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

F Bonetti, Sakura Schäfer‐Nameki+1 more|arXiv (Cornell University)|Mar 6, 2024

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper proposes a novel correspondence between 3d topological orders—characterized by modular tensor categories (MTCs)—and Seifert three-manifolds equipped with ADE gauge groups. It constructs MTCs from flat connections on Seifert manifolds, conjecturing modularity when the first homology with coefficients in the gauge group's center vanishes, and successfully realizes all known MTCs up to rank 5 within this framework.

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 $G$.

Motivation & Objective

• To establish a new framework for classifying 3d topological orders using Seifert three-manifolds and ADE gauge groups.
• To provide a constructive method for computing modular data (S- and T-matrices), R- and F-symbols, and anyon spins from topological data.
• To test whether this construction realizes all known modular tensor categories up to rank 5, including both unitary and non-unitary cases.
• To identify a topological criterion—vanishing first homology with coefficients in the gauge group's center—for the resulting category to be modular.
• To explore whether this correspondence offers an alternative, geometric classification of MTCs beyond traditional modular data.

Proposed method

• The construction maps each Seifert manifold, specified by n pairs of coprime integers (p_i, q_i) encoding singular fibers, to a fusion category via a graded Deligne product of pre-modular categories C(sl(N), p_i, q_i).
• Anyons are modeled as anyonic flat G_C connections on the Seifert manifold, with topological invariants derived from Chern-Simons theory and Reidemeister torsion.
• The modular S- and T-matrices are computed from the flat connection data, and the F- and R-symbols are reconstructed from simple building blocks.
• The conjecture is that the resulting category is modular if and only if H₁(M₃, Z_G) = 0, where Z_G is the center of the gauge group G.
• The framework is tested by realizing all known MTCs of rank r ≤ 5 using Seifert manifolds and simply-laced Lie groups G of ADE type.
• The method leverages known results on modular data classification and compares the output with existing tables of MTCs, including non-unitary examples.

Experimental results

Research questions

• RQ1 Can all known 3d topological orders, including non-unitary ones, be realized via flat connections on Seifert three-manifolds with ADE gauge groups?
• RQ2 What topological condition on the Seifert manifold ensures that the resulting fusion category is modular?
• RQ3 How do the S- and T-matrices, spins, and braiding statistics of anyons emerge from the geometry of the Seifert manifold and gauge group?
• RQ4 Does this construction provide a complete and geometric classification of MTCs up to rank 5?
• RQ5 Why does this framework succeed where previous constructions (e.g., Cho et al.) fail to realize all rank 5 MTCs?

Key findings

• All known modular tensor categories of rank r ≤ 5—both unitary and non-unitary—are successfully realized within the MTC[M₃, G] framework using Seifert manifolds and ADE gauge groups.
• The construction produces the correct S- and T-matrices for all such MTCs, with explicit formulas for anyon spins and braiding statistics.
• The conjecture that MTC[M₃, G] is modular if and only if H₁(M₃, Z_G) = 0 is strongly supported by the realization of all low-rank MTCs under this condition.
• The framework reproduces 192 distinct MTCs of rank 6, including previously missing models that were not realizable in earlier proposals such as Cho:2020ljj.
• The method provides a constructive path to computing F- and R-symbols from topological data, offering a geometric origin for the full anyonic data.
• The results suggest that the Seifert manifold–Gauge group correspondence may offer a complete, geometric classification of MTCs, surpassing prior approaches in coverage.