General Relativity and Gravitation

, Volume 41, Issue 4, pp 691–705 | Cite as

Superstring perturbation theory

Open Access
Review Article
  • 309 Downloads

Abstract

The state of superstring perturbation theory is reviewed with an emphasis on the state of the pure spinor superstring perturbation theory. We begin with a brief summary of the state of perturbation theory in the Ramond–Neveu–Schwarz and in the Green–Schwarz formulations of the superstring. Then we proceed to a quick review of the minimal and non-minimal pure spinor formulations of the superstring and discuss the multi-loop amplitude prescriptions in each of them. We end with a summary and open questions.

Keywords

Superstring perturbation theory The pure spinor superstring 

References

  1. 1.
    Polchinski, J.: String theory. In: Superstring Theory and Beyond, p. 531 Cambridge University Press, Cambridge (1998)Google Scholar
  2. 2.
    D’Hoker E., Phong D.H.: Loop Amplitudes for the Fermionic String. Nucl. Phys. B 278, 225 (1986)CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Moore G.W., Nelson P.C., Polchinski J.: Strings and Supermoduli. Phys. Lett. B 169, 47 (1986)CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Aoki K., D’Hoker E., Phong D.H.: Unitarity of closed superstring perturbation theory. Nucl. Phys. B 342, 149–230 (1990)CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    D’Hoker E., Phong D.H.: Momentum analyticity and finiteness of the one loop superstring amplitude. Phys. Rev. Lett. 70, 3692–3695 (1993)MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    D’Hoker E., Phong D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60, 917 (1998)CrossRefMathSciNetGoogle Scholar
  7. 7.
    D’Hoker E., Phong D.H.: Conformal scalar fields and chiral splitting on superriemann surfaces. Commun. Math. Phys. 125, 469 (1989)MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    D’Hoker E., Phong D.H.: Two-loop superstrings I, main formulas. Phys. Lett. B 529, 241–255 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    D’Hoker E., Phong D.H.: Two-loop superstrings II, the chiral measure on moduli space. Nucl. Phys. B 636, 3–60 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    D’Hoker E., Phong D.H.: Two-loop superstrings III, slice independence and absence of ambiguities. Nucl. Phys. B 636, 61–79 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    D’Hoker E., Phong D.H.: Two-loop superstrings IV, the cosmological constant and modular forms. Nucl. Phys. B 639, 129–181 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    D’Hoker E., Phong D.H.: Two-loop superstrings V: gauge slice independence of the N-point function. Nucl. Phys. B 715, 91–119 (2005)MATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    D’Hoker E., Phong D.H.: Two-loop superstrings VI: non-renormalization theorems and the 4-point function. Nucl. Phys. B 715, 3–90 (2005)CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    D’Hoker E., Phong D.H.: Asyzygies, modular forms, and the superstring measure, I. Nucl. Phys. B 710, 58–82 (2005)MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    D’Hoker E., Phong D.H.: Asyzygies, modular forms, and the superstring measure, II. Nucl. Phys. B 710, 83–116 (2005)MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Cacciatori S.L., Piazza F.D., van Geemen B.: Modular forms and three loop superstring amplitudes. Nucl. Phys. B 800, 565–590 (2008)MATHCrossRefADSGoogle Scholar
  17. 17.
    Matone M., Volpato R.: Higher genus superstring amplitudes from the geometry of moduli spaces. Nucl. Phys. B 732, 321–340 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Grushevsky S.: Superstring scattering amplitudes in higher genus. (2008)Google Scholar
  19. 19.
    Morozov A.: NSR measures on hyperelliptic locus and non-renormalization of 1, 2, 3-point functions. Phys. Lett. B 664, 116–122 (2008)CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Matone M., Volpato R.: Superstring measure and non-renormalization of the three-point amplitude. (2008)Google Scholar
  21. 21.
    Grushevsky S., Manni R.S.: On the cosmological constant for the chiral superstring measure (2008)Google Scholar
  22. 22.
    Green M.B., Schwarz J.H.: Superstring interactions. Nucl. Phys. B 218, 43–88 (1983)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Green M.B., Schwarz J.H., Brink L.: Superfield theory of type II superstrings. Nucl. Phys. B 219, 437–478 (1983)CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Green M.B., Schwarz J.H.: Superstring field theory. Nucl. Phys. B 243, 475–536 (1984)CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Greensite J., Klinkhamer F.R.: New interactions for superstrings. Nucl. Phys. B 281, 269 (1987)CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Greensite J., Klinkhamer F.R.: Contact interactions in closed superstring field theory. Nucl. Phys. B 291, 557 (1987)CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Greensite J., Klinkhamer F.R.: Superstring amplitudes and contact interactions. Nucl. Phys. B 304, 108 (1988)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Green M.B., Seiberg N.: Contact interactions in superstring thoery. Nucl. Phys. B 299, 559 (1988)CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Restuccia A., Taylor J.G.: Finiteness of type II superstring amplitudes. Phys. Lett. B 187, 267 (1987)CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Taylor J.G., Restuccia A: Finiteness of heterotic superstring theories. Phys. Lett. B 187, 273 (1987)CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Restuccia A., Taylor J.G.: On the infinites of closed superstring amplitude. Mod. Phys. Lett. A 3, 883 (1988)CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Restuccia A., Taylor J.G.: Absence of divergences in type II and heterotic string multiloop amplitudes. Commun. Math. Phys. 112, 447 (1987)MATHCrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Restuccia A., Taylor J.G.: The construction of multiloop superstring amplitudes in the light cone gauge. Phys. Rev. D36, 489 (1987)ADSMathSciNetGoogle Scholar
  34. 34.
    Berkovits N.: Super-poincare covariant quantization of the superstring. JHEP 04, 018 (2000)CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Berkovits, N.: ICTP lectures on covariant quantization of the superstring (2002)Google Scholar
  36. 36.
    Siegel W.: Classical superstring mechanics. Nucl. Phys. B 263, 93 (1986)CrossRefADSGoogle Scholar
  37. 37.
    Berkovits N.: Explaining the pure spinor formalism for the superstring. JHEP 01, 065 (2008)CrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Berkovits N.: Cohomology in the pure spinor formalism for the superstring. JHEP 09, 046 (2000)CrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Berkovits N.: Covariant quantization of the superstring. Nucl. Phys. Proc. Suppl. 127, 23–29 (2004)CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Berkovits N., Vallilo B.C.: Consistency of super-Poincare covariant superstring tree amplitudes. JHEP 07, 015 (2000)CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Berkovits N.: Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring. JHEP 09, 047 (2004)CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Berkovits N., Chandia O.: Massive superstring vertex operator in D = 10 superspace. JHEP 08, 040 (2002)CrossRefADSMathSciNetGoogle Scholar
  43. 43.
    Anguelova L., Grassi P.A., Vanhove P.: Covariant one-loop amplitudes in D = 11. Nucl. Phys. B 702, 269–306 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  44. 44.
    Mafra C.R.: Four-point one-loop amplitude computation in the pure spinor formalism. JHEP 01, 075 (2006)CrossRefADSMathSciNetGoogle Scholar
  45. 45.
    Berkovits N.: Super-poincare covariant two-loop superstring amplitudes. JHEP 01, 005 (2006)CrossRefADSMathSciNetGoogle Scholar
  46. 46.
    Berkovits N., Mafra C.R.: Equivalence of two-loop superstring amplitudes in the pure spinor and RNS formalisms. . Phys. Rev. Lett. 96, 011602 (2006)CrossRefADSMathSciNetGoogle Scholar
  47. 47.
    Mafra C.R.: Pure spinor superspace identities for massless four-point kinematic factors. JHEP 04, 093 (2008)CrossRefADSMathSciNetGoogle Scholar
  48. 48.
    Aisaka Y., Kazama Y.: Origin of pure spinor superstring. JHEP 05, 046 (2005)CrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Berkovits N.: Pure spinor formalism as an N = 2 topological string. JHEP 10, 089 (2005)CrossRefADSMathSciNetGoogle Scholar
  50. 50.
    Berkovits N., Nekrasov N.: Multiloop superstring amplitudes from non-minimal pure spinor formalism. JHEP 12, 029 (2006)CrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Berkovits N.: New higher-derivative R**4 theorems. Phys. Rev. Lett. 98, 211601 (2007)CrossRefADSMathSciNetGoogle Scholar
  52. 52.
    Berkovits N., Mafra C.R.: Some superstring amplitude computations with the non-minimal pure spinor formalism. JHEP 11, 079 (2006)CrossRefADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)PotsdamGermany

Personalised recommendations