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Decomposition in 2d non-invertible gaugings

Alonso Perez-Lona

Abstract

We extend the decomposition conjecture to 2d quantum field theories with a gauged $\text{Rep}(H)$ symmetry category for $H$ a finite-dimensional semisimple Hopf algebra with $\text{Rep}(G)$ trivially-acting and $\text{Vec}(Γ)$ the remaining symmetry, for $G,Γ$ finite groups. We check our extension by explicitly computing partition functions, and by verifying that previous results arise as special cases. Furthermore, we compute the topological operators responsible for enforcing the decomposition. Then, drawing from these results, we formulate a plausible decomposition conjecture for the even more general case of $\text{Rep}(H'')$ trivially-acting and $\text{Rep}(H')$ the remaining symmetry, for $H',H''$ Hopf algebras, not necessarily associated with groups.

Decomposition in 2d non-invertible gaugings

Abstract

We extend the decomposition conjecture to 2d quantum field theories with a gauged symmetry category for a finite-dimensional semisimple Hopf algebra with trivially-acting and the remaining symmetry, for finite groups. We check our extension by explicitly computing partition functions, and by verifying that previous results arise as special cases. Furthermore, we compute the topological operators responsible for enforcing the decomposition. Then, drawing from these results, we formulate a plausible decomposition conjecture for the even more general case of trivially-acting and the remaining symmetry, for Hopf algebras, not necessarily associated with groups.

Paper Structure

This paper contains 39 sections, 169 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. General conjecture
  4. Mathematical background
  5. Exact sequences of fusion categories
  6. Exact sequences of Hopf algebras
  7. Consistency check: partition function computations
  8. Gaugeable algebra
  9. Partition function computation
  10. Module decomposition
  11. Torus partition function
  12. Topological point operators and projectors
  13. Review of the invertible case
  14. Projection operators for Hopf algebra abelian extensions
  15. Examples
  16. ...and 24 more sections

Figures (6)

  • Figure 1: A trivially-acting Topological Defect Line (TDL) $L_k$ is connected to the trivial TDL $L_1$ by a Topological Point Operator (TPO) $\sigma_k$.
  • Figure 2: The TPO $\sigma_k$ from Figure \ref{['diagram:tpo-definition']} is equivalently depicted as an operator living at the end of a semi-infinite trivially-acting TDL $L_k$.
  • Figure 3: Action of TDL $L_{\gamma}$ on TPO $\sigma_k$.
  • Figure 4: Action of the effectively-acting TDL $L_{g}\in {\rm ob}({\rm Vec}(G))$ on TPO $\sigma_k$.
  • Figure 5: The TPO's $\sigma_{k_i}$ inherit a product from a choice of Frobenius multiplication $\mu:H \otimes H \to H$ via the inclusion $L_{k_i} \subset H=\bigoplus_{\gamma \in \Gamma}L_{\gamma}$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Conjecture