Advertisement

Communications in Mathematical Physics

, Volume 129, Issue 2, pp 393–429 | Cite as

Topological gauge theories and group cohomology

  • Robbert Dijkgraaf
  • Edward Witten
Article

Abstract

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH4(BG,Z). In a similar way, possible Wess-Zumino interactions of such a groupG are classified byH3(G,Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map fromH4(BG,Z) toH3(G,Z). We generalize this correspondence to topological “spin” theories, which are defined on three manifolds with spin structure, and are related to what might be calledZ2 graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.

Keywords

Neural Network Manifold Statistical Physic Complex System Gauge 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989)CrossRefGoogle Scholar
  2. 2.
    Moore, G., Seiberg, N.: Taming the conformal zoo. Phys. Lett.220B, 422 (1989)Google Scholar
  3. 3.
    Chern, S.-S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math.99, 48–69 (1974)Google Scholar
  4. 4.
    Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and topology. Lecture Notes in Mathematics vol.1167. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  5. 5.
    Witten, E.: Non-Abelian Bosonization in two dimensions. Commun. Math. Phys.92, 455 (1984)CrossRefGoogle Scholar
  6. 6.
    Segal, G.: Lecture at the IAMP Congress (Swansea, July 1988), and Oxford University preprint (to appear)Google Scholar
  7. 7.
    Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: Operator algebra of orbifold models. Commun. Math. Phys.123, 485 (1989)CrossRefGoogle Scholar
  8. 8.
    Borel, A.: Topology of Lie groups and characteristic classes. Bull. A.M.S.61, 397–432, (1955)Google Scholar
  9. 9.
    Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces I. Am. J. Math.80, 458–538 (1958)Google Scholar
  10. 10.
    Milnor, J., Stasheff, J.: Characteristic classes. Annals of Mathematics Studies vol.76. Princeton, NJ: Princeton University Press 1974Google Scholar
  11. 11.
    Madsen, I., Milgram, R.J.: The classifying spaces for surgery and cobordism of manifolds. Annals of Mathematics Studies vol.92. Princeton, NJ: Princeton University Press 1979Google Scholar
  12. 12.
    Brown, K.S.: Cohomology of groups. Graduate Texts in Mathematics vol.87. Berlin, Heidelberg, New York: Springer 1982Google Scholar
  13. 13.
    Freed, D.S.: Determinants, torsion, and strings. Commun. Math. Phys.107,483–514 (1986)CrossRefGoogle Scholar
  14. 14.
    Milnor, J.: Construction of universal bundles II. Ann. Math.63, 430–436 (1956)Google Scholar
  15. 15.
    Stong, R.E.: Notes on cobordism theory. Mathematical Notes. Princeton, NJ: Princeton University Press 1968Google Scholar
  16. 16.
    Conner, P.E., Floyd, E.E.: Differentiable periodic maps. Bull. Am. Math. Soc.68, 76–86 (1962)Google Scholar
  17. 17.
    Narasinhan, H.S., Ramanan, S.: Existence of universal connections. Am. J. Math.83, 563–572 (1961);85, 223–231 (1963)Google Scholar
  18. 18.
    Felder, G., Gawedzki, K., Kupiainen, A.: Spectra of Wess-Zumino-Witten models with arbitrary simple groups. Commun. Math. Phys.117, 127–158 (1988)CrossRefGoogle Scholar
  19. 19.
    Gepner, D., Witten, E.: String theory on group manifolds. Nucl. Phys.B278, 493–549 (1986)CrossRefGoogle Scholar
  20. 20.
    Elitzur, S., Moore, G., Schwimmer, A., Seiberg, N.: Remarks on the canonical quantization of the Chern-Simons-Witten theory. Preprint IASSNS-HEP-89/20Google Scholar
  21. 21.
    Freed, D.S., Uhlenbeck, K.K.: Instantons and four-manifolds. Math. Sci. Res. Inst. Publ. vol.1, Berlin, Heidelberg, New York: Springer 1984Google Scholar
  22. 22.
    't Hooft, G.: Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus. Commun. Math. Phys.81, 167–275 (1981); van Baal, P.: Some results forSU(N) gauge fields on the hypertorus. Commun. Math. Phys.85, 529 (1982)Google Scholar
  23. 23.
    Schellekens, A.N., Yankielowicz, S.: Extended Chiral algebras and modular invariant partition functions. Preprint CERN-TH5344/89Google Scholar
  24. 24.
    Karpilovsky, G.: Projective representations of finite groups. New York: Marcel Dekker 1985Google Scholar
  25. 25.
    Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 360 (1988)CrossRefGoogle Scholar
  26. 26.
    Hempel, J.: 3-Manifolds. Annals of Mathematics Studies vol.86. Princeton, NJ: Princeton University Press 1976Google Scholar
  27. 27.
    Vafa, C.: Modular invariance and discrete torsion on orbifolds. Nucl. Phys. B273, 592 (1986)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Robbert Dijkgraaf
    • 1
  • Edward Witten
    • 2
  1. 1.Institute for Theoretical PhysicsUniversity of UtrechtThe Netherlands
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonUSA

Personalised recommendations