### 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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Journal | Letters in Mathematical Physics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Letters in Mathematical Physics*. https://doi.org/10.1007/s11005-018-01152-w

**Maps from a surface into a compact Lie group and curvature.** / Larrain-Hubach, Andres; Pickrell, Douglas M.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Maps from a surface into a compact Lie group and curvature

AU - Larrain-Hubach, Andres

AU - Pickrell, Douglas M

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Infinite dimensional Lie groups

KW - Pseudo differential operators

KW - Wodzicki residue

UR - http://www.scopus.com/inward/record.url?scp=85060202110&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060202110&partnerID=8YFLogxK

U2 - 10.1007/s11005-018-01152-w

DO - 10.1007/s11005-018-01152-w

M3 - Article

AN - SCOPUS:85060202110

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

ER -