Variational problems for gauge fields. (English) Zbl 0562.53059

Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 585-591 (1984).
Summary: [For the entire collection see Zbl 0553.00001.]
This talk will be a brief survey of recent results in the study of elliptic systems of partial differential equations of gauge field theory. The equations of this talk are the classical Euler-Lagrange equations corresponding to current quantum field theory models. It is not clear to what extent any of these results is interesting in physics; however, these equations coming from physics are now very important in mathematics itself. At the Congress in 1978, A. Jaffe gave a talk on gauge theories which included some classical theory [Proc. int. Congr. Math., Helsinki 1978, Vol. 2, 905-916 (1980; Zbl 0425.58023)]. Progress in the last five years can be measured by reading his article. The main progress has been the step from considering special, scalar forms of the equations to handling the full gauge theory. Even more remarkable are the applications which are turning up in pure mathematics. Special note should be made of the contributions of Clifford Taubes and Simon Donaldson to the subject.


53C80 Applications of global differential geometry to the sciences
81T08 Constructive quantum field theory
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory