Maps from a surface into a compact Lie group and curvature

Andres Larrain-Hubach, Doug Pickrell

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)1257-1267
Number of pages11
JournalLetters in Mathematical Physics
Volume109
Issue number5
DOIs
StatePublished - May 1 2019

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Fingerprint Dive into the research topics of 'Maps from a surface into a compact Lie group and curvature'. Together they form a unique fingerprint.

Cite this