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MTC$[M_3, G]$: 3d Topological Order Labeled by Seifert Manifolds

Federico Bonetti, Sakura Schafer-Nameki, Jingxiang Wu

Abstract

We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group $G$. Topological order in 2+1d is known to be characterized in terms of modular tensor categories (MTCs), and we thus propose a relation between MTCs and Seifert three-manifolds. The correspondence defines for every Seifert manifold and choice of $G$ a fusion category, which we conjecture to be modular whenever the Seifert manifold has trivial first homology group with coefficients in the center of $G$. The construction determines the spins of anyons and their S-matrix, and provides a constructive way to determine the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realizing all MTCs (unitary or non-unitary) with rank $r\leq 5$ in terms of Seifert manifolds and a choice of Lie group $G$.

MTC$[M_3, G]$: 3d Topological Order Labeled by Seifert Manifolds

Abstract

We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group . Topological order in 2+1d is known to be characterized in terms of modular tensor categories (MTCs), and we thus propose a relation between MTCs and Seifert three-manifolds. The correspondence defines for every Seifert manifold and choice of a fusion category, which we conjecture to be modular whenever the Seifert manifold has trivial first homology group with coefficients in the center of . The construction determines the spins of anyons and their S-matrix, and provides a constructive way to determine the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realizing all MTCs (unitary or non-unitary) with rank in terms of Seifert manifolds and a choice of Lie group .
Paper Structure (15 sections, 1 theorem, 90 equations, 2 figures, 10 tables)

This paper contains 15 sections, 1 theorem, 90 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Seifert Manifolds and Flat Connections
  3. Seifert 3-manifolds
  4. Flat Connections
  5. Anyonic Flat Connections
  6. 3d Topological Order and MTCs
  7. MTCs from Seifert Manifolds
  8. All low rank MTCs from MTC(g,M3)
  9. More on Seifert Manifolds
  10. Mapping Anyons to Anyonic Flat Connections
  11. The case where gcd for all Seifert fibers
  12. The case N prime with some Seifert fibers with gcd
  13. Fundamental Weyl Alcove for Simply-laced g
  14. Tables of MTCs from M3 for Rank r<=5
  15. Rank 6

Key Result

Proposition 1

Fix $N\ge 2$ and consider the Seifert manifolds where $\gcd(N,q_k)=1$ for $k=1,\dots,n$, and $1\le t <n$, with $p_k>N$ for $k=1,\dots,t$. Then $\mathsf{RFC} ( \mathfrak{sl}(N) , M_3 )$ and $\mathsf{RFC} ( \mathfrak{sl}(N) , \widetilde{M}_3 )$ have the same modular data.

Figures (2)

  • Figure 1: Fundamental affine Weyl alcove ${\Delta}_{3,7}$ for $\mathfrak{sl}(3)$ with $\phi(\lambda) = 0,1,2$ for blue, green and orange respectively. Here $s_i$ are Weyl reflections, which label all the different alcoves. We will restrict to the fundamental one (labeled by 1).
  • Figure 2: Fundamental Weyl alcove for $\mathfrak{sl}(2)$ with $p=3, 3,3,4$ respectively. $\mathbb{Z}_2$-grading is depicted by the color blue (even) and green (odd).

Theorems & Definitions (4)

  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Proposition 1