TY - JOUR
T1 - Maps from a surface into a compact Lie group and curvature
AU - Larrain-Hubach, Andres
AU - Pickrell, Doug
N1 - Publisher Copyright:
© 2019, Springer Nature B.V.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2019/5/1
Y1 - 2019/5/1
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 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.
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 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.
KW - Infinite dimensional Lie groups
KW - Pseudo differential operators
KW - Wodzicki residue
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U2 - 10.1007/s11005-018-01152-w
DO - 10.1007/s11005-018-01152-w
M3 - Article
AN - SCOPUS:85060202110
VL - 109
SP - 1257
EP - 1267
JO - Letters in Mathematical Physics
JF - Letters in Mathematical Physics
SN - 0377-9017
IS - 5
ER -