Skip to main content

Introduction to String Compactification

  • Chapter
  • First Online:
Book cover Geometric and Topological Methods for Quantum Field Theory

Part of the book series: Lecture Notes in Physics ((LNP,volume 668))

Abstract

We present an elementary introduction to compactifications with unbroken supersymmetry. After explaining how this requirement leads to internal spaces of special holonomy we describe Calabi-Yau manifolds in detail. We also discuss orbifolds as examples of solvable string compactifications.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. Berkovits, ICTP lectures on covariant quantization of the superstring, hep-th/0209059.

    Google Scholar 

  2. P. Candelas, G. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B256 (1985) 46–74.

    Article  Google Scholar 

  3. L. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B261 (1985) 678–686; Strings on Orbifolds II, Nucl. Phys. B274 (1986) 285–314.

    Google Scholar 

  4. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vols. I and II, Cambridge University Press, 1987.

    Google Scholar 

  5. D. Lüst and S. Theisen, Lectures On String Theory, Lect. Notes Phys. 346, Springer-Verlag, 1989.

    Google Scholar 

  6. J. Polchinski, String Theory, Vols. I and II, Cambridge University Press, 1998.

    Google Scholar 

  7. tt http://www.aei.mpg.de/~theisen.

    Google Scholar 

  8. M. Duff, B.E.W. Nilsson and C. Pope, Kaluza-Klein Supergravity Phys. Rept. 130 (1986) 1–142.

    Article  Google Scholar 

  9. J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, 1992; J. D. Lykken, Introduction to supersymmetry, hep-th/9612114.

    Google Scholar 

  10. N. Polonsky, Supersymmetry: Structure and phenomena. Extensions of the standard model, Lect. Notes Phys. M68, Springer-Verlag, 2001 (hep-ph/0108236).

    Google Scholar 

  11. V. A. Rubakov and M. E. Shaposhnikov, Extra Space-Time Dimensions: Towards a Solution to the Cosmological Constant Problem}, Phys. Lett. B125 (1983) 139–143.

    Google Scholar 

  12. A.L. Besse, Einstein Manifolds, Springer-Verlag, 1987.

    Google Scholar 

  13. D.D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.

    Google Scholar 

  14. P. Candelas, A. M. Dale, C. A. Lütken and R. Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B298 (988) 493–525; P. Candelas, M. Lynker and R. Schimmrigk, Calabi-Yau Manifolds in Weighted P4, Nucl. Phys. B341 (1990) 383–402; A. Klemm and R. Schimmrigk, Landau-Ginzburg string vacua, Nucl. Phys. B411 (1994) 559–583, hep-th/9204060; M. Kreuzer and H. Skarke, No mirror symmetry in Landau-Ginzburg spectra!, Nucl. Phys. B388 (1992) 113–130, hep-th/9205004; see also http://hep.itp.tuwien.ac.at/ kreuzer/CY/.

    Google Scholar 

  15. C. P. Burgess, A. Font and F. Quevedo, Low-Energy Effective Action for the Superstring, Nucl. Phys. B272 (1986) 661; A. Font and F. Quevedo, N=1 Supersymmetric Truncations and the Superstring Low-Energy Effective Theory, Phys. Lett. B184 (1987) 45–48.

    Google Scholar 

  16. R.C. Gunning, Lectures on Riemann Surfaces. Mathematical Notes, Princeton Univ. Press, Princeton 1967;M. Nakahara, Geometry, Topology and Physics, Adam Hilger, 1990.

    Google Scholar 

  17. A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77–202, hep-th/9401139.

    Article  Google Scholar 

  18. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, Mirror Symmetry, Clay Mathematics Monographs, Vol. 1, AMS 2003.

    Google Scholar 

  19. G. Horowitz, What is a Calabi-Yau Space?, in Proceedings of the workshop on Unified String Theories, Santa Barbara 1985, M. Green and D. Gross, eds.

    Google Scholar 

  20. T. Hübsch, Calabi-Yau Manifolds, World-Scientific, 1992.

    Google Scholar 

  21. S. Hosono, A. Klemm and S. Theisen,Lectures on Mirror Symmetry, hep-th/9403096.

    Google Scholar 

  22. B. Greene, String theory on Calabi-Yau manifolds, hep-th/9702155.

    Google Scholar 

  23. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley 1978.

    Google Scholar 

  24. S. Chern, Complex Manifolds without Potential Theory, Springer-Verlag, 1979.

    Google Scholar 

  25. R.O. Wells, Differential Analysis on Complex Manifolds, Springer-Verlag, 1979.

    Google Scholar 

  26. K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer-Verlag, 1986.

    Google Scholar 

  27. G. Tian, Canonical Metrics in Kähler Goemetry, Birkhäuser, 2000.

    Google Scholar 

  28. P. Candelas, Lectures on Complex Manifolds, published in Superstrings’ 87, Proceedings of the 1987 ICTP Spring School, pp. 1–88.

    Google Scholar 

  29. A. Dimca, Singularities and Topology of Hypersurfaces, Springer-Verlag, 1992; I. Dolgachev, Weighted Projective Varieties, in “Group Actions and Vector Fields, Proceedings 1981”, LNM 959, Springer-Verlag, Berlin, 1982, pp 34–71.

    Google Scholar 

  30. M. Reid, Young Persons Guide to Canonical Singularities, Proc. Sym. Pure Math 46, AMS, 1987, 345–414; Canonical 3-Folds, in “Journées de géometrique algébrique d’Angers”, (A. Beauville, ed.) 1990, 273–310.

    Google Scholar 

  31. W. Fulton, Introduction to Toric Varieties, Princeton University Press, 1993.

    Google Scholar 

  32. D.A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, Vol. 68, AMS, 1999.

    Google Scholar 

  33. P. Candelas and X. C. de la Ossa, Comments on Conifolds, Nucl. Phys. B342 (1990) 246–268.

    Article  CAS  Google Scholar 

  34. P.S. Howe, G. Papadopoulos and K.S. Stelle, Quantizing The N=2 Super Sigma Model In Two-Dimensions, Phys. Lett. B176 (1986) 405–410.

    Google Scholar 

  35. A. Lichnerowicz, Global theory of connections and holonomy groups, Noordhoff International Publishing, 1976.

    Google Scholar 

  36. P. Griffiths, On the periods of certain rational integrals I,II, Ass. of Math. 90 (1969) 460–495, 498–541.

    Google Scholar 

  37. D. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces in Mirror Symmetry I (s.-T. Yau, ed.), AMS and International Press, 1998,p. 185–199, alg-geom/9202026.

    Google Scholar 

  38. P. Candelas, Yukawa couplings between (2,1)-forms, Nucl. Phys. B298 (1988) 458–492.

    Article  CAS  Google Scholar 

  39. P. Mayr,Mirror Symmetry, N = 1 Superpotentials and Tensionless Strings on Calabi-Yau Fourfolds, Nucl. Phys. B494 (1997) 489–545, hep-th/9610162; A. Klemm, B. Lian, S.S. Roan and S.T. Yau, Calabi-Yau Fourfolds for M-Theory and F-Theory Compactifications, Nucl. Phys. B518 (1998) 515–574, hep-th/9701023.

    Google Scholar 

  40. V. Batyrev, Variations of the Mixed Hodge Structure of Affine Hypersurfaces in Algebraic Tori}, Duke Math. J. 69 (1993) 349–409; Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties}, J. Algebraic Geometry 3 (1994) 493–535, alg-geom/9310003; Quantum Cohomology Rings of Toric Manifolds, Astérisque 218 (1993) 9–34, alg-geom/9310004.

    Google Scholar 

  41. P. Candelas, E. Derrick and L. Parkes, Generalized Calabi-Yau manifolds and the mirror of a rigid manifold, Nucl. Phys. B407 (1993) 115–154, hep-th/9304045.

    Article  Google Scholar 

  42. P. Candelas and X. de la Ossa, Moduli Space of Calabi-Yau Manifolds, Nucl. Phys. B355 (1991) 455–481.

    Article  Google Scholar 

  43. S. Ferrara and S. Theisen, Moduli Spaces, Effective Actions and Duality Symmetry in String Compactification}, Proceedings of the 3rd Hellenic School on Elementary Particle Physics, Corfu 1989 (E.N. Argyres, N. Tracas and G. Zoupanos, eds.), p. 620–656.

    Google Scholar 

  44. P. S. Aspinwall, K3 surfaces and string duality, hep-th/9611137.

    Google Scholar 

  45. N. Barth and S. Christensen, Quantizing Fourth Order Gravity Theories. 1. The Functional Integral, Phys. Rev. D28 (1983) 1876–1893; G.’t Hooft and M. Veltman, One Loop Divergencies In The Theory Of Gravitation, Annales Poincaré Phys. Theor. A20 (1974) 69–94.

    Google Scholar 

  46. G. Tian, Smoothness of the Universal Deformation Space of Compact Calabi-Yau Manifolds and its Peterson-Weil Metric}, in S.T. Yau (ed), Mathematical Aspects of String Theory, World Scientific, 1987, p. 629–647.

    Google Scholar 

  47. A. Todorov. The Weyl-Petersson Geometry of the Moduli-Space of SU(n⪚3) (Calabi-Yau) Manifolds I}, Commun. Math. Phys. 126 (1989) 325–346.

    Google Scholar 

  48. P. Candelas, P. S. Green and T. Hübsch, Rolling Among Calabi-Yau Vacua, Nucl. Phys. B330 (1990) 49–102.

    Article  CAS  Google Scholar 

  49. M. Dine, P. Huet and N. Seiberg, Large and Small Radius in String Theory, Nucl. Phys. B322 (1989) 301–316; J. Dai, R.G. Leigh and J. Polchinski, New Connections between String Theories, Mod. Phys. Lett. A4 (1989).

    Google Scholar 

  50. N. Seiberg, Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys. B303 (1988) 286–304.

    Article  CAS  Google Scholar 

  51. M. Bodner, A. C. Cadavid and S. Ferrara, (2,2) Vacuum Configurations for Type IIA Superstrings: N=2 Supergravity Lagrangians and Algebraic Geometry}, Class. Quantum Grav. 8 (1991) 789–808.

    Article  CAS  Google Scholar 

  52. D. Morrison, Mirror Symmetry and the Type II String, Nucl.Phys.Proc.Suppl. 46 (1996) 146–155, hep-th/9512016.

    Article  CAS  Google Scholar 

  53. B. de Wit and A. Van Proeyen, Potentials and Symmetries of General Gauged N=2 Supergravity - Yang-Mills Models}, Nucl. Phys. B245 (1984) 89–117; B. de Wit, P. G. Lauwers and A. Van Proeyen, Lagrangians Of N=2 Supergravity – Matter Systems, Nucl. Phys. B255 (1985) 569–608; E. Cremmer, C. Kounnas, A. Van Proeyen, J. P. Derendinger, S. Ferrara, B. de Wit and L. Girardello, Vector Multiplets Coupled To N=2 Supergravity: Superhiggs Effect, Flat Potentials and Geometric Structure}, Nucl. Phys. B250 (1985) 385–426.

    Google Scholar 

  54. J. Bagger and E. Witten, Matter Couplings In N=2 Supergravity, Nucl. Phys. B222 (1983) 1–10.

    Article  Google Scholar 

  55. S. Cecotti, S. Ferrara and L. Girardello, Geometry of Type II Superstrings and the Moduli of Superconformal Field Theories}, Int. J. Mod. Phys. A4 (1989) 2475–2529.

    Google Scholar 

  56. P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, A Pair of Calabi-Yau Manifolds as an Exactly Soluble Superconformal Theory, Nucl. Phys. B359 (1991) 21–74.

    Article  Google Scholar 

  57. P. Candelas, T. Hübsch and R. Schimmrigk, Relation Between the Weil-Petersson and Zamolodchikov Metrics, Nucl. Phys. B329 (1990) 583–590.

    Article  CAS  Google Scholar 

  58. L. J. Dixon, V. Kaplunovsky and J. Louis, On Effective Field Theories Describing (2,2) Vacua Of The Heterotic String, Nucl. Phys. B329 (1990) 27–82.

    Google Scholar 

  59. I. Antoniadis, E. Gava, K. S. Narain and T. R. Taylor, Superstring threshold corrections to Yukawa couplings, Nucl. Phys. B407 (1993) 706–724, hep-th/9212045; J. P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, On loop corrections to string effective field theories: Field dependent gauge couplings and sigma model anomalies}, Nucl. Phys. B372 (1992) 145–188; G. Lopes Cardoso and B. A. Ovrut, Coordinate and Kähler Sigma Model Anomalies and their Cancellation in String Effective Field Theories}, Nucl. Phys. B392 (1993) 315–344, hep-th/9205009.

    Google Scholar 

  60. R. Bryant and P. Griffiths, Some Observations on the Infinitesimal Period Relations for Regular Threefolds with Trivial Canonical Bundle}, in Progress in Mathemtaics 36, (M. Artin and J. Tate, eds.), Birkhäuser 1983.

    Google Scholar 

  61. A. Strominger, Special Geometry, Commun. Math. Phys. 133 (1990) 163–180.

    Google Scholar 

  62. B. Craps, F. Roose, W. Troost and A. Van Proeyen, The definitions of special geometry, hep-th/9606073 and What is special Kaehler geometry?, Nucl. Phys. B503 (1997) 565–613, hep-th/9703082.

    Article  Google Scholar 

  63. P. Mayr, On supersymmetry breaking in string theory and its realization in brane worlds, Nucl. Phys. B593 (2001) 99–126, hep-th/0003198.

    Article  CAS  Google Scholar 

  64. I. Antoniadis, S. Ferrara, R. Minasian and K. S. Narain, R4 couplings in M- and type II theories on Calabi-Yau spaces, Nucl. Phys. B507 (1997) 571–588, hep-th/9707013.

    Article  CAS  Google Scholar 

  65. I. Antoniadis, R. Minasian, S. Theisen and P. Vanhove, String loop corrections to the universal hypermultiplet, Class. Quant. Grav. 20 (2003) 5079–5102, hep-th/0307268.

    Article  Google Scholar 

  66. L. J. Dixon, Some World Sheet Properties of Superstring Compactifications, on Orbifolds and Otherwise}, published in Superstrings, Unified Theories and Cosmology 1987, Proceedings of the 1987 ICTP Summer Workshop, pp. 67–126.

    Google Scholar 

  67. J. Erler and A. Klemm, Comment on the Generation Number in Orbifold Compactifications, Commun. Math. Phys. 153 (1993) 579–604, hep-th/9207111.

    Google Scholar 

  68. D. G. Markushevich, M. A. Olshanetsky and A. M. Perelomov, Description of a Class of Superstring Compactifications Related to Semisimple Lie Algebras}, Commun. Math. Phys. 111 (1987) 247–274.

    Article  Google Scholar 

  69. D.N. Page, A Physical Picture of the K3 Gravitational Instanton, Phys. Lett. B80 (1978) 55–57.

    Google Scholar 

  70. L.E. Ibáñez, J. Mas, H.P. Nilles and F. Quevedo, Heterotic Strings in Symmetric and Asymmetric Orbifold Backgrounds, Nucl. Phys. B301 (1988) 157–196; A. Font, L.E. Ibáñez, F. Quevedo and A. Sierra, The Construction of “Realistic” Four-Dimensional Strings through Orbifolds, Nucl. Phys. B331 (1990) 421–474.

    Google Scholar 

  71. S.-S. Roan, On the Generalization of Kummer Surfaces, J. Diff. Geom. 30 (1989) 523–537; G. Tian and S.-T. Yau, Complete Kähler Manifolds with Zero Ricci Curvature I, Invent. math. 106 (1991) 27–60.

    Google Scholar 

  72. P. Townsend, A New Anomaly-Free Chiral Supergravity Theory from Compactification on K3, Phys. Lett. B139 (1984) 283–287.

    Google Scholar 

  73. S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four-folds, Nucl. Phys. B584 (2000) 69–108, hep-th/9906070.

    Article  CAS  Google Scholar 

  74. A. Strominger, Superstrings With Torsion, Nucl. Phys. B274 (1986) 253–284.

    Article  Google Scholar 

  75. S. Gurrieri, J. Louis, A. Micu and D. Waldram, Mirror symmetry in generalized Calabi-Yau compactifications, Nucl. Phys. B654 (2003) 61–113, hep-th/0211102.; G. L. Cardoso, G. Curio, G. Dall’Agata, D. Lüst, P. Manousselis and G. Zoupanos, Non-Kaehler string backgrounds and their five torsion classes, Nucl. Phys. B652 (2003) 5–34, hep-th/0211118.; K. Becker, M. Becker, K. Dasgupta and P. S. Green, Compactifications of heterotic theory on non-Kaehler complex manifolds. I, JHEP 0304 (2003) 007, hep-th/0301161; J. P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, hep-th/0302158; K. Becker, M. Becker, P. S. Green, K. Dasgupta and E. Sharpe, Compactifications of heterotic strings on non-Kaehler complex manifolds. II, hep-th/0310058; S. Fidanza, R. Minasian and A. Tomasiello, Mirror symmetric SU(3)-structure manifolds with NS fluxes, hep-th/0311122.

    Google Scholar 

  76. S. B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D66 (2002) 106006(16), hep-th/0105097.

    Google Scholar 

  77. M. Grana and J. Polchinski, Supersymmetric three-form flux perturbations on AdS5, Phys. Rev. D63 (2001) 026001(8), hep-th/0009211.

    Google Scholar 

  78. S. Kachru, M. B. Schulz and S. Trivedi, Moduli stabilization from fluxes in a simple IIB orientifold, JHEP 0310 (2003) 007, hep-th/0201028; R. Blumenhagen, D. Lüst and T. R. Taylor, Moduli stabilization in chiral type IIB orientifold models with fluxes, Nucl. Phys. B663 (2003) 319–342, hep-th/0303016; J. F. G. Cascales and A. M. Uranga, Chiral 4d N = 1 string vacua with D-branes and NSNS and RR fluxes, JHEP 0305 (2003) 011, hep-th/0303024.

    Google Scholar 

  79. B. S. Acharya, M Theory, G2-Manifolds and Four-Dimensional Physics, Class. Quant. Grav. 19 (2002) 5619–5653.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Hernán Ocampo Sylvie Paycha Andrés Vargas

Rights and permissions

Reprints and permissions

About this chapter

Cite this chapter

Font, A., Theisen, S. Introduction to String Compactification. In: Ocampo, H., Paycha, S., Vargas, A. (eds) Geometric and Topological Methods for Quantum Field Theory. Lecture Notes in Physics, vol 668. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11374060_3

Download citation

Publish with us

Policies and ethics