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The Freed--Quinn line bundle from higher geometry

Daniel Berwick-Evans, Emily Cliff, Laura Murray

TL;DR

This work provides a concrete higher-geometric construction of the Freed--Quinn line bundle on the moduli of $G$-bundles over a surface, by factoring through a categorical central extension $\mathcal{G}(G,A,\alpha)$ and the bicategory ${\sf Bun}_{\mathcal{G}}(\Sigma)$. By carefully truncating along fiberwise isomorphisms, the authors produce a ${\rm U}(1)$-bundle $\mathscr{P}_{\mathcal{G}}$ over ${\sf Bun}_G(\Sigma)$ that is shown to be isomorphic to Freed--Quinn’s line bundle $\mathcal{L}_G^\alpha$, with a compatible action of the mapping class group. The construction yields explicit cocycles for three presentations (Čech, triangulation, holonomy) and proves the isomorphism by comparing these cocycles across presentations, thereby connecting higher-categorical transfer to classical transgression. In addition, the paper explores Klein forms and mapping-class group representations via group cohomology of $\pi_1(\Sigma)$, illustrating how higher-geometry data encode modular phenomena. Overall, the approach offers a transparent, combinatorial route to higher-geometric realizations of CS-type line bundles and their symmetries, with potential implications for equivariant elliptic cohomology and extended TQFTs.

Abstract

For a finite group $G$, and level $α\in Z^3(BG;{\rm U}(1))$, 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.

The Freed--Quinn line bundle from higher geometry

TL;DR

This work provides a concrete higher-geometric construction of the Freed--Quinn line bundle on the moduli of -bundles over a surface, by factoring through a categorical central extension and the bicategory . By carefully truncating along fiberwise isomorphisms, the authors produce a -bundle over that is shown to be isomorphic to Freed--Quinn’s line bundle , with a compatible action of the mapping class group. The construction yields explicit cocycles for three presentations (Čech, triangulation, holonomy) and proves the isomorphism by comparing these cocycles across presentations, thereby connecting higher-categorical transfer to classical transgression. In addition, the paper explores Klein forms and mapping-class group representations via group cohomology of , illustrating how higher-geometry data encode modular phenomena. Overall, the approach offers a transparent, combinatorial route to higher-geometric realizations of CS-type line bundles and their symmetries, with potential implications for equivariant elliptic cohomology and extended TQFTs.

Abstract

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

Paper Structure

This paper contains 27 sections, 38 theorems, 153 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The key players
  3. String structures determine trivializations of Chern--Simons theory
  4. The geometry of 2-group principal bundles
  5. Acknowledgements
  6. Background on (higher) principal bundles
  7. Groupoids and Lie groupoids
  8. Covers and presentations of groupoids
  9. Presentations of ${\sf Bun}_G(X)$
  10. 2-groups and 2-group bundles
  11. Constructing a ${\rm U}(1)$-bundle over ${\sf Bun}_G(\Sigma)$ from the geometry of 2-group bundles
  12. The fibers of $\pi\colon {\sf Bun}_\mathcal{G}(X)\to {\sf Bun}_G(X)$
  13. A groupoid $\mathscr{P}_\mathcal{G}$ over ${\sf Bun}_G(\Sigma)$ constructed from ${\sf Bun}_{\mathcal{G}}(\Sigma)$
  14. The ${\rm U}(1)$-action on $\mathscr{P}_\mathcal{G}$
  15. The mapping class group action on $\mathscr{P}_\mathcal{G}$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Fix a finite group $G$, degree 3 cocycle $\alpha\in Z^3(BG;{\rm U}(1))$, and oriented surface $\Sigma$. There is a canonical isomorphism of line bundles over ${\sf Bun}_G(\Sigma)$ between the Freed--Quinn bundle and eq:LGquotient, Furthermore, this isomorphism is equivariant for the action of the mapping class group of $\Sigma$ on ${\sf Bun}_G(\Sigma)$.

Figures (1)

  • Figure 1: A $\mathcal{G}$-bundle over a surface $\Sigma$ determines an element of ${\rm U}(1)$ witnessing the relation $\prod_{i=1}^{\sf g} [g_i,h_i]\stackrel{\sigma}{\to} 1$.

Theorems & Definitions (136)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 1.4
  • proof : Proof
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • proof : Proof
  • ...and 126 more