
Journal of Lie Theory 27 (2017), No. 4, 11511177 Copyright Heldermann Verlag 2017 QuasiPeriodic Paths and a String 2Group Model from the Free Loop Group Michael Murray School of Mathematical Sciences, University of Adelaide, Australia michael.murray@adelaide.edu.au David M. Roberts School of Mathematical Sciences, University of Adelaide, Australia david.roberts@adelaide.edu.au Christoph Wockel Department of Mathematics, University of Hamburg, Germany christoph@wockel.eu We address the question of the existence of a model for the string 2group as a strict Lie2group using the free loop group LSpin (or more generally LG for compact simple simplyconnected Lie groups G). BaezCransStevensonSchreiber constructed a model for the string 2group using a based loop group. This has the deficiency that it does not admit an action of the circle group S^{1}, which is of crucial importance, for instance in the construction of a (hypothetical) S^{1}equivariant index of (higher) differential operators. The present paper shows that there are in fact obstructions for constructing a strict model for the string 2group using LG. We show that a certain infinitedimensional 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 2group 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. Keywords: Lie 2group, string 2group, loop group, Lie groupoid. MSC: 22E67, 18D35, 22A22, 53C08, 81T30. [ Fulltextpdf (413 KB)] for subscribers only. 