On classification of extremal non-holomorphic conformal field theories. (English) Zbl 1362.81086
Summary: Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category \(\mathcal{C}\) and a central charge \(c\). A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category \(\mathcal{C}\), there exists a unitary chiral conformal field theory \(\mathcal{V}\) so that its modular tensor category \({{\mathcal{C}}_{\mathcal{V}}}\) is \(\mathcal{C}\). In this paper, we initiate a mathematical program in and around this conjecture. We define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge, and non-trivial representation theory. We show that there are finitely many different characters of extremal vertex operator algebras \(\mathcal{V}\) possessing at most three different irreducible modules. Moreover, we list all of the possible characters for such vertex operator algebras with \(c\leqslant 48\) .
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
References:
| [1] | Bantay P and Gannon T 2007 Vector-valued modular functions for the modular group and the hypergeometric equation Commun. Number Theory Phys.1 651-80 · Zbl 1215.11041 · doi:10.4310/CNTP.2007.v1.n4.a2 |
| [2] | Bruillard P, Ng S-H, Rowell E C and Wang Z 2016 Rank-finiteness for modular categories J. Am. Math. Soc.29 857-81 · Zbl 1344.18008 · doi:10.1090/jams/842 |
| [3] | Cano J, Cheng M, Mulligan M, Nayak C, Plamadeala E and Yard J 2014 Bulk-edge correspondence in (2 + 1)-dimensional abelian topological phases Phys. Rev. B 89 115116 · doi:10.1103/PhysRevB.89.115116 |
| [4] | Carpi S, Kawahigashi Y, Longo R and Weiner M 2015 From vertex operator algebras to conformal nets and back Mem. Amer. Math. Soc. at press (arXiv:1503.01260 [math.OA]) |
| [5] | Freedman M, Kitaev A, Larsen M and Wang Z 2003 Topological quantum computation Bull. Am. Math. Soc.40 31-8 · Zbl 1019.81008 · doi:10.1090/S0273-0979-02-00964-3 |
| [6] | Gannon T 2014 The theory of vector-valued modular forms for the modular group 2014 Conformal Field Theory, Automorphic Forms and Related Topics (Berlin: Springer) pp 247-86 · Zbl 1377.11055 |
| [7] | Gannon T 2016 Vector-valued modular forms and modular tensor categories Talk at Modular Categories – their Representations, Classification, and Applications (Oaxaca, Mexico, 14-19 August 2016) |
| [8] | Gaberdiel M R, Hampapura H R and Mukhi S 2016 Cosets of meromorphic CFTs and modular differential equations J. High Energy Phys.JHEP04(2016) 156 · Zbl 1388.81662 · doi:10.1007/JHEP04(2016)156 |
| [9] | Hampapura H R and Mukhi S 2016 On 2d conformal field theories with two characters J. High Energy Phys.JHEP01(2016) 005 · Zbl 1388.81215 · doi:10.1007/JHEP01(2016)005 |
| [10] | Höhn G 1955 Selbstduale Vertexoperatorsuperalgebren und das Babymonster PhD Thesis Universität Bonn (see: Bonner Mathematische Schriften 286) |
| [11] | Höhn G 2003 Genera of vertex operator algebras and three-dimensional topological quantum field theories Fields Inst. Commun.39 89-107 · Zbl 1103.17008 · doi:10.1090/fic/039/05 |
| [12] | Huang Y-Z 2005 Vertex operator algebras, the Verlinde conjecture, and modular tensor categories Proc. Natl Acad. Sci. USA102 5352-6 · Zbl 1112.17029 · doi:10.1073/pnas.0409901102 |
| [13] | Junla N 2014 Classification of certain genera of codes, lattices and vertex operator algebras PhD Thesis Kansas State University |
| [14] | Mason G 2007 Vector-valued modular forms and linear differential operators Int. J. Number Theory3 377-90 · Zbl 1197.11054 · doi:10.1142/S1793042107000973 |
| [15] | Mathur S D, Mukhi S and Sen A 1988 On the classification of rational conformal field theories Phys. Lett. B 213 303-8 · doi:10.1016/0370-2693(88)91765-0 |
| [16] | Read N 2009 Conformal invariance of chiral edge theories Phys. Rev B 79 245304 · doi:10.1103/PhysRevB.79.245304 |
| [17] | Rowell E, Stong R and Wang Z 2009 On classification of modular tensor categories Comm. Math. Phys.292 343-89 · Zbl 1186.18005 · doi:10.1007/s00220-009-0908-z |
| [18] | Schellekens A N 1993 Meromorphic c = 24 conformal field theories Commun. Math. Phys.153 159-85 · Zbl 0782.17014 · doi:10.1007/BF02099044 |
| [19] | Tener J E 2016 Geometric realization of algebraic conformal field theories (arXiv:1611.01176 [math-ph]) · Zbl 1418.81081 |
| [20] | Turaev V G 1994 Quantum Invariants of Knots and 3-Manifolds(de Gruyter Studies in Mathematics vol 18) (Berlin: W de Gruyter & Co) · Zbl 0812.57003 |
| [21] | Wang Z 2010 Topological Quantum Computation vol 112 (Providence, RI: American Mathematical Society) · Zbl 1239.81005 · doi:10.1090/cbms/112 |
| [22] | Wen X-G 1992 Theory of the edge states in fractional quantum hall effects Int. J. Mod. Phys. B 6 1711-62 · doi:10.1142/S0217979292000840 |
| [23] | Zhu Y 1996 Modular invariance of characters of vertex operator algebras J. Amer. Math. Soc.9 237-302 · Zbl 0854.17034 · doi:10.1090/S0894-0347-96-00182-8 |
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