Communications in Mathematical Physics

, Volume 118, Issue 3, pp 411–449 | Cite as

Topological sigma models

  • Edward Witten


A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surfaceΣ to an arbitrary almost complex manifoldM. It possesses a fermionic BRST-like symmetry, conserved for arbitraryΣ, and obeyingQ2=0. In a suitable version, the quantum ground states are the 1+1 dimensional Floer groups. The correlation functions of the BRST-invariant operators are invariants (depending only on the homotopy type of the almost complex structure ofM) similar to those that have entered in recent work of Gromov on symplectic geometry. The model can be coupled to dynamical gravitational or gauge fields while preserving the fermionic symmetry; some observations by Atiyah suggest that the latter coupling may be related to the Jones polynomial of knot theory. From the point of view of string theory, the main novelty of this type of sigma model is that the graviton vertex operator is a BRST commutator. Thus, models of this type may correspond to a realization at the level of string theory of an unbroken phase of quantum gravity.


Vertex Operator Sigma Model Gauge Field Homotopy Type Symplectic Geometry 
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 of Smooth Four-Manifolds, Oxford preprintGoogle Scholar
  2. 2.
    Floer, A.: An instanton invariant for three manifolds. Courant Institute preprint (1987); Morse theory for fixed points of symplectic diffeomorphisms. Bull. Am. Math. Soc.16 279 (1987)Google Scholar
  3. 3.
    Atiyah, M. F.: New invariants of three and four dimensional manifolds. To appear in the proceedings of the Symposium on the Mathematical Heritage of Hermann Weyl (Chapel Hill, May, 1987), ed. R. Wells et. al.Google Scholar
  4. 4.
    Witten, E.: Topological quantum field theory. Commun. Math. Phys. (in press)Google Scholar
  5. 5.
    Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math.82 307 (1985); and in the Proceedings of the International Congress of Mathematicians, Berkeley (August, 1986), p. 81Google Scholar
  6. 6.
    Jones, V.: A polynomial invariant for knots via von neumann algebras. Bull. Am. Math. Soc.82 103 (1985)Google Scholar
  7. 7.
    Witten, W.: Superconducting strings. Nucl. Phys.B249 557 (1985)Google Scholar
  8. 8.
    Gross, D. J., Harvey, J. A., Martinec, E., Rohm, R.: Heterotic string theory. Nucl. Phys.B256 253 (1985)Google Scholar
  9. 9.
    Witten, E.: Cosmic superstrings. Phys. Lett.153B 243 (1985)Google Scholar
  10. 10.
    Witten, E.: Topological gravity. IAS preprint (February 1988)Google Scholar
  11. 11.
    Horowitz, G. T., Lykken, J., Rohm, R., Strominger, A.: Phys. Rev. Lett.57 162 (1978)Google Scholar
  12. 12.
    Strominger, A.: Lectures on closed string field theory. To appear in the proceedings of the ICTP spring workshop, 1987Google Scholar
  13. 13.
    Gross, D. J.: High energy symmetries of string theory, Princeton preprint 1988Google Scholar
  14. 14.
    Friedan, D., Martinec, E., Shenker, S.: Nucl. Phys.B271 93 (1986)Google Scholar
  15. 15.
    Peskin, M.: Introduction to string and superstring theory. SLAC-PUB-4251 (1987)Google Scholar
  16. 16.
    Atiyah, M. F.: Concluding remarks at the Schloss Ringberg meeting (March, 1987)Google Scholar
  17. 17.
    Hooft, G. ‘t: Computation of the quantum effects due to a four dimensional pseudoparticle. Phys. Rev.D14 3432 (1976)Google Scholar
  18. 18.
    Witten, E.: Topological gravity: IAS preprint, February, 1988Google Scholar
  19. 19.
    Atiyah, M. F., Bott, R.: The moment map and equivariant cohomology. Topology23 1 (1984)Google Scholar
  20. 20.
    Mathai, V., Quillen, D.: Superconnections, thom classes, and equivariant differential forms: Topology25A 85 (1986)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Edward Witten
    • 1
  1. 1.School of Natural SciencesInstitute for Advanced Study, Olden LanePrincetonUSA

Personalised recommendations