[Paper Review] Classification of Rank 6 Modular Categories with Galois Group $\langle (012)(345) angle$

David J. Green|arXiv (Cornell University)|Aug 20, 2019

Algebraic structures and combinatorial models8 references4 citations

TL;DR

This paper classifies rank 6 modular tensor categories with Galois group ⟨(012)(345)⟩ using Galois theory and computational algebraic geometry. It proves that all such categories have modular data conjugate to either the product of the semion category with (A₁,5)₁/₂ or a specific subcategory of 𝒞(𝔰𝔬₅,9,eʲπi/9) with gcd(18,j)=1, and identifies the fusion rules of PSO(5)₃/₂ as the smallest with no unitary realization.

ABSTRACT

Modular Tensor Categories (MTC's) arise in the study of certain condensed matter systems. There is an ongoing program to classify MTC's of low rank, up to modular data. We present an overview of the methods to classify modular tensor categories of low rank, applied to the specific case of a rank 6 category with Galois group $\langle(012)(345) angle$, and show that certain symmetries in this case imply nonunitarizable (hence, nonphysical) MTC's. We show that all the rank 6 MTC's with this Galois group have modular data conjugate to either the product of the semion category with $(A_1, 5)_{\frac{1}{2}}$ or a certain modular subcategory of $\mathcal{C}(\mathfrak{so}_5, 9, e^{jπi/9})$ with gcd$(18, j) = 1$.

Motivation & Objective

• To complete the classification of rank 6 modular tensor categories (MTCs) with a specific Galois group, ⟨(012)(345)⟩, extending prior work on low-rank MTCs.
• To determine whether such MTCs admit unitary realizations, addressing physical realizability in condensed matter systems.
• To identify the modular data of these categories up to Galois conjugacy, leveraging symmetries and algebraic constraints.
• To establish that the fusion rules of PSO(5)₃/₂ are the smallest known with no unitary realization, contributing to the understanding of nonunitarizable MTCs.

Proposed method

• Applies Galois theory to modular data, using the action of Galois automorphisms on the S-matrix and T-matrix to constrain possible solutions.
• Employs computational algebraic geometry via Gröbner bases to solve systems of polynomial equations derived from S-matrix orthogonality and twist relations.
• Uses Lemma 3.2 to classify fusion rules under the given Galois group, reducing the problem to solving constrained systems over number fields.
• Imposes constraints from admissible modular data: unitarity of S, finitely ordered T, integrality of fusion rules, and cyclotomic field conditions.
• Analyzes sign choices and initial ideals in polynomial systems to systematically eliminate inconsistent cases and reduce to known solutions.
• Verifies that all solutions correspond to modular data conjugate to either the semion category tensored with (A₁,5)₁/₂ or a subcategory of 𝒞(𝔰𝔬₅,9,eʲπi/9) with gcd(18,j)=1.

Experimental results

Research questions

• RQ1 What are the complete modular data for rank 6 modular tensor categories with Galois group ⟨(012)(345)⟩?
• RQ2 Which of these categories are unitary, and which are nonunitarizable, particularly in the context of physical realizability?
• RQ3 Can the fusion rules of PSO(5)₃/₂ be shown to be the smallest known with no unitary realization?
• RQ4 How do symmetries under the Galois group ⟨(012)(345)⟩ constrain the possible S and T matrices in rank 6 MTCs?
• RQ5 Are all such MTCs with this Galois group Galois-conjugate to known categories like the semion category or subcategories of 𝒞(𝔰𝔬₅,9,eʲπi/9)?

Key findings

• All rank 6 modular tensor categories with Galois group ⟨(012)(345)⟩ have modular data conjugate to either the product of the semion category with (A₁,5)₁/₂ or a modular subcategory of 𝒞(𝔰𝔬₅,9,eʲπi/9) with gcd(18,j)=1.
• The fusion rules of PSO(5)₃/₂, constructed in [12], are the smallest known to have no unitary realization, as established by this classification.
• The classification shows that certain symmetries in the Galois group imply nonunitarizable MTCs, thus excluding physical realizability.
• The T-matrix is fully determined by the S-matrix and the system of polynomial relations, including θ₁+θ₂+1=0 and θ₂²+θ₂+1=0, which fix the root of unity structure.
• The solution space reduces to known categories after eliminating all other sign choices and initial ideals via Gröbner basis computation.
• The Galois group action ensures that all solutions are constrained to two families: the semion product and the subcategory of 𝒞(𝔰𝔬₅,9,eʲπi/9) with gcd(18,j)=1.