Contemporary Mathematics
Volume 54, 1986
THE IMPACT OF GAUGE THEORIES ON NONLINEAR INFINITE DIMENSIONAL ANALYSIS
Melvyn S. Berger
ABSTRACT. Nonlinear Gauge Theories are currently having a profound
effect on many areas of mathematics. One need only consider recent
applications in 4he topology of four dimensional manifolds, exotic
structures on R , deep studies in the calculus of variations in the
large, new algebraic approaches to algebraic geometry by instantons.
However, less well-known are the advances in nonlinear analysis brought
about by a close consideration of the new structures inherent in
Gauge Theories. It is the purpose of this article to discuss these
advances as they relate to infinite dimensional analysis and non-
linear problems.
1. INGREDIENTS OF A GAUGE THEORY
There are four steps in setting up a Gauge Theory in infinite dimensional
analysis. Of course these four steps are based on ideas connected with
Maxwell's equations of electromagnetism.
(i) First one chooses a real valued functional I(A) defined as the
integral over Rn of a real valued function G(A) and we write
(1) I(A)
=
fRn
G(A).
Here the functional I(A) is called the action and A is a one-
form, or vector field usually defined as the vector potential. In
the case of Maxwell's equations the function G(A) is a quadratic
function of A and the associated Euler-Lagrange equations for the
functional I(A) are linear equations. The nonlinear aspect of the
theory enters when the real valued function G(A) is a nonquadratic
function of A.
(ii) One chooses a Gauge Group G usually a compact Lie group in such a
way that the action functional I(A) is invariant under the group G.
(iii) The Mathematical Problem. The mathematical problem involves the
determination of the critical points of the functional I(A) as
precisely as possible.
1
© 1986 American Mathematical Society
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http://dx.doi.org/10.1090/conm/054/849791
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