pith. sign in

arxiv: 2503.23761 · v2 · submitted 2025-03-31 · ✦ hep-th

Two approaches to the holomorphic modular bootstrap

Suresh Govindarajan , Jagannath Santara This is my paper

Pith reviewed 2026-05-22 22:38 UTC · model grok-4.3

classification ✦ hep-th
keywords holomorphic modular bootstrapvector-valued modular formsrational conformal field theoryadmissible charactersWronskian indexmultiplier systemcharacter vectors
0
0 comments X

The pith

Vector-valued modular forms with shared multipliers yield new admissible character sets for rational CFTs through linear combinations.

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

The paper presents an approach to the holomorphic modular bootstrap that starts from a known rational conformal field theory and constructs new vector-valued modular forms sharing its multiplier system. These new forms are generated using existing results from the theory of vector-valued modular forms, after which particular linear combinations are tested for admissibility. The method recovers every previously known admissible solution in the two-character case with Wronskian indices 6 and 8, and it produces examples with up to six characters. It is offered as a practical route when the direct enumeration of characters becomes intractable for larger numbers.

Core claim

Given a rational CFT, its characters can be assembled into a vector-valued modular form with multiplier; known results on such forms then produce additional solutions with the identical multiplier, and admissible linear combinations of these solutions furnish new candidate character vectors that satisfy the necessary positivity and integrality conditions.

What carries the argument

Vector-valued modular forms with a fixed multiplier, generated from a seed RCFT and subjected to linear combinations that enforce admissibility.

If this is right

  • All known admissible two-character solutions with Wronskian indices 6 and 8 are recovered exactly.
  • New admissible solutions appear in explicit examples with three to six characters.
  • The construction supplies candidate character vectors without enumerating all possible solutions from scratch.
  • The same multiplier is preserved throughout, restricting the search to a single orbit.

Where Pith is reading between the lines

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

  • The approach may scale to theories with more than six characters where direct bootstrap becomes combinatorially expensive.
  • It supplies an independent source of candidate characters that could be fed into other consistency checks such as fusion-rule closure.
  • If many admissible combinations turn out to lack corresponding CFTs, the method would highlight a gap between mathematical solutions and physically realized theories.

Load-bearing premise

That linear combinations satisfying the mathematical admissibility conditions actually describe characters of physical rational CFTs rather than formal artifacts.

What would settle it

A linear combination that meets all stated admissibility criteria (non-negative coefficients, correct leading q-exponents, integrality) yet fails to produce a consistent modular-invariant partition function or fusion rules when assembled into a full theory.

read the original abstract

The holomorphic bootstrap attempts to classify rational conformal field theories. The straight ahead approach is hard to implement when the number of characters become large. We combine all characters of an RCFT to form a vector valued modular form with multiplier. Using known results from the theory of vector valued modular forms, given a known RCFT, we obtain new vector valued modular forms that share the same multiplier as the original RCFT. By taking particular linear combinations of the new solutions, we look for and find new admissible solutions. In the well-studied two character case, we reproduce all known admissible solutions with Wronskian indices $6$ and $8$. The method is illustrated with examples with up to six characters. The method using vector valued modular forms thus provides a new approach to the holomorphic modular bootstrap.

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 / 1 minor

Summary. The paper proposes combining characters of known RCFTs into vector-valued modular forms (VVMFs) with multipliers, then using established VVMF theory to generate new forms sharing the same multiplier. Particular linear combinations are taken to produce new admissible solutions (non-negative integer q-coefficients, correct leading exponents, modular covariance). In the two-character case this reproduces all known admissible solutions with Wronskian indices 6 and 8; the method is illustrated with examples up to six characters, offering an alternative to direct holomorphic bootstrap when the number of characters is large.

Significance. The concrete reproduction of all known two-character solutions supplies direct evidence that the VVMF generation and linear-combination pipeline works in a controlled setting. If the admissible solutions are shown to satisfy full RCFT consistency conditions (Verlinde integrality, non-negative fusion coefficients), the approach would provide a systematic route to new character sets when brute-force modular bootstrap becomes intractable. The use of established theorems on VVMFs rather than ad-hoc fitting is a methodological strength.

major comments (2)
  1. [Section describing six-character examples] The central claim that the procedure yields a viable bootstrap method for RCFTs rests on the unverified step that mathematically admissible linear combinations of generated VVMFs correspond to consistent rational CFTs. For the six-character examples, no checks are reported for Verlinde integrality of fusion coefficients or non-negative structure constants, which are load-bearing for the physical interpretation.
  2. [Method section on linear combinations of VVMFs] The method description states that 'particular linear combinations' are chosen to obtain new admissible solutions, but supplies no explicit, reproducible algorithm or selection criteria for the coefficients. This omission is load-bearing for the claim that the approach systematically finds new solutions beyond the two-character reproduction.
minor comments (1)
  1. The abstract and main text should include a compact table listing the new admissible solutions found (central charges, leading exponents, Wronskian indices) so that readers can immediately compare them with the known two-character list.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the revisions we intend to make.

read point-by-point responses

  1. Referee: [Section describing six-character examples] The central claim that the procedure yields a viable bootstrap method for RCFTs rests on the unverified step that mathematically admissible linear combinations of generated VVMFs correspond to consistent rational CFTs. For the six-character examples, no checks are reported for Verlinde integrality of fusion coefficients or non-negative structure constants, which are load-bearing for the physical interpretation.

    Authors: We agree that the six-character examples are presented solely as admissible solutions (non-negative integer coefficients and correct leading exponents) without Verlinde or fusion-coefficient checks. The paper's scope is the generation of mathematically admissible character sets via VVMF methods as a first step in the holomorphic bootstrap; full RCFT consistency conditions are computationally demanding for six characters and lie outside the present work. We will add an explicit statement in the discussion section clarifying this limitation and noting that the generated solutions are candidates requiring further physical validation. revision: yes

  2. Referee: [Method section on linear combinations of VVMFs] The method description states that 'particular linear combinations' are chosen to obtain new admissible solutions, but supplies no explicit, reproducible algorithm or selection criteria for the coefficients. This omission is load-bearing for the claim that the approach systematically finds new solutions beyond the two-character reproduction.

    Authors: The linear combinations are chosen by searching for small-integer coefficient vectors that produce resulting q-expansions with non-negative integer coefficients and the required leading exponents. In the two-character case this is exhaustive; for higher characters we perform targeted enumeration. We acknowledge that the current text does not spell out the precise search procedure. We will revise the methods section to include an explicit description of the coefficient-selection algorithm together with the specific combinations used in each example, thereby making the procedure reproducible. revision: yes

Circularity Check

0 steps flagged

No circularity: method generates candidates from external VVMF theory and validates on known RCFTs.

full rationale

The derivation begins with known RCFT characters assembled into VVMFs, applies independently established results on vector-valued modular forms to produce new forms with the same multiplier, and searches for admissible linear combinations. Reproduction of all known two-character solutions with Wronskian indices 6 and 8 functions as an external consistency check rather than a tautological outcome. No quoted step reduces by definition or fitted parameter to the target result, and cited VVMF theorems are treated as external mathematical input without load-bearing self-citation chains that collapse the argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the applicability of known theorems about vector-valued modular forms to RCFT characters and on the assumption that admissible linear combinations exist and can be identified.

axioms (1)
  • domain assumption Known results from the theory of vector valued modular forms apply directly to the characters of an RCFT and generate new forms with the same multiplier.
    Invoked to obtain the new vector valued modular forms from a given RCFT.

pith-pipeline@v0.9.0 · 5655 in / 1183 out tokens · 64604 ms · 2026-05-22T22:38:46.462086+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

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

  1. Quasi-Characters for three-character Rational Conformal Field Theories

    hep-th 2025-10 unverdicted novelty 6.0

    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...

  2. 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.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 2 Pith papers · 10 internal anchors

  1. [1]

    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]

    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]

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

    work page 1989

  4. [4]

    On 2d Conformal Field Theories with Two Characters

    H. R. Hampapura and S. Mukhi, “On 2d Conformal Field Theor ies with Two Char- acters,” JHEP 01 (2016) 005 [[arXiv:1510.04478]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  5. [5]

    Towards a Classification of Two-Character Rational Conformal Field Theories

    A. R. Chandra and S. Mukhi, “Towards a Classification of Tw o-Character Rational Conformal Field Theories,” JHEP 04 (2019), 153 [arXiv:1810.09472 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  6. [6]

    Mukhi and B

    S. Mukhi and B. C. Rayhaun, “Classification of Unitary RCF Ts with Two Primaries and Central Charge Less Than 25,” [arXiv:2208.05486 [hep-t h]]

    work page arXiv

  7. [7]

    Modular Differential Equations with Movable Poles and Admissible RCFT Characters,

    A. Das, C. N. Gowdigere, S. Mukhi and J. Santara, “Modular Differential Equations with Movable Poles and Admissible RCFT Characters,” [arXiv :2308.00069 [hep-th]]

    work page arXiv

  8. [8]

    Fourier Coefficients of Vector-val ued Modular Forms of Dimension 2,

    C. Franc and G. Mason, “Fourier Coefficients of Vector-val ued Modular Forms of Dimension 2,” Can. Math. Bull. 57 (2014) 485-494 [ arXiv:1304.4288 [math,NT]]

    work page arXiv 2014

  9. [9]

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

    C. Franc, and G. Mason, “Hypergeometric series, modular linear differential equa- tions and vector-valued modular forms.” The Ramanujan Jour nal 41 (2016): 233-267 [arXiv:1503.05519 [math.NT]] 24

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  10. [10]

    Classification of some vertex op erator algebras of rank 3

    C. Franc, and G. Mason, “Classification of some vertex op erator algebras of rank 3.” Algebra & Number Theory 14 (6) (2020): 1613-1667 [arXiv:190 5.07500 [math.QA]]

    work page 2020

  11. [11]

    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 [arXiv:1602.01022 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  12. [12]

    Rational CFT with thre e characters: the quasi- character approach,

    S. Mukhi, R. Poddar and P. Singh, “Rational CFT with thre e characters: the quasi- character approach,” JHEP 05 (2020), 003 [arXiv:2002.01949 [hep-th]]

    work page arXiv 2020

  13. [13]

    Classifying thr ee-character RCFTs with Wronskian index equalling 0 or 2,

    A. Das, C. N. Gowdigere and J. Santara, “Classifying thr ee-character RCFTs with Wronskian index equalling 0 or 2,” JHEP 11 (2021), 195 [arXiv:2108.01060 [hep-th]]

    work page arXiv 2021

  14. [14]

    Classifying th ree-character RCFTs with Wronskian index equalling 3 or 4,

    C. N. Gowdigere, S. Kala and J. Santara, “Classifying th ree-character RCFTs with Wronskian index equalling 3 or 4,” [arXiv:2308.01149 [hep- th]]

    work page arXiv

  15. [15]

    Fermioni c rational conformal field theories and modular linear differential equations,

    J. B. Bae, Z. Duan, K. Lee, S. Lee and M. Sarkis, “Fermioni c rational conformal field theories and modular linear differential equations,” PTEP 2021 (2021) no.8, 08B104 [arXiv:2010.12392 [hep-th]]

    work page arXiv 2021

  16. [16]

    Bootstra pping fermionic rational CFTs with three characters,

    J. B. Bae, Z. Duan, K. Lee, S. Lee and M. Sarkis, “Bootstra pping fermionic rational CFTs with three characters,” JHEP 01 (2022), 089 [arXiv:2108.01647 [hep-th]]

    work page arXiv 2022

  17. [17]

    Holomorphic quasi-modular bootstr ap,

    Y. Pan and C. Zeng, “Holomorphic quasi-modular bootstr ap,” [arXiv:2409.01095 [hep- th]]

    work page arXiv

  18. [18]

    Bosonic rational conformal field theori es in small genera, chiral fermionization, and symmetry/subalgebra duality,

    B. C. Rayhaun, “Bosonic rational conformal field theori es in small genera, chiral fermionization, and symmetry/subalgebra duality,” J. Mat h. Phys. 65 (2024) no.5, 052301 [arXiv:2303.16921 [hep-th]]

    work page arXiv 2024

  19. [19]

    Wronskian Indic es and Rational Conformal Field Theories,

    A. Das, C. N. Gowdigere and J. Santara, “Wronskian Indic es and Rational Conformal Field Theories,” JHEP 04 (2021), 294 [arXiv:2012.14939 [hep-th]]

    work page arXiv 2021

  20. [20]

    Classification of RCFT from Holomorphic Modu lar Bootstrap: A Status Report,

    S. Mukhi, “Classification of RCFT from Holomorphic Modu lar Bootstrap: A Status Report,” [arXiv:1910.02973 [hep-th]]

    work page arXiv 1910

  21. [21]

    Holomorphi c modular bootstrap revis- ited,

    J. Kaidi, Y.-H. Lin and J. Parra-Martinez, “Holomorphi c modular bootstrap revis- ited,” JHEP 12 (2021) 151 [ 2107.13557]

    work page arXiv 2021

  22. [22]

    Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular T ensor Categories,

    J. A. Harvey, Y. Hu and Y. Wu, “Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular T ensor Categories,” J. Phys. A 53 (2020) no.33, 334003 [arXiv:1912.11955 [hep-th]]

    work page arXiv 2020

  23. [23]

    Hecke relations, cosets and t he classification of 2d RCFTs,

    Z. Duan, K. Lee and K. Sun, “Hecke relations, cosets and t he classification of 2d RCFTs,” JHEP 09 (2022), 202 [arXiv:2206.07478 [hep-th]]. 25

    work page arXiv 2022

  24. [24]

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

    P. Bantay, “The Kernel of the modular representation an d the Galois action in RCFT,” Commun. Math. Phys. 233 (2003), 423-438 [arXiv:math/0102149 [math]]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  25. [25]

    Conformal characters and the modular representation

    P. Bantay and T. Gannon, “Conformal characters and the m odular representation,” JHEP 02 (2006), 005 [arXiv:hep-th/0512011 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  26. [26]

    Modular differential equations for characters of RCFT

    P. Bantay, “Modular differential equations for characte rs of RCFT,” JHEP 06 (2010), 021 [arXiv:1004.2579 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  27. [27]

    The theory of vector-modular forms for the modular group

    T. Gannon, “The theory of vector-modular forms for the m odular group,” Contrib. Math. Comput. Sci. 8 (2014), 247-286 [arXiv:1310.4458 [math.NT]]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  28. [28]

    Kaneko, K

    M. Kaneko, K. Nagatomo and Y. Sakai, Lett. Math. Phys. 103 (2013), 439-453

    work page 2013

  29. [29]

    A. R. Chandra and S. Mukhi, SciPost Phys. 6 (2019) no.5, 053 [arXiv:1812.05109 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  30. [30]

    The Sage Developers, SageMath, the Sage Mathematics So ftware System, v10.5, https://www.sagemath.org (2025)

    work page 2025

  31. [31]

    Akhila, S

    S. Akhila, S. Govindarajan and J. Santara, work in progr ess

    work page

  32. [32]

    On classification of mod ular tensor categories,

    E. Rowell, R. Stong and Z. Wang, “On classification of mod ular tensor categories,” Commun. Math. Phys., 292 (2),(2009) 343-389

    work page 2009

  33. [33]

    On the classification of fusion rings

    D. Gepner and A. Kapustin, Phys. Lett. B 349 (1995), 71-75 [arXiv:hep-th/9410089 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  34. [34]

    Introduction to modular forms,

    D. Zagier,“Introduction to modular forms,” in From number theory to physics (pp. 238-291). Berlin, Heidelberg: Springer Berlin Heidelberg

    work page

  35. [35]

    Elliptic modular forms and their applicati ons

    D. Zagier, “Elliptic modular forms and their applicati ons”, in J. H .Bruinier et al. The 1-2-3 of modular forms: Lectures at a summer school in Nordfj ordeid, Norway (2008): 1-103

    work page 2008

  36. [36]

    Quantum black ho les, wall crossing, and mock modular forms,

    A. Dabholkar, S. Murthy and D. Zagier, “Quantum black ho les, wall crossing, and mock modular forms,” (2012) [arXiv preprint arXiv:1208.40 74]. 26

    work page 2012