We address the question of the existence of a model for the string 2-group as a strict Lie-2-group using the free loop group L-Spin (or more generally LG for compact simple simply-connected Lie groups G). Baez-Crans-Stevenson-Schreiber constructed a model for the string 2-group using a based loop group. This has the deficiency that it does not admit an action of the circle group S1, which is of crucial importance, for instance in the construction of a (hypothetical) S1-equivariant index of (higher) differential operators. The present paper shows that there are in fact obstructions for constructing a strict model for the string 2-group using LG. We show that a certain infinite-dimensional manifold of smooth paths admits no Lie group structure, and that there are no nontrivial Lie crossed modules analogous to the BCSS model using the universal central extension of the free loop group. Afterwards, we construct the next best thing, namely a coherent model for the string 2-group using the free loop group, with explicit formulas for all structure. This is in particular important for the expected representation theory of the string group that we discuss briefly in the end.
Michael Murray
1
;
David M. Roberts
1
;
Christoph Wockel
2
1
School of Mathematical Sciences, University of Adelaide, Australia
2
Department of Mathematics, University of Hamburg, Germany
Michael Murray; David M. Roberts; Christoph Wockel. Quasi-Periodic Paths and a String 2-Group Model from the Free Loop Group. Journal of Lie Theory, Tome 27 (2017) no. 4, pp. 1151-1177. http://geodesic.mathdoc.fr/item/JOLT_2017_27_4_a14/
@article{JOLT_2017_27_4_a14,
author = {Michael Murray and David M. Roberts and Christoph Wockel},
title = {Quasi-Periodic {Paths} and a {String} {2-Group} {Model} from the {Free} {Loop} {Group}},
journal = {Journal of Lie Theory},
pages = {1151--1177},
year = {2017},
volume = {27},
number = {4},
url = {http://geodesic.mathdoc.fr/item/JOLT_2017_27_4_a14/}
}
TY - JOUR
AU - Michael Murray
AU - David M. Roberts
AU - Christoph Wockel
TI - Quasi-Periodic Paths and a String 2-Group Model from the Free Loop Group
JO - Journal of Lie Theory
PY - 2017
SP - 1151
EP - 1177
VL - 27
IS - 4
UR - http://geodesic.mathdoc.fr/item/JOLT_2017_27_4_a14/
ID - JOLT_2017_27_4_a14
ER -
%0 Journal Article
%A Michael Murray
%A David M. Roberts
%A Christoph Wockel
%T Quasi-Periodic Paths and a String 2-Group Model from the Free Loop Group
%J Journal of Lie Theory
%D 2017
%P 1151-1177
%V 27
%N 4
%U http://geodesic.mathdoc.fr/item/JOLT_2017_27_4_a14/
%F JOLT_2017_27_4_a14
