The Feynman in string theory

  • Edward Witten
Open Access
Regular Article - Theoretical Physics


The Feynman is an important ingredient in defining perturbative scattering amplitudes in field theory. Here we describe its analog in string theory. Roughly one takes the string worldsheet to have Lorentz signature when a string is going on-shell although it has Euclidean signature generically. (This article is based on a talk presented at the Zuminofest in Berkeley, California, May 2-4, 2013.)


Superstrings and Heterotic Strings Bosonic Strings 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    D.J. Gross, Superstrings and unification, in 24th International Congress on High Energy Physics, R. Kotthaus and J.H. Kuhn eds., Springer-Verlag, (1989) [INSPIRE].
  2. [2]
    A. Berera, Unitary string amplitudes, Nucl. Phys. B 411 (1994) 157 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S. Mandelstam, Factorization in dual models and functional integration in string theory, in The birth of string theory, A. Cappelli et al. eds., Cambridge (2012) [arXiv:0811.1247] [INSPIRE].
  4. [4]
    L. Brink and D.I. Olive, Recalculation of the the unitary single planar dual loop in the critical dimension of space time, Nucl. Phys. B 58 (1973) 237 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    K. Amano, A Finite String Loop Amplitude in a Finite Form, Nucl. Phys. B 328 (1989) 510 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    J.L. Montag and W.I. Weisberger, A Finite representation for a superstring scattering amplitude and its low-energy limit, Nucl. Phys. B 363 (1991) 527 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    E. D’Hoker and D.H. Phong, The Box graph in superstring theory, Nucl. Phys. B 440 (1995) 24 [hep-th/9410152] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    L.A. Pochhammer, Zur theorie der Eulerschen integrale, Math. Ann. 35 (1890) 495.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    A.J. Hanson and J.-P. Sha, A Contour Integral Representation For The Dual Five-Point Function And A Symmetry Of The Genus Four Surface in6, J. Phys. A 39 (2006) 2509 [math-ph/0510064] [INSPIRE].ADSMATHGoogle Scholar
  10. [10]
    H. Kawai, D.C. Lewellen and S.H.H. Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Mandelstam, The interacting-string picture and fundamental interactions, in Unified String Theories proceedings of the Workshop on Unified String Theories, Santa Barbara, California U.S.A., July 29 - August 16 1985, M.B. Green and D.J. Gross eds., World Scientific (1985), pg. 46-102 [INSPIRE].
  12. [12]
    B. Zwiebach, Closed string field theory: an introduction, hep-th/9305026 [INSPIRE].
  13. [13]
    E. Witten, Notes on super Riemann surfaces and their moduli, arXiv:1209.2459 [INSPIRE].
  14. [14]
    K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
  15. [15]
    E. Witten, Notes on supermanifolds and integration, arXiv:1209.2199 [INSPIRE].
  16. [16]
    E. Witten, More on superstring perturbation theory, arXiv:1304.2832 [INSPIRE].
  17. [17]
    E. Witten, Superstring perturbation theory revisited, arXiv:1209.5461 [INSPIRE].
  18. [18]
    L. Magnea, S. Playle, R. Russo and S. Sciuto, Multi-loop open string amplitudes and their field theory limit, JHEP 09 (2013) 081 [arXiv:1305.6631] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    E. Witten, Noncommutative geometry and string field theory, Nucl. Phys. B 268 (1986) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    S.B. Giddings, E.J. Martinec and E. Witten, Modular invariance in string field theory, Phys. Lett. B 176 (1986) 362 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    S. Giddings, Conformal techniques in string theory and string field theory, Phys. Rept. 170 (1988) 167 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    K. Aoki, E. D’Hoker and D.H. Phong, Unitarity of Closed Superstring Perturbation Theory, Nucl. Phys. B 342 (1990) 149 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    G. ’t Hooft and M. Veltman, Diagrammar, NATO Adv. Study Inst. Ser. B Phys. 4 (1974) 177.Google Scholar
  24. [24]
    J.D. Bjorken and S. Drell, Relativistic quantum fields, McGraw-Hill, New York U.S.A. (1965).MATHGoogle Scholar
  25. [25]
    E. D’Hoker and D.H. Phong, Lectures on two loop superstrings, Conf. Proc. C 0208124 (2002) 85 [hep-th/0211111] [INSPIRE].MATHGoogle Scholar
  26. [26]
    D.J. Gross and P.F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    E. Witten, Analytic continuation of Chern-Simons theory, in Chern-Simons Gauge Theory: 20 Years After, J.E. Andersen et al. eds., arXiv:1001.2933 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonUnited States

Personalised recommendations