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Rational CFT with three characters: the quasi-character approach. (English) Zbl 1437.81081

Summary: Quasi-characters are vector-valued modular functions having an integral, but not necessarily positive, \(q\)-expansion. Using modular differential equations, a complete classification has been provided in arXiv:1810.09472 for the case of two characters. These in turn generate all possible admissible characters, of arbitrary Wronskian index, in order two. Here we initiate a study of the three-character case. We conjecture several infinite families of quasi-characters and show in examples that their linear combinations can generate admissible characters with arbitrarily large Wronskian index. The structure is completely different from the order two case, and the novel coset construction of arXiv:1602.01022 plays a key role in discovering the appropriate families. Using even unimodular lattices, we construct some explicit three-character CFT corresponding to the new admissible characters.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R12 Groups and algebras in quantum theory and relations with integrable systems
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[1] Mathur, SD; Mukhi, S.; Sen, A., On the classification of rational conformal field theories, Phys. Lett., B 213, 303 (1988) · doi:10.1016/0370-2693(88)91765-0
[2] Mathur, SD; Mukhi, S.; Sen, A., Reconstruction of conformal field theories from modular geometry on the torus, Nucl. Phys., B 318, 483 (1989) · doi:10.1016/0550-3213(89)90615-9
[3] Naculich, SG, Differential equations for rational conformal characters, Nucl. Phys., B 323, 423 (1989) · doi:10.1016/0550-3213(89)90150-8
[4] Kiritsis, EB, Fuchsian differential equations for characters on the torus: a classification, Nucl. Phys., B 324, 475 (1989) · doi:10.1016/0550-3213(89)90475-6
[5] Kiritsis, EB, Analytic aspects of rational conformal field theories, Nucl. Phys., B 329, 591 (1990) · doi:10.1016/0550-3213(90)90073-M
[6] Durganandini, P.; Panda, S.; Sen, A., Some properties of supercharacters in superconformal field theories, Nucl. Phys., B 332, 433 (1990) · doi:10.1016/0550-3213(90)90104-L
[7] Kaneko, M.; Koike, M., On modular forms arising from a differential equation of hypergeometric type, Ramanujan J., 7, 145 (2003) · Zbl 1050.11047 · doi:10.1023/A:1026291027692
[8] 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, Germany (2006). · Zbl 1197.11050
[9] Mason, G., Vector-valued modular forms and linear differential operators, Int. J. Number Theor., 03, 377 (2007) · Zbl 1197.11054 · doi:10.1142/S1793042107000973
[10] Mason, G., 2-dimensional vector-valued modular forms, Ramanujan J., 17, 405 (2008) · Zbl 1233.11043 · doi:10.1007/s11139-007-9054-4
[11] Tuite, MP, Exceptional vertex operator algebras and the Virasoro algebra, Contemp. Math., 497, 213 (2009) · Zbl 1225.17034 · doi:10.1090/conm/497/09780
[12] Kaneko, M.; Nagatomo, K.; Sakai, Y., Modular forms and second order ordinary differential equations: Applications to vertex operator algebras, Lett. Math. Phys., 103, 439 (2013) · Zbl 1283.11070 · doi:10.1007/s11005-012-0602-5
[13] Hampapura, HR; Mukhi, S., On 2d conformal field theories with two characters, JHEP, 01, 005 (2016) · Zbl 1388.81215 · doi:10.1007/JHEP01(2016)005
[14] Gaberdiel, MR; Hampapura, HR; Mukhi, S., Cosets of meromorphic CFTs and modular differential equations, JHEP, 04, 156 (2016) · Zbl 1388.81662
[15] Hampapura, HR; Mukhi, S., Two-dimensional RCFT’s without Kac-Moody symmetry, JHEP, 07, 138 (2016) · Zbl 1390.81598 · doi:10.1007/JHEP07(2016)138
[16] Arike, Y.; Kaneko, M.; Nagatomo, K.; Sakai, Y., Affine vertex operator algebras and modular linear differential equations, Lett. Math. Phys., 106, 693 (2016) · Zbl 1338.81344 · doi:10.1007/s11005-016-0837-7
[17] Franc, C.; Mason, G., Hypergeometric series, modular linear differential equations, and vector-valued modular forms, Ramanujan J., 41, 233 (2016) · Zbl 1418.11064 · doi:10.1007/s11139-014-9644-x
[18] Tener, JE; Wang, Z., On classification of extremal non-holomorphic conformal field theories, J. Phys., A 50, 115204 (2017) · Zbl 1362.81086
[19] G. Mason, K. Nagatomo and Y. Sakai, Vertex operator algebras with two simple modules — The Mathur-Mukhi-Sen theorem revisited, arXiv:1803.11281.
[20] Chandra, AR; Mukhi, S., Towards a classification of two-character rational conformal field theories, JHEP, 04, 153 (2019) · Zbl 1415.81074 · doi:10.1007/JHEP04(2019)153
[21] C. Franc and G. Mason, Classification of some three-dimensional vertex operator algebras, (2019).
[22] Christe, P.; Ravanini, F., A new tool in the classification of rational conformal field theories, Phys. Lett., B 217, 252 (1989) · Zbl 0696.17013 · doi:10.1016/0370-2693(89)90861-7
[23] Mathur, SD; Sen, A., Group theoretic classification of rotational conformal field theories with algebraic characters, Nucl. Phys., B 327, 725 (1989) · doi:10.1016/0550-3213(89)90312-X
[24] V. Ostrik, Fusion categories of rank 2, Math. Res. Lett.10 (2002) 177 [math/0203255]. · Zbl 1040.18003
[25] Bantay, P.; Gannon, T., Conformal characters and the modular representation, JHEP, 02, 005 (2006) · doi:10.1088/1126-6708/2006/02/005
[26] Bantay, P.; Gannon, T., Vector-valued modular functions for the modular group and the hypergeometric equation, Commun. Num. Theor. Phys., 1, 651 (2007) · Zbl 1215.11041 · doi:10.4310/CNTP.2007.v1.n4.a2
[27] Gannon, T., The theory of vector-modular forms for the modular group, Contrib. Math. Comput. Sci., 8, 247 (2014) · Zbl 1377.11055 · doi:10.1007/978-3-662-43831-2_9
[28] Harvey, JA; Wu, Y., Hecke relations in rational conformal field theory, JHEP, 09, 032 (2018) · Zbl 1398.81211 · doi:10.1007/JHEP09(2018)032
[29] Mukhi, S.; Panda, S.; Sen, A., Contour integral representations for the characters of rational conformal field theories, Nucl. Phys., B 326, 351 (1989) · doi:10.1016/0550-3213(89)90136-3
[30] S. Mukhi, R. Poddar and P. Singh, Contour integrals and the modular S-matrix, arXiv:1912.04298 [INSPIRE].
[31] Schellekens, AN, Meromorphic c = 24 conformal field theories, Commun. Math. Phys., 153, 159 (1993) · Zbl 0782.17014 · doi:10.1007/BF02099044
[32] Chandra, AR; Mukhi, S., Curiosities above c = 24, SciPost Phys., 6, 053 (2019) · doi:10.21468/SciPostPhys.6.5.053
[33] Belavin, AA; Polyakov, AM; Zamolodchikov, AB, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys., B 241, 333 (1984) · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X
[34] Knizhnik, VG; Zamolodchikov, AB, Current algebra and Wess-Zumino model in two-dimensions, Nucl. Phys., B 247, 83 (1984) · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2
[35] J.A. Harvey, Y. Hu and Y. Wu, Galois symmetry induced by Hecke relations in rational conformal field theory and associated modular tensor categories, arXiv:1912.11955 [INSPIRE].
[36] Kaneko, M.; Zagier, D., Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, AMS/IP Studies Adv. Math., 7, 97 (1998) · Zbl 0955.11018 · doi:10.1090/amsip/007/05
[37] Verlinde, EP, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys., B 300, 360 (1988) · Zbl 1180.81120 · doi:10.1016/0550-3213(88)90603-7
[38] Dotsenko, VS; Fateev, VA, Four point correlation functions and the operator algebra in the two-dimensional conformal invariant theories with the central charge c < 1, Nucl. Phys., B 251, 691 (1985) · doi:10.1016/S0550-3213(85)80004-3
[39] Dotsenko, VS; Fateev, VA, Conformal algebra and multipoint correlation functions in two-dimensional statistical models, Nucl. Phys., B 240, 312 (1984) · doi:10.1016/0550-3213(84)90269-4
[40] Kervaire, M., Unimodular lattices with a complete root system, Enseign. Math., 40, 59 (1994) · Zbl 0808.11024
[41] O.D. King, A mass formula for unimodular lattices with no roots, Math. Comput.72 (2003) 839 [math/0012231]. · Zbl 1099.11035
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