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 W 1 / 2 (S 1 , K) / K is a (nonnegative 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 Σ , W 1 (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 W s (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