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# String structures, 2-group bundles, and a categorification of the Freed-Quinn line bundle

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## Abstract

For a 2-group constructed from a finite group and 3-cocycle, we provide an explicit description of the bicategory of flat 2-group bundles on an oriented surface in terms of weak representations of the fundamental group. We show that this bicategory encodes (flat) string structures. Furthermore, we identify the space of isomorphism classes of objects with Freed and Quinn's line bundle appearing in Chern-Simons theory of a finite group.

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Higher gauge theory, Categories in Algebra
• J Baez
• U Schreiber
J. Baez and U. Schreiber, Higher gauge theory, Categories in Algebra, Geometry and Mathematical Physics, eds. A. Davydov et al, Contemp. Math. 431 (2007).
A geometric construction of the Witten genus II
, A geometric construction of the Witten genus II, ArXiv preprint, http://arxiv.org/ abs/1112.0816 (2011).
Topological modular forms and conformal nets, Mathematical Foundations of QFT and Perturbative String Theory
• C Douglas
• A Henriques
C. Douglas and A. Henriques, Topological modular forms and conformal nets, Mathematical Foundations of QFT and Perturbative String Theory, Proc. Sympos. Pure Math. 83 (2011).
• P Hu
• I Kriz
P. Hu and I. Kriz, Conformal field theory and elliptic cohomology, Advances in Mathematics 189 (2004), no. 2, 325-412.
Principal ∞-bundles: Presentations
, Principal ∞-bundles: Presentations, Journal of Homotopy and Related Structures 10 (2015).
Lifting problems and transgression for non-abelian gerbes
, Lifting problems and transgression for non-abelian gerbes, Advances in Mathematics 242 (2013).
• G Segal
G. Segal, Elliptic cohomology, Séminaire N. Bourbaki 695 (1988).
The definition of conformal field theory, Topology, geometry and quantum field theory
, The definition of conformal field theory, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 421-577.
Schommer-Pries, Central extensions of smooth 2-groups and a finite-dimensional string 2-group
C. J. Schommer-Pries, Central extensions of smooth 2-groups and a finite-dimensional string 2-group, Geometry & Topology 15 (2011).
• U Schreiber
• K Waldorf
U. Schreiber and K. Waldorf, Smooth functors vs. differential forms, Homology Homotopy Appl. 13 (2011).
String geometry vs. spin geometry on loop spaces
, String geometry vs. spin geometry on loop spaces, J. Geom. Phys. 97 (2015).