This is a brief note on curvature properties of Sobolev Lie groups of maps from a Riemann surface into a compact Lie group K. Freed showed that, in a necessarily qualified sense, the quotient space W1 / 2(S1, K) / K is a (nonnegative constant) Einstein “manifold” with respect to the essentially unique PSU (1 , 1) -invariant metric, where Ws denotes maps of L2 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 Σ , W1(Σ , K) / K is a (nonnegative constant) Einstein “manifold” with respect to the essentially unique conformally invariant metric. Because of the qualifications involved in these statements, in practice it is necessary to consider curvature for Ws(Σ , K) for s above the critical exponent, and limits.
- Infinite dimensional Lie groups
- Pseudo differential operators
- Wodzicki residue
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics