Smooth loop stacks of differentiable stacks and gerbes
Abstract
R\'esum\'e. Nous d\'efinissons un groupo\"ide de Fr\'echet-Lie Map(S^1,X) d'ana-foncteurs du cercle vers un groupo\"ide de Lie X. Ceci fournit une pr\'esentation du Hom-champ Hom(S^1,\cX), o\`u \cX est le champ diff\'erentiable associ\'e \`a X. Nous appliquons cette construction au groupo\"ide de Lie sous-jacent au `gerbe fibr\'e' d'une vari\'et\'e diff\'erentiable M; le r\'esultat est un gerbe fibr\'e au-dessus de l'espace des lacets LM de M. Abstract. We define a Fr\'echet--Lie groupoid Map(S^1,X) of anafunctors from the circle into a Lie groupoid X. This provides a presentation of the Hom-stack Hom(S^1,\cX), where \cX is the differentiable stack associated to X. We apply this construction to the Lie groupoid underlying a bundle gerbe on a manifold M; the result is a bundle gerbe on the loop space LM of M.
Cite
@article{arxiv.1602.07973,
title = {Smooth loop stacks of differentiable stacks and gerbes},
author = {David Michael Roberts and Raymond F. Vozzo},
journal= {arXiv preprint arXiv:1602.07973},
year = {2021}
}
Comments
In English. v2 27 pages. Added sections 6 and 7 dealing with applications. v3 Lemma 6.5 (=Lemma 6.4 in v2) corrected. 31 pages, version to appear in Cahiers Top. Geom. Diff. Cat