pith. sign in

arxiv: 2507.07170 · v4 · submitted 2025-07-09 · ✦ hep-th · math-ph· math.MP· math.NT· math.QA· math.RT

Signs, growth and admissibility of quasi-characters and the holomorphic modular bootstrap for RCFT

Arpit Das , Sunil Mukhi This is my paper

Pith reviewed 2026-05-19 05:34 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPmath.NTmath.QAmath.RT
keywords quasi-charactersmodular linear differential equationsrational conformal field theoriesholomorphic modular bootstrapWronskian indexadmissible charactersFrobenius recursioncoefficient signs
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The pith

Quasi-characters for rank-2 RCFTs have coefficients with alternating signs that stabilize at n around c/12, estimated via Frobenius recursion on modular differential equations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the coefficients in a complete basis of rank-2 quasi-characters exhibit rigorously alternating signs that settle to one fixed sign in the regime n ~ c/12, together with a practical estimate of their growth with central charge. This regime lies beyond the reach of standard Cardy asymptotics, so the result supplies the missing control needed to decide when linear combinations of these quasi-characters yield admissible characters with non-negative integer coefficients. Because admissible characters are the building blocks of holomorphic partition functions, the sign and growth information directly constrains which candidate RCFTs can exist at arbitrary Wronskian index.

Core claim

Using the Frobenius recursion relations satisfied by the solutions of modular linear differential equations, the coefficients a_n of rank-2 quasi-characters are shown to alternate in sign and to approach a stable sign pattern once n reaches order c/12; the same recursion supplies an explicit leading growth law for these coefficients in the same range.

What carries the argument

Frobenius recursion relations for the coefficients of solutions to rank-2 modular linear differential equations; these relations encode the leading large-n behavior and thereby determine both the growth with c and the eventual sign stabilization.

If this is right

  • Candidate RCFT partition functions can now be assembled at arbitrarily large Wronskian index by combining quasi-characters whose coefficients satisfy the proved sign and growth conditions.
  • The holomorphic modular bootstrap is extended from the Cardy regime into the intermediate n ~ c/12 window for all rank-2 cases.
  • Any linear combination that would produce a negative coefficient at this order is immediately ruled out as an admissible character.
  • The same recursion technique supplies a uniform method to check admissibility for every entry in the classified list of quasi-characters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The recursion method may generalize directly to higher-rank quasi-characters once their modular differential equations are classified.
  • Sign stabilization at this order could be used as a quick numerical filter before attempting full modular invariance checks on candidate partition functions.
  • The growth estimate might be compared with explicit spectra of known RCFTs (such as minimal models) to test consistency outside the asymptotic regime.

Load-bearing premise

The assumption that the known complete list of rank-2 quasi-characters is exhaustive and that the modular linear differential equation plus its Frobenius recursion fully capture the leading asymptotics without further constraints from modular invariance at higher orders.

What would settle it

Explicit computation of the first few dozen coefficients for any rank-2 quasi-character at a central charge where n ≈ c/12; the signs must alternate and then lock to one value, and the magnitude must follow the predicted growth law.

read the original abstract

Rational conformal field theories in 2d have partition functions built from holomorphic characters, whose classification can be addressed via the holomorphic modular bootstrap. This is facilitated by a special basis of ``quasi-characters'' that has been completely classified for rank-2. Suitably combining these to form admissible characters with non-negative integral coefficients $a_n$ depends crucially on the signs and growth of the quasi-character coefficients. We use Frobenius recursion relations for Modular Linear Differential Equations to estimate the growth with $c$ of these coefficients in the region $n\sim\frac{c}{12}$ that is inaccessible to Cardy asymptotics, and to prove rigorously that they have alternating signs that stabilise to a fixed sign at this order. This provides a practical path to obtain candidate RCFT partition functions at arbitrary Wronskian index.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that for the completely classified rank-2 quasi-characters of RCFTs, the Frobenius recursion relations obtained from the associated Modular Linear Differential Equations can be used both to estimate the growth of the coefficients a_n with central charge c in the regime n ∼ c/12 (inaccessible to Cardy asymptotics) and to prove rigorously that these coefficients alternate in sign and stabilize to a fixed sign at this order. The resulting sign and growth information is then applied to construct admissible linear combinations that yield candidate RCFT partition functions with non-negative integral coefficients at arbitrary Wronskian index.

Significance. If the central claims hold, the work supplies an analytic handle on the holomorphic modular bootstrap that is complementary to both Cardy asymptotics and numerical fitting. The explicit use of recursion relations derived from the indicial equation, together with the complete classification of rank-2 quasi-characters, provides a parameter-free route to sign stabilization that can be checked against the already-classified basis; this is a concrete strength for constructing admissible characters beyond low Wronskian index.

major comments (2)
  1. [§4.2, Eq. (4.7)] §4.2, Eq. (4.7): the recursion is derived from the indicial equation of the rank-2 MLDE and alternation is shown for the leading large-n term, yet the stabilization argument does not supply an explicit bound on the remainder arising from the requirement that the linear combination remain invariant under the full SL(2,ℤ) action at higher orders in q. Without such a bound it is not immediate that the leading-term sign dominates for all admissible combinations when n ∼ c/12 and c is finite but large.
  2. [§5.1, Table 2] §5.1, Table 2: the numerical verification for the first few quasi-characters shows clear alternation up to n ≈ 100, but the table does not report the size of the next-to-leading correction relative to the leading term; this datum is needed to assess whether the stabilization proven for the leading coefficient survives the full modular-invariance constraint at the order relevant to the central claim.
minor comments (2)
  1. The definition of the Wronskian index is introduced only in §2; a brief reminder in the abstract or introduction would improve readability for readers outside the immediate subfield.
  2. [§3] Reference to the prior complete classification of rank-2 quasi-characters (cited in §3) should include the explicit arXiv number or journal reference in the text rather than only in the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised concern the rigor of the sign-stabilization proof and the quantitative support from numerics. We address each below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4.2, Eq. (4.7)] the recursion is derived from the indicial equation of the rank-2 MLDE and alternation is shown for the leading large-n term, yet the stabilization argument does not supply an explicit bound on the remainder arising from the requirement that the linear combination remain invariant under the full SL(2,ℤ) action at higher orders in q. Without such a bound it is not immediate that the leading-term sign dominates for all admissible combinations when n ∼ c/12 and c is finite but large.

    Authors: The Frobenius recursion is obtained directly from the MLDE satisfied by the quasi-characters, which already incorporates the modular transformation properties under the generators of SL(2,ℤ). For admissible linear combinations, the full invariance is enforced by the choice of coefficients that cancel non-invariant terms. We agree, however, that an explicit remainder bound would make the dominance of the leading sign fully transparent for finite but large c. In the revision we will add a lemma deriving such a bound from the Wronskian index and the already-established growth estimates, confirming that the leading alternating term controls the sign for n ≳ c/12. revision: yes

  2. Referee: [§5.1, Table 2] the numerical verification for the first few quasi-characters shows clear alternation up to n ≈ 100, but the table does not report the size of the next-to-leading correction relative to the leading term; this datum is needed to assess whether the stabilization proven for the leading coefficient survives the full modular-invariance constraint at the order relevant to the central claim.

    Authors: We thank the referee for this observation. While the existing table demonstrates clear sign alternation, including the relative magnitude of sub-leading terms will better quantify how rapidly the leading behavior dominates. In the revised manuscript we will extend Table 2 (or add a companion table) to report the ratio of the next-to-leading to leading coefficients for each listed quasi-character, thereby providing direct numerical evidence that the stabilization persists under the modular constraints at the relevant orders. revision: yes

Circularity Check

0 steps flagged

No circularity: signs and growth derived independently via Frobenius recursion on classified inputs

full rationale

The paper takes the complete classification of rank-2 quasi-characters as an input from prior literature and then applies Frobenius recursion relations extracted from the indicial equation of the associated modular linear differential equations. This recursion directly yields the leading large-n asymptotic behavior and the alternating sign pattern for coefficients in the n ∼ c/12 regime. Because the recursion is a standard, parameter-free consequence of the MLDE and produces new quantitative statements (growth rate with c and stabilized sign) that are not presupposed by the classification, the central claims do not reduce to the inputs by construction. No fitted parameters are renamed as predictions, no self-citation is invoked to forbid alternatives, and no ansatz is smuggled in; the derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior complete classification of rank-2 quasi-characters and on the standard properties of modular linear differential equations; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The quasi-characters for rank-2 RCFTs have already been completely classified in prior work.
    Invoked implicitly when the authors state that suitably combining these quasi-characters yields admissible characters.
  • domain assumption Frobenius recursion relations derived from modular linear differential equations remain valid and capture the leading large-n asymptotics in the window n∼c/12.
    This is the mathematical engine used to prove sign stabilization and growth estimates.

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Forward citations

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · cited by 1 Pith paper · 11 internal anchors

  1. [1]

    Towards a Classification of Two-Character Rational Conformal Field Theories

    A.R. Chandra and S. Mukhi, Towards a Classification of Two-Character Rational Conformal Field Theories, JHEP 04 (2019) 153 [ 1810.09472]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  2. [2]

    Mathur, S

    S.D. Mathur, S. Mukhi and A. Sen, On the Classification of Rational Conformal Field Theories, Phys. Lett. B213 (1988) 303

    work page 1988

  3. [3]

    Mathur, S

    S.D. Mathur, S. Mukhi and A. Sen, Reconstruction of Conformal Field Theories From Modular Geometry on the Torus , Nucl. Phys. B318 (1989) 483

    work page 1989

  4. [4]

    Kaneko and D

    M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, AMS/IP Studies in Advanced Mathematics 7 (1998) 97

    work page 1998

  5. [5]

    Kaneko and M

    M. Kaneko and M. Koike, On Modular Forms Arising from a Differential Equation of Hypergeometric Type, The Ramanujan Journal 7 (2003) 145

    work page 2003

  6. [6]

    Kaneko, On Modular forms of Weight (6n + 1)/5 Satisfying a Certain Differential Equation, in Number Theory, W

    M. Kaneko, On Modular forms of Weight (6n + 1)/5 Satisfying a Certain Differential Equation, in Number Theory, W. Zhang and Y. Tanigawa, eds., (Boston, MA), pp. 97–102, Springer US, 2006

    work page 2006

  7. [7]

    On classification of modular tensor categories

    E. Rowell, R. Stong and Z. Wang, On classification of modular tensor categories , Communications in Mathematical Physics 292 (2009) 343 [ 0712.1377]. 20Numerically, we find that for j = 1 and n = 2 M − 1 with large 2 ≤ M ≤ 200, β∗ ∈ (0, 1). We conclude this by noting that for β = 0, ˜r(0, M) < 0 and |˜r(0, M)| increases with M. Additionally, for β = 1, ˜r...

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  8. [8]

    Conformal characters and the modular representation

    P. Bantay and T. Gannon, Conformal characters and the modular representation , JHEP 02 (2006) 005 [ hep-th/0512011]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  9. [9]

    Bantay and T

    P. Bantay and T. Gannon, Vector-valued modular functions for the modular group and the hypergeometric equation, Commun. Num. Theor. Phys. 1 (2007) 651

    work page 2007

  10. [10]

    The theory of vector-modular forms for the modular group

    T. Gannon, The theory of vector-modular forms for the modular group , Contrib. Math. Comput. Sci. 8 (2014) 247 [ 1310.4458]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  11. [11]

    Das, C.N

    A. Das, C.N. Gowdigere, S. Mukhi and J. Santara, Modular differential equations with movable poles and admissible RCFT characters , JHEP 12 (2023) 143 [ 2308.00069]

    work page arXiv 2023

  12. [12]

    Mukhi, R

    S. Mukhi, R. Poddar and P. Singh, Rational CFT with three characters: the quasi-character approach, JHEP 05 (2020) 003 [ 2002.01949]

    work page arXiv 2020

  13. [13]

    Meromorphic c=24 Conformal Field Theories

    A.N. Schellekens, Meromorphic c = 24 conformal field theories , Commun. Math. Phys. 153 (1993) 159 [ hep-th/9205072]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  14. [14]

    Hecke Relations in Rational Conformal Field Theory

    J.A. Harvey and Y. Wu, Hecke Relations in Rational Conformal Field Theory , JHEP 09 (2018) 032 [ 1804.06860]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  15. [15]

    Curiosities above c = 24

    A.R. Chandra and S. Mukhi, Curiosities above c = 24 , SciPost Phys. 6 (2019) 053 [1812.05109]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  16. [16]

    Das, C.N

    A. Das, C.N. Gowdigere and S. Mukhi, Meromorphic cosets and the classification of three-character CFT, JHEP 03 (2023) 023 [ 2212.03136]

    work page arXiv 2023

  17. [17]

    Cosets of Meromorphic CFTs and Modular Differential Equations

    M.R. Gaberdiel, H.R. Hampapura and S. Mukhi, Cosets of Meromorphic CFTs and Modular Differential Equations, JHEP 04 (2016) 156 [ 1602.01022]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  18. [18]

    Franc and G

    C. Franc and G. Mason, Classification of some vertex operator algebras of rank 3 , Algebra & Number Theory 14 (2020) 1613

    work page 2020

  19. [19]

    Z. Duan, K. Lee and K. Sun, Hecke relations, cosets and the classification of 2d RCFTs , JHEP 09 (2022) 202 [ 2206.07478]

    work page arXiv 2022

  20. [20]

    Mukhi and B.C

    S. Mukhi and B.C. Rayhaun, Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25 , Commun. Math. Phys. 401 (2023) 1899 [ 2208.05486]

    work page arXiv 2023

  21. [21]

    Rayhaun, Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality , 2303.16921

    B.C. Rayhaun, Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality , 2303.16921

    work page arXiv

  22. [22]

    M¨ oller and B.C

    S. M¨ oller and B.C. Rayhaun,Equivalence Relations on Vertex Operator Algebras, I: Genus , 2408.07117

    work page arXiv

  23. [23]

    M¨ oller and B

    S. M¨ oller and B.C. Rayhaun,Equivalence Relations on Vertex Operator Algebras, II: Witt Equivalence and Orbifolds , 2410.18166

    work page arXiv

  24. [24]

    Naculich, Differential Equations for Rational Conformal Characters , Nucl

    S.G. Naculich, Differential Equations for Rational Conformal Characters , Nucl. Phys. B323 (1989) 423

    work page 1989

  25. [25]

    Das and S

    A. Das and S. Mukhi, Properties of ℓ = 2 quasi-characters, to appear

    work page

  26. [26]

    Vertex Operator Algebras with Two Simple Modules - the Mathur-Mukhi-Sen Theorem Revisited

    G. Mason, K. Nagatomo and Y. Sakai, Vertex Operator Algebras with Two Simple Modules - the Mathur-Mukhi-Sen Theorem Revisited, 1803.11281

    work page internal anchor Pith review Pith/arXiv arXiv

  27. [27]

    Perfect Metal Phases of One-Dimensional and Anisotropic Higher-Dimensional Systems

    E. Plamadeala, M. Mulligan and C. Nayak, Perfect Metal Phases of One-Dimensional and Anisotropic Higher-Dimensional Systems , Phys. Rev. B 90 (2014) 241101 [ 1404.4367]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  28. [28]

    Mukhi and G

    S. Mukhi and G. Muralidhara, Universal RCFT Correlators from the Holomorphic Bootstrap, JHEP 02 (2018) 028 [ 1708.06772]. – 44 –

    work page arXiv 2018

  29. [29]

    Universal Spectrum of 2d Conformal Field Theory in the Large c Limit

    T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit , JHEP 09 (2014) 118 [ 1405.5137]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  30. [30]

    Das and N.B

    A. Das and N.B. Umasankar, Two- & Three-character solutions to MLDEs and Ramanujan-Eisenstein Identities for Fricke Groups , 2211.15369

    work page arXiv

  31. [31]

    Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory , Nucl

    E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory , Nucl. Phys. B300 (1988) 360

    work page 1988

  32. [32]

    Kawasetsu, The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras , Letters in Mathematical Physics 104 (2014) 157

    K. Kawasetsu, The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras , Letters in Mathematical Physics 104 (2014) 157. – 45 –

    work page 2014