[Paper Review] Fermionic Rational Conformal Field Theories and Modular Linear Differential Equations

Jin-Beom Bae, Zhihao Duan+3 more|arXiv (Cornell University)|Oct 23, 2020

Algebraic structures and combinatorial models51 references7 citations

TL;DR

This paper extends the Modular Linear Differential Equation (MLDE) method to classify fermionic Rational Conformal Field Theories (RCFTs) with non-negative integer coefficients in their q-series characters. By formulating MLDEs for level-two congruence subgroups Γθ, Γ⁰(2), and Γ₀(2), corresponding to different spin structures on the torus, the authors derive first- and second-order holomorphic MLDEs without poles, identifying a large class of fermionic RCFTs, some of which are supersymmetric, and providing a foundation for classifying fermionic Modular Tensor Categories.

ABSTRACT

We define Modular Linear Differential Equations (MLDE) for the level-two congruence subgroups $Γ_\vartheta$, $Γ^0(2)$ and $Γ_0(2)$ of $ ext{SL}_2(\mathbb Z)$. Each subgroup corresponds to one of the spin structures on the torus. The pole structures of the fermionic MLDEs are investigated by exploiting the valence formula for the level-two congruence subgroups. We focus on the first and second order holomorphic MLDEs without poles and use them to find a large class of `Fermionic Rational Conformal Field Theories', which have non-negative integer coefficients in the $q$-series expansion of their characters. We study the detailed properties of these fermionic RCFTs, some of which are supersymmetric. This work also provides a starting point for the classification of the fermionic Modular Tensor Category.

Motivation & Objective

• To systematically classify fermionic Rational Conformal Field Theories (RCFTs) with non-negative integer coefficients in their q-series characters.
• To extend the Modular Linear Differential Equation (MLDE) method to level-two congruence subgroups Γθ, Γ⁰(2), and Γ₀(2), each associated with a distinct spin structure on the torus.
• To identify a large class of fermionic RCFTs, including supersymmetric examples, through holomorphic MLDEs without poles.
• To provide a starting point for the classification of fermionic Modular Tensor Categories (MTCs), which are essential for classifying fermionic topological phases of matter.

Proposed method

• Formulate Modular Linear Differential Equations (MLDEs) for the level-two congruence subgroups Γθ, Γ⁰(2), and Γ₀(2), corresponding to NS, ṼNS, and R spin structures on the torus.
• Use the valence formula for level-two congruence subgroups to analyze the pole structures of the fermionic MLDEs.
• Focus on first- and second-order holomorphic MLDEs without poles to ensure physical consistency and integer character coefficients.
• Solve the MLDEs to obtain q-series expansions of characters with non-negative integer coefficients, identifying candidate fermionic RCFTs.
• Classify solutions into BPS and non-BPS types based on conformal weight and central charge, with detailed tables of solutions for various central charges.
• Relate the resulting fermionic RCFTs to potential fermionic Modular Tensor Categories (MTCs), noting that different RCFTs with same fusion rules correspond to the same MTC.

Experimental results

Research questions

• RQ1 How can the MLDE method be extended to classify fermionic RCFTs associated with level-two congruence subgroups of SL₂(ℤ)?
• RQ2 What are the constraints on the pole structure of fermionic MLDEs derived from the valence formula for level-two congruence subgroups?
• RQ3 Which holomorphic MLDEs without poles yield fermionic RCFTs with non-negative integer coefficients in their q-series expansions?
• RQ4 Which of the identified fermionic RCFTs are supersymmetric, and how do they relate to known superconformal models?
• RQ5 How do the solutions of the fermionic MLDEs contribute to the classification of fermionic Modular Tensor Categories?

Key findings

• The authors derive and solve first- and second-order holomorphic MLDEs for the level-two congruence subgroups Γθ, Γ⁰(2), and Γ₀(2), yielding a large class of fermionic RCFTs with non-negative integer coefficients in their q-series characters.
• Solutions are tabulated for central charges c = 10 to c = 47/2, with explicit q-series expansions for NS and R sector characters, such as c = 10: q⁻¹/⁶(5 + 1004q + 20510q² + ...), and c = 47/2: q⁻⁴⁷/⁴⁸(1 + 4371q³/² + 96256q² + ...).
• For c = 11/2, the R-sector character is q⁻¹/¹²(11 + 2026q + 45067q² + ...), and for c = 23/2, it is q⁻²³/²⁴(1 + 23q + 4600q³/² + ...), showing consistent non-negative integer coefficients.
• The NS-sector solutions include c = 10: q⁻¹/⁶(5 + 1004q + 20510q² + ...), and c = 23: q⁻²³/²⁴(1 + 23q + 4600q³/² + ...), with conformal weights and coefficients satisfying the modular invariance and integrality conditions.
• The classification includes both BPS and non-BPS types, with specific μ₁, μ₂, μ₃ parameters for each solution, such as (3/22, -15/176, -10/99) for c = 10 in non-BPS type IV.
• The results provide a systematic starting point for classifying fermionic Modular Tensor Categories, as different fermionic RCFTs with identical fusion rules correspond to the same fermionic MTC.