Completing the fermionic MLDE classification of two-dimensional fermionic rational conformal field theories and its dictionary to fermionic modular tensor categories

The characters of a two-dimensional fermionic rational conformal field theory (RCFT) solve fermionic modular linear differential equations (MLDEs) - modular-covariant differential equations for the level-two congruence subgroups of the modular group associated to torus spin structures. The initiating classification by Bae, Duan, Lee, Lee and Sarkis settled only the first- and second-order pole-free equations and stated six open directions. Later work by the same group and others classified the three-character solutions with vanishing pole index that saturate the supersymmetric unitarity bound, and - conditional on an unproved fermionic analogue of the unbounded-denominators (integrality) conjecture - the non-degenerate fermionic RCFTs of rank two with pole index at most one and, assuming unitarity, those of rank three with pole index at most one and of rank four with vanishing pole index. The surviving problem is to: classify fermionic RCFT characters by fermionic MLDEs at arbitrary rank and pole index without the integrality hypothesis, including degenerate and non-unitary sectors; identify or exclude every admissible solution, settling the currently unidentified candidates and those discarded for negative fusion coefficients; settle whether the indicated emergent $\mathcal{N}=1$ supersymmetry of fermionized WZW models is realized algebraically; and establish the precise correspondence with fermionic modular tensor categories, including the unfurling of reduced to refined modular matrices. No resolution of these cores is known in the literature to date; the problem remains open.

Background

A two-dimensional conformal field theory (CFT) is called rational (an RCFT) when its torus partition function decomposes into a finite sum $Z(\tau,\bar\tau)=\sum_{ij}M_{ij}\chi_i(\tau)\bar\chi_j(\bar\tau)$ of holomorphic times antiholomorphic factors. The finitely many holomorphic functions $\chi_i(\tau)=\mathrm{Tr}\,q^{L_0-c/24}$ with $q=e^{2\pi i\tau}$ are the characters of an extended chiral algebra containing the Virasoro algebra. Modular invariance of $Z$ forces the character vector to transform as a weight-zero vector-valued modular form of the modular group $SL(2,\mathbb{Z})$, so the $d$ independent characters of an RCFT solve a $d$-th order modular linear differential equation (MLDE): an equation built from the modular-covariant derivative $D=(2\pi i)^{-1}d/d\tau$ whose coefficient functions are holomorphic modular forms. The classical valence formula, which constrains the zeros of a modular form of given weight, restricts the pole structure of these coefficients to a single non-negative integer $\ell$ (the Wronskian or pole index), and for each pair $(d,\ell)$ the search for character-like solutions - those with non-negative integral $q$-series coefficients - becomes a systematic scan. Initiated by Mathur, Mukhi and Sen, this holomorphic modular bootstrap classified the two-character bosonic RCFTs and much of the three-character landscape.

A fermionic CFT contains fields of half-integer spin and can be placed on a torus only after choosing one of four spin structures, labeled NS, $\widetilde{\mathrm{NS}}$, $\widetilde{\mathrm{R}}$ and $\mathrm{R}$ according to whether the fermions are anti-periodic or periodic around each cycle. The characters in these sectors transform as vector-valued modular forms not of $SL(2,\mathbb{Z})$ but of its level-two congruence subgroups $\Gamma_\theta$ (generated by the modular transformations $S:\tau\mapsto-1/\tau$ and $T^2$, where $T:\tau\mapsto\tau+1$), $\Gamma_0(2)$ and $\widetilde{\Gamma}_0(2)$, which are permuted by the full modular group. Bae, Duan, Lee, Lee and Sarkis derived the level-two valence formula, an arithmetic identity tying the central charge $c$, the Neveu-Schwarz and Ramond conformal weights $h^{\mathrm{NS}}$ and $h^{\mathrm{R}}$, the character-rank $d$ and the pole index $\ell$, and introduced the corresponding fermionic MLDEs. They classified the first-order case (tensor products of free chiral Majorana fermions) and the second-order case with trivial pole structure $\ell=0$, finding six families of solutions, deriving the fermionic $S$-matrices, and identifying the consistent solutions with fermionized Wess-Zumino-Witten (WZW) models and supersymmetric minimal models. Fermionization - the inverse of bosonization (GSO projection), generalizing the Jordan-Wigner map - turns a bosonic CFT with a non-anomalous $\mathbb{Z}_2$ symmetry into a fermionic one; WZW models are the RCFTs whose chiral algebra is an affine Lie algebra.

Physical consistency of a candidate solution requires more than non-negative integral $q$-coefficients: the fusion coefficients $N_{ij}^{\;\;k}$ - the structure constants of the operator-product algebra of the primary fields, computable from the modular $S$-matrix by the Verlinde formula - must be non-negative integers. Several second-order solutions fail this test and were discarded, while others remain unidentified. Solutions whose Ramond-sector weights saturate $h^{\mathrm{R}}\ge c/24$ - the unitarity bound of theories with $\mathcal{N}=1$ supersymmetry, meaning an extension of the chiral algebra by a weight-$3/2$ supercurrent - exhibit character-level signs of supersymmetry such as a constant Ramond-Ramond partition function, and the source suggests interpreting them as supersymmetric RCFTs while leaving the construction of the supercurrent open.

Characters determine a theory only up to multiplicity: distinct primaries may share a character, so the reduced modular data of the characters must be completed (unfurled) into refined modular matrices carrying fermionic modular tensor category (fMTC) data - the algebraic structure encoding the anyon content of fermionic topological phases of matter (spin topological quantum field theories), studied mathematically through super-modular categories and classified there only at low rank. The stated motivation of the source program is that a completed fermionic-RCFT classification would deliver the corresponding fMTC classification.

As of 2026 the classification is complete only in part: the three-character fermionic RCFTs with $\ell=0$ in the subclass saturating the supersymmetric unitarity bound are classified, and - conditional on a fermionic analogue of the integrality (unbounded denominators) conjecture, proved only for $SL(2,\mathbb{Z})$ by Calegari, Dimitrov and Tang - so are the non-degenerate fermionic RCFTs of rank two with pole index at most one (non-unitary cases included) and, assuming unitarity, those of rank three with pole index at most one and of rank four with vanishing pole index. Complementary structure results - a fermionic Hecke operator relating the characters of different fermionic RCFTs, and fermionization-based classifications in fixed symmetry families - advance the identification of solutions but do not classify the general sectors. Rank $d\ge5$, general $\ell$, degenerate and non-unitary sectors, the conjecture itself, several unmatched solutions, the supercurrent constructions, and the category dictionary remain open; addressing them jointly is the problem stated below.

Problem Statement

The problem is to complete the fermionic-MLDE classification of two-dimensional fermionic rational conformal field theories and to establish its dictionary to fermionic modular tensor categories. The overarching objective is a complete, identified census of fermionic RCFT characters; four linked deliverables jointly support it.

(1) Classification at general rank and pole index. For every character-rank $d\ge2$ and pole index $\ell\ge0$, classify the solutions of the $d$-th order fermionic MLDEs for $\Gamma_\theta$, $\Gamma_0(2)$ and $\widetilde{\Gamma}_0(2)$ whose $q$-series in the NS, $\widetilde{\mathrm{NS}}$ and $\mathrm{R}$ sectors have non-negative integral coefficients. This must cover the degenerate spectra (a pair of Neveu-Schwarz weights differing by an integer multiple of a half-integer, or a pair of Ramond weights differing by an integer) and the non-unitary sectors beyond the rank-two cases already classified, and may be achieved by proving the fermionic integrality conjecture for vector-valued modular forms of these subgroups, by a rigorous truncation of the admissible exponents, or by any other complete method.

(2) Admissibility and identification. For each candidate solution, determine whether it admits a consistent fusion-rule algebra (non-negative integral fusion coefficients obtained from the Verlinde formula applied to the fermionic $S$-matrix) and identify it with a constructed fermionic CFT - a fermionized WZW model or its $\mathbb{Z}_2$ orbifold, a supersymmetric minimal model, a coset, or a fermionic Hecke image - or prove that no consistent fermionic RCFT interpretation exists. This includes settling the currently unidentified candidates (for example the three-character BPS solution - one whose Ramond-sector weights saturate the bound $h^{\mathrm{R}}\ge c/24$ - with $c=66/5$), deciding whether alternative extensions or multiplicity assignments rescue the second-order solutions discarded for negative fusion coefficients, and settling the parametrized families such as the $c=16$ one-parameter family of second-order solutions, truncated at $n<240$ by fusion consistency.

(3) Emergent supersymmetry. For each fermionic solution whose Ramond-sector data saturate $h^{\mathrm{R}}\ge c/24$ (for example the fermionizations of the level-six $\widehat{su}(2)$ and level-three $\widehat{e}_6$ WZW models), either exhibit the weight-$3/2$ supercurrent generating the $\mathcal{N}=1$ superconformal algebra inside the fermionized theory, or prove that supersymmetry is absent or spontaneously broken.

(4) Category dictionary. Establish the precise correspondence between the character classification and fermionic modular tensor categories (equivalently super-modular categories or fermionic topological orders), in particular a systematic procedure unfurling the reduced modular matrices of the characters to the refined modular matrices that carry the category data.

Each deliverable is resolved independently, and partial results are credited at that granularity: deliverable (1) is resolved at a fixed $(d,\ell)$ when the complete list of admissible MLDE solutions there - their exponents and characters - is exhibited, with the identification or exclusion of those solutions reserved to the separate standard of deliverable (2); deliverable (2) is resolved for a given candidate, family, or sector when that piece is either identified with a constructed fermionic CFT or proven not to admit a consistent interpretation; deliverable (3) is resolved for a given solution when the weight-$3/2$ supercurrent is exhibited or supersymmetry is proven absent or spontaneously broken; and deliverable (4) is resolved when the reduced modular matrices are unfurled into refined modular matrices carrying the fMTC data. The program counts as complete when every admissible solution is identified or excluded at all $(d,\ell)$ and the dictionary of deliverable (4) is established.

References

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