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Differential equations and logarithmic intertwining operators for strongly graded vertex algebras

Jinwei Yang

Department of Mathematics, University of Notre Dame, 255 Hurley Building, Notre Dame, IN 46556-4618, USA

https://doi.org/10.1142/S0219199716500097Cited by:2 (Source: Crossref)

Abstract

We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded vertex algebra under a certain finiteness condition and a condition related to the horizontal gradings. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.

Keywords:

AMSC: 17B69

References

1. R. E. Borcherds, Monstrous moonshine and the monstrous Lie superalgebras, Invent. Math. 109 (1992) 405–444. Crossref, Web of Science, Google Scholar
2. C. Dong, Vertex algebras associated with even lattices, J. Algebra 161 (1993) 245–265. Crossref, Web of Science, Google Scholar
3. C. Dong, H. Li and G. Mason, Regularity of rational vertex operator algebras, Adv. Math. 132 (1997) 148–166. Crossref, Web of Science, Google Scholar
4. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Memoirs of the American Mathematical Society, Vol. 104, No. 494 (American Mathematical Society, Providence, RI, 1993). Google Scholar
5. I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, Vol. 134 (Academic Press, Boston, 1988). Google Scholar
6. Y.-Z. Huang, Differential equations and intertwining operators, Comm. Contemp. Math. 7 (2005) 375–400. Link, Web of Science, Google Scholar
7. Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929. Google Scholar
8. Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra, preprint (2011); arXiv:1110.1931. Google Scholar
9. Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories, Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248. Google Scholar
10. A. W. Knapp, Representation Theory of Semisimple Groups (Princeton University Press, Princeton, NJ, 1986). Crossref, Google Scholar
11. J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics, Vol. 227 (Birkhäuser, Boston, 2003). Google Scholar
12. A. Milas, Weak modules and logarithmic intertwining operators for vertex operator algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, eds. S. Berman, P. Fendley, Y.-Z. Huang, K. Misra and B. Parshall, Contemporary Mathematics, Vol. 297 (American Mathematical Society, Providence, RI, 2002), pp. 201–225. Crossref, Google Scholar
13. A. Milas, Logarithmic intertwining operators and vertex operators, Comm. Math. Phys. 277 (2008) 497–529. Crossref, Web of Science, Google Scholar
14. J. Yang, Tensor products of strongly graded vertex algebras and their modules, J. Pure Appl. Alg. 217 (2013) 348–363. Crossref, Web of Science, Google Scholar
15. J. Yang, Some results in the representation theory of strongly graded vertex algebras, Ph.D. thesis, Rutgers University (2014). Google Scholar
16. J. Yang, Vertex algebra associated to the abelian current Lie algebras, preprint (2015); arXiv:1508.01923. Google Scholar