Communications in Mathematical Physics

, Volume 117, Issue 3, pp 353–386 | Cite as

Topological quantum field theory

  • Edward Witten


A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in topology of low dimensional manifolds; the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The model may also be interesting from a physical viewpoint; it is in a sense a generally covariant quantum field theory, albeit one in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.


Neural Network Manifold Covariance Gauge Theory Quantum Field Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Donaldson, S.: An application of gauge theory to the topology of four manifolds. J. Differ. Geom.18, 269 (1983); The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differ. Geom.26, 397 (1987); Polynomial invariants for smooth four-manifolds. Oxford preprintGoogle Scholar
  2. 2.
    Freed, D., Uhlenbeck, K.: Instantons and four manifolds. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  3. 3.
    Belavin, A., Polyakov, A., Schwartz, A., Tyupkin, Y.: Phys. Lett. B59, 85 (1975)Google Scholar
  4. 4.
    Taubes, C.: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differ. Geom.17, 139 (1982)Google Scholar
  5. 5.
    Uhlenbeck, K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31 (1982). Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11 (1982)Google Scholar
  6. 6.
    Floer, A.: An instanton invariant for three manifolds. Courant Institute preprint (1987); Morse theory for fixed points of symplectic diffeomorphisms. Bull. AMS16, 279 (1987)Google Scholar
  7. 7.
    Atiyah, M.F.: New invariants of three and four dimensional manifolds. In: The Symposium on the Mathematical Heritage of Hermann Weyl, Wells, R. et al. (eds.). (Univ. of North Carolina, May, 1987)Google Scholar
  8. 8.
    Braam, P.J.: Floer homology groups for homology three spheres. University of Utrecht Mathematics preprint 484, November, 1987Google Scholar
  9. 9.
    Witten, E.: Supersymmetry and morse theory. J. Differ. Geom.17, 661 (1982)Google Scholar
  10. 10.
    't Hooft, G.: Computation of the quantum effects due to a four dimensional pseudoparticle. Phys. Rev. D14, 3432 (1976)Google Scholar
  11. 11.
    Jackiw, R., Rebbi, C.: Phys. Rev. Lett.37, 172 (1976)Google Scholar
  12. 12.
    Callan, C.G., Dashen, R., Gross, D.J.: Phys. Lett.63 B, 334 (1976)Google Scholar
  13. 13.
    Atiyah, M.F., Hitchin, N., Singer, I.: Self-duality in Riemannian geometry. Proc. Roy. Soc. London A362, 425 (1978)Google Scholar
  14. 14.
    Affleck, I., Dine, M., Seiberg, N.: Dynamical supersymmetry breaking in supersymmetric QCD. Nucl. Phys. B241, 493 (1984); Dynamical supersymmetry breaking in four dimensions and its phenomenological implications. Nucl. Phys. B256, 557 (1985)Google Scholar
  15. 15.
    Seiberg, N.: IAS preprint (to appear)Google Scholar
  16. 16.
    Novikov, V.A., Shifman, M.A., Vainshtein, A.I., Zakharov, V.I.: Nucl. Phys. B229, 407 (1983)Google Scholar
  17. 16a.
    Amati, D., Konishi, K., Meurice, Y., Rossi, G.C., Veneziano, G.: Non-perturbative aspects in supersymmetric gauge theories. Physics Reports (to appear)Google Scholar
  18. 17.
    Friedan, D., Martinec, E., Shenker, S.: Nucl. Phys. B271, 93 (1986)Google Scholar
  19. 18.
    Peskin, M.: Introduction to string and superstring theory. SLAC-PUB-4251 (1987)Google Scholar
  20. 19.
    Green, M.B., Schwarz, J.H., Witten, E.: Superstring theory. Cambridge: Cambridge University Press 1987Google Scholar
  21. 20.
    Witten, E.: Global anomalies in string theory. In: Symposium on anomalies, geometry, and topology. White, A., Bardeen, W. (eds.), especially pp. 90–95. Singapore: World Scientific 1985Google Scholar
  22. 21.
    Becchi, C., Rouet, A., Stora, R.: The abelian Higgs-Kibble model, unitarity of theS-operator. Phys. Lett.69 B, 309 (1974); Renormalization of gauge theories. Ann. Phys.98, 287 (1976)Google Scholar
  23. 22.
    Tyupin, I.V.: Gauge invariance in field theory and in statistical physics in the operator formalism. Lebedev preprint FIAN No. 39 (1975), unpublishedGoogle Scholar
  24. 23.
    Kugo, T., Ojima, I.: Manifestly covariant canonical formulation of Yang-Mills theories. Phys. Lett.73 B, 459 (1978); Local covariant operator formalism of non-abelian gauge theories and quark confinement problem. Supp. Prog. Theor. Phys.66, 1 (1979)Google Scholar
  25. 24.
    Polchinski, J.: Scale and conformal invariance in quantum field theory. Univ. of Texas preprint UTTG-22-87Google Scholar
  26. 25.
    D'Adda, A., DiVecchia, P.: Supersymmetry and instantons. Phys. Lett.73 B, 162 (1978)Google Scholar
  27. 26.
    Witten, E.: AnSU(2) anomaly. Phys. Lett.117 B, 432 (1982)Google Scholar
  28. 27.
    Segal, G.: Oxford preprint (to appear)Google Scholar
  29. 28.
    Horowitz, G.T., Lykken, J., Rohm, R., Strominger, A.: Phys. Rev. Lett.57, 283 (1986)Google Scholar
  30. 29.
    Witten, E.: Topological gravity. IAS preprint, February, 1988Google Scholar
  31. 30.
    Witten, E.: Topological sigma models. Commun. Math. Phys. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Edward Witten
    • 1
  1. 1.School of Natural Sciences, Institute for Advanced StudyPrincetonUSA

Personalised recommendations