Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Brauer groups of symmetries of abelian Dijkgraaf-Witten theories

Jürgen Fuchs, Jan Priel, Christoph Schweigert, Alessandro Valentino

TL;DR

The work identifies and realizes the full Brauer-Picard group of symmetries for abelian Dijkgraaf–Witten theories by embedding the abstract Brauer–Picard data into gauge-theoretic surface defects. It shows that the symmetry group is generated by Aut$(A)$ (kinematical), B-fields (dynamical), and partial electric–magnetic dualities, and that each symmetry is encoded by a defect whose transmission functor acts on bulk Wilson lines via a braided auto-equivalence of ${ m D}(A) ext{-mod}$, i.e. by an element of ${ m O}_q(Aigoplus A^*)$. A central tool is the transmission functor, arising from cylinders with defects, which provides a concrete field-theoretic realization of the bijection between invertible bimodule categories and braided auto-equivalences of the center. The paper also proves that these generators suffice to generate ${ m O}_q(Aigoplus A^*)$, thereby classifying all symmetries of the bulk theory in terms of concrete gauge-theoretic data and clarifying the action on bulk Wilson lines. $

Abstract

Symmetries of three-dimensional topological field theories are naturally defined in terms of invertible topological surface defects. Symmetry groups are thus Brauer-Picard groups. We present a gauge theoretic realization of all symmetries of abelian Dijkgraaf-Witten theories. The symmetry group for a Dijkgraaf-Witten theory with gauge group a finite abelian group $A$, and with vanishing 3-cocycle, is generated by group automorphisms of $A$, by automorphisms of the trivial Chern-Simons 2-gerbe on the stack of $A$-bundles, and by partial e-m dualities. We show that transmission functors naturally extracted from extended topological field theories with surface defects give a physical realization of the bijection between invertible bimodule categories of a fusion category and braided auto-equivalences of its Drinfeld center. The latter provides the labels for bulk Wilson lines; it follows that a symmetry is completely characterized by its action on bulk Wilson lines.

On the Brauer groups of symmetries of abelian Dijkgraaf-Witten theories

TL;DR

The work identifies and realizes the full Brauer-Picard group of symmetries for abelian Dijkgraaf–Witten theories by embedding the abstract Brauer–Picard data into gauge-theoretic surface defects. It shows that the symmetry group is generated by Aut (kinematical), B-fields (dynamical), and partial electric–magnetic dualities, and that each symmetry is encoded by a defect whose transmission functor acts on bulk Wilson lines via a braided auto-equivalence of , i.e. by an element of . A central tool is the transmission functor, arising from cylinders with defects, which provides a concrete field-theoretic realization of the bijection between invertible bimodule categories and braided auto-equivalences of the center. The paper also proves that these generators suffice to generate , thereby classifying all symmetries of the bulk theory in terms of concrete gauge-theoretic data and clarifying the action on bulk Wilson lines. $

Abstract

Symmetries of three-dimensional topological field theories are naturally defined in terms of invertible topological surface defects. Symmetry groups are thus Brauer-Picard groups. We present a gauge theoretic realization of all symmetries of abelian Dijkgraaf-Witten theories. The symmetry group for a Dijkgraaf-Witten theory with gauge group a finite abelian group , and with vanishing 3-cocycle, is generated by group automorphisms of , by automorphisms of the trivial Chern-Simons 2-gerbe on the stack of -bundles, and by partial e-m dualities. We show that transmission functors naturally extracted from extended topological field theories with surface defects give a physical realization of the bijection between invertible bimodule categories of a fusion category and braided auto-equivalences of its Drinfeld center. The latter provides the labels for bulk Wilson lines; it follows that a symmetry is completely characterized by its action on bulk Wilson lines.

Paper Structure

This paper contains 18 sections, 1 theorem, 68 equations.

Table of Contents

  1. Symmetries of abelian Dijkgraaf-Witten theories
  2. Braided auto-equivalences of bulk Wilson lines.
  3. Universal kinematical symmetries.
  4. Universal dynamical symmetries.
  5. Electric-magnetic dualities.
  6. Topological surface defects and bimodule categories.
  7. The transmission functor.
  8. Plan of the paper.
  9. Surface defects in DW theories and the transmission functor
  10. Surface defects in Dijkgraaf-Witten theories
  11. The transmission functor
  12. Transmission functors for Dijkgraaf-Witten theories
  13. Action of the transmission functor on simple objects
  14. Realizing the symmetries
  15. Kinematical symmetries: group automorphisms
  16. ...and 3 more sections

Key Result

Corollary 3.2

The $A\text{-}\mathrm{vect}$-bimodule category associated with the pair $(H,\theta)$ is invertible iff and the restriction of the commutator cocycle $\omega_\theta$commcoc to is non-degenerate.

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 3.1
  • Corollary 3.2
  • proof
  • Example 4.2
  • Example 4.3