Communications in Mathematical Physics

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

Topological gauge theories and group cohomology

  • Robbert Dijkgraaf
  • Edward Witten


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.


Neural Network Manifold Statistical Physic Complex System Gauge Theory 
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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

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