The Freed--Quinn line bundle from higher geometry

2025/10/12 by Daniel Berwick-Evans, Berwick-Evans, Daniel, Emily Cliff +3
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2510.10773

Abstract

For a finite group G, and level α∈ Z³(BG;\rm U(1)), Freed and Quinn construct a line bundle over the moduli space of G-bundles on surfaces. Global sections determine the values of Chern--Simons theory at level α on surfaces. In this paper, we provide an alternate construction using tools from higher geometry: the pair (G,α) determines a 2-group group, and the Freed--Quinn line arises as a categorical truncation of the bicategory of 2-group bundles.

Citations

• A new approach to (3+1)-dimensional TQFTs via topological modular forms
• Elliptic cohomology and quantum field theory
• Flat principal 2-group bundles and flat string structures
• Bicommutant categories from conformal nets
• String structures and trivialisations of a Pfaffian line bundle
• Topological Quantum Field Theories from Compact Lie Groups
• The Classifying Space of a Topological 2-Group
• Hecke operators in equivariant elliptic cohomology and generalized moonshine

Related

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• A Wilson Line Realisation of Quantum Groups
• Surfaces isogenous to a product of curves, braid groups and mapping\n class groups
• On the Morgan-Shalen compactification of the SL(2,C) character varieties of surface groups
• The moduli stack of G-bundles