Several noteworthy results in mathematics have recently been obtained
by studying gauge theory on a two-dimensional spacetime (or Riemann
surface); these results (formulas for correlation functions in
two-dimensional Yang-Mills theory) were discovered by E. Witten
(1991-92). Yang-Mills theory involves a gauge field (or connection),
and the Yang-Mills action is the norm squared of its curvature. This
gauge field is a generalization of the vector potential from
electromagnetism, and the curvature corresponds to the tensor formed
from the electric and magnetic fields. The two-dimensional case is a
prototype for the four-dimensional case, and can be solved exactly.
The mathematical interpretation of Witten's results is that they can be
used to obtain formulas for the multiplication in the cohomology of
certain moduli spaces (spaces parametrizing flat connections on Riemann
surfaces). For an abelian gauge group the moduli space is a quotient of
the first cohomology group of the Riemann surface. We will talk
about the nonabelian case which arises in Witten's work.
Witten's formulas have been given a mathematically rigorous proof
(L. Jeffrey and F. Kirwan 1998) using methods from symplectic geometry,
which is the natural mathematical framework for the Hamiltonian
formulation of classical mechanics. On the way to outlining these
results, we describe the dictionary relating the objects that appear in
the physics literature and those that appear in the mathematics
literature.
