Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
vertex algebra; Riemann-Hilbert correspondence; D-module; KZ-equations; WZW-model
vertex algebra; Riemann-Hilbert correspondence; D-module; KZ-equations; WZW-model
Summary:
We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a tree algebra. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\mathbb{C}$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\mathbb{Q}$.
References:
[1] Beilinson, A., Drinfeld, V.: Chiral algebras. Amer. Math. Soc. Colloq. Publ. 51 (2004). MR 2058353 | Zbl 1138.17300
[2] Borcherds, R. E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109 (1992), 405–444. DOI 10.1007/BF01232032 | MR 1172696 | Zbl 0799.17014
[3] Borel, A. (ed.), : Algebraic D–modules. Perspective in Math., vol. 2, Academic Press, 1987. MR 0882000 | Zbl 0642.32001
[4] Deligne, P.: Équations différentielles à points singuliers réguliers (French). Lecture Notes in Math., vol. 163, Springer Verlag, 1970. MR 0417174
[5] Frenkel, I., Lepowsky, I., Meurman, A.: Vertex operator algebras and the Monster. Academic Press, Boston, MA, 1988. MR 0996026 | Zbl 0674.17001
[6] Grauert, H.: Analytische Faserungen über holomorph–vollständigen Räumen. Math. Ann. 135 (1958), 263–273. DOI 10.1007/BF01351803 | MR 0098199 | Zbl 0081.07401
[7] Griffin, P. A., Hernandez, O. F.: Structure of irreducible $SU(2)$ parafermion modules derived vie the Feigin–Fuchs construction. Internat. J. Modern Phys. A 7 (1992), 1233–1265. DOI 10.1142/S0217751X92000533 | MR 1146819
[8] Grothendieck, A.: Esquisse d’un programme. Geometric Galois Actions (L.Schneps, Lochak, P., eds.), London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997. MR 1483107 | Zbl 0901.14001
[9] Hortsch, R., Kriz, I., Pultr, A.: A universal approach to vertex algebras. J. Algebra 324 (7) (2010), 1731–1753. DOI 10.1016/j.jalgebra.2010.05.012 | MR 2673758
[10] Hu, P., Kriz, I.: On modular functors and the ideal Teichmüller tower. Pure Appl. Math. Q. 1 1 (3), part 2 (2005), 665–682. DOI 10.4310/PAMQ.2005.v1.n3.a7 | MR 2201329 | Zbl 1149.55015
[11] Huang, Y. Z.: Two-dimensional conformal geometry and vertex operator algebras. Prog. Math. 148, Birkhäuser, Boston, 1997. MR 1448404 | Zbl 0884.17021
[12] Huang, Y. Z.: Generalized rationality and a “Jacobi identity" for intertwining operator algebras. Selecta Math. (N.S.) 6 (3) (2000), 22–267. MR 1817614 | Zbl 1013.17026
[13] Huang, Y. Z.: Differential equations, duality and intertwining operators. Comm. Contemp. Math. 7 (2005), 649–706. DOI 10.1142/S021919970500191X | MR 2175093
[14] Huang, Y. Z., Lepowsky, J.: Tensor products of modules for a vertex operator algebra and vertex tensor categories. Lie theory and geometry, Prog. Math. 123, Birkhäuser, Boston, 1994, pp. 349–383. MR 1327541 | Zbl 0848.17031
[15] Huang, Y. Z., Lepowsky, J.: Intertwining operator algebras and vertex tensor categories for affine Lie algebras. Duke J. Math. 99 (1) (1999), 113–134. DOI 10.1215/S0012-7094-99-09905-2 | MR 1700743 | Zbl 0953.17016
[16] Huang, Y.Z.: Representations of vertex operator algebras and braided finite tensor categories. Vertex algebras and related areas, Contemp. Math. 497, Amer. Math. Soc., Providence, RI, 2009, pp. 97–111. MR 2568402
[17] Segal, G.: The definition of conformal field theory. Topology, geometry and quantum field theory, London Math. Soc. Lecture Ser. 308, Cambridge Univ. Press, Cambridge, 2004, preprint in the 1980's, pp. 421–577. MR 2079383
Search
Partner of
