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Vertex operator algebras and their representations are basic
ingredients in conformal field theory, a theory playing important
roles in both condensed matter physics and string theory. For a vertex
operator algebra satisfying certain reductivity and finiteness
conditions, a theorem of Yongchang Zhu says that certain traces of
vertex operators on the representations of the algebra form a basis of
a module for the modular group SL(2, Z). In particular, for vertex
operator algebras associated to affine Lie algebras, the Virasoro
algebra, lattices and the moonshine module for the Monster group, we
have this modular invariance property. I will discuss a proof of this
theorem, various concepts and tools needed in this proof and some
generalizations and applications.
Prerequisites: I will assume that the students have some basic knowledge in algebra and complex variables, as covered in the first-year graduate courses. Text: I will use several research papers, including, Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237--307 and Yi-Zhi Huang, Differential equations, duality and modular invariance, Comm. Contemp. Math. 7 (2005), 649--706. |