ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Lust, Intersecting brane worlds: a path to the standard model? Class. Quant. Grav. 21, S1399 (2004). arXiv:hep-th/0401156 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P. Candelas, Introduction to Complex Manifolds (Lectures at the 1987 Trieste Spring School)Google Scholar
  20. 20.
    P. Candelas, X. de la Ossa, Moduli space of Calabi-Yau manifolds. Nucl. Phys. B 355, 455 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    S. Salamon, Riemannian Geometry and Holonomy Groups. Pitman Research Notes in Mathematics, vol. 201 (Longman, Harlow, 1989). D. Joyce, Compact Manifolds with Special Holonomy (Oxford University Press, Oxford, 2000)Google Scholar
  22. 22.
    M. Graña, R. Minasian, M. Petrini, A. Tomasiello, Supersymmetric backgrounds from generalized Calabi-Yau manifolds. JHEP 0408, 046 (2004). arXiv:hep-th/0406137 ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Grãna, R. Minasian, M. Petrini, A. Tomasiello, Generalized structures of N = 1 vacua. arXiv:hep-th/0505212
  24. 24.
    U. Lindstrom, R. Minasian, A. Tomasiello, M. Zabzine, Generalized complex manifolds and supersymmetry. Commun. Math. Phys. 257, 235 (2005). arXiv:hep-th/0405085. U. Lindstrom, M. Rocek, R. von Unge, M. Zabzine, Generalized Kaehler geometry and manifest N = (2, 2) supersymmetric nonlinear sigma-models. arXiv:hep-th/0411186. M. Zabzine, Hamiltonian perspective on generalized complex structure. arXiv:hep-th/0502137
  25. 25.
    C. Jeschek, F. Witt, Generalised G(2)-structures and type IIB superstrings. JHEP 0503, 053 (2005). arXiv:arXiv:hep-th/0412280 ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    S. Kachru, M.B. Schulz, S. Trivedi, Moduli stabilization from fluxes in a simple IIB orientifold. JHEP 0310, 007 (2003). arXiv:hep-th/0201028 ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    S. Kachru, R. Kallosh, A. Linde, S.P. Trivedi, De Sitter vacua in string theory. Phys. Rev. D 68, 046005 (2003). arXiv:hep-th/0301240 ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Institut de Physique ThéoriqueCEA/SaclayGif-sur-Yvette CedexFrance
  2. 2.Imperial College LondonLondonUK

Personalised recommendations