Modular Data: The Algebraic Combinatorics of Conformal Field Theory
Terry Gannon
Department of Mathematical Sciences University of Alberta Edmonton Canada T6G 1G8
DOI: 10.1007/s10801-005-2514-2
Abstract
This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It tries to refine, modernise, and bridge the gap between papers [6] and [55]. Our paper is essentially self-contained, apart from some of the background motivation (Section 1) and examples (Section 3) which are included to give the reader a sense of the context. Detailed proofs will appear elsewhere. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems.
Pages: 211–250
Keywords: keywords fusion ring; modular data; conformal field theory; affine Kac-Moody algebra
Full Text: PDF
References
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23. P. Bouwknegt, P. Dawson, and D. Ridout, “D-branes on group manifolds and fusion rings,” J. High Energy Phys. 0212 (2002), 065.
24. A.S. Buch, A. Kresch, and H. Tamvakis, “Gromov-Witten invariants on Grassmannians,” J. Amer. Math. Soc. 16 (2003), 901-915.
25. A. Cappelli, C. Itzykson, and J.-B. Zuber, “The A-D-E classification of minimal and A(1) conformal invariant 1 theories,” Commun. Math. Phys. 113 (1987), 1-26.
26. J. Cardy, “Boundary conditions, fusion rules and the Verlinde formula,” Nucl. Phys. B324 (1989), 581-596.
27. M. Caselle and G. Ponzano, “Analyticity, modular invariance and the classification of three operator fusion algebras,” Phys. Lett. B242 (1990), 52-58.
28. H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, Springer, New York, 1988.
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30. A. Coste and T. Gannon, “Remarks on Galois in rational conformal field theories,” Phys. Lett. B323 (1994), 316-321.
31. A. Coste and T. Gannon, “Congruence subgroups and conformal field theory,” preprint (math.QA/0002044).
32. A. Coste, T. Gannon, and P. Ruelle, “Finite group modular data,” Nucl. Phys. B581 (2000), 679-717.
33. L. Crane and D.N. Yetter, “Deformations of (bi)tensor categories,” Cah. Top. Géom. Diff. Catég. 39 (1998), 163-180.
34. P. Di Francesco and J.-B. Zuber, “SU(N ) lattice integrable models associated with graphs,” Nucl. Phys. B338 (1990), 602-646.
35. P. Di Francesco and J.-B. Zuber, “SU(N ) lattice integrable models and modular invariance,” in Recent Developments in Conformal Field Theory, World Scientific, (1990), pp. 179-215.
36. Ph. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer, New York, 1997.
37. R. Dijkgraaf, “The mathematics of fivebranes,” in Proc. Intern. Congr. Math., Berlin, Vol. III, (1998), pp. 133-142. GANNON
38. R. Dijkgraaf, C. Vafa, E. Verlinde, and H. Verlinde, “The operator algebra of orbifold models,” Commun. Math. Phys. 123 (1989), 485-526.
39. R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology,” Commun. Math. Phys. 129 (1990), 393-429.
40. C. Dong, “Vertex algebras associated with even lattices,” J. Alg. 160 (1993), 245-265.
41. C. Dong, H. Li, and G. Mason, “Simple currents and extensions of vertex operator algebras,” Commun. Math. Phys. 180 (1996), 671-707.
42. C. Dong, H. Li, and G. Mason, “Vertex operator algebras associated to admissible representations of \hat sl2,” Commun. Math. Phys. 184 (1997), 65-93.
43. C. Dong, H. Li, and G. Mason, “Modular-invariance of trace functions in orbifold theory and generalised moonshine,” Commun. Math. Phys. 214 (2000), 1-56.
44. W. Eholzer, “On the classification of modular fusion algebras,” Commun. Math. Phys. 172 (1995), 623-660.
45. F. Englert, L. Houart, A. Taormina, and P. West, “The symmetry of M-theories,” J. High Energy Phys. 9 (2003), 020.
46. P. Etingof and S. Gelaki, “Isocategorical groups,” Intern. Math. Res. Notices (2) (2001), 59-76.
47. D.E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford University Press, 1998.
48. G. Faltings, “A proof for the Verlinde formula,” J. Alg. Geom. 3 (1994), 347-374.
49. B. Feigin and F. Malikov, “Modular functor and representation theory of \hat sl(2) at a rational level,” in Operads, Contemp. Math. 202, Amer. Math. Soc., Providence, 1997, pp. 357-405.
50. A.J. Feingold and M.D. Weiner, “Type A fusion rules from elementary group theory,” in Recent Developments in Infinite-Dimensional Lie algebras and Conformal Field Theory, S. Berman et al. (Eds.), Amer. Math. Soc., Providence, 2002, pp. 97-115.
51. M. Finkelberg, “An equivalence of fusion categories,” Geom. Funct. Anal. 6 (1996), 249-267.
52. D.S. Freed, M.J. Hopkins, and C. Teleman, “Twisted K-theory and loop group representations,” math.AT/0312155.
53. I.B. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math, Vol. 134, Academic Press, London-New York, 1988.
54. I.B. Frenkel and Y. Zhu, “Vertex operator algebras associated to representations of affine and Virasoro algebras,” Duke Math. J. 66 (1992), 123-168.
55. J. Fuchs, “Fusion rules in conformal field theory,” Fortsch. Phys. 42 (1994), 1-48.
56. J. Fuchs, B. Gato-Rivera, B. Schellekens, and C. Schweigert, “Modular invariants and fusion rule automorphisms from Galois theory,” Phys. Lett. B334 (1994), 113-120.
57. J. Fuchs, S. Hwang, A.M. Semikhatov, and I.Y.U. Tipunin, “Nonsemisimple fusion algebras and the Verlinde formula,” hep-th/0306274.
58. J. Fuchs, A.N. Schellekens, and C. Schweigert, “From Dynkin diagram symmetries to fixed point structures,” Commun. Math. Phys. 180 (1996), 39-97.
59. J. Fuchs and C. Schweigert, “Branes: from free fields to general backgrounds,” Nucl. Phys. B530 (1998), 99-136.
60. L. Funar, “On the TQFT representations of the mapping class groups,” Pac. J. Math. 188 (1999), 251-274.
61. P. Furlan, A. Ganchev, and V. Petkova, “Quantum groups and fusion rules multiplicitites,” Nucl. Phys. B343 (1990), 205-227.
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