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 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.
| Original language | English (US) |
|---|---|
| Journal | Letters in Mathematical Physics |
| DOIs | |
| State | Accepted/In press - Jan 1 2019 |
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Keywords
- Infinite dimensional Lie groups
- Pseudo differential operators
- Wodzicki residue
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
Cite this
Maps from a surface into a compact Lie group and curvature. / Larrain-Hubach, Andres; Pickrell, Douglas M.
In: Letters in Mathematical Physics, 01.01.2019.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 -