  These are notes on seminal work of Freed, and subsequent developments, on the
curvature properties of (Sobolev Lie) groups of maps from a Riemannian manifold
into a compact Lie group. We are mainly interested in critical cases which are
relevant to quantum field theory. For example Freed showed that, in a
necessarily qualified sense, the quotient space $W^{1/2}(S^1,K)/K$ is a
(positive constant) Einstein `manifold' with respect to the essentially unique
PSU(1,1) invariant metric, where $W^{s}$ denotes maps of $L^2$ Sobolev order s.
In a similarly qualified sense, and in addition making use of the Dixmier
trace/Wodzicki residue, we show that for a Riemann surface Sigma,
$W^1(\Sigma,K)/K$ is a (positive constant) Einstein `manifold' with respect to
the essentially unique conformally invariant metric. As in the one dimensional
case, invariance implies Einstein, but the sign of the Ricci curvature has to
be computed. Because of the qualifications involved in these statements, in
practice it is necessary to consider curvature for $W^s(\Sigma,K)$ for s above
the critical exponent, and limits. The formula we obtain is surprisingly
simple.
