Abstract
In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers [n,l]. n is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and l, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study [3,0] and [3,2] MLDEs in order to classify the corresponding CFTs. We reduce the problem to a “finite” problem: to classify CFTs with central charge 0 < c ≤ 96, we need to perform 6, 720 computations for the former and 20, 160 for the latter. Each computation involves (i) first finding a simultaneous solution to a pair of Diophantine equations and (ii) computing Fourier coefficients to a high order and checking for positivity.
In the [3,0] case, for 0 < c ≤ 96, we obtain many character-like solutions: two infinite classes and a discrete set of 303. After accounting for various categories of known solutions, including Virasoro minimal models, WZW CFTs, Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto unknown character-like solutions which could potentially give new CFTs. We also classify [3,2] CFTs for 0 < c ≤ 96: each CFT in this case is obtained by adjoining a constant character to a [2,0] CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York U.S.A. (1997).
G. W. Moore and N. Seiberg, Lectures on RCFT, RU-89-32 (1989).
J. Fuchs, I. Runkel and C. Schweigert, Twenty-five years of two-dimensional rational conformal field theory, J. Math. Phys. 51 (2010) 015210 [arXiv:0910.3145] [INSPIRE].
M. R. Gaberdiel, An Introduction to conformal field theory, Rept. Prog. Phys. 63 (2000) 607 [hep-th/9910156] [INSPIRE].
L. Kilford, Modular Forms: A classical and computational introduction, Imperial College Press, London U.K. (2015).
M. Ram Murty, M. Dewar and H. Graves, Problems in the Theory of Modular Forms, Hindustan Book Agency, New Delhi India (2016).
G. Anderson and G. W. Moore, Rationality in Conformal Field Theory, Commun. Math. Phys. 117 (1988) 441 [INSPIRE].
T. Eguchi and H. Ooguri, Differential Equations for Conformal Characters in Moduli Space, Phys. Lett. B 203 (1988) 44 [INSPIRE].
S. D. Mathur, S. Mukhi and A. Sen, On the Classification of Rational Conformal Field Theories, Phys. Lett. B 213 (1988) 303 [INSPIRE].
S. D. Mathur, S. Mukhi and A. Sen, Reconstruction of Conformal Field Theories From Modular Geometry on the Torus, Nucl. Phys. B 318 (1989) 483 [INSPIRE].
S. G. Naculich, Differential equations for rational conformal characters, Nucl. Phys. B 323 (1989) 423 [INSPIRE].
H. R. Hampapura and S. Mukhi, On 2d Conformal Field Theories with Two Characters, JHEP 01 (2016) 005 [arXiv:1510.04478] [INSPIRE].
M. R. Gaberdiel, H. R. Hampapura and S. Mukhi, Cosets of Meromorphic CFTs and Modular Differential Equations, JHEP 04 (2016) 156 [arXiv:1602.01022] [INSPIRE].
H. R. Hampapura and S. Mukhi, Two-dimensional RCFT’s without Kac-Moody symmetry, JHEP 07 (2016) 138 [arXiv:1605.03314] [INSPIRE].
A. R. Chandra and S. Mukhi, Towards a Classification of Two-Character Rational Conformal Field Theories, JHEP 04 (2019) 153 [arXiv:1810.09472] [INSPIRE].
A. R. Chandra and S. Mukhi, Curiosities above c = 24, SciPost Phys. 6 (2019) 053 [arXiv:1812.05109] [INSPIRE].
S. Mukhi, Classification of RCFT from Holomorphic Modular Bootstrap: A Status Report, in Pollica Summer Workshop 2019: Mathematical and Geometric Tools for Conformal Field Theories, 10, 2019 [arXiv:1910.02973] [INSPIRE].
S. Mukhi, R. Poddar and P. Singh, Rational CFT with three characters: the quasi-character approach, JHEP 05 (2020) 003 [arXiv:2002.01949] [INSPIRE].
S. Mukhi and R. Poddar, Universal correlators and novel cosets in 2d RCFT, JHEP 02 (2021) 158 [arXiv:2011.09487] [INSPIRE].
P. Bantay, Modular differential equations for characters of RCFT, JHEP 06 (2010) 021 [arXiv:1004.2579] [INSPIRE].
J. E. Tener and Z. Wang, On classification of extremal non-holomorphic conformal field theories, J. Phys. A 50 (2017) 115204 [arXiv:1611.04071] [INSPIRE].
J. A. Harvey and Y. Wu, Hecke Relations in Rational Conformal Field Theory, JHEP 09 (2018) 032 [arXiv:1804.06860] [INSPIRE].
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. A 53 (2020) 334003 [arXiv:1912.11955] [INSPIRE].
J.-B. Bae, S. Lee and J. Song, Modular Constraints on Conformal Field Theories with Currents, JHEP 12 (2017) 045 [arXiv:1708.08815] [INSPIRE].
J.-B. Bae, Z. Duan, K. Lee, S. Lee and M. Sarkis, Fermionic Rational Conformal Field Theories and Modular Linear Differential Equations, arXiv:2010.12392 [INSPIRE].
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, AMS/IP Studies Adv. Math. 7 (1998) 97.
M. Kaneko and M. Koike, On Modular Forms Arising from a Differential Equation of Hypergeometric Type, Ramanujan J. 7 (2003) 145 [math/0206022].
M. Kaneko, On Modular forms of Weight (6n + 1)/5 Satisfying a Certain Differential Equation, in: Number Theory, W. Zhang and Y. Tanigawa, eds., Springer, Boston U.S.A. (2006), pg. 97.
M. Kaneko, K. Nagatomo and Y. Sakai, Modular forms and second order ordinary differential equations: Applications to vertex operator algebras, Lett. Math. Phys. 103 (2013) 439 [INSPIRE].
T. Gannon, The theory of vector-modular forms for the modular group, Contrib. Math. Comput. Sci. 8 (2014) 247 [arXiv:1310.4458] [INSPIRE].
Y. Arike, M. Kaneko, K. Nagatomo and Y. Sakai, Affine Vertex Operator Algebras and Modular Linear Differential Equations, Lett. Math. Phys. 106 (2016) 693 [INSPIRE].
C. Franc and G. Mason, Hypergeometric Series, Modular Linear Differential Equations and Vector-valued Modular Forms, Ramanujan J. 41 (2016) 233 [arXiv:1503.05519].
M. Kaneko, K. Nagatomo and Y. Sakai, The Third Order Modular Linear Differential Equations, J. Algebra 485 (2017) 332.
G. Mason, K. Nagatomo and Y. Sakai, Vertex Operator Algebras with Two Simple Modules — the Mathur-Mukhi-Sen Theorem Revisited, arXiv:1803.11281.
N. H. Abel, Précis d’une théorie des fonctions elliptiques, J. Reine Angew. Math. 4 (1829) 309.
O. A. Castro-Alvaredo, B. Doyon and F. Ravanini, Irreversibility of the renormalization group flow in non-unitary quantum field theory, J. Phys. A 50 (2017) 424002 [arXiv:1706.01871] [INSPIRE].
A. Das, C. N. Gowdigere and J. Santara, Wronskian Indices and Rational Conformal Field Theories, JHEP 04 (2021) 294 [arXiv:2012.14939] [INSPIRE].
J. Kaidi and E. Perlmutter, Discreteness and integrality in Conformal Field Theory, JHEP 02 (2021) 064 [arXiv:2008.02190] [INSPIRE].
J. Kaidi, Y.-H. Lin and J. Parra-Martinez, Holomorphic modular bootstrap revisited, arXiv:2107.13557 [INSPIRE].
J.-B. Bae, Z. Duan, K. Lee, S. Lee and M. Sarkis, Bootstrapping Fermionic Rational CFTs with Three Characters, arXiv:2108.01647 [INSPIRE].
C. Franc and G. Mason, Classification of some vertex operator algebras of rank 3, Alg. Number Theory 14 (2020) 1613 [arXiv:1905.07500].
A. N. Schellekens, Meromorphic c = 24 conformal field theories, Commun. Math. Phys. 153 (1993) 159 [hep-th/9205072] [INSPIRE].
C. Marks, Irreducible vector-valued modular forms of dimension less than six, Illinois J. Math. 55 (2011) 1267 [arXiv:1004.3019].
P. Bruillard, S.-H. Ng, E. C. Rowell and Z. Wang, On classification of modular categories by rank, Int. Math. Res. Not. 2016 (2016) 7546 [arXiv:1507.05139].
J. M. Landsberg and L. Manivel, The sextonions and \( {E}_{7\frac{1}{2}} \), Adv. Math. 201 (2006) 143 [math/0402157].
A. Das, C. N. Gowdigere and J. Santara, Studying three-parameter MLDEs, work in progress.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2108.01060
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Das, A., Gowdigere, C.N. & Santara, J. Classifying three-character RCFTs with Wronskian index equalling 0 or 2. J. High Energ. Phys. 2021, 195 (2021). https://doi.org/10.1007/JHEP11(2021)195
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2021)195