Journal of Lie Theory 27 (2017), No. 4, 1151--1177
Copyright Heldermann Verlag 2017
Quasi-Periodic Paths and a String 2-Group 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 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.
Keywords: Lie 2-group, string 2-group, loop group, Lie groupoid.
MSC: 22E67, 18D35, 22A22, 53C08, 81T30.
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