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Three-Dimensional Spin TFTs from Gauging Line Defects

Jannik Gröne, Ingo Runkel

Abstract

From the input of an oriented three-dimensional TFT with framed line defects and a commutative $Δ$-separable Frobenius algebra $A$ in the ribbon category of these line defects, we construct a three-dimensional spin TFT. The framed line defects of the spin TFT are labelled by certain equivariant modules over $A$, and the spin structure may or may not extend to a given line defect. Physically the spin TFT can be interpreted as the result of gauging a one-form symmetry in the original oriented TFT. This spin TFT extends earlier constructions in Blanchet-Masbaum (1996) and Blanchet (2005) [arXiv:math/0303240], and it reproduces the classification of abelian spin Chern-Simons theories in Belov-Moore (2005) [arXiv:hep-th/0505235].

Three-Dimensional Spin TFTs from Gauging Line Defects

Abstract

From the input of an oriented three-dimensional TFT with framed line defects and a commutative -separable Frobenius algebra in the ribbon category of these line defects, we construct a three-dimensional spin TFT. The framed line defects of the spin TFT are labelled by certain equivariant modules over , and the spin structure may or may not extend to a given line defect. Physically the spin TFT can be interpreted as the result of gauging a one-form symmetry in the original oriented TFT. This spin TFT extends earlier constructions in Blanchet-Masbaum (1996) and Blanchet (2005) [arXiv:math/0303240], and it reproduces the classification of abelian spin Chern-Simons theories in Belov-Moore (2005) [arXiv:hep-th/0505235].

Paper Structure

This paper contains 18 sections, 27 theorems, 79 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Spin Geometry
  3. PLCW Spaces
  4. Spin Structures
  5. Commutative Frobenius Algebras
  6. Algebras and Modules
  7. Commutative Frobenius Algebras in Ribbon Categories
  8. Nakayama-Local Modules
  9. The Gauging Construction
  10. Spin and Oriented Bordism Categories
  11. Framed 1-Skeleta and Ribbon Graphs
  12. Spin Structures as (Coloured) Ribbon Graphs
  13. Construction of the Spin TFT
  14. Line Defects in the spin TFT
  15. Spin Refinements
  16. ...and 3 more sections

Key Result

Theorem 1

thm:zspin_is_a_TFT. Let $\mathcal{S}$ be an idempotent-complete symmetric monoidal category (such as $\textsf{Vect}$ or $\textsf{sVect}$) and let $\mathcal{C}$ be a ribbon category with $A\in\mathcal{C}$ a commutative $\Delta$-separable Frobenius algebra. Then from a symmetric monoidal functor (com one can construct a symmetric monoidal functor

Figures (14)

  • Figure 1: Simple examples of subdivisions of 1-, 2- and 3-cells.
  • Figure 2: The two spin structures on $S^1$ for a $SO(n)$ bundle with $n\ge3$. We only draw the first two vectors of each frame for visual clarity. For the spin structure on the left the frame is constant and can be extended over the disk by contracting the circle. For the one on the right this is not possible.
  • Figure 3: The generators for the set of relations on connections on the coupon. The direction of the ribbons may be either incoming or outgoing
  • Figure 4: The relation for reversing a ribbon.
  • Figure 5: Taking the 2-2-2-stratification of the sphere $S^2$, we can draw an example of the (neighbourhood) of the $S^2\times[0,\epsilon]$ part. The outer sphere goes on to the rest of the bordism, while on the inner sphere, i.e. at $S^2\times\{0\}$ we have punctures inserted for the distinguished coupon in $S^2\times\{\epsilon\}$.
  • ...and 9 more figures

Theorems & Definitions (85)

  • Theorem
  • Theorem
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Theorem 2.1.4: Kir12, Thm. 8.1
  • Remark 2.2.1
  • Definition 3.1.1
  • Definition 3.1.2
  • Definition 3.1.3
  • ...and 75 more