## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1257-1267 |

Number of pages | 11 |

Journal | Letters in Mathematical Physics |

Volume | 109 |

Issue number | 5 |

DOIs | |

State | Published - May 1 2019 |

## Keywords

- Infinite dimensional Lie groups
- Pseudo differential operators
- Wodzicki residue

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics