arxiv: 2604.11277 · v2 · submitted 2026-04-13 · ✦ hep-th

Recognition: unknown

Updating the holomorphic modular bootstrap

Suresh Govindarajan , Akhila Sadanandan

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:29 UTC · model grok-4.3

classification ✦ hep-th

keywords holomorphic modular bootstrapmodular linear differential equationsadmissible solutionstenable solutionsfusion rulesconformal field theorymodular tensor categoriesWronskian index

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The pith

Admissible solutions to all modular linear differential equations with up to six characters are found for effective central charges up to 24.

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

The paper updates the holomorphic modular bootstrap by incorporating a recent exact S-matrix computation within the modular linear differential equation framework. It systematically solves for admissible solutions to MLDEs with one to six characters, Wronskian index below 6, and one accessory parameter when the effective central charge is at most 24. From these it selects the tenable solutions that obey good fusion rules and matches them where possible to known conformal field theories and their modular tensor category classes in unitary cases. A sympathetic reader would care because the work provides a more complete catalog of candidate rational CFTs that satisfy modular invariance and consistency conditions in a bounded range.

Core claim

By solving the modular linear differential equations with up to six characters and incorporating the exact S-matrix, the authors obtain all admissible solutions with Wronskian index less than 6 and one accessory parameter for c_eff less than or equal to 24. They further filter these for tenable solutions that satisfy good fusion rules and match them to known CFTs and MTC classes in the unitary cases.

What carries the argument

The modular linear differential equation (MLDE) whose solutions give the characters, with its Wronskian index and accessory parameters fixing the form and the exact S-matrix fixing the modular transformations used to check fusion rules.

If this is right

All MLDEs under the given bounds on characters, Wronskian index, accessory parameters, and c_eff have been solved for admissible solutions.
Tenable solutions are distinguished from merely admissible ones by the presence of good fusion rules.
Unitary tenable solutions are associated with specific modular tensor category classes.
The finite list allows direct identification of many known CFTs within the enumerated solutions.

Where Pith is reading between the lines

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

The same enumeration procedure could be run at higher character numbers or more accessory parameters to locate additional candidate CFTs.
If every tenable solution turns out to describe a physical theory, the list would constitute a near-complete classification of rational CFTs with small numbers of characters and c_eff at most 24.
Explicit construction of models realizing the unidentified tenable solutions could be attempted to test whether they yield consistent CFTs.

Load-bearing premise

That any solution to the MLDE satisfying the stated admissibility conditions on exponents and having good fusion rules corresponds to a consistent physical conformal field theory.

What would settle it

A known unitary CFT with six or fewer characters that does not appear among the enumerated admissible solutions, or a tenable solution that produces negative fusion coefficients when used to compute higher correlation functions.

read the original abstract

We update the holomorphic modular bootstrap incorporating a recent result that computes the exact S-matrix within the Modular Linear Differential Equation (MLDE) setting. Further, using knowledge of the allowed exponents modulo one, we obtain admissible solutions to all MLDE's with up to six characters and Wronskian index < 6 and one accessory parameter with c_eff <= 24. We then identify which of the admissible solutions have good fusion rules -- we call such solutions tenable. When possible, we identify the CFT and in the unitary cases the MTC class they belong to.

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.

Desk Editor's Note private letter to a colleague

This paper updates the holomorphic modular bootstrap with a recent S-matrix result and enumerates admissible plus tenable MLDE solutions up to six characters, though the physical consistency of the tenable list still needs explicit checks on q-series coefficients.

read the letter

The key takeaway is that this paper performs a systematic enumeration of solutions to modular linear differential equations in the holomorphic setting, incorporating a recent exact S-matrix computation. They go through all MLDEs with up to six characters, Wronskian index less than six, one accessory parameter, and effective central charge up to 24, find the admissible ones based on exponents, then filter for those with good fusion rules which they call tenable, and match some to known theories or MTC classes. They do a solid job extending the prior literature by making the scan more complete with the new S-matrix input. The enumeration produces a list of candidates that can be checked against existing classifications, and identifying the MTC classes for unitary cases adds a nice connection to topological data. If the code or methods are clear, this could serve as a reference for low-character holomorphic CFTs. The soft spot is the one the stress-test note flags. Having admissible exponents and satisfying the fusion rules from Verlinde does not automatically ensure the full character expansions have non-negative integer coefficients or that the theory is consistent at higher levels. The paper claims to obtain admissible solutions and then tenable ones, but if they did not explicitly verify the integrality and positivity of the q-series coefficients for all cases, or check the S-matrix behavior across the accessory parameter, then some listed tenable solutions might not correspond to actual CFTs. That would make the classification less reliable than it appears. This is really for experts in 2d CFT classification, modular bootstrap techniques, and related math physics. Someone working on holomorphic theories or looking for new examples in string theory or stat mech contexts would find the list useful. It deserves serious peer review because the work is a concrete update in an established program, and referees can check the computational details and the strength of the consistency arguments.

Referee Report

1 major / 2 minor

Summary. The paper updates the holomorphic modular bootstrap by incorporating a recent exact S-matrix computation in the MLDE framework. It enumerates all admissible solutions to MLDEs with up to six characters, Wronskian index <6, one accessory parameter, and c_eff ≤24, then filters them to 'tenable' solutions with good fusion rules, identifying the corresponding CFT (and MTC class in unitary cases) where possible.

Significance. If the tenable solutions are consistent, this provides a valuable systematic catalog of holomorphic RCFT candidates within the given bounds, strengthening prior bootstrap classifications through exact S-matrix control and fusion-rule filtering. The enumeration itself is a concrete contribution that can guide further searches for new theories or verification of known ones.

major comments (1)

[Results on tenable solutions] In the section on admissible and tenable solutions: the filtering to solutions with good fusion rules (via Verlinde) is load-bearing for the central claim, yet the manuscript does not appear to include an explicit check that all character q-expansions have non-negative integer coefficients. This integrality is required to exclude spurious solutions and is not automatically ensured by the exponent conditions or the incorporated S-matrix result.

minor comments (2)

The abstract contains a minor grammatical issue: 'MLDE's' should read 'MLDEs'.
A summary table listing all tenable solutions (with c_eff, exponents, fusion data, and CFT identification) would improve readability and allow quick cross-reference with prior classifications.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their careful reading of our manuscript and for recommending minor revision. We address the major comment below.

read point-by-point responses

Referee: In the section on admissible and tenable solutions: the filtering to solutions with good fusion rules (via Verlinde) is load-bearing for the central claim, yet the manuscript does not appear to include an explicit check that all character q-expansions have non-negative integer coefficients. This integrality is required to exclude spurious solutions and is not automatically ensured by the exponent conditions or the incorporated S-matrix result.

Authors: We thank the referee for pointing this out. The admissible solutions are derived from the MLDE framework with the given constraints on the number of characters, Wronskian index, accessory parameter, and effective central charge. For each solution, the q-expansions are computed, and we ensure that the coefficients are non-negative integers as a prerequisite for considering them as potential characters before checking the fusion rules. This step, although performed in our analysis, was not explicitly stated in the text. We will update the manuscript to include a clear description of this integrality verification in the section on admissible and tenable solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; enumeration of admissible MLDE solutions is self-contained via external S-matrix input and standard modular constraints

full rationale

The paper's central procedure solves MLDEs under explicit constraints (up to six characters, Wronskian index <6, one accessory parameter, c_eff <=24), filters admissible solutions using allowed exponents modulo one and Verlinde-derived fusion rules, and identifies tenable cases against known CFTs/MTCs. This chain rests on an incorporated external exact S-matrix result plus standard modular invariance assumptions rather than any fitted parameter being relabeled as a prediction or any self-definitional reduction. No load-bearing step reduces by construction to the paper's own inputs; the derivation is computational enumeration against external benchmarks and is therefore self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard MLDE framework for CFT characters, a recent external S-matrix result, and definitions of admissibility and tenability. One accessory parameter is treated as a variable in the equations considered.

free parameters (1)

accessory parameter
MLDEs are considered with one accessory parameter as part of the enumeration setup.

axioms (2)

domain assumption Partition functions of CFTs are modular invariant under SL(2,Z).
Core assumption enabling the use of MLDEs in the holomorphic bootstrap.
domain assumption Admissible MLDE solutions with positive coefficients and good fusion rules correspond to consistent CFTs.
Links mathematical solutions to physical theories in the bootstrap program.

pith-pipeline@v0.9.0 · 5377 in / 1534 out tokens · 85937 ms · 2026-05-10T16:29:53.549938+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 22 canonical work pages · 2 internal anchors

[1]

CFT and modular forms

T. Gannon, “CFT and modular forms.” Presentation at the BIRS workshop on Recent Developments in Logarithmic Conformal Field Theory, July, 2025

2025
[2]

Anderson and G

G. Anderson and G. W. Moore,Rationality in Conformal Field Theory,Commun. Math. Phys.117(1988) 441

1988
[3]

Gannon,The theory of vector-modular forms for the modular group,Contrib

T. Gannon,The theory of vector-modular forms for the modular group,Contrib. Math. Comput. Sci.8(2014) 247–286, [1310.4458]

work page arXiv 2014
[4]

S. D. Mathur, S. Mukhi and A. Sen,Reconstruction of Conformal Field Theories From Modular Geometry on the Torus,Nucl. Phys. B318(1989) 483–540. – 39 –

1989
[5]

S. D. Mathur, S. Mukhi and A. Sen,On the Classification of Rational Conformal Field Theories,Phys. Lett. B213(1988) 303–308

1988
[6]

S. G. Naculich,Differential equations for rational conformal characters,Nucl. Phys. B323 (1989) 423–440

1989
[7]

H. M. Hampapura and S. Mukhi,On 2d conformal field theories with two characters,JHEP 01(2016) 005, [1510.04478]

work page arXiv 2016
[8]

A. R. Chandra and S. Mukhi,Towards a Classification of Two-Character Rational Conformal Field Theories,JHEP04(2019) 153, [1810.09472]

work page Pith review arXiv 2019
[9]

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) 1899–1949, [2208.05486]

work page arXiv 2023
[10]

Franc and G

C. Franc and G. Mason,Hypergeometric series, modular linear differential equations and vector-valued modular forms,The Ramanujan Journal41(2016) 233–267, [1503.05519]

work page arXiv 2016
[11]

Franc and G

C. Franc and G. Mason,Classification of some vertex operator algebras of rank 3,Algebra & Number Theory14(2020) 1613–1667, [1905,07500]

2020
[12]

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
[13]

Kaidi, Y.-H

J. Kaidi, Y.-H. Lin and J. Parra-Martinez,Holomorphic modular bootstrap revisited,JHEP 12(2021) 151, [2107.13557]

work page arXiv 2021
[14]

J.-B. Bae, Z. Duan, K. Lee, S. Lee and M. Sarkis,Bootstrapping fermionic rational CFTs with three characters,JHEP01(2022) 089, [2108.01647]

work page arXiv 2022
[15]

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

work page arXiv
[16]

S.-H. Ng, E. C. Rowell and X.-G. Wen,Classification of modular data up to rank 12, 2023

2023
[17]

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) 651–680

2007
[18]

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
[19]

Beukers and G

F. Beukers and G. Heckman,Monodromy for the hypergeometric function nFn−1,Inventiones mathematicae95(1989) 325–354

1989
[20]

S. D. Mathur and A. Sen,Group Theoretic Classification of Rational Conformal Field Theories With Algebraic Characters,Nucl. Phys. B327(1989) 725–743

1989
[21]

Govindarajan, A

S. Govindarajan, A. Jain, A. Sadanandan and A. Kidambi,S-matrices in the holomorphic modular bootstrap approach,2602.14665

work page arXiv
[22]

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
[23]

Eholzer,On the classification of modular fusion algebras,Commun

W. Eholzer,On the classification of modular fusion algebras,Commun. Math. Phys.172 (1995) 623–659, [hep-th/9408160]

work page arXiv 1995
[24]

Nobs,Die irreduziblen Darstellungen der GruppenSL 2(Zp), insbesondereSL 2(Z2)

A. Nobs,Die irreduziblen Darstellungen der GruppenSL 2(Zp), insbesondereSL 2(Z2). I, Comment. Math. Helv.51(1976) 465–489. – 40 –

1976
[25]

Nobs and J

A. Nobs and J. Wolfart,Die irreduziblen Darstellungen der GruppenSL 2(Zp), insbesondere SL2(Zp). II,Comment. Math. Helv.51(1976) 491–526

1976
[26]

GAP – Groups, Algorithms, and Programming, Version 4.15.1

The GAP Group, “GAP – Groups, Algorithms, and Programming, Version 4.15.1.” https://www.gap-system.org, 2025

2025
[27]

Gannon,Comments on nonunitary conformal field theories,Nucl

T. Gannon,Comments on nonunitary conformal field theories,Nucl. Phys. B670(2003) 335–358, [hep-th/0305070]

work page arXiv 2003
[28]

De Boer and J

J. De Boer and J. Goeree,Markov traces and II(1) factors in conformal field theory, Commun. Math. Phys.139(1991) 267–304

1991
[29]

Ng and P

S.-H. Ng and P. Schauenberg,Frobenius–Schur indicators and exponents of spherical categories,Adv. in Mathematics211(2007) 34–71, [arXiv:math/0601012]

work page arXiv 2007
[30]

Bordeaux,PARI/GP version2.15.4, 2023

The PARI Group, Univ. Bordeaux,PARI/GP version2.15.4, 2023

2023
[31]

M. R. Gaberdiel, H. R. Hampapura and S. Mukhi,Cosets of Meromorphic CFTs and Modular Differential Equations,JHEP04(2016) 156, [1602.01022]

work page Pith review arXiv 2016
[32]

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
[33]

J. A. Harvey and Y. Wu,Hecke Relations in Rational Conformal Field Theory,JHEP09 (2018) 032, [1804.06860]

work page arXiv 2018
[34]

Bantay,The Kernel of the modular representation and the Galois action in RCFT, Commun

P. Bantay,The Kernel of the modular representation and the Galois action in RCFT, Commun. Math. Phys.233(2003) 423–438, [math/0102149]

work page arXiv 2003
[35]

Two approaches to the holomorphic modular bootstrap

S. Govindarajan and J. Santara,Two approaches to the holomorphic modular bootstrap, JHEP10(3, 2025) 181, [2503.23761]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[36]

Quasi-Characters for three-character Rational Conformal Field Theories

S. Govindarajan, A. Sadanandan and J. Santara,Quasi-Characters for three-character Rational Conformal Field Theories,2510.24248

work page internal anchor Pith review Pith/arXiv arXiv
[37]

K. Lee, K. Sun and H. Wang,On intermediate Lie algebraE 7+1/2,Lett. Math. Phys.114 (2024) 13, [2306.09230]

work page arXiv 2024
[38]

Quasi-characters for four and five character rational conformal field theories

S. Govindarajan and A. Sadanandan, “Quasi-characters for four and five character rational conformal field theories.” 2026. – 41 –

2026