Maps from a surface into a compact Lie group and curvature (Q2000942)

Let $M$ be a Riemann surface and let $K$ be a compact Lie group. Let $G$ be a group of maps from $M$ to $K$ endowed with a suitably chosen left invariant Riemannian metric. For $s>\frac12\dim\{M\}=1$, one could consider the space of continuous maps which have $s$ $L^2$ derivatives using the Sobolov norm; this is a Hilbert Lie group. For the critical exponent $s=\frac12\dim\{M\}=1$, the elements are no longer necessarily continuous and one obtains a topological group but not necessarily a Lie group. The author considers the curvature of this situation motivated by quantum field Theory. If $s>\frac12\dim\{M\}=1$, the authors show that the curvature operator is a pseudo-differential operator of order at most $-2$. To define the Ricci curvature, one replaces the ordinary trace by the Wodzicki residue. Section 1 of the paper provides an introduction and puts matters in context. Section 2 recalls some of the theory of general Lie groups. Section 3 deals with Sobolev Lie groups. The order of the curvature operator is studied and the Ricci curvature is examined. Section 4 deals with the Wodzicki residue regularization of the Ricci curvature.