Rational CFT & the modular bootstrap

• Introduces a differential Θ-map relating modular differential equations and quasi-characters at different Wronskian indices.
• Builds rank-two and rank-three constructions that connect higher-index sectors to simpler seed equations with vanishing Wronskian index.
• Tracks modular transformation properties and integrality through the maps, giving a systematic way to organize families across the Wronskian tower.
• Uses the map together with earlier sign estimates to establish previously conjectured sign and growth properties of rank-two quasi-characters with Wronskian index ℓ = 2.

Scope. The reduction is made within the non-logarithmic, irreducible class treated in the paper. Positivity is not generally preserved, so mapped quasi-characters still need admissibility checks and CFT realization is a separate question.

• Proves the alternating-sign patterns and eventual sign stabilization of rank-two quasi-character families with vanishing Wronskian index.
• Develops both a controlled large-parameter analysis and an exact induction argument from the MLDE coefficient recursions.
• Estimates coefficient growth near n ∼ |c|/12, where the signs stabilize and the usual fixed-central-charge Cardy asymptotics do not directly apply.
• Provides quantitative ingredients for combining quasi-characters into spectra with non-negative integer degeneracies.

Scope. The paper studies rank-two, ℓ = 0 quasi-characters. An admissible spectrum is a candidate for an RCFT; it does not by itself establish a complete conformal field theory.

• Parametrizes two-character modular differential equations with arbitrary numbers of poles at movable points of moduli space.
• Derives accessory-parameter constraints by excluding logarithmic singularities in the character solutions.
• Obtains recursion relations between sectors with different numbers of poles by studying limits in which a pole moves to infinity.
• Solves the one- and two-movable-pole cases in detail, finds agreement with quasi-character constructions, and identifies genuine CFT examples.

Scope. The general derivation assumes generic pole locations, with coincident poles discussed separately. The existence of infinitely many admissible character vectors does not imply infinitely many CFT realizations at fixed index.

• Uses bilinear coset-type pairings with meromorphic characters to test which admissible three-character spectra can describe CFTs.
• Classifies the unitary, vanishing-Wronskian-index three-character theories outside the exceptional central charges c = 8 and c = 16: two infinite affine series and 45 additional theories.
• Separates consistent CFT realizations from inadmissible multiplicities, excluded CFT candidates and possible intermediate vertex operator algebras.
• Derives constraints on meromorphic theories through pairings at total central charges 8, 16, 24, 32 and 40.

Scope. The c = 8 and c = 16 sectors are excluded from the classification. The paper also explicitly notes representation-theoretic and embedding subtleties in turning all physics-level coset arguments into rigorous VOA statements.

• Reverses the usual coset logic: uses lower-central-charge CFT pairings to predict new meromorphic theories at higher central charge.
• Proposes 34 infinite families and 46 examples at c = 32 and c = 40 through three-character coset relations.
• Identifies proposed non-lattice theories whose current algebras include non-simply-laced factors or levels greater than one.
• Shows how periodic modular data in the B- and D-series supports a repeatable construction across increasing central charge.

Scope. These are proposals backed by detailed physics-level evidence; the paper does not claim a complete classification or a rigorous construction of every proposed VOA.

• Turns the search for three-character solutions with Wronskian index zero or two and 0 < c ≤ 96 into a finite set of Diophantine computations.
• Combines integrality constraints with high-order Fourier-series checks to organize admissible and character-like solutions.
• Identifies seven then-new candidate spectra in the zero-index sector, alongside known affine and coset examples.
• Shows that the index-two admissible solutions in the studied range arise by adjoining a constant character to a two-character, zero-index solution.

Scope. The stated search range is 0 < c ≤ 96. Candidate spectra are distinguished from proven CFTs, and the later coset-classification paper further tests their realization.

• Computes character counts and Wronskian indices for broad families of WZW, Virasoro-minimal and N = 1 super-Virasoro theories.
• Derives analytic formulas in tractable rank and level regimes and supplements them with systematic computations.
• Shows that level-one WZW theories with at least two characters have vanishing Wronskian index; the one-character E₈ theory is an explicit exception.
• Builds a catalogue of known theories in MLDE parameter space, providing benchmarks and partial classifications for bootstrap searches.

Scope. This organizes known families and gives partial classifications; it does not classify every RCFT for a given character count and Wronskian index.

• Constructs a Serre–Ramanujan-type derivative for Fricke groups of prime level and uses it to formulate higher-order MLDEs.
• Derives Ramanujan–Eisenstein identities adapted to these groups.
• Finds one-, two- and three-character admissible solutions at levels two and three, including examples related to McKay–Thompson series.
• Develops bilinear identities and candidate partition functions, connecting the solutions to modular forms and possible lattice interpretations.

Scope. Admissible series and putative partition functions are not all established as physical CFTs. The concrete solution analysis concentrates on Fricke levels two and three.

• Re-derives the constraint that meromorphic CFT central charges are multiples of eight using the one-character MLDE framework.
• Sets up the differential equation for arbitrary Wronskian index and analyzes the modular properties of its solutions.
• Uses Rademacher asymptotics to show why odd-index solutions fail the required admissibility conditions.

Scope. This is a new route to an established constraint in the meromorphic, one-character setting, rather than a classification of all such CFTs.