Modular invariance plays an important role in the representation theory of infinite-dimensional Lie algebras, the representation theory of vertex operator algebras, conformal field theory and many other areas. This is an introductory course on the modular invariance in conformal field theory, The following topics will be covered in the course:
- Modular invariance of graded dimensions of suitable modules for Heisenberg algebras, affine Lie algebras and the VIrasoro algebra.
- Vertex operator algebras, modules and intertwining operators.
- q-traces and pseudo-traces.
- Differential equations.
- Associative algebras and vertex operator algebras.
- Modular invariance theorem.
- Applications (Verlinde formula, rigidity of tensor categories of modules for vertex operator algebras and so on).
Text: Lecture notes:
pdf file
Main references:
- The reference [H1] in the lecture notes above
proves the operator product expansion (associativity) of intertwining operators.
- The reference [H2] in the lecture notes above proves the modular invariance of intertwining operators. This is the main reference for this course.
- The reference [H3] in the lecture notes above in the lecture notes above proves the Verlinde conjecture and Verlinde formula.
- The reference [H4] proves the rigidity and modularity (nondegeneracy property).
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