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arxiv: 2510.24248 · v2 · submitted 2025-10-28 · ✦ hep-th

Quasi-Characters for three-character Rational Conformal Field Theories

Suresh Govindarajan , Akhila Sadanandan , Jagannath Santara This is my paper

Pith reviewed 2026-05-18 03:27 UTC · model grok-4.3

classification ✦ hep-th
keywords solutionsadmissibleconstructarisefusionknownmldepoints
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The pith

All (3,0) admissible solutions are expressed via a universal _3F_2 hypergeometric formula; (3,3) solutions are built from them using Bantay-Gannon duality with only 7 of 15 having proper fusion rules, and further (3,6) and (3,9) solutions are generated as integer points on a polytope via quasi-char

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

Rational conformal field theories model physical systems with conformal symmetry, such as critical points in statistical mechanics or strings in theoretical physics. Their characters are functions that encode the spectrum of states and must obey modular invariance to ensure consistency under transformations of the torus. For theories with exactly three characters, these functions satisfy a third-order modular linear differential equation (MLDE). The paper shows that every (3,0) solution can be written with one universal expression involving the generalized hypergeometric function _3F_2, chosen so that it correctly reproduces the monodromy behavior around elliptic points in the moduli space. Using a known duality that maps (3,0) solutions to (3,3) solutions, the authors compute the modular S-matrix and fusion rules for fifteen previously known (3,3) cases and find that only seven satisfy the requirement of non-negative integer fusion coefficients. They next employ the matrix version of the MLDE to generate quasi-characters that share the same multiplier system as the original solutions. Linear combinations of these quasi-characters produce new admissible solutions, which the authors observe correspond to integer lattice points inside a polytope whose facets are fixed by the MLDE coefficients. This geometric picture lets them enumerate all admissible solutions for Wronskian indices 6 and 9. In a few cases the resulting characters match known RCFTs; the method also yields a large additional family with still higher Wronskian index. The work therefore supplies both a closed-form description of an entire class of solutions and a systematic generator for new ones.

Core claim

We show that all (3,0) solutions can be written in terms of a universal formula involving the _3F_2 hypergeometric function that takes into account the monodromy at the elliptic points.

Load-bearing premise

The assumption that the Bantay-Gannon duality maps every (3,0) admissible solution to a (3,3) solution whose modular properties and fusion rules can be computed directly from the original data without additional consistency conditions.

Figures

Figures reproduced from arXiv: 2510.24248 by Akhila Sadanandan, Jagannath Santara, Suresh Govindarajan.

Figure 1
Figure 1. Figure 1: 3.4.2 Admissible characters of type W W2(b, c1, c2) =   q α0−1 (1 + b q + O(q 2 )) q α1−1 (c1 + p1,1(b, c1, c2) q + p1,2(b1, b2) q 2 + O(q 3 )) q α2−1 (c2 + p2,1(b, c1, c2) q + p2,2(b, c1, c2) q 2 + O(q 3 ))   , (3.20) We list the various possibilities that depend on specific choices for the free parameters (b, c1, c2). 1. Since P a δαa = −3, we have δℓ = 18 when c1, c2 ̸= 0. These are of type W2. 18 … view at source ↗
Figure 1
Figure 1. Figure 1: Example of type U admissible characters starting with a c = 23, h1 = 3/2, h2 = 15/8 RCFT. They occur as integral points on the interior and boundary of a quadrilateral. We see two other (3, 0) theories at the corners (b1, b2) = (46, 0) and (b1, b2) = (0, 575). These are shown as red dots and connected by green dots which are (3, 6) admissible characters. We show the (3, 12) points that lie on the boundarie… view at source ↗
Figure 2
Figure 2. Figure 2: In this type W1,1 example, we obtain (3, 6) theories on integral points that lie on the line 1 46 (c1 − 8742). All integral points inside the polytope and on the c1 = 0 line are (3, 12) admissible solutions. There are no (3, 0) solutions. 2. When one of (s1, s2) is zero with the other being non-zero, we have δℓ = 18. When s1 > 0, we look for additional conditions on the constants (s2, b1, b2) such that p2,… view at source ↗
read the original abstract

We revisit (3,0) and (3,3) admissible solutions obtained using the MLDE method. We show that all $(3,0)$ solutions can be written in terms of a universal formula involving the ${}_3F_2$ hypergeometric function that takes into account the monodromy at the elliptic points. We construct $(3,3)$ admissible solutions from (3,0) CFTs using a duality due to Bantay and Gannon. This enables us to compute their modular properties such as the S-matrix and the fusion rules. We find that only 7 of the 15 known (3,3) admissible solutions have proper fusion rules. Using the theory of matrix MLDE, starting with a known (3,0) and (3,3) solutions, we construct two other solutions, that are typically quasi-characters that share the same multiplier as the original solution. We then construct linear combinations that lead to new admissible solutions. We observe that admissible solutions arise as integer points that lie on a polytope. We construct all possible (3,6) and (3,9) admissible solutions that arise in this fashion. In some cases, we identify RCFT that arise from our (3,6) admissible solutions. In addition, we obtain a large family of admissible solutions with higher 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

3 major / 2 minor

Summary. The manuscript revisits admissible solutions for three-character RCFTs obtained via the MLDE method. It claims that every (3,0) solution admits a universal expression in terms of the _3F_2 hypergeometric function that incorporates monodromy at the elliptic points. Using Bantay-Gannon duality, (3,3) solutions are constructed from the (3,0) data; their S-matrices and fusion rules are computed, yielding the result that only 7 of the 15 known (3,3) solutions possess proper fusion rules. Matrix MLDE is then employed to generate quasi-characters sharing the original multiplier; linear combinations of these produce new admissible solutions. The authors observe that admissible solutions correspond to integer points inside a polytope defined by the MLDE coefficient constraints, construct all (3,6) and (3,9) solutions arising this way, identify some as RCFTs, and obtain a large family with higher Wronskian index.

Significance. If the universal hypergeometric formula and the duality constructions are rigorously verified, the work would supply a systematic, parameter-efficient route to admissible solutions at higher Wronskian indices and a concrete count of fusion-rule-valid (3,3) theories. The polytope observation, if shown to be independent of the enumerated solutions, could illuminate the geometry of the space of MLDE coefficients. The explicit identification of RCFTs among the (3,6) solutions would constitute a tangible advance in classification.

major comments (3)
  1. [universal formula section] Abstract and the section presenting the universal formula: the claim that every (3,0) admissible solution is captured by a single _3F_2 expression accounting for elliptic-point monodromy is stated without the explicit hypergeometric identity or the step-by-step reduction from the MLDE; this derivation is load-bearing for the completeness assertion.
  2. [Bantay-Gannon duality construction] The paragraph on Bantay-Gannon duality and the 7/15 count: the mapping is asserted to produce (3,3) solutions whose fusion rules follow directly from the original (3,0) data, yet no explicit verification is supplied that the resulting characters close under the Verlinde formula or satisfy positivity without additional modular-invariance checks; this assumption underpins both the count and the claim that duality introduces no hidden obstructions.
  3. [polytope and admissible solutions] The polytope observation and the axiom that admissible solutions are exactly the integer points inside the MLDE-coefficient polytope: the statement appears derived from the same set of enumerated solutions rather than from an independent constraint, rendering the observation potentially circular and weakening its use as a classification tool.
minor comments (2)
  1. Notation for the Wronskian index and the multiplier system should be defined at first use and kept consistent across the matrix-MLDE and duality sections.
  2. Tables listing the 15 (3,3) solutions and the 7 valid fusion-rule cases would benefit from an additional column indicating which solutions arise from the duality map versus direct MLDE search.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will incorporate to improve clarity and rigor.

read point-by-point responses

  1. Referee: [universal formula section] Abstract and the section presenting the universal formula: the claim that every (3,0) admissible solution is captured by a single _3F_2 expression accounting for elliptic-point monodromy is stated without the explicit hypergeometric identity or the step-by-step reduction from the MLDE; this derivation is load-bearing for the completeness assertion.

    Authors: We agree that the explicit derivation is essential for substantiating the completeness claim. In the revised manuscript we will insert a dedicated subsection deriving the universal _3F_2 expression directly from the third-order MLDE, including the precise hypergeometric identity that encodes the monodromy at the elliptic points and the reduction steps that show every admissible (3,0) solution is recovered from this single formula. revision: yes

  2. Referee: [Bantay-Gannon duality construction] The paragraph on Bantay-Gannon duality and the 7/15 count: the mapping is asserted to produce (3,3) solutions whose fusion rules follow directly from the original (3,0) data, yet no explicit verification is supplied that the resulting characters close under the Verlinde formula or satisfy positivity without additional modular-invariance checks; this assumption underpins both the count and the claim that duality introduces no hidden obstructions.

    Authors: The Bantay-Gannon duality supplies the characters and the associated S-matrix by a direct linear transformation of the (3,0) data; the fusion coefficients are then obtained from the Verlinde formula applied to this S-matrix. We have verified closure, integrality and positivity for all fifteen cases, finding that only seven satisfy the required conditions. To make the verification transparent we will add, in the revision, an explicit worked example of the S-matrix and fusion-rule table for one representative solution together with a summary confirming the checks for the full set. revision: partial

  3. Referee: [polytope and admissible solutions] The polytope observation and the axiom that admissible solutions are exactly the integer points inside the MLDE-coefficient polytope: the statement appears derived from the same set of enumerated solutions rather than from an independent constraint, rendering the observation potentially circular and weakening its use as a classification tool.

    Authors: The polytope is defined by the linear inequalities on the MLDE coefficients that enforce admissibility (non-negative leading coefficients in the q-expansions and integrality of the Wronskian index). These inequalities follow directly from the general structure of the MLDE and are independent of any particular solution set. The integer points inside the polytope then generate the admissible solutions. We will revise the relevant section to state the defining inequalities explicitly and to clarify that the polytope is a consequence of the MLDE constraints rather than an a-posteriori fit to the enumerated solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations build on external MLDE framework and Bantay-Gannon duality

full rationale

The paper derives a universal _3F_2 expression for (3,0) solutions by solving the standard modular linear differential equations (MLDE) while incorporating elliptic-point monodromy, then applies the externally cited Bantay-Gannon duality to generate (3,3) solutions and compute their S-matrix and fusion rules. Subsequent constructions of quasi-characters via matrix MLDE, linear combinations yielding new admissible solutions, and the observation that solutions appear as integer points on a polytope are presented as empirical outcomes from these methods rather than definitional inputs or self-referential fits. No load-bearing step reduces by construction to the paper's own fitted parameters or prior self-citations; the Bantay-Gannon reference is to independent prior work, and the polytope remark functions as a descriptive pattern rather than a forcing constraint. The overall chain remains self-contained against external RCFT benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claims rest on the MLDE method for generating admissible solutions, the Bantay-Gannon duality for mapping solution types, and the definition of quasi-characters as functions sharing the same multiplier; the polytope structure is presented as an observed geometric property of the constructed solutions.

free parameters (1)
  • MLDE coefficients chosen for admissibility
    Parameters in the differential equation are selected so that the resulting q-expansions have non-negative integer coefficients.
axioms (2)
  • domain assumption The Bantay-Gannon duality preserves modular properties sufficiently to allow direct computation of S-matrix and fusion rules for the image solutions.
    Invoked to construct all (3,3) solutions from (3,0) ones.
  • ad hoc to paper Admissible solutions correspond exactly to the integer points lying inside the polytope defined by the MLDE coefficient constraints.
    Used to enumerate all (3,6) and (3,9) solutions.
invented entities (1)
  • quasi-characters no independent evidence
    purpose: Functions that share the multiplier of the original characters and serve as building blocks for linear combinations yielding new admissible solutions.
    Introduced via the matrix MLDE construction; no independent physical realization is provided in the abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Updating the holomorphic modular bootstrap

    hep-th 2026-04 unverdicted novelty 5.0

    Admissible solutions to MLDEs with ≤6 characters and c_eff ≤24 are enumerated; tenable ones with good fusion rules are identified, with some linked to specific CFTs and MTC classes.

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Works this paper leans on

38 extracted references · 38 canonical work pages · cited by 1 Pith paper · 13 internal anchors

  1. [1]

    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

  2. [2]

    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

  3. [3]

    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

  4. [4]

    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

  5. [5]

    King,A mass formula for unimodular lattices with no roots,Mathematics of Computation72(2003) 839 [0012231]

    O.D. King,A mass formula for unimodular lattices with no roots,Mathematics of Computation72(2003) 839 [0012231]

    work page 2003

  6. [6]

    On 2d Conformal Field Theories with Two Characters

    H.M. Hampapura and S. Mukhi,On 2d conformal field theories with two characters, JHEP01(2016) 005 [1510.04478]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  7. [7]

    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

  8. [8]

    Das, C.N

    A. Das, C.N. Gowdigere and J. Santara,Classifying three-character RCFTs with Wronskian index equalling 0 or 2,JHEP11(2021) 195 [2108.01060]

    work page arXiv 2021

  9. [9]

    Gowdigere, S

    C.N. Gowdigere, S. Kala and J. Santara,Classifying three-character RCFTs with Wronskian index equalling 3 or 4,2308.01149. 44

    work page arXiv

  10. [10]

    Das, C.N

    A. Das, C.N. Gowdigere and S. Mukhi,Meromorphic cosets and the classification of three-character CFT,JHEP03(2023) 023 [2212.03136]

    work page arXiv 2023

  11. [11]

    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,JHEP04(2019) 153 [1810.09472]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  12. [12]

    Mukhi, R

    S. Mukhi, R. Poddar and P. Singh,Rational CFT with three characters: the quasi-character approach,JHEP05(2020) 003 [2002.01949]

    work page arXiv 2020

  13. [13]

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

    A. Das and S. Mukhi,Holomorphic bootstrap for RCFT: signs and bounds for quasi-characters,2507.07170

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    Kaidi, Y.-H

    J. Kaidi, Y.-H. Lin and J. Parra-Martinez,Holomorphic modular bootstrap revisited, JHEP12(2021) 151 [2107.13557]

    work page arXiv 2021

  15. [15]

    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

  16. [16]

    Harvey, Y

    J.A. Harvey, Y. Hu and Y. Wu,Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular Tensor Categories,J. Phys. A53(2020) 334003 [1912.11955]

    work page arXiv 2020

  17. [17]

    Z. Duan, K. Lee and K. Sun,Hecke relations, cosets and the classification of 2d RCFTs,JHEP09(2022) 202 [2206.07478]

    work page arXiv 2022

  18. [18]

    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,JHEP04(2016) 156 [1602.01022]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  19. [19]

    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

  20. [20]

    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

  21. [21]

    Two approaches to the holomorphic modular bootstrap

    S. Govindarajan and J. Santara,Two approaches to the holomorphic modular bootstrap,JHEP10(2025) 181 [2503.23761]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  22. [22]

    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,J. Math. Phys.65(2024) 052301 [2303.16921]

    work page arXiv 2024

  23. [23]

    The kernel of the modular representation and the Galois action in RCFT

    P. Bantay,The Kernel of the modular representation and the Galois action in RCFT,Commun. Math. Phys.233(2003) 423 [math/0102149]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  24. [24]

    Conformal characters and the modular representation

    P. Bantay and T. Gannon,Conformal characters and the modular representation, JHEP02(2006) 005 [hep-th/0512011]. 45

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  25. [25]

    Modular differential equations for characters of RCFT

    P. Bantay,Modular differential equations for characters of RCFT,JHEP06(2010) 021 [1004.2579]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  26. [26]

    Beukers and G

    F. Beukers and G. Heckman,Monodromy for the hypergeometric function nFn−1, Inventiones mathematicae95(1989) 325

    work page 1989

  27. [27]

    Hypergeometric series, modular linear differential equations, and vector-valued modular forms

    C. Franc and G. Mason,Hypergeometric series, modular linear differential equations and vector-valued modular forms,The Ramanujan Journal41(2016) 233 [1503.05519]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  28. [28]

    Franc and G

    C. Franc and G. Mason,Classification of some vertex operator algebras of rank 3, Algebra & Number Theory14(2020) 1613 [1905,07500]

    work page 2020

  29. [29]

    Di Francesco, P

    P. Di Francesco, P. Mathieu and D. Senechal,Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997), 10.1007/978-1-4612-2256-9

    work page doi:10.1007/978-1-4612-2256-9 1997

  30. [30]

    Mukhi, R

    S. Mukhi, R. Poddar and P. Singh,Contour integrals and the modularS-matrix, JHEP07(2020) 045 [1912.04298]

    work page arXiv 2020

  31. [31]

    Studies of [3,3] characters

    C.N. Gowdigere, S. Kala and A. Rawat, “Studies of [3,3] characters.” 2025

    work page 2025

  32. [32]

    Das, C.N

    A. Das, C.N. Gowdigere, S. Mukhi and J. Santara,Modular differential equations with movable poles and admissible RCFT characters,JHEP12(2023) 143 [2308.00069]

    work page arXiv 2023

  33. [33]

    Moore and N

    G.W. Moore and N. Seiberg,Classical and Quantum Conformal Field Theory, Commun. Math. Phys.123(1989) 177

    work page 1989

  34. [34]

    Tables of (3,6) RCFTs obtained as generalized coset duals of (3,0) RCFTs fromc= 32 Kervaire self-dual lattices

    A. Das, C.N. Gowdigere and J. Santara, “Tables of (3,6) RCFTs obtained as generalized coset duals of (3,0) RCFTs fromc= 32 Kervaire self-dual lattices.” 2023

    work page 2023

  35. [35]

    On classification of modular tensor categories

    E. Rowell, R. Stong and Z. Wang,On classification of modular tensor categories, Communications in Mathematical Physics292(2009) 343 [0712.1377]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  36. [36]

    Work in progress

    S. Govindarajan and A. Sadanandan, “Work in progress.”

    work page

  37. [37]

    Zagier,Introduction to modular forms, inFrom number theory to physics, pp

    D. Zagier,Introduction to modular forms, inFrom number theory to physics, pp. 238–291, Springer (1992)

    work page 1992

  38. [38]

    Zagier,Elliptic modular forms and their applications, inThe 1-2-3 of modular forms: Lectures at a summer school in Nordfjordeid, Norway, pp

    D. Zagier,Elliptic modular forms and their applications, inThe 1-2-3 of modular forms: Lectures at a summer school in Nordfjordeid, Norway, pp. 1–103, Springer (2008). 46

    work page 2008